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I’ve been curious about Chorus for a while, since I’ve been working on and off with chorus design myself. There were a few things I didn’t understand, like what the relationshp is between the modulation LFO’s waveshape and the frequency modulation of signals going through the chorus. You’d think that if you use a sinewave to modulate the BBD clock, you’d get a sinewave modulation of frequency, right? Wrong! So what do you get? At this point, I realised that it was a bit more complicated than I was giving it credit for and I’d have to really think about it. This page is some of the results of those studies.
Simulation of BBD chorus
So what did I do? Well, I did what I always do when I don’t understand something. I wrote a simulation of it. If I can write a sim of a situation, then I know that I understand it. If the sim doesn’t work, there’s something I’m still missing. Often the process of thinking about how to simulate something, and seeing how and why the simulation doesn’t work gives me an insight into the real situation. If that doesn’t work, playing with the simulation gives me a useful way to perform repeatable experiments that would often be awkward to do in reality.
My simulated delayline has 1024 stages, just like the SAD1024, MN3007, or MN3207 chip. Of these, only the last one is still available, so it’s the chip likely to turn up in modern analog chorus designs.
My chorus has linear clock modulation like the typical chorus effects. More about the problems this causes later – for now, we’re just simulating a typical chorus. The clock frequency at any given moment is:
$clock_freq = $clock_centre_freq + ($lfo * $mod_depth * 10000); // 10KHz mod range
Here’s the LFO waveform (blue) and the Clock rate (green). The horizontal green line is the clock centre frequency with no modulation applied. You can see that with a modulation depth of 20KHz, the clock frequency goes up to 60KHz, and down to 20KHz. The LFO rate and Clock rate are just chosen to give a reasonable display. We can see 400msecs of signal here.
Ok, so far, so good. Let’s feed some audio into it and see if we can see anything happening. Remember this is just the signal through the BBD (the “Wet only” signal), so it’s not a full chorus. In fact, it’s just a vibrato unit. But mixing the dry and wet signals together is easy, and that’s not the bit I don’t understand, so I’m ignoring the dry signal. This next graph adds a low audio ramp wave (red). You can easily see the pitch change caused by the clock modulation. Notice the initial delay before any signal comes through the delay line is clearly visible on the left hand side of the graph.
Now this is where things start to get interesting. What do you think is happening to that audio frequency? It should be varying up and down, following the changes in the clock rate, right? Let’s have a look. The next graph uses the slope of the output ramp wave to give an estimate of the audio output frequency at each moment. This is only possible with ramp waves of fixed amplitude, but it’s handy for our demo. This is plotted in red. I’ve taken the LFO off to stop it getting too cluttered.
Interesting, don’t you think? The frequency modulation produced isn’t a simple sine wave – it’s distorted. Let’s have better look at that without the audio:
So what’s going on here? Think about how the BBD makes a pitch change. When the LFO output is rising, the clock frequency is increasing, and samples are being read out faster than they were read in – their pitch is shifted up. Likewise, when the LFO is falling, the clock frequency is decreasing, and samples are read out more slowly than they were read in – the pitch is decreased. The key here is that the pitch shift is not caused by a high clock frequency or a low clock frequency, but by an increasing or decreasing clock frequency. It’s the rate–of–change of clock frequency that’s important.
For a sine wave LFO input like we’ve been using, the point of maximum increase is the middle of the upwards slope, where our clock rate crosses the horizontal red line. Similarly, the maximum decrease is on the downwards slope where it crosses the red line. If we plot this rate of increase, we’re plotting the differential of sin(x), which is just cos(x) – the same thing shifted forwards a bit.
But hang on a minute! Our frequency modulation isn’t following cos(x) for our sin(x) LFO! That curve isn’t a cosine curve any more than it’s a sine curve. Where’s that distortion coming from?
Why does the modulation get distorted?
Let’s back up for a moment and consider the delay. How would we measure the delay at any moment in time? Well, the total amount of delay is just the amount of delay provided by each bucket, all added together, and how much delay you get depends on how fast the clock was going for each of the previous buckets. The delay for a single bucket is the length of time since the last clock pulse – e.g. the clock period. The total delay is the sum of all the last X clock periods, where X is the number of BBD stages. Put another way, we’re integrating the area under the clock period curve for the last X samples.
Let’s plot the clock period curve. Here it is, added to the graph in green:
The first thing to notice about this is that it’s already slightly distorted, since we’re now looking at 1/sin(x) rather than sin(x). But the distortion is all in the vertical direction, not in the time axis. To say that another way, the waveform is still symmetrical left-to-right. Ok, now let’s see what shape curve the delay makes if we add up the last X periods of that clock period graph. Here it is in blue:
Now we see where the rest of the distortion comes from. Although the clock frequency matches the LFO, the actual delay doesn’t directly, because it is the sum of all the clock periods for the whole length of the delay line. Incidentally, with the clock frequency we choose originally, the blue line goes from slightly below 10msecs to slighty above 20msecs, so we’re pretty much in chorus territory here.
But it still doesn’t look like the frequency modulation!
Well, no, true. It doesn’t. But you remember when we talked about it being the rate-of-change of delay that was important? We were thinking that for a sin(x) LFO, we’d see a cos(x) frequency modulation? It isn’t that simple, since as we’ve shown, the total delay follows a much more complicated curve than the clock frequency. But the point still stands – it’s the rate-of-change of delay that matters. Here’s the plot with the rate-of-change added in pale blue:
Now at last we’ve got a waveform that looks like our frequency modulation! The frequency modulation follows the rate-of-change of the total delay, and that waveform isn’t anything like the modulation LFO’s waveform.
Great! So what frequency modulation do I get with other LFO waveforms?
Ok, let’s have a look at a few in isolation. First we’ve got the sine wave LFO that we’ve just seen:
A triangle wave LFO is probably even more common, since they’re easy to build:
And square wave LFOs are even easier to build, but rarely used for chorus:
Now, there’s an interesting result! Although the LFO jumps between two levels, the output frequency jumps between three!
Linear versus Exponential clock modulation for BBD chorus
One of the problems with a typical chorus unit is the pitch modulation gets deeper as the BBD clock rate is reduced (e.g. as the delay is made longer). This makes the pitch variation very obvious – “seasick” or “warbley” are words often used to describe the sound. A typical chorus unit uses an LFO to modify its clock frequency, and that clock modulation is linear, so a given modulation depth will give (for example) +/-25KHz of clock modulation. This is how our simulation has operated thus far. When I considered this, it seemed to me that must be the reason why the depth seems to go up. If you consider a high clock frequency of 200KHz, a modulation of +/-25KHz is about 12%, or about 2 semitones. If you then consider what happens at a low clock frequency of 50KHz, the same modulation of +/-25KHz now shifts the clock by about 50%, or roughly an octave (50-25 = 25, which is 50% of 50KHz, 50+25 = 75, which is 150% of 50KHz).
So what’s the solution? Use exponential frequency modulation like a synth VCO of course! Then the LFO mod depth would be specified as “an octave” or “4 semitones” and an octave shift at 50KHz is the same as an octave shift at 200KHz.
An Update – some further thoughts
A while after posting this article, I had an email discussion about it with Brian Neunaber of Neunaber Audio Effects. Brian was initially slightly sceptical about the effect I claimed to have found and thought it might be an effect of the simulation or the frequency measurement method. This challenge pushed me to ensure that the method and results were sound. Once I’d convinced him that the effect was real, he wrote out the equations for what I’ve stated above and modelled them in Wolfram Alpha. I’ll reproduce his working below.
Firstly, we know that the total delay is related to the number of stages and the clock frequency:
total_delay = 1024 / (2 * clock_freq)
Note that clock_freq doesn’t have to be a constant. It could vary.
We also know that the change in pitch is related to the rate-of-change of the total delay:
change_in_pitch = d/dt (1024 / (2*clock_freq) )
Now, how about we make our clock have a base frequency of 40KHz, and modulate by +/-20KHz:
clock_freq = (40000 + 20000 * sin(2*pi*2*t) )
Put that into the pitch change equation:
change in pitch = d/dt (1024 / (2 * (40000 + 20000 * sin(2*pi*2*t) ) ) )
We can plot that in Wolfram Alpha. This shows us our by-now-familiar distorted curve. Brian wondered how much actual pitch change that represents, so he also plotted log2(1+x) in Wolfram Alpha to see how much shift in octaves that is. The maximum pitch change is around 0.2 octaves, or approximately 2.4 semitones. That’s quite a lot for a chorus, and another plot shows that if we reduce the modulation depth, the distortion also decreases. It seems reasonable to me to suspect that slower LFO rates also decrease the distortion, since it usually reduces the waveform’s rate of change (though not for a sharp square wave). On this basis, the effect won’t show up on slow, shallow chorus waveforms like I initially thought, but will definitely be present on deep, fast flangers. Perhaps this article should have been titled “Investigations into what a BBD Flanger unit *really* does” instead!
My thanks to Brian for the discussion and his thoughts on the matter. |
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In this article we will once again dive in the work of Sir Isaac Newton. I already wrote an article about this one (The Beginning of Physics: Newton) but, he does deserve something more. This time we’ll explore his work in more detail than his life.
This is the eleventh article in the Beginning of Physics series. If you didn’t read the previous part you should definitely do so, specially because Newton’s life is in there:
So where do we start? Good question, and the answer is optics (because I want so)
Newton on Optics
In 1666 Newton discovered some important stuff about light. Like its nature and composition, you know, basic stuff.
This year, as previously said in the last article, was his miraculus year, where he wrote his theory of light and colors.
He observed that when light passes through a prism, different colors are refracted in different angles. This lead Newton to the conclusion that color is a property intrinsic to light itself. This might not seem such a huge deal today (because it’s obvious), but back then this was a serious debate, and Newton ended it!
He also showed that colored-light doesn’t change its properties. At all. Try it yourself: You may reflect it, scatter it or even transmit it. The light remains the same color. From this Newton came to the brilliant conclusion that color is a result of objects interacting with already-colored-light, rather than being generated by the object themselves. And this is true!!
It should be pretty obvious be now that Newton believed that light was a particle. In fact, people who later believed light was a particle showed Newton’s theories as a proof. This caused a lot of debate on Newton’s theories (like Robert Hooke) but also caused Newton’s entrance on the Royal Academy in 1672.
Newton on Math
Newton’s work has been said to “distinctly advance every branch of mathematics then studied“. His most known work is calculus. Yes, the stuff which is being way too hard for you to learn.
Newton wrote some brief stuff on calculus in 1666 and later worked it hard while on planetary motion (more on that later). He used integrals and derivatives to calculate the motion of planets (because these can be described by the change in velocity, acceleration and other properties).
But now comes the big Revelation: Newton wasn’t the first one to use calculus. I know right, didn’t expect this one… Newton’s Rival was Gotfried Wilbelm Leibniz, born in Leipzig, former Holy Roman Empire, in 1646. He, kind of like Newton, worked basically in every branch of science. Inventor of the two wheeled mechanical calculator, the binary notation (later used on computers) and a major figure in philosophy, Leibniz was quite a badass as well.
Gotfried worked out elements of calculus as far back as 1675, a decade before Newton’s Principia. In this year, Leibniz did what no human being had ever done before: he calculated the area under the graph of a function using integrals! You might not think this is something amazingly awesome, but it really is.
While in is work on calculus, Leibniz made up the symbols of differentials (δ) and the integral symbol, summa (∫). We still use this symbols, because we use Leibniz system of calculus, not Newton’s! You also didn’t know this one, did you?
So, why is this dispute still a thing really? Because Leibniz would only publish his full version in 1693 in the “Fundamental Theorem of Calculus”. For the rest of Leibniz life, he would fight to prove that he invented calculus first and independently of Newton. Only on more recent years can we give the credits he deserved.
Now, we may all love a good fight such as this one but, we’re forgetting the most important thing: two persons independently came to calculus! What other proof do you need to believe math is the language of the universe?
Now, back to Newton.
Newton on Gravity
As already wrote in the last article (), Newton was talking to Edmund Halley when asked “Why do planets move in ellipsis?”. Newton thought for a second and said “Hold my bear please” and 18 months later came back with the answer: Universal Theory of Gravity.
In his work Newton stated his three laws of motion, laying down the foundation for classical mechanics. He also came to the conclusion of some really important stuff. He would provide another proof for heliocentrism, showing that according to his theory, the sun must be the center of the Solar System. However, Newton would also realize the Sun cannot be center of the Solar System. What I mean by this is that Newton believed no body could be at rest, and so a “center” of anything. Newton rather thought it as “the common center of gravity of the Earth, the Sun and all the planets is to be the esteemed the center of the World” (which is very close to the Sun).
Now finally, let’s get technical!
Newton’s first Law of Motion – Inertia
In states that “an object in motion will remain in motion, and an object at rest will remain at rest, unless acted upon by a force“. This basically means that you need a force to take an object out of its initial state (at rest for example). How hard it is to move the object depends on the object’s inertia. You can measure inertia via the objects mass. The more mass an object has, the harder it is to move!
This is better explained in the second law of motion.
Newton’s second Law of Motion
It states that “net force is equal to mass time acceleration” or as an equation:
Where F is the net force applied, m is the mass of the body, and a is the body’s acceleration. Thus, a net force applied to a body produces acceleration. In the same way, when a object is accelerating it means a force is being applied to it.
Probably the most common and intuitive case of a net force producing acceleration is the gravitational force. Imagine you through a 2 kilogram coconut (because, who doesn’t like coconuts?) straight up in the air. After a second or two, the coconut will start falling due to the gravitational force with an acceleration of about 9.81 m/s^2 (if there is no wind and of course we neglect air resistance).
So, if gravity is the only force acting on the coconut, we can calculate the force of gravity by using F=ma. So the formula will became
Where Fg is now the gravitational force and g the rate of acceleration, 9.81 m/s^2. Now we can calculate the force of gravity:
Fg=mg = 2kg (9.81 m/s^2) = 19.62 kg(m)/s^2 = 19.62 N
And this is how you determine the force of gravity, or the weight of something. Now, those units are a bit too much, so we just call it Newtons (N, as you saw above) in honor of Sir Isaac Newton.
Now, usually gravity isn’t the only force on action, so we must take into account other forces. This’s where we get to a force that tends to show up a lot, which is explained by Newton’s third Law.
Newton’s third Law of Motion
“For every action, there is an equal but opposite reaction“. You should know this one from about ten thousand memes right? But there is more to this law than just that.
We call this reaction force the normal force (N), because it’s perpendicular to whatever surface your object is resting on.
This reaction force is different from other forces (like gravity) however. It’s kind of special. The thing is, the magnitude of the reaction force changes.
Imagine a box on the ground. The box has a weight of 10 N, so it pushes on the ground with a force of 10 N. Now, why doesn’t the box fall through the ground? Because of the reaction force, which pushes back on the box with a equal force (10 N).
If the weight of this box was 20 N, the ground would push back with the same magnitude. This will happen on and on, until the ground can’t counteract anymore and it breaks.
I hope you have liked this article. If you did (or if you didn’t) please comment on your thoughts about it! |
Distance And Displacement Worksheet
Title distance and displacement worksheet author last modified by workstation created date pm other titles distance and displacement worksheet contains basic conceptual questions about distance, displacement, speed, and velocity.whats included in this and editable student worksheet and word digital version for use in google drive prepared This worksheet contains basic conceptual questions about distance, displacement, speed, and velocity.
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Towards an Effective Particle-String Resolution
of the Cosmological Constant Problem
Raman Sundrum Department of Physics
Boston, MA 02215, USA
The Cosmological Constant Problem is re-examined from an effective field theory perspective. While the connection between gravity and particle physics has not been experimentally probed in the quantum regime, it is severely constrained by the successes of Standard Model quantum field theory at short distances, and classical General Relativity at large distances. At first sight, it appears that combining particle physics and gravity inevitably leads to an effective field theory below the weak scale which suffers from large radiative corrections to the cosmological constant. Consequently, this parameter must be very finely tuned to lie within the experimental bounds. An analog of just this type of predicament, and its resolution, are described in some detail using only familiar quantum field theory. The loop-hole abstracted from the analogy is the possibility of graviton “compositeness” at a scale less than eV, which cuts off the large contributions to the cosmological constant from standard model physics. Experimentally, this would show up as a dramatic breakdown of Newton’s Law in upcoming sub-centimeter tests of gravity. Currently, strings are the only known example of such compositeness. It is proposed that the gravitational sector comprises strings of very low tension, which couple to a stringy “halo” surrounding each point-like standard model particle.
A naturalness problem is like the sight of a needle standing upright on a table; it is consistent to assume a delicate balance, but one strongly suspects an invisible stabilizing force. The balance one must explain has the form of an extremely fine cancellation between large virtual contributions to an observable from physics at very different length scales. The grandest and most baffling of all the naturalness problems in fundamental physics is the Cosmological Constant Problem (CCP). It emerges upon putting together the two separately successful parts of our physical understanding: classical General Relativity and the quantum field theoretic Standard Model (SM). Ref. provides a good review. The problem is so tightly constrained that one can hope that its final resolution will reveal an essentially unique and qualitatively new stabilizing mechanism.
Here is an outline of the problem. The classical theory of general relativity that has been tested at long distances can be thought of as the result of integrating out short distance quantum fluctuations from both the SM and gravitational sectors. Einstein’s equations describing the leading long-distance behavior of the metric field, , are,
Here and are the curvature tensor and scalar respectively, is the classical energy-momentum tensor for the matter and radiation in the universe, Newton’s constant has been written in terms of the Planck mass, , and is the cosmological constant.111This definition of the cosmological constant differs by a factor of from the astrophysical convention, in order to give the units of energy-density. is very sensitive to the short distance physics which has been integrated out. The SM contributes its “vacuum energy”, very roughly given by,
where is the scale of electroweak symmetry breaking. The other contributions are from less well-understood sources, namely short-distance quantum gravity and particle physics beyond the standard model.222My language assumes that the standard model (possibly with the exclusion of the physical Higgs degree of freedom) is just an effective theory valid below roughly , and is superseded by some other theory at higher energies. However, as long as these exotic contributions do not unexpectedly finely cancel the SM contribution, we must have the following rough lower bound on the cosmological constant,
On the other hand, in solutions to eq. (1), contributes to the cosmological expansion rate. This permits a conservative bound to be put on by using the measured expansion rate of the universe and estimates of . With high confidence, the experimental bound is given by
Now, eqs. (3) and (4) are in wild disagreement. To lower the bound in eq. (3) sufficiently to accord with eq. (4) requires an unbelievably fine cancellation between the contributions to from quantum fluctuations below and above the weak scale. This is the CCP.
The attitude taken in this paper is that some part of the preceding story is simply wrong, and the true story must eliminate the need for fine-tuning in order to obtain an acceptably small value for . It is frequently believed that the true account cannot be understood by conventional means. According to this view, the resolution of the CCP may not be expressible in terms of the local quanta and interactions of a relativistic quantum theory. This is not the viewpoint of the present paper; the fundamental principles of relativity, quantum mechanics and locality are central to the understanding of the CCP, and the proposed resolution does not transcend them.
However, the CCP as described above is extremely robust, based only on the co-existence of gravity and mass scales of order (and supersymmetry-breaking of at least the same magnitude). The CCP then follows by elementary power-counting. Section 2 of this paper describes the CCP in greater detail using effective field theory methodology. Effective field theory provides a clear and economical separation of the facts and principles which we have already tested experimentally, from the physics which is still beyond our reach, both in the gravitational and particle physics sectors. It is a useful language for examining assumptions which we may need to discard, as well as for evaluating new proposals for solving the CCP.
In order to solve the CCP we must change the power-counting which determines how sensitive to SM mass scales the long-distance theory of eq. (1) is. In quantum field theory, whenever the physics of a large mass scale is integrated out, the sensitivity of the low-energy effective theory is determined by power-counting for the weakly-coupled degrees of freedom at that mass scale, not just the degrees of freedom at the lowest energies. To use this observation in the case of the CCP, we must ensure that at particle physics energies, the gravitational degrees of freedom are profoundly different from those in eq. (1). We can loosely speak of the graviton as a “composite” of these new degrees of freedom. How can this crucial new physics be right underfoot without our having noticed, and how exactly can compositeness help with the CCP? Section 3 provides a detailed analogy of the CCP where these questions, and others, can be understood in the context of a simple toy universe. This serves as a useful warm-up because the resolution of the toy naturalness problem is based on completely familiar physics.
Finally, in section 4, a possible new mechanism for stabilizing an acceptably small cosmological constant is put forward. Below the weak scale, it consists of a gravitational sector made of extremely low-tension strings, with string-scale less than eV, interacting with the stringy “halos” carried by SM particles. The low string tension cuts off the virtual contributions to so that it naturally satisfies eq. (4). At very large distances only the massless string mode, namely the graviton, is relevant, and the dynamics reduces to general relativity. In accelerator experiments, the macroscopic string halo carried by particles is unobserved because the stringy gravitational physics is too weakly coupled to compete with point-like SM interactions. The detailed structure of such an effective particle-string theory has not yet been worked out, but I discuss its necessary properties as well as possible directions towards its construction. If this scenario is correct there will be a striking experimental signature: Newton’s Law will completely break down when gravity is tested at sub-centimeter distances!
This proposal may appear heretical from the view of traditional field theory and string theory. However, recent developments in string theory offer some encouragement. There is evidence that strings can co-exist with objects of different dimensionality, D-branes, including -branes which are point-like states. For a review see ref. . There have already been several calculations of the scattering of D-branes with strings and with each other, which reveal a stringy halo about the D-branes . These results may be useful for constructing effective particle-string theories. Refs. are some initial forays in this direction.
However, I wish to point out an important difference between the goal of the present work and the goal of most of the string literature. The recent string theory upheaval is part of a very ambitious program aimed at a non-perturbative understanding of fundamental interactions at at the highest energies. On the other hand, the CCP is a puzzle whose answer lies at present-day energies, but is presumably hidden from view because of the weakness of the gravitational force. The purpose here is to develop an effective theory which has an ultraviolet cutoff given by the weak scale, and whose parameters can naturally be fit to experiment. The effective theory is permitted to break down above the weak scale, and be replaced by a more fundamental description there.
Sections 2, 3 and 4 may be read in any order depending on the background and interests of the reader. Section 5 provides the conclusions.
I make use of rough estimates in several places. It is customary when power-counting to keep track of factors of that arise from the dimensionality of space-time. In this paper, I will consider all such factors to be order one since the Cosmological Constant Problem involves such large numbers that, by comparsion, factors are unremarkable. When estimating Feynman diagrams, dimensional regularization is implicit for simplicity. This will not remove any important physics (for example it does not eliminate the CCP) because the important mass scales will be explict and will not need to be represented by a dimensionful cutoff.
2 The Problem in Context
2.1 The standard effective theory of particles and gravity
The most straightforward way to put together the SM and general relativity is to write the lagrangian
where appears in , minimally coupled to maintain general covariance.333To be more precise, for fermions we must work in terms of the vierbein, but this detail is inessential in this paper. In order to compute quantum mechanical fluctuations of the metric around a Minkowski space vacuum we note that,
where is the Minkowski metric, and is the canonically normalized spin- graviton field. For most regions of spacetime, this weak field expansion about a Minkowski metric is justified. The broad perspective of general relativity adopted in this paper, as a phenomenological theory of gravity, is detailed in refs. . It is based on the generally observed principles of relativity and quantum mechanical unitarity.
As is well-known, the inclusion of gravity renders the lagrangian non-renormalizeable by elementary power-counting. This means that the resultant theory cannot be a fundamental description of nature at all energies (at least perturbatively). However, the lagrangian is a sensible basis for a quantum theory effective at energies far below the Planck scale, . Recall how this works in a general non-renormalizeable theory. Technically a non-renormalizeable theory requires an infinite number of counterterms, which at first sight appears disasterous. The situation greatly improves if we restrict ourselves to physical processes at energies, , far below the (smallest) mass scale suppressing the non-renormalizeable interactions, . This allows us to work to any fixed order in the small parameter , say . To this order only the finite number of interactions and counterterms of dimension less than or equal to are relevant. While this statement is rather obvious at tree-level, non-trivially it survives loops and renormalization. For we thereby obtain a well-defined and predictive effective field theory. The effective theory must give way to a more fundamental description at some scale below , or perhaps be sensible but strongly-coupled above . The best known example of a non-renormalizeable effective field theory is the chiral lagrangian description of pions, treated as Nambu-Goldstone bosons of chiral symmetry breaking. For a review see ref. . The typical scale appearing in the non-renormalizeable interactions is the hadronic scale, GeV. The effective field theory is therefore sensible and weakly-coupled for GeV. For GeV, the effective theory fails completely and must be replaced by the more fundamental QCD description.
In the case at hand, the scale suppressing the non-renormalizeable interactions is . Therefore the theory given by eq. (5) makes sense at energies . In fact let us take the ultraviolet cutoff of our effective theory to be the weak scale, as denoted by the appearing on the left-hand side of eq. (5). This allows us to remain agnostic about the nature of physics beyond the weak scale. The ellipsis in eq. (5) can contain higher-dimension gauge and coordinate invariant interactions, whose effects are small at energies far below the weak scale.444More precisely, they are irrelevant in that their dominant effects can be absorbed into finite renormalizations of the lower dimension interactions.
As far as accelerator experiments are concerned, eq. (5), provides a very economical summary of what has been actually observed. They overwhelmingly confirm a relativistic quantum field theory given by the SM for energies below the weak scale, with gravitational forces being negligible. Thus the “laboratory tested” part of eqs. (5, 6) is given by the formal limit,
At macroscopic distances, with large amounts of matter and radiation, the SM forces are effectively neutralized and gravity dominates. Because of the large distances, masses and numbers of quanta, the classical approximation is justified. Conceptually, one arrives at a classical effective theory for this regime by integrating out all quantum fluctuations from eq. (5). The result must have the form of classical general relativity, eq. (1). This is because eq. (1) is the most general form consistent with the general covariance of our starting point, eq. (5), up to terms involving higher-dimension metric invariants which are irrelevant at macroscopic distances.
It is important to note that only the classical macroscopic effective theory, rather than the full effective quantum field theory of eq. (5), has been tested gravitationally. This is in contrast to the SM sector, where the full quantum field theoretic implications of eq. (5) have been tested. Therefore we must bear in mind that while eq. (5) is in accord with all gravitational tests since it reduces to eq. (1), the “bare” parameter allowing us to fit the experimental bound of eq. (4), eq. (5) may not be unique in this respect.
It is somewhat of a nuisance that eq. (1) combines two steps in its derivation, the integrating out of microscopic physics and the classical limit for large numbers of quanta. It is useful to separate the two issues by considering a long distance effective theory in a simplified setting, involving just a few SM particles, but treated fully quantum field theoretically. I will develop such an effective theory in the next subsection. It will provide a useful point of contact when we discuss the analogy in Section 3.
2.2 A macroscopic quantum effective lagrangian with gravity
Consider a few stable massive spin- particles, , with relative momenta only of order , interacting with soft gravitons with energies of order . The length scale, will act as our short-distance cutoff. We can take mm, which is less than the shortest range over which gravity has been tested. One can imagine to be a ground state hydrogen atom say, whose compositeness cannot be resolved by the long wavelength gravitons. Alternatively we can take to be just a proxy for more fundamental particles like an electron, neglecting the complications of spin and charge. For simplicity I will also neglect the other soft massless particles, photons and neutrinos. We therefore have an isolated sector which should be described by an effective lagrangian containing only the and fields. This type of theory is entirely analogous to the heavy particle effective theories used in studying the strong interactions, where soft pions interact with a massive hadron, or gluons interact with a heavy quark. This is reviewed in ref. . I will simply take over the methodology to the case at hand.
The first observation is that since , the -velocity of , , is approximately conserved in collisions with soft gravitons, to within . Thus the momenta have the form,
For the simple case considered here, is common to all the particles involved, since their relative momenta were assumed to be of order . We perform a field redefinition of the scalar field to remove the large fixed component of the momentum, ,
The field thereby has residual momentum , just like the gravitons. Because is a Minkowski space vector, not a generally covariant vector, it is important to note that the generally covariant derivative for is , rather than just for .
The general form of the -scale effective lagrangian in this sector is,
The effective lagrangian is manifestly generally coordinate invariant, a symmetry of the starting point, eq. (5). The ellipsis contains higher dimension operators constructed from and , including local self-interactions, whose effects are small at large distances (but can be systematically included). If , we can expand the quantum theory about a Minkowski vacuum, . In the frame of the particles, given by , eq. (10) then becomes,
describing non-relativistic particles coupled to gravity, predominantly through their gravitational “charge”, . After gauge-fixing the gravitational fields (see for example ref. ), we can integrate out -graviton exchange which dominates the interactions, to obtain a non-local Newtonian potential interaction between particles. If , the field theory must be expanded about an “expanding universe” metric rather than Minkowski space.
Since eq. (10) describes a quantum field theory, we may ask if can naturally be as small as eq. (4) under quantum corrections within this effective theory. The answer is yes! Conceptually, all the fields in eq. (10) have their momenta cut off at , because of the field redefinition, eq. (9). Thus although by power-counting we estimate that should get radiative corrections of order four powers of the cutoff, this is just , which for mm, is well within the experimental bound, eq. (4). In fact if we adopt dimensional regularization, the -loop corrections to vanish.
The effective field theory described above reproduces some familiar phenomena of classical general relativity, such as the Newtonian force between non-relativistic masses, and gravitational radiation. On the other hand the effective theory is fully quantum mechanical and unitary in its domain of validity, and predicts inherently quantum corrections to the classical approximation. (An example of such corrections is described in ref. .) Yet, it has a naturally small cosmological constant. For these reasons it is a useful conceptual link between microscopic physics and classical general relativity.
2.3 The Cosmological Constant Problem
The -scale effective theory given by eq. (10) and the classical effective theory of eq. (1) both share the same cosmological constant, . I will focus on eq. (10) since it is a straightforward quantum effective field theory, though analogous statements follow for classical general relativity, eq. (1). If we do not look beyond the effective theory of eq. (10), can naturally satisfy the experimental bound, eq. (4), as pointed out above. However, our present point of view is that is determined by matching the effective theory of eq. (10) with the more fundamental theory of eq. (5), or conceptually, by integrating out the physics below . If one fixes a particular regularization and renormalization scheme, say dimensional regularization with minimal subtraction, one can actually perform the matching computations. Here we only require the results of simple power-counting, which shows that is negligibly renormalized in matching at , while by contrast, is quartically sensitive to the mass scales of the SM, so that,
We have no way of understanding why the physics above the weak scale, which determines , should so precisely cancel against the SM contributions, in order for eq. (4) to hold.
Thus I conclude, although eq. (5) reduces to the SM at short distances, (eq. (7)), reproducing all accelerator experiments, and though it reduces to eqs. (1) and (10) at macroscopic distances, thereby accomodating all gravitational measurements, it is not the correct effective theory below the weak scale because it involves a fantastic and inexplicable fine-tuning of . We must therefore see what room we have for changing the weak scale theory without destroying its highly non-trivial theoretical consistency and agreement with experiment.
What seems highly significant to me is this. The cosmological constant is usefully thought of as a non-derivative graviton self-coupling (which de-stabilizes Minkowski spacetime). Quantum corrections to come from loops of massive SM states, coupled to external graviton lines at essentially zero momentum. Therefore necessarily, these massive SM states are far off-shell. On the other hand, experimentally we have only tested the gravitational couplings of SM states which are very nearly on-shell.555By contrast note that accelerator experiments have very successfully probed highly virtual, purely SM effects, in the form of running couplings and precision electroweak tests. For example, the particles of the previous subsection are very nearly on-shell in the domain of validity of the -scale effective theory. It follows that all the large quantum corrections to from the weak-scale theory of eq. (5) come from a tremendous theoretical extrapolation to the regime where gravitons couple to SM particles which are far off-shell. We can hold out some hope that the CCP can be avoided by a different weak scale effective theory, which however still reduces to eqs. (1) and (10) in the domain of on-shell SM matter coupled to soft gravitons.
2.4 Constraints on alternative weak-scale theories
In thinking about alternative effective theories, it is crucial to observe two powerful fundamental principles, at least as far as physics below the weak scale is concerned. First, to quite large distances, spacetime appears as a Minkowski continuum. It also appears to be true down to distances of order , since the highly successful SM loop computations depend sensitively on this assumption. Secondly, nature is quantum mechanical, at least up to weak scale energies. Furthermore, it is difficult to perturb the quantum principle withut leading to physical absurdities. Therefore it would appear that we cannot seriously doubt the principles of (local) special relativity and quantum mechanics in the gravitational sector below the weak scale. These two principles impose very severe constraints on model-building. Taken with the experimental success of general relativity at large distances they necessarily imply the existence of a massless spin-two particle, the graviton, which must underlie any effective theory of gravity. Furthermore, this effective theory must obey the gauge symmetry of general coordinate invariance . This is similar to the case of light spin-one particles, where a gauge symmetry is needed to decouple unphysical degrees of freedom, but in the case of spin-two the gauge symmetry is unique!666There have been suggestions that general coordinate invariance can be replaced by restricted invariance under coordinate transformations with unit Jacobian. However, both classically and quantum mechanically this is precisely equivalent to a generally invariant theory with an arbitrary (but not naturally small) cosmological constant. See ref. for a brief review, plus references.
Now, if we restrict ourselves to the minimal particle content, namely the SM particles and the graviton, the form of the effective theory is given by eq. (5), this being the most general invariant form that reduces to the SM when , and containing the kinetic term for the graviton field. But eq. (5) is just the effective theory we are trying to avoid. Thus we conclude that new particles associated with gravity must be present. They must be very light indeed in order to remain in the effective theory down to the very low energies necessary to protect the cosmological constant, as has recently been emphasized in ref. .
Unfortunately, all proposals to couple extra particles to eq. (5) have failed to cure the naturalness problem. Generic addition of extra light particles does not evade the simple power-counting which says that the cosmological constant is quartically sensitive to the highest mass scales in the theory. Supersymmetrizing eq. (5) does in fact stabilize a suitably small . However this requires supersymmetry to be unbroken in the SM sector to very high precision, in order to suitably reduce the contribution in eq. (12). Experimentally however, we know that supersymmetry is badly broken in this sector. Other than supersymmetry the only other special symmetry that can control the cosmological constant is conformal symmetry. This is also badly broken in nature, but there have been several attempts to make a dynamical field that relaxes to zero as a consequence of the conformal anomaly, similarly to the way an axion can relax the strong interactions to a CP-conserving vacuum in the presence of a angle, as a consequence of the axial anomaly. For the CCP all such attempts have failed for the general reason described in ref. . To summarize, while we can always weakly couple eq. (5) to new light particles, there is no reason for these to significantly reduce the SM loop contributions to .
The seemingly impossible predicament posed by the CCP has given rise to proposals which play by different rules from those we have adopted. They cannot be evaluated within any local effective theory and are difficult to test experimentally. It is possible that one of these proposals is nevertheless true. Perhaps the best-known is Coleman’s wormhole proposal . Here, wormhole physics, just below the Planck scale, gives rise to peculiar non-local effects (from the viewpoint of our macroscopic spacetime), whereby the fundamental “constants” of nature become dynamical variables, but without any local spacetime variations. The relevant path integral is infinitely peaked at values of these constants such that the bottom-line cosmological constant vanishes, .
The present paper describes a deliberately restricted search for a resolution of the CCP which can be described by a natural effective theory, expressed in terms of local degrees of freedom. This is the time-honored approach taken towards other naturalness problems such as the the Strong CP problem or the Higgs naturalness problem. However, the arguments of this section seem to suggest that we are at an impasse. There may be a way out though, as suggested by the following parable.
3 An Analogy
In this section I describe a naturalness problem, analogous to the CCP, which occurs within a toy model universe. This toy problem has the advantage of involving only the familiar quantum field theory of particles with spins less than or equal to one. Nevertheless, the resolution sheds light on how the CCP might be resolved. The model consists of two sectors, a toy “Standard Model” (TSM) accounting for short-distance “laboratory” physics, and a toy “gravity” (TG) only noticeable at very large distances.
The TSM is simply the quantum electrodynamics of eight identical flavors of charged fermions, , . I will cut off the electromagnetic interactions at larger than laboratory distances by giving the photon a very small mass.777Of course, in the real world the electromagnetic force is negligible on large distance scales because of the neutrality of massive gravitating objects, like planets and stars. The toy photon mass makes for a simpler story. Recall that for an abelian gauge field, a mass term is both renormalizable and naturally small (only receiving logarithmic quantum corrections). The renormalizeable and natural TSM theory is then given by
with . We will consider the TSM to have been tested at lab momenta, very roughly of order (where the photon mass is negligible), and to a precision given by . For example, our momentum resolution is of order , and we are insensitive to -loop QED effects for such that . Nevertheless I will consider to be small enough that eq. (13) has been non-trivially tested as a quantum field theory.
On the other hand, TG corresponds to the observation of a very weak classical scalar Yukawa force, , between non-relativistic particles888Unlike the real world, in the toy universe the photon does not “gravitate”. over very large distances and times,
Notice this implies that the exponential suppression is always turned on in the Yukawa force, but clearly it is still the dominant force at very large distances. The TG force is too weak to be observed at short distances in the lab, against the background of electromagnetism, but is seen outside the photon range. To be concrete let us take,
The mass scale is extremely small,
At the purely classical level this is acceptable, as is a very small or zero cosmological constant in classical general relativity.
The minimal relativistic quantum field theory incorporating both the TSM and TG necessarily associates a scalar field, , with the Yukawa force,
where the scalar coupling is included for renormalizability, though it is too small to observe and plays no further role. Eq. (17) is the analog of eq. (5). Like eq. (5), it suffers from a naturalness problem. Here, the problem is why the scalar mass, , is so small, despite much larger quantum corrections coming from TSM loops. Standard power-counting and eqs. (15, 16) give,
Physicists of the toy universe may note that a small scalar mass is stabilized by supersymmetry. But the fact that no superpartners have been observed for energies well above means that supersymmetry is badly broken, and eq. (18) still holds. This is closely analogous to the situation with the cosmological constant and supersymmetry in the real world.
Other than supersymmetry there is no mechanism by which a weakly coupled fundamental scalar can naturally avoid corrections like eq. (18). One might think that the spin- particle could be fundamental and light if it is a Nambu-Goldstone boson of a spontaneously broken symmetry, but this possibility can be ruled out as well. Even though fundamental Nambu-Goldstone bosons are naturally massless, their “decay constants” are naturally of order the highest scale in the theory, in the present case, . The fact that the spin- particle has non-derivative couplings to the ’s means that the spontaneously broken symmetry must also be explicitly broken. The same explicit breaking which gives rise to a Yukawa coupling, , naturally gives rise to a pseudo-Nambu-Goldstone boson mass-squared of order , which is incompatible with eq. (16).
The only remaining means of obtaining a naturally light spin- particle is to make it a composite, like a hadron, with a very low compositeness scale. Even this approach offers no comfort at first sight. The basic reason is that the self-energy estimate due to TSM loops, yielding eq. (18), is performed at essentially zero external momentum, and so is completely insensitive to whether is composite or fundamental. Thus in any model where has a low enough compositeness scale to naturally satisfy eq. (16), it will be impossible to arrange for a Yukawa coupling as large as in eq. (15). I will illustrate this with a specific example. Suppose we try to make a scalar glueball of a Yang-Mills sector, with a confinement scale , which sets the glueball mass. To obtain a Yukawa coupling to the ’s we can use a higher dimension interaction,
where is a dimensionless coupling, and tr is the Yang-Mills operator that interpolates the glueball, ,
Recall that a non-renormalizable interaction such as eq. (19) is acceptable within effective field theory. Taking the effective theory cutoff to be of order , the energy scale probed in the lab, the theory remains weakly coupled at the cutoff provided . (In fact we must have in order for the -gluon interactions to have not been directly seen in the lab.) Therefore we arrive at the unsatisfactory result,
in contradiction to eq. (15).
I hope to have convinced the reader that, like the CCP, this toy naturalness problem seems to leave no room for manoeuvre. However, this is a false impression.
Fortunately, compositeness does allow the resolution of the naturalness problem. In order to invalidate the reasoning behind the large quantum corrections to from the TSM, we must not only take to be a composite light hadron, but we must also consider the massive particles it interacts with in TG to be heavy hadrons containing the TSM particles as heavy quarks! The specific resolution I have in mind is given by,
where I have introduced an gauge theory, under which the and quarks are triplets, and the eight TSM fields form an adjoint representation. Only the ’s are electrically charged however.
The is a particular combination of the three QCD pions, , made out of and quarks, while the heavy fermion it interacts with hadronically in TG, , is a composite of the adjoint quark and glue. The light quark masses are needed to produce non-derivative Yukawa couplings of to , and to generate a small . The Yukawa coupling we need, , breaks isospin symmetry under which the pions form a triplet, whereas is a singlet. The requisite isospin breaking is arranged by taking . Another technicality is that QCD is normally a parity-conserving theory, so a single pseudo-scalar pion cannot couple to the scalar as required. I have therefore added an order one CP-violating -term.999In fact, even for , the Yukawa coupling is not generated at first order in because of a vacuum re-alignment induced by . However, the Yukawa coupling is generated at higher order in . The resulting Yukawa coupling is then of order a small power of , while
By taking , we can consistently choose so that and , as desired!
There are three issues we would like to understand better: (i) Why is the composite QCD structure not already observed in the long distance TG sector? (ii) Why is the composite structure not visible in the lab? (iii) How does the composite structure cure the mass of extreme sensitivity to the TSM mass scale, ? Some of the discussion will be similar to that of ref. , where a model with very small was also considered.
(i) The interactions of very low-energy pions with slow heavy hadrons can be described using a Heavy Hadron Effective Lagrangian (reviewed in ref. ),
Recall that is some linear combination of the fields depending on and . The ellipsis contains terms whose effects on are negligible at the very low momentum transfers, , corresponding to eq. (14). These include the higher dimension couplings (suppressed by powers of ) of to itself and to the , and all couplings involving fields other than . Eq. (24) is just the analog of eq. (11). Integrating out -exchange, which dominates the interactions at long distance, yields the simple Yukawa potential. The next lightest state above the that can be exchanged between ’s is a two-pion state. But in the regime of eq. (14), even the two-pion exchanges are exponentially suppressed relative to single- exchange.
As far as eq. (24) is concerned the scalar mass is naturally small because of the very low cutoff on the effective theory. Compositeness effects are invisible because the compositeness scale is too high compared with the (virtual) momenta corresponding to eq. (14). We see that the first-conjectured form of the TG field theory is wrong. does not interact with the quarks, but rather with the hadronic “brown muck” of the heavy hadron.
One might worry that there can be excited composites, which have different Yukawa couplings , but over the time scales of eq. (14) such states would decay to the lowest stable state. A minor technical dynamical assumption that must be made (but which fortunately has no analog in the real CCP) is that any exotic composites of and two or more light quarks are heavy enough to decay to via pion emission, so that their possibly different Yukawa couplings are not seen.
(ii) Typical lab momenta are of order , where the running QCD coupling is weak. In the limit where it vanishes, the TSM sector completely decouples from the QCD sector. We can work out the actual value of the coupling renormalized at the laboratory momentum resolution, , using the one-loop QCD -function and the fact that we have already chosen . The result is,
Therefore QCD-induced momentum transfers larger than have amplitudes suppressed by , so they are too small to be seen against TSM interactions. On the other hand, amplitudes where the QCD-induced momentum transfers are less than remain unsuppressed, corresponding to soft radiation of light hadrons and excitation of the resonances. But such momentum transfers are smaller than our momentum resolution. The QCD sector is therefore invisible in the lab! Note, it is only the particles which feel the electromagnetic force and determine the outcome of lab experiments.
In the absence of the electromagnetic interaction the ’s would also form heavy quarkonium bound states, but with electromagnetism the QCD interactions will only negligibly perturb the electromagnetically bound states.
(iii) We now see that the unnaturally large quantum corrections in eq. (18) arose because of loops, where the appears far off-shell, with a Yukawa coupling to . But this simple coupling is only valid in eq. (24), where the is nearly on-shell. The extrapolation off-shell is completely invalid since the compositeness scale is very low. The true quantum corrections to the mass from the TSM sector require knowledge of the full QCD dynamics. We will now correctly compute the sensitivity to . To make the question precise let us fix some ultraviolet cutoff, , relative to which we can measure masses. This could be the scale of some new physics beyond the toy standard model. We also fix and ask how changes as a function of . We already have the mass formula for pseudo-Nambu-Goldstone bosons, eq. (23). We can integrate out the effects of the very heavy (adjoint) quark because of the asymptotic freedom of QCD. The dominant behavior follows from the one-loop renormalization group. The infrared renormalized quark mass parameters that appear in eq. (23) are the result of running down from . To one-loop order however, this mass renormalization is independent of the heavy quark mass, . Only is changed at one loop because the heavy quark slows down the running of between and . A standard perturbative matching computation then leads to,
We see that is not unnaturally sensitive to . Doubling only leads to a doubling of , as compared to the extreme and unnatural sensitivity implied by the naive result, eq. (18). The quadratic sensitivity of scalar radiative corrections to the ultraviolet scale has been eliminated by having no scalar degree of freedom present at , only quarks and gluons. These constituents of the scalar are only logarithmically sensitive to .
To summarize, we were able to resolve the toy naturalness problem by giving the TG sector a very low compositeness scale and making the “gravitating” TSM particles into constituents of composite states. The toy composite “graviton” interacts with the compositeness “halo” that surrounds the TSM particles. At very low momenta this is indistinguishable from a direct coupling to the nearly on-shell TSM particles. If this is extrapolated to when the TSM particles are far off-shell, one runs into the naturalness problem. In reality though, the off-shell contributions are cut off by compositeness. The naive extrapolation misses this composite softness of the interactions in the TG dynamics. On the other hand the compositeness interactions are invisible in the lab, compared with the much stronger hard interactions of the TSM. The obvious regime to discover the compositeness dynamics is at intermediate distances, where TSM interactions are still neutralized but compositeness effects are unsuppressed in the TG dynamics.
4 The Effective Particle-String Scenario
The moral of the previous section is that the power-counting that points to the inevitability of the CCP only holds if the graviton is fundamental, not if it is “composite”. To exploit this observation we must ensure that there simply is no graviton at the energies at which we integrate out SM particles, en route to obtaining the long-distance theory of gravity. At these SM energies there should only be the degrees of freedom which will bind into the graviton at much lower energies. A second requirement is that the SM particles must couple to composite gravity, and yet their couplings to other SM particles must be point-like at least down to distances. Unlike the case of the scalar in the toy model though, the compositeness of the graviton cannot be accomplished within the ordinary Minkowski space quantum field theory of point particles. This is due to the following very general theorem : a theory in Minkowski space which admits a well-behaved, conserved energy-momentum tensor cannot have a graviton in its spectrum.
Fortunately, string theory evades this theorem and gives a sensible meaning to graviton compositeness. Though formulated in Minkowski space, its energy-momentum tensor is not “well-behaved” and there is a massless graviton in the spectrum, as discussed in ref. . In terms of the well-known similarity between string theory and QCD (which of course was important historically for the discovery of string theory ), the graviton can be thought of as a massless “glueball” of string theory. Now in QCD there are sum rules that can be derived in terms of the fundamental description which look miraculous or finely-tuned in terms of the hadronic description. They are not enforced by any symmetry but by the special nature of the dynamics. The same is true in string theory with respect to the cosmological constant.
For simplicity let us consider the case of the perturbative bosonic string in 26 dimensions . There are effectively two parameters, the string mass-scale, , which plays the role of the graviton compositeness scale, and the string coupling, . For , the string spectrum corresponds to an infinite number of “composite” particle-modes of varying spins and masses, including a graviton. The Planck scale is very large, . If we ignore the string principle, we can compute the -loop contributions to the cosmological constant from each of the particles. Each contribution is quartically sensitive to the particle mass, and there are an infinite number of such contributions. Clearly we must introduce an ultraviolet cutoff, which cuts off both the infinity of contributions and the infinity in each contribution. This still leaves many large contributions to the cosmological constant. If we want the renormalized cosmological constant to come out very small, we must also add a counterterm chosen very precisely to finely cancel the large -loop contributions. Of course, in string theory the sum of one-loop diagrams plus counterterm is not calculated in this piecemeal fashion, but rather at one stroke. The result is an ultraviolet finite cosmological constant, . Note, this is just the power-counting dependence on an ultraviolet cutoff, , in 26-dimensional general relativity. This illustrates our expectation that the compositeness scale should cut off the divergences of general relativity. Since the -dimensional Planck scale is given by
can be made arbitrarily small compared to by taking small enough . In Planck units this corresponds to a very low tension string theory.
However, in the usual string formulation of particle physics, the SM particles are also identified as string vibrational modes, and we must have so that the stringy excitations of the SM particles are too massive to appear in present-day experiments. Strings with such a large string-scale cannot solve the CCP. In fact we can integrate out the excited string states and return to eq. (5) and its unpleasant consequences. Instead, we wish to pursue the possibility that there are strings in the gravitational sector with extremely low string-scale, , but the SM particles are not themselves made of these strings. Instead SM particles are point-like, at least up to the weak scale. Just as the heavy quarks of the last section were surrounded by a light hadronic halo to form a heavy meson, the SM particles may be surrounded by a stringy halo with which the graviton string mode interacts.
As yet there are no fully realisitic candidates known within string theory, which are point-like on the string scale and can be identified with the SM particles. However the recently discovered solitonic D-branes do possess some promising qualitative features. For example, 0-branes are point-like objects with masses much larger than , which can probe a continuum spacetime down to distances much shorter than . At long distance their interactions conform to general relativistic expectations in terms of graviton exchange. At distances smaller than the composite graviton is an entirely inappropriate degree of freedom, and the force between 0-branes becomes intrinisically stringy .
I therefore propose that in nature the SM particles are dynamically more akin to 0-branes than they are to perturbative string modes such as the graviton. Since string theories are only consistently formulated with supersymmetry, it remains a problem to explain how supersymmetry ends up badly broken in the SM sector. Nevertheless, supposing this is possible, we would like to explicitly understand how the cosmological constant can be cut off by the scale of graviton compositeness, , rather than being sensitive to the much larger SM masses. Below I offer a picture of how this might work. I can make no pretence of rigor.
4.1 How the particle-string might solve the CCP
Let us consider a simple, abstracted version of our problem. To eliminate the complication of supersymmetry-breaking and compactification, let us simply work within bosonic string theory in 26 euclidean dimensions (turning a blind eye to the existence of a tachyon). This will be our gravitational string sector. Let the 0-brane of this theory represent a “SM particle”, with mass . For , the 0-branes are much more massive than the string scale. Strings are permitted to end on the 0-brane worldline, the attached string constituting a string “halo”. Closed strings, including the graviton, are emitted and absorbed by this halo, inducing gravitational interactions for the 0-branes. We want to estimate the contributions of virtual 0-brane loops to the cosmological constant, .101010Strictly speaking, the notion of a 0-brane perturbative loop expansion is ill-defined, since these 0-branes are so massive that their gravitational couplings are large. I will however use this language since it is the most familiar one, and because it is likely to apply to a more realistic construction. The naive power-counting guess, ignoring the string principle, would be . I will argue that the cosmological constant is instead set by the graviton compositeness scale, , so that .
The simplifying consideration is that the euclidean action for a particle of mass, , will suppress 0-brane world-line loops which are much bigger than . Thus on the string scale they are essentially point-like events in spacetime, to which string worldsheet boundaries can attach.111111This is very much like ordinary quantum field theory, where integrating out a massive field introduces local interactions for the light fields. In the string literature, such events are known as “D-instantons”. The sized strings should be insensitive to the tiny -scale structure.
The contribution to the cosmological constant due to D-instantons has been computed and the result is finite and of order . This result can be understood as follows. The cosmological constant correction is given by the sum of (first-quantized) connected string diagrams with no vertex operators, where the string worldsheet boundaries attach to the D-instanton. The dominant contribution from a single worldsheet is of order , corresponding to a disk topology, more complicated topologies being suppressed by powers of . The sign of this contribution requires a detailed calculation and is negative. The dominant contribution from worldsheets is given by identical disks, whose boundaries attach to the D-instanton. Their contribution is just the th power of the single-disk result, divided by a symmetry factor of !. Summing over gives the factor .
Thus we expect that the contribution to the cosmological constant from 0-brane loops is suppressed by (without being careful about the prefactor) and is therefore negligible for ! Therefore the cosmological constant is dominated by the string-loop correction discussed earlier,
4.2 Phenomenological aspects of this scenario
A fully realistic effective particle-string theory has not yet been constructed. I will just list some important features that it ought to have.
The theory must contain SM particles and critical strings. The particles must live in four spacetime dimensions and be point-like at least down to distances. The string length scale and compactification radii can however be much larger. Examples of four-dimensional particle-like behavior co-existing with strings, and large compactification radii “seen” only by the strings, have been found and discussed in refs. .
The theory must be unitary below the weak scale. It is permitted to break down above the weak scale, since we are not trying to guess the very high energy physics.
The spin-2 graviton must be the only massless non-SM state with couplings to matter (unless they are even weaker than gravity). Then unitarity ensures that at long distances the dynamics reduces to general relativity . For distances of order or smaller, the massive string physics will become important and general relativity must break down. The fact that gravity has already been tested at distances of a few centimeters without deviation from Newton’s Law, gives the bound,
The compositeness of gravity must make the cosmological constant insensitive to the large SM masses, its size being set instead by the compositeness scale, ,
This is also the power-counting result that follows from thinking of as an ultraviolet cutoff for the effective theory of general relativity.
To satisfy the bound of eq. (4), we must have,
If the string compactification radii are of order , the string coupling is given by,
This may seem absurdly small, but recall that in string theory, , and the stabilization of the dilaton vev is still not understood. It may be related to the other absurdly small number in nature, . In any case, small is not technically unnatural.
The new stringy physics must be negligible in SM experiments. While the strings have typical length mm, their couplings are so incredibly weak that they should not interefere with the SM interactions. At lab momenta, the strings should form an insubstantial cloud about the SM particles. In particular, they should not upset the theoretical agreement with SM experiments which are sensitive to very small mass splittings, such as kaon mixing or atomic structure. This may be of concern given eq. (29).
String theories are presently formulated with supersymmetry as an essential ingredient for full consistency, yet supersymmetry must appear broken by at least in the particle sector. This suggests a minimal supersymmetry breaking in the string sector of order . This scale may set the minimal permissible string-scale. If so, eV.
Finally, let us consider how this scenario might be experimentally tested. We expect that the nature of the gravitational force should dramatically change for distances smaller than the compositeness length scale, , in a manner which cannot be described by the exchange of a finite variety of massive particles (such as the light scalars discussed in refs. or , for example). For example, the interaction between a pair of 0-branes is described at long distance in terms of the exchange of massless closed string modes such as the graviton, while at short distance it is described by open strings connecting the 0-branes . In this regime the gravitational force can become weaker with shorter distances!
Eqs. (29) and (31) narrowly constrain the compositeness length scale at which the radical departures from general relativity (Newton’s Law) must occur,
It is therefore our very good fortune that this is just the range over which gravity will be sensitively tested in the experiment proposed in ref. . If composite gravity resolves the CCP as proposed here, it will show itself in this experiment and be quite distinct from any other “fifth force” phenomenon which can be described within field theory!
The Cosmological Constant Problem was argued to be intractable as a naturalness problem in effective field theory unless the graviton was “composite” with a scale of compositeness below eV. The standard model particles must also participate in this compositeness and yet retain their point-like behavior in accelerator experiments up to very high energies. The only sensible version of graviton compositeness that is known, is string theory. It was proposed that the standard model particles inherit their gravitational interactions by virtue of their their stringy “halos”, coupled to a gravitational string sector. The string-scale plays the role of the compositeness scale. It was argued that this stringiness can acceptably cut off contributions to the cosmological constant from ultraviolet mass scales. The mechanism is reminiscent of the relative insensitivity of light hadron masses to heavy quark masses in QCD. This was the basis for the detailed analogy discussed in the paper.
If this particle-string scenario is realized in nature, it will lead to a dramatic breakdown of Newton’s Law on the millimeter scale, which will be experimentally probed. On the theoretical side, much work still remains in order to construct a fully realistic effective particle-string theory and demonstrate its requisite properties. The particle-string scenario considered here would obviously also have deep implications for physics at the highest energy scales.
Finally, it is worth keeping in mind that there may be other, presently undiscovered, manifestations of graviton compositeness that can also reduce the sensitivity of the cosmological constant to ultraviolet mass scales. Fortunately, independently of the form of graviton compositeness which resolves the CCP, power-counting suggests that eq. (33) constrains the compositeness length scale. Therefore, the composite behavior should still show up in upcoming experimental tests of gravity at short distances.
This research was supported by the U.S. Department of Energy under grant #DE-FG02-94ER40818. I wish to thank Tom Banks, Sekhar Chivukula, Andrew Cohen, Nick Evans, Shamit Kachru, Martin Schmaltz and especially my father, R. M. Sundrum, and my wife, Jamuna Sundrum, for useful conversations on the subject of this paper.
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- John Stewart Bell
John Stewart Bell (
June 28 1928– October 1 1990) was a physicist, and the originator of Bell's Theorem, one of the most important theorems in quantum physics.
Life and work
He was born in
Belfast, Northern Ireland, and graduated in experimental physics at the Queen's University of Belfast, in 1948. He went on to complete a PhD at the University of Birmingham, specialising in nuclear physicsand quantum field theory. His career began with the British Atomic Energy Agency, in Malvern, Britain's, then Harwell Laboratory. After several years he moved to the European Center for Nuclear Research ( CERN, "Conseil Européen pour la Recherche Nucléaire"). Here he worked almost exclusively on theoretical particle physicsand on accelerator design, but found time to pursue a major avocation, investigating the fundamentals of quantum theory.
In 1964, after a year's leave from CERN that he spent at
Stanford University, the University of Wisconsin-Madisonand Brandeis University, he wrote a paper entitled "On the Einstein-Podolsky-Rosen Paradox" [John Bell, "Speakable and Unspeakable in Quantum Mechanics", p. 14] . In this work, he showed that the carrying forward EPR's analysis [Einstein, "et al.", "Can Quantum Mechanical Description of Physical Reality Be Considered Complete?"] permits one to derive the famous Bell's inequality. This inequality, derived from some basic philosophical assumptions, conflicts with the predictions of quantum mechanics.
There is some disagreement regarding what Bell's inequality—in conjunction with the EPR paradox—can be said to imply. Bell held that not only local hidden variables, but any and all local theoretical explanations must conflict with quantum theory: "It is known that with Bohm's example of EPR correlations, involving particles with spin, there is an irreducible
nonlocality." [Bell, p. 196 ] According to an alternate interpretation, not all local theories in general, but only local hidden variables have shown incompatibility with quantum theory.
Despite the fact that hidden variable schemes are often associated with the issue of indeterminism, or uncertainty, Bell was instead concerned with the fact that orthodox quantum mechanics is a subjective theory, and the concept of measurement figures prominently in its formulation. It was not that Bell found measurement unacceptable in itself. He objected to its appearance at quantum mechanics' most fundamental theoretical level, which he insisted must be concerned only with sharply-defined mathematical quantities and unambiguous physical concepts.
In Bell's words: "The concept of 'measurement' becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level... does not any analysis of measurement require concepts more fundamental than measurement? And should not the fundamental theory be about these more fundamental concepts?" [Bell, p. 117 ]
Bell was impressed that within Bohm’s
nonlocal hidden variable theory, reference to this concept was not needed, and it was this which sparked his interest in the field of research.
But if he were to thoroughly explore the viability of Bohm's theory, Bell needed to answer the challenge of the so-called impossibility proofs against hidden variables. Bell addressed these in a paper entitled "On the Problem of Hidden Variables in Quantum Mechanics". [ Bell, p.1 ] Here he showed that von Neumann’s argument [John von Neumann, "Mathematical Foundations of Quantum Mechanics"] does not prove impossibility, as it claims. The argument fails in this regard due to its reliance on a physically unreasonable assumption. In this same work, Bell showed that a stronger effort at such a proof (based upon
Gleason's theorem) also fails to eliminate the hidden variables program. (The flaw in von Neumann's proof was previously discovered by Grete Hermannin 1935, but did not become common knowledge until rediscovered by Bell.)
If these attempts to disprove hidden variables failed, can Bell's resolution of the EPR paradox be considered a success? According to Bell's interpretation, quantum mechanics itself has been demonstrated to be irreducibly nonlocal. Therefore, one cannot fault a hidden variables scheme if, as in the pilot wave theory of de Broglie and Bohm, it includes "
superluminal signalling", i.e., nonlocality.
In 1972 the first of many experiments that have shown (under the extrapolation to ideal detector efficiencies) a violation of Bell's Inequality was conducted. Bell himself concludes from these experiments that "It now seems that the non-locality is deeply rooted in quantum mechanics itself and will persist in any completion." [Bell, p. 132] This, according to Bell, also implied that quantum mechanics cannot be embedded into a locally causal hidden variables theory.
Bell remained interested in objective 'observer-free' quantum mechanics. He stressed that at the most fundamental level, physical theories ought not to be concerned with observables, but with 'be-ables': "The beables of the theory are those elements which might correspond to elements of reality, to things which exist. Their existence does not depend on 'observation'." [Bell, p. 174] He remained impressed with Bohm's hidden variables as an example of such a scheme and he attacked the more subjective alternatives such as the Copenhagen and Everett "many-worlds" interpretations. [Bell, p. 92, 133, 181]
Bell seemed to be quite comfortable with the notion that future experiments would continue to agree with quantum mechanics and violate his inequalities. Referring to the
Bell test experiments, he remarked:
::"It is difficult for me to believe that quantum mechanics, working very well for currently practical set-ups, will nevertheless fail badly with improvements in counter efficiency ..." [Bell, p. 109]
Some people continue to believe that agreement with Bell's inequalities might yet be saved. They argue that in the future much more precise experiments could reveal that one of the known loopholes, for example the so-called "fair sampling loophole", had been biasing the interpretations. This latter loophole, first publicized by Philip Pearle in 1970 [Philip Pearle, "Hidden-Variable Example Based upon Data Rejection"] , is such that "increases" in counter efficiency "decrease" the measured quantum correlation, eventually destroying the empirical match with quantum mechanics. Most mainstream physicists are highly skeptical about all these "loopholes", admitting their existence but continuing to believe that Bell's inequalities must fail.
Bell died unexpectedly of a
cerebral hemorrhagein Belfast in 1990. His contribution to the issues raised by EPR was significant. Some regard him as having demonstrated the failure of local realism (local hidden variables). Bell's own interpretation is that locality itself met its demise.
Bell's theorem, published in the mid-1960s
Bell's spaceship paradox
EPR paradox, a thought experiment by Einstein, Podolsky, and Rosen published in 1935 as an attack on quantum theory
CHSH Bell test, an application of Bell's theorem
Quantum mechanical Bell test prediction
Local hidden variable theory
*Aczel, Amir D, "Entanglement: The Greatest Mystery in Physics" (2001), New York: Four Walls Eight Windows
*Bell, John S, "Speakable and Unspeakable in Quantum Mechanics" (1987), Cambridge University Press, ISBN 0-521-36869-3, 2004 edition with introduction by
Alain Aspectand two additional papers: ISBN 0-521-52338-9
*Einstein, Podolsky, Rosen, "Can Quantum Mechanical Description of Physical Reality Be Considered Complete?", "Phys. Rev." 47, 777 (1935).
*von Neumann, John, "Mathematical Foundations of Quantum Mechanics" (1932), Princeton University Press 1996 edition: ISBN 0-691-02893-1
*Pearle, Philip, "Hidden-Variable Example Based upon Data Rejection", Physical Review D, 2, 1418-25 (1970)
* [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bell_John.html MacTutor profile (University of St. Andrews)]
* [http://physicsweb.org/articles/world/11/12/8 John Bell and the most profound discovery of science (December 1998)]
* [http://www.rds.ie/home/index.aspx?id=1755 The Most Profound Discovery of Science (September 2006)]
Wikimedia Foundation. 2010. |
The transient air mass balance equation for the change in the zone humidity ratio = sum of internal scheduled latent loads + infiltration + system + multizone airflows + convection to the zone surfaces may be expressed as follows:
CW = humidity capacity multiplier (See the InputOutput Reference for additional information on the object ZoneCapacitanceMultiplier:ResearchSpecial)
In the same manner as described above for zone air temperature (ref. Basis for the Zone and Air System Integration), the solution algorithms provided in the ZoneAirHeatBalanceAlgorithm object are also applied to zone air moisture calculations.
In order to calculate the derivative term with respect to time, the first order backward finite difference method, defined as the EulerMethod in the ZoneAirHeatBalanceAlgorithm object, may be used:
The zone air humidity ratio update at the current time step using the EulerMethod may be expressed as follows:
To preserve the stability of the calculation of the zone humidity ratio, the third order differential approximation, derived by a Taylor Series and used in the calculation of the next time step’s zone air temperature, is also applied to the zone air humidity ratio calculations. This algorithm is the default choice and is defined as 3rdOrderBackwardDifference in the ZoneAirHeatBalanceAlgorithm object.
The third order derivative derived from a Taylor Series expansion is defined as:
The coefficients of the approximated derivative are very close to the coefficients of the analogous Adams-Bashforth algorithm. Then the approximated derivative is substituted into the mass balance and the terms with the humidity ratio at past time steps are all put on the right hand side of the equation. This third order derivative zone humidity ratio update increases the number of previous time steps that are used in calculating the new zone humidity ratio, and decreases the dependence on the most recent. The higher order derivative approximations have the potential to allow the use of larger time steps by smoothing transitions through sudden changes in zone operating conditions.
This gives us the basic air mass balance equation that will be solved two different ways, one way for the predict step and one way for the correct step.
Since the third choice of solution algorithms uses an integration approach, defined as AnalyticalSolution in the ZoneAirHeatBalanceAlgorithm object, it does not require any approximations and has no truncation errors. The solutions in both prediction and correction are provided below in detail.
For the moisture prediction case the equation is solved for the anticipated system response as shown below.
Since the program provides three solution algorithms, the moisture prediction from each solution algorithm is given below.
For this solution algorithm, the air mass balance for the predicted air system load or response is:
For this solution algorithm, the air mass balance for the predicted system load or response is given below:
Then, using the following substitutions, the air mass balance equation becomes:
For this solution algorithm, the air mass balance for the predicted air system load or response is given below:
PredictedSystemLoad[kgWater/sec]=[Nsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙miWzi+˙minf]∗⎡⎢ ⎢ ⎢⎣Wtsetpoint−Wt−δtz∗exp⎛⎜ ⎜ ⎜⎝−Nsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙miWzi+˙minfρairVzCWδt⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥⎦∗⎡⎢ ⎢ ⎢⎣1−exp⎛⎜ ⎜ ⎜⎝−Nsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙mi+˙minfρairVzCWδt⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥⎦−1−(Nsl∑i=1kgmassschedloadNsurfaces∑i=1AihmiρairzWsurfsi+Nzones∑i=1˙miWzi+˙minfW∞)
At the prediction point in the simulation, the system air mass flows are not known; therefore, the system response is approximated. The predicted air system moisture load is then used in the system simulation to achieve the best results possible. The system simulation components that have moisture control will try to meet this predicted moisture load. For example, humidifiers will look for positive moisture loads and add moisture at the specified rate to achieve the relative humidity setpoint. Likewise, dehumidification processes will try to remove moisture at the specified negative predicted moisture load to meet the relative humidity setpoint.
After the system simulation is completed the actual response from the air system is used in the moisture correction of step, which is shown next.
For the correct step the expanded air mass balance equation is solved for the final zone humidity ratio at the current time step. When the air system is operating, the mass flow for the system outlet includes the infiltration mass flow rate, therefore the infiltration mass flow rate is not included as a separate term in the air mass balance equation. But when the air system is off, the infiltration mass flow in is then exhausted out of the zone directly.
In the same manner as described above for predicting the moisture load to be met by the air system, the zone air moisture correction calculation will be described individually for the three solution algorithms.
Using the same A, B, and C parameters from the prediction step modified with actual zone mass flows with the air system ON and OFF result in:
If (ZoneSupplyAirMassFlowRate > 0.0) Then
Else If (ZoneSupplyAirMassFlowRate < = 0.0) Then
Inserting in the parameters A, B and C above in the air mass balance equation, it simplifies to:
Wtz=⎡⎢ ⎢⎣B+C∗(3Wt−δtz−32Wt−2δtz+13Wt−3δtz)(116)∗C+A⎤⎥ ⎥⎦
Wtz=⎡⎢ ⎢ ⎢⎣Wt−δtz−Nsl∑i=1kgmassschedload+Nsurfaces∑i=1AihmiρairzWsurfsi+Nzones∑i=1˙miWzi+˙minfW∞+˙msysWsupNsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙mi+˙minf+˙msys⎤⎥ ⎥ ⎥⎦∗exp⎛⎜ ⎜ ⎜⎝−Nsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙mi+˙minf+˙msysρairVzCWδt⎞⎟ ⎟ ⎟⎠+Nsl∑i=1kgmassschedload+Nsurfaces∑i=1AihmiρairzWsurfsi+Nzones∑i=1˙miWzi+˙minfW∞+˙msysWsupNsurfaces∑i=1Aihmiρairz+Nzones∑i=1˙mi+˙minf+˙msys
The above solutions are implemented in the Correct Zone Air Humidity Ratio step in EnergyPlus. This moisture update equation is used for the Conduction Transfer Function (CTF) heat balance algorithm, in addition to the effective moisture penetration depth (EMPD) with conduction transfer function heat balance algorithm. The equations are identical except that the convection to the zone surfaces is non-zero for the moisture penetration depth case. This moisture update allows both methods to be updated in the same way, with the only difference being the additional moisture capacitance of the zone surfaces for the Effective Moisture Penetration Depth (EMPD) solution approach.
When the HAMT (Combined Heat And Moisture Finite Element) defined in the HeatBalanceAlgorithm object is applied, the moisture update equations are also the same as the equations used in the effective moisture penetration depth (EMPD) with conduction transfer function solution algorithm.
Which moisture buffering model is best?[LINK]
The ’correct’ moisture buffering model depends on the questions being answered by the building energy simulation. Previous research (Woods et al., 2013a) has shown that using the effective capacitance model to account for moisture buffering of materials will provide a good estimate of energy use when humidity is not being actively controlled. See the InputOutput Reference for additional information on the object ZoneCapacitanceMultiplier:ResearchSpecial. This model has some limitations (Woods et al., 2013b):
- it will not accurately predict indoor humidity (or thermal comfort),
it will not accurately predict energy use when humidity is being actively controlled, and
it will not provide insight into the moisture content and potential moisture problems associated with a specific wall construction.
The effective moisture penetration depth (EMPD) model will address the first two concerns above: it can accurately predict indoor humidity, and can accurately predict energy use associated with controlling humidity. The EMPD model requires more user input than the effective capacitance model, specifically some of the moisture properties of the materials in the building. For more information, see the Effective Moisture Penetration Depth Model section in this document.
Like the EMPD model, the combined heat, air, and moisture transfer (HAMT) model addresses the first two issues discussed above for the effective capacitance model. It also addresses the third, by providing temperature and moisture profiles through composite building walls, and helping to identify surfaces with high surface humidity. The HAMT model requires a few more user inputs on moisture properties of materials than the EMPD model, and this model also increases the required simulation time by an order of magnitude. For more information on this model, see the Combined Heat and Moisture Transfer (HAMT) Model section in this document.
Note that the EMPD and HAMT models above ensure accurate calculations of the effect of moisture buffering, but it will only be accurate relative to reality when given appropriate inputs for the material properties.
Woods, J., J. Winkler, D. Christensen, Moisture modeling: Effective moisture penetration depth versus effective capacitance, in Thermal Performance of the Exterior Envelopes of Whole Buildings XII International Conference. 2013a: Clearwater, FL.
Woods, J., Winkler, J, and Christensen, D. Evaluation of the Effective Moisture Penetration Depth Model for Estimating Moisture Buffering in Buildings, NREL/TP-5500-57441, 2013b. |
At IGotAnOffer, we have helped more than 30,000 candidates prepare for their consulting interviews. Students who go through our full training programme are a happy bunch: more than 80% of them get an offer at McKinsey, BCG or Bain.
Developing fast and accurate maths skills is a big part of being successful at case interviews. In the following guide we've listed a number of free tools, formulas and tips you can use to become much faster at maths and radically improve your chances of getting an offer.
Part 1: Case maths apps and tools
Mental maths is a muscle. But if you are like us, you probably haven't exercised that muscle much since you left high school. As a consequence, your case interview preparation should include some maths training.
If you don't remember how to calculate basic additions, substractions, divisions and multiplications without a calculator that's what you should focus on first. Our McKinsey and BCG & Bain case interview programmes both include a refresher on the topic.
In addition, Khan Academy has also put together helpful resources. Here are the ones we recommend taking a look at if you need an in-depth arithmetic refresher:
Once you're feeling comfortable with the basics you'll need to regularly exercise your mental maths muscle in order to become as fast and accurate as possible. Our McKinsey and BCG & Bain case interview programmes include a calculation workbook PDF with maths drills. We recommend doing a few everyday so you get more and more comfortable over time.
In addition, you can also use the following resources. We haven't tested all of them but some of the candidates we work with have used them in the past and found them helpful.
- Preplounge's maths tool. This web tool is very helpful to practice additions, subtractions, multiplications, divisions and percentages. You can both sharpen your precise and estimation maths with it.
- Victor Cheng's maths tool. This tool is similar to the Preplounge one but the user experience is less smooth in our opinion.
- Magoosh's mental maths app (iOS + Android). If you want to practice your mental maths on the go this free mobile app is great. It lets you work on different types of calculations using mental maths flashcards. You can also track your progress as you study.
- Mental math cards challenge app (iOS). This mobile app let's you work on your mental maths in a similar way to the previous one. Don't let the old school graphics deter you from using it. The app itself is actually very good.
- Mental math games (Android). If you're an Android user this one is a good substitute to the mental math cards challenge one on iOS.
Part 2: Case maths formulas
The links we have listed above should go a long way in helping you bring your maths skills to a good level. In addition, you will also need to learn the formulas for the main business and finance concepts you will come across in your interviews.
We've put together a list of the important maths formulas for you with concepts that you should really master for your interviews and concepts which are optional in our experience.
2.1. Must-know maths formulas
Revenue = Volume x Price
Cost = Fixed costs + Variable costs
Profit = Revenue - Cost
Profit margin (aka Profitability) = Profit / Revenue
Return on investment (ROI) = Annual profit / Initial investment
Breakeven (aka Payback period) = Initial investment / Annual profit
If you have any questions about the formulas above you can ask them at the bottom of this page and our team will answer them. Alternatively, you can also read our in-depth articles about finance concepts for case interviews and for the McKinsey PST or watch this video where we explain these concepts in great detail.
2.2. Optional maths formulas
Having an in-depth knowledge of the business terms below and their corresponding formula is NOT required to get offers at McKinsey, BCG, Bain and other firms in our experience. But having a rough idea of what they are can be handy.
EBITDA = Earnings Before Interest Tax Depreciation and Amortisation
EBIDTA is essentially profits with interest, taxes, depreciation and amortization added back to it. It's useful to compare companies across industries as it takes out the accounting effects of debt and taxes which vary widely between say Facebook (little to no debt) and ExxonMobil (tons of debt to finance infrastructure projects). More here.
NPV = Net Present Value
Let's say you invest $1,000 in project A and $1,000 in project B. You expect to receive your initial investment + $500 from A in one week. And you expect to receive your initial investment + $500 from B in 5 years. Intuitively you probably feel that A is more valuable than B as you'll get the same amount of money but quicker. NPV aims to adjust future cashflows so different investments such as A and B can easily be compared. More here.
Return on equity = Profits / Shareholder equity
Return on equity (ROE) is a measure of financial performance similar to ROI. ROI is usually used for standalone projects while ROE is used for companies. More here.
Return on assets = Profits / Total assets
Return on assets (ROA) is an alternative measure to ROE and a good indicator of how profitable a company is compared to its total assets. More here.
Part 3: Fast maths tips and tricks
The standard long divisions and multiplications approaches are great because they're generic and you can use them for any calculation. But they are also extremely slow. In our experience, you can become MUCH faster at maths by using non-standard approaches we've listed below.
All these approaches have ONE thing in common: they aim at rearranging and simplifying calculations to find the EASIEST path to the result. Let's step through each of them one by one.
3.1. Rounding numbers
The first step towards becoming faster is to round numbers whenever you can. 365 days becomes 350. The US population of 326m becomes 300m. Etc. You get the idea.
The tricky thing about rounding numbers is that if you round them too much you risk a) distorting the final result / finding, and b) your interviewer telling you to round the numbers less.
Rounding numbers is more of an art than a science, but in our experience the following two tips tend to work well:
- We usually recommend to not round numbers by more than +/- 10%. This is a rough rule of thumb but gives good results based on conversations with past candidates.
- You also need to alternate between rounding up and rounding down so the effects cancel out. For instance, if you're calculating A x B, we would recommend rounding A UP, and rounding B DOWN so the roundings compensate each other.
Note you won't always be able to round numbers. In addition, even after you round numbers the calculations could still be difficult. So let's go through a few tips that can help in these situations.
3.2. Handling large numbers
Large numbers are difficult to deal with because of all the 0s. To be faster you need to use notations that enable you to get rid of these annoying 0s. We recommend you use labels and the scientific notation if you aren't already doing so.
Labels (k, m, b)
Use labels for thousand (k), million (m) and billion (b). You'll write numbers faster and it will force you to simplify calculations. Let's use 20,000 x 6,000,000 as an example.
- No labels: 20,000 x 60,000,000 = ... ???
- Labels: 20k x 6m = 120k x m = 120b
This approach also works for divisions. Let's try 480,000,000,000 divided by 240,000,000.
- No labels: 480,000,000,000 / 240,000,000 = ... ???
- Labes: 480b / 240m = 480k / 240 = 2k
When you can't use labels, the scientific notation is a good alternative. If you're not sure what this is, you're really missing out. But fortunately Khan Academy has put together a good primer on the topic here.
- Multiplication example: 600 x 500 = 6 x 5 x 102 X 102 = 30 x 104 = 300,000 = 300k
- Division example: (720,000 / 1,200) / 30 = (72 / (12 x 3)) x (104 / (102 x 10)) = (72 / 36) x (10) = 20
When you're comfortable with labels and the scientific notation you can even start mixing them:
- 200k x 600k = 2 x 6 x 104 x m = 2 x 6 x 10 x b = 120b
To be fast at maths, you need to avoid writing down long divisions and multiplications as they take a LOT of time. In our experience, doing multiple easy calculations is faster and leads to less errors than doing one big long calculation.
A great way to achieve this is to factor and expand expressions to create simpler calculations. If you're not sure what the basics of factoring and expanding are, you can use Khan Academy again here and here. Let's start with factoring.
Simple numbers: 5, 15, 25, 50, 75, etc.
In case interviews and tests like the McKinsey PST or BCG Potential Test some numbers come up very frequently and it's useful to know shortcuts to handle them. Here are some of these numbers: 5, 15, 25, 50, 75, etc. These numbers are frequent but not particularly easy to deal with.
For instance, consider 36 x 25. It's not obvious what the result is. And a lot of people would need to write down the multiplication on paper to find the answer. However there's a MUCH faster way based on the fact that 25 = 100 / 4. Here's the fast way to get to the answer:
- 36 x 25 = (36 / 4) x 100 = 9 x 100 = 900
- 68 x 25 = (68 / 4) x 100 = 17 x 100 = 1,700
- 2,600 / 25 = (2,600 / 100) x 4 = 26 x 4 = 104
- 1,625 / 25 = (1,625 / 100) x 4 = 16.25 x 4 = 65
- 2.5 = 10 / 4
- 5 = 10 / 2
- 7.5 = 10 x 3 / 4
- 15 = 10 x 3 / 2
- 25 = 100 / 4
- 50 = 100 / 2
- 75 = 100 x 3 / 4
Once you're comfortable using this approach you can also mix it with the scientific notation on numbers such as 0.75, 0.5, 0.25, etc.
Factoring the numerator / denominator
For divisions, if there are no simple numbers (e.g. 5, 25, 50, etc.), the next best thing you can do is to try to factor the numerator and / or denominator to simplify the calculations. Here are a few examples:
- Factoring the numerator: 300 / 4 = 3 x 100 / 4 = 3 x 25 = 75
- Factoring the denominator: 432 / 12 = (432 / 4) / 3 = 108 / 3 = 36
- Looking for common factors: 90 / 42 = 6 x 15 / 6 x 7 = 15 / 7
Another easy way to avoid writing down long divisions and multiplications is to expand calculations into simple expressions.
Expanding with additions
Expanding with additions is intuitive to most people. The idea is to break down one of the terms into two simpler numbers (e.g. 5; 10; 25; etc.) so the calculations become easier. Here are a couple of examples:
- Multiplication: 68 x 35 = 68 x (10 + 25) = 680 + 68 x 100 / 4 = 680 + 1,700 = 2,380
- Division: 705 / 15 = (600 + 105) / 15 = (15 x 40) / 15 + 105 / 15 = 40 + 7 = 47
Notice that when expanding 35 we've carefully chosen to expand to 25 so that we could use the helpful tip we learned in the factoring section. You should keep that in mind when expanding expressions.
Expanding with subtractions
Expanding with subtractions is less intuitive to most people. But it's actually extremely effective, especially if one of the terms you are dealing with ends with a high digit like 7, 8 or 9. Here are a couple of examples:
- Multiplication: 68 x 35 = (70 - 2) x 35 = 70 x 35 - 70 = 70 x 100 / 4 + 700 - 70 = 1,750 + 630 = 2,380
- Division: 570 / 30 = (600 - 30) / 30 = 20 - 1= 19
3.5. Growth rates
Finally, you will also often have to deal with growth rates in case interviews. These can lead to extremely time-consuming calculations so it's important that you learn how to deal with them efficiently.
Multiply growth rates together
Let's imagine your client's revenue is $100m. You estimate it will grow by 20% next year and 10% the year after that. In that situation, the revenues in two years will be equal to:
- Revenue in two years = $100m x (1 + 20%) x (1 + 10%) = $100m x 1.2 x 1.1 = $100m x (1.2 + 0.12) = $100m x 1.32 = $132m
Growing at 20% for one year followed by 10% for another year therefore corresponds to growing by 32% overall. To find the compound growth you simply need to multiply them together and subtract one: (1.1 x 1.2) - 1= 1.32 - 1 = 0.32 = 32%. This is the quickest way to calculate compound growth rates precisely.
Note that this approach also works perfectly with negative growth rates. Let's imagine for instance that sales grow by 20% next year, and then decrease by 20% the following year. Here's the corresponding compound growth rate:
- Compound growth rate = (1.2 x 0.8) - 1 = 0.96 - 1 = -0.04 = -4%
Note how growing by 20% and then shrinking by 20% is not equal to flat growth (0%). This is an important result to keep in mind.
Estimate compound growth rates
Multiplying growth rates is a really efficient approach when calculating compound growth over a short period of time (e.g. 2 or 3 years). But let's imagine you want to calculate the effect of 7% growth over five years. The precise calculation you would need to do is:
- Precise growth rate: 1.07 x 1.07 x 1.07 x 1.07 x 1.07 - 1 = ... ???
- Estimate growth rate = Growth rate x Number of years
In our example:
- Estimate growth rate: 7% x 5 years = 35%
In reality if you do the precise calculation (1.075 - 1) you will find that the actual growth rate is 40%. The estimation method therefore gives a result that's actually quite close. In case interviews your interviewer will always be happy with you taking that shortcut as doing the precise calculation takes too much time.
If you would like to fast track your case interview preparation and maximise your chances of getting an offer at McKinsey, BCG or Bain, come and train with us. More than 80% of the candidates training with our programmes end up getting an offer at their target firm. We know this because we give half of their money back to people who don't.
McKinsey Case Interview Training Programme
BCG & Bain Case Interview Training Programme
Any questions about case interview maths?
If you have any questions about case interview maths, do not hesitate to ask them below and we will be more than happy to answer them. All questions are good questions, so go ahead!
The IGotAnOffer team |
Compass Surveying and Theodolite
Download Solution PDF
The line of sight generates a vertical plane when:
This question was previously asked in
MP Sub Engg Official Civil Paper Held on 9th July 2017 - Shift 2
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Vertical axis is parallel to horizontal axis
Vertical axis is perpendicular to horizontal axis
Horizontal axis is parallel to vertical axis
Horizontal axis is perpendicular to vertical axis
(Detailed Solution Below)
Option 4 : Horizontal axis is perpendicular to vertical axis
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Building Material & Concrete Technology
Download Solution PDF
Line of sight:
It is the
passing through the
intersection of the crosshair on the diaphragm
optical center of the objective lens.
when a line of sight comes in a horizontal plane, it is called a
line of collimation.
line of sight
horizontal axis is perpendicular to the vertical axis.
Fundamental lines of theodolite:
The fundamental lines of a theodolite are the vertical axis, the axis of plate levels, the line of collimation, the horizontal axis, and the bubble line of altitude.
When the theodolite is in proper adjustment, the following conditions should be satisfied.
Horizontal circle perpendicular to the vertical axis.
Vertical circle perpendicular to a horizontal axis
The vertical axis must pass through the center of the graduated horizontal circle.
The horizontal axis must pass through the center of the vertical circle.
Tangent to plate bubble must be perpendicular to the vertical axis.
Line of sight must be perpendicular to the transit axis (trunnion axis)
Transit axis(Horizontal axis or trunnion axis) must be perpendicular to the vertical axis.
For the horizontal position of the telescope and for the altitude bubble at the center, the reading on the vertical circle must be zero.
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More Compass Surveying and Theodolite Questions
Reiteration method is also called as
In theodolites, the axis of rotation of telescope in the vertical plane indicates:
Which of the following statements is/are incorrect about the Prismatic Compass? A. The needle is broad but it does not act as an index. B. The graduated ring is attached with the needle. This does not rotate along with the line of sight. C. The readings are taken directly seeing through the top of the glass.
The angle between true meridian and magnetic meridian is termed as:
The Theodolite is an instrument used for measuring very accurately
Agate cap is fitted with a-
In order to measure the magnetic bearing of a line, the theodolite should be provided with-
In a quadrantal bearing system the angle is N44° - 30' W in whole bearing system it will be:
What is the whole circle bearing of a quadrant bearing N 15° 28' W?
Agonic line is the line joining points having declination-
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Choose the correct statement. a) The sum of the measured interior angles should be equal (2N - 4) right angles. b) If the exterior angles are measured, their sum should be equal to (2N + 4) right angle. Where N is the number of sides of the traverse.
Convert 327°24' whole circle bearing to quadrantal bearing. Select the correct option.
A flagpole appears in two successive photographs taken at an altitude of 2,000 m above datum. The focal length of the camera is 120 mm and the length of the air base is 200 m. The parallax for the top of the pole is 52.52 mm and for the bottom is 48.27 mm. Find the difference in elevation between the top and the bottom of the pole.
______ is the process of rephotographing an aerial photograph so that the effects of tilt are eliminated.
The point on the upper portion of the celestial sphere marked by the plumb line above the observer is called the:
Instrument used for ocean sounding where the depth of water is too much, and to make a continuous and accurate record of the depth of water below the boat or ship at which it is installed, is called as:
The process of determining the differences of elevations of stations from observed vertical angles and known distances is known as:
A surveyor measured the distance between two points on the plan drawn to a scale 1 cm = 40 m and the result was 468 m. Later however, he discovered that he had used a scale of 1 cm = 20 m. Find the true distance between the two points.
The line passing through the intersection of the horizontal and vertical cross hairs and optical centre of the subject glass and its continuation is called:
The magnetic bearing of a line AB is 48°24. Calculate the true bearing if the magnetic declination is E 5°38.
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Integrable and superintegrable systems with spin in three-dimensional Euclidean space
A systematic search for superintegrable quantum Hamiltonians describing the interaction between two particles with spin and , is performed. We restrict to integrals of motion that are first-order (matrix) polynomials in the components of linear momentum. Several such systems are found and for one non-trivial example we show how superintegrability leads to exact solvability: we obtain exact (nonperturbative) bound state energy formulas and exact expressions for the wave functions in terms of products of Laguerre and Jacobi polynomials.
PACS numbers: 02.30.Ik, 03.65.-w, 11.30.-j, 25.80.Dj
The purpose of this research program is to perform a systematic study of integrability and superintegrability in the interaction of two particles with spin. Specifically in this article we consider a system of two nonrelativistic particles, one with spin (e.g. a nucleon) the other with spin (e.g. a pion), moving in the three-dimensional Euclidean space .
The Pauli-Schrödinger equation in this case will have the form
where the term represents the spin-orbital interaction. We use the notation
for the linear momentum, angular momentum and Pauli matrices, respectively. The curly bracket in (1.1) denotes an anticommutator.
For spinless particles the Hamiltonian
is a scalar operator, whereas in (1.1) is a matrix operator and is a two component spinor.
In the spinless case (1.5) the Hamiltonian is integrable if there exists a pair of commuting integrals of motion , that are well-defined quantum mechanical operators, such that , and are algebraically independent. If further algebraically independent integrals exist, the system is superintegrable. The best known superintegrable systems in are the hydrogen atom and the harmonic oscillator. Each of them is maximally superintegrable with independent integrals, generating an and an algebra, respectively [1, 2].
A systematic search for quantum and classical superintegrable scalar potentials in (1.5) with integrals that are first- and second-order polynomials in the momenta was performed some time ago [3, 4, 5]. First-order integrals correspond to geometrical symmetries of the potential, second-order ones are directly related to the separation of variables in the Schrödinger equation or Hamilton-Jacobi equation in the classical case [3, 6, 7, 8].
First- and second-order integrals of motion are rather easy to find for Hamiltonians of the type (1.5) in Euclidean space. The situation with third- and higher-order integrals is much more difficult [9, 10, 11].
If a vector potential term, corresponding e.g. to a magnetic field is added, the problem becomes much more difficult and the existence of second-order integrals no longer implies the separation of variables [12, 13].
The case (1.1) with a spin-orbital interaction turns out to be quite rich and rather difficult to treat systematically. In a previous article we have considered the same problem in . Here we concentrate on the Hamiltonian (1.1) in but restrict to first-order integrals. Thus we search for integrals of motion of the form
where , , , and , , , () are all scalar functions of .
In Section 2 we show that a spin-orbital interaction of the form
can be induced by a gauge transformation from a purely scalar potential (in particular from ). In Section 3 we derive and discuss the determining equations for the existence of first-order integrals. In Section 4 we restrict to rotationally invariant potentials and and classify the integrals of motion into multiplets. Solutions of the determining equations are obtained in Section 5. Superintegrable potentials are discussed in Section 6. In Section 7 we solve the Pauli-Schrödinger equation for one superintegrable system explicitly and exactly. Finally, the conclusions and outlook are given in Section 8.
2 Spin-orbital interaction induced by a gauge transformation
In this Subsection we show that a spin-orbit term could be gauge induced from a scalar Hamiltonian (1.5) by a gauge transformation. The transformation matrix must be an element of
where () are real functions of (). It is seen that in order to generate a spin-orbit term we need to have
where is an arbitrary real scalar function of (). Equation (2.2) implies first-order partial differential equations for and , three of which are , . Hence, without loss of generality we choose and then write the remaining equations as
which could be solved for the highest-order derivatives of (i.e. , and ). Then, the compatibility conditions of these give
which implies that . Hence, we conclude that is gauge induced and it is the only potential which could be generated from a scalar Hamiltonian by a gauge transformation.
The explicit form of the gauge transformation is found as
where , and are the following constants
With this transformation matrix the transformed Hamiltonian is found to be
2.2 Integrals for and
The potential is gauge induced from a Hamiltonian of the form (1.5) (though each term is multiplied by a identity matrix). Hence the integrals for this case are just the gauge transforms of the integrals of motion of this Hamiltonian (i.e. and ). They can be written as
and satisfy the following commutation relations
The Lie algebra is isomorphic to a direct sum of the algebra with itself
2.3 Integrals for and
Since these potentials are gauge induced from a free Hamiltonian, the integrals are just the gauge transforms of , and , which can be written as
They satisfy the following commutation relations
Hence the -dimensional Lie algebra is isomorphic to a direct sum of the Euclidean Lie algebra with the algebra
3 Determining equations for an integral of motion
In this Subsection we give the full set of determining equations obtained from the commutativity condition , where is the Hamiltonian given in (1.1) and is the most general first-order integral of motion given in (1.6). This commutator has second-, first- and zero-order terms in the momenta. By setting the coefficients of different powers of the momenta equal to zero in each entry of this matrix we obtain the following determining equations. Since, the Planck constant enters into the determiming equations in a nontrivial way we keep it throughout the whole set of determining equations. However, after giving the determining equations we set for simplicity.
i) Determining equations coming from the second-order terms
From the diagonal elements it is immediately found that , and are linear functions and are expressed for any potentials and as
where and () are real constants. After introducing (3.1) into the rest of the coefficients of the second-order terms and separating the imaginary and real parts of the coefficients coming from the off-diagonal elements we are left with an overdetermined system of eighteen partial differential equations for , , () and . These are,
ii) Determining equations coming from the first-order terms
After introducing (3.1) and separating the real and imaginary parts, we have the following twelve partial differential equations
where , and are given in (3.1). There are also nine other second-order partial differential equations for , , and , coming from the coefficients of the first-order terms. However, these are differential consequences of (3.2) so we do not present them here.
iii) Determining equations coming from the zero-order terms
Setting the coefficients of the zero-order terms in each entry of the commutation relation equal to zero and separating the real and imaginary parts, we have the following four partial differential equations
3.2 Discussion of solution in general case
In general the solution of the determining equations (3.2)-(3.5) for the unknowns , , and , , , () turns out to be a difficult problem. However, it is seen that the determining equations (3.3) do not involve , , , () and and hence could be analyzed separately.
In order to determine the unknown functions and , we express the first-order derivatives of ’s from (3.3) and require the compatibility of the mixed partial derivatives. This requirement gives us another equations for ’s and first-order derivatives of them. Now, if we introduce the first-order derivatives of ’s from (3.3) into this system, we get a system of algebraic equations for ’s (). This system of algebraic equations can be written in the following way:
where is a matrix and and are and vectors, respectively. The matrix can be written as:
where () are defined as follows
The vector is and the entries of the vector are given as follows: |
Steps to declare the Math Major (Declaration Periods: Fall 9/15- 12/9 & Spring 2/1 - 4/17)
- Complete the calculus requirement (Math 1400 and Math 1410+2400 or Math 1610+2600) and one proof-based major course.
- Go to Path @ Penn to formally request to add or remove your major to the Math Office (otherwise it won't show on your transcripts).
- Use this link to complete the top portion of our course plan worksheet and email it to the major advisor we assign to you after we process your Path @ Penn request.
- After consulting with your advisor, he/she will review your worksheet for processing.
- Allow 15 business days for your major to appear on your transcript. If its not visible by day 15th email me at email@example.com
- Contact your math advisor once a semester. Email your advisor to answer questions or make changes to your course plan.
- Once you enter your semester of graduation, you must email your math major advisor to certify your math major worksheet for completion.
On This Page
- Admission to the Major Program Entrance Requirements for the Major.
- Background: The Major and its goals.
- What can I do with a math major?
- Advanced Placement Potential Credit for incoming freshmen.
- The Major Program Requirements A minimum of 13 credits are needed.
- Planning your Mathematics Major Our suggested plan for completing your major.
- Other Useful Experiences for a Math Major
- Mathematics Major: Biological Concentration
- The Mathematics Minor
- Further Recommendations Advice for students planning graduate study in mathematics or related fields.
- The Honors Program Requirements for a degree with honors.
- The Master's Program Submatriculation into graduate study at Penn.
- Advice General advice regarding the major.
- External links.
- Current Course Descriptions (for reference in course planning)
- Courses offered next semester
Permission to major in mathematics is normally obtained by the end of the sophomore year, but planning for it should begin as early as possible. It is important that majors entering their junior year commence satisfying the algebra and analysis requirements.
To be admitted to the major, a student must have completed successfully (i.e., with grades of C or better) the calculus requirement as well as one proof-based math class (such as Math 2020, 2030, 1610, 2600, 3140) in the freshman and sophomore years. A higher-level proof-based class may be substituted at the discretion of a math major advisor.
Students who plan to have math as their second major should have a cumulative GPA of at least 3.0, an average of at least 3.0 in their math courses, and no math grades lower than B-.
The Major is open to SEAS undergraduates (as a second major) as well as to students in the College.
Mathematical training allows one to take a problem, abstract its essential features, and investigate them further. This ability can assist greatly in such diverse fields as economics, law, medicine, engineering, and computer science -- as well as in the more traditional activities of research and teaching.
The goals of the major program are to assist students in acquiring both an understanding of mathematics and an ability to use it. We wish to inspire the discovery of new mathematics as well as the application of mathematics to other fields.
The mathematics major provides a solid foundation for graduate study in mathematics as well as background for study in economics, the biological sciences, the physical sciences and engineering, as well as many non-traditional areas. This flexibility is available through an appropriate choice of electives within the major. A variety of electives are offered. They are designed to serve the needs of mathematics majors and others who want more advanced training in mathematics and its applications. Most of these courses presume our basic two year calculus sequence.
The mathematics major is also excellent training for students interested in elementary and secondary education. For information on the elementary education undergraduate major or the secondary education submatriculation program which leads to a Master's degree, students should consult the Undergraduate Chair as well as the Director of Teacher Education in the Graduate School of Education.
Highly qualified and motivated students should note the possibility of obtaining both the B.A. and M.A. degrees in four years. This is discussed below.
Given the widening role of mathematics, students with special interests and needs may wish to consider the possibility of an individualized program of study, perhaps in conjunction with a major in another field. The Major Coordinator should be consulted about this.
How to Plan a Mathematics Major Prospective majors should first check the information listed under Advanced Placement. We strongly encourage students to master the basic material as early as possible, and AP credit is equal to credit for a course taken at Penn. Students are urged to read the Major Program Requirements carefully, and use it as a guideline to plot the plan. You should also read Other Useful Experience and Further Recommendations for a complete overview.
See Careers in Mathematics . It is maintained by the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.
See Math in the Media for pointers to current general articles involving mathematics and its applications.
Note: The AP policy and more details on this subject (and on "transfer credit") can be found on our web page AP/Transfer Credit Information.
We strongly encourage students to master the basic material as early as possible. It is our policy to waive prerequisite course requirements for those students who can pass an examination that demonstrates that they know the material. These remarks apply especially to the first-year calculus courses. For these, a student may receive credit towards the degree (in addition to the waiving of prerequisites) by either of the following methods:
- Passing the external Advanced Placement BC Exam administered by the College Entrance Examination Board with a score of 5 gives credit for Math 1400. Lower scores on the BC Exam receive no course credit. No credit is given for the AB Exam. Students taking first semester calculus, math 1400, are expected to have had an AB calculus course in high school.
- Passing the internal Advanced Placement Examination administered in the first week of the fall and spring semester by the mathematics department. A student may take the examination regardless of whether he or she took the external exam described under (1) above.
Those receiving advanced placement and planning to enroll in more advanced courses should see the Major Coordinator, who will help them plan a program of study.
The Mathematics Major comprises 13 courses organized into eight basic requirements. Each of the 13 courses must be taken for a grade (i.e., not pass/fail), and must be completed with a grade of C (2.0) or better. (A student who receives a grade lower than C in a requirement consisting of more than one course may still count that course toward the major by achieving a grade of C or above in a more advanced course within the same requirement.) The math department also expects the completion of at least one proof-based math class in the freshman or sophomore year in order to be admitted into the major (usually Math 2020, 2030, 1600, 2600, or 3140), or permission of a math major advisor. Courses taken on a pass/fail basis will not count toward fulfilling the following requirements.
- Three Semester Calculus Requirement. This is satisfied by any of the three sequences 1400-1410-2400 or 10400-1610-2600. The 10400 requirement can be satisfied by AP credit for the Calculus BC exam with a score of 5. The courses 1610-2600 are proof based and provide the best preparation for higher mathematics, and in particular for Math 3600 and 3610. Math 1300 does not count toward the Math major.
- Advanced Calculus Requirement Math majors must take either a fourth semester of calculus, Math 2410, or partial differential equations Math 4250.
- Complex Analysis Requirement All math majors must take Complex Analysis Math 4100.
- Seminar Requirement This is satisfied by taking either Math 2020 (intro to analysis) or Math 2030 (intro to algebra). These courses carry one-unit of credit and are intended to be taken concurrently with calculus. For students taking honors calculus (Math 1610-2400) the seminar requirement is replaced by a higher math elective course.
Students who begin with Math 1400 in their freshman year usually postpone this requirement until their second year. Students who have already taken one of Math 2410, Math 4100 or Math 4250 can substitute a higher math elective course for the Seminar Requirement. Under exceptional circumstances, other students may also make such a substitution with the permission of the Undergraduate Chair. In general, though, we recommend that prospective math majors take a freshman seminar to gain an overview of the subject.
- Linear algebra requirement.
Math Majors must take Advanced Linear Algebra Math 3140. Math 3140 is a prerequisite for Math 3700 and Math 5020.
- Algebra Requirement This is satisfied by taking the sequence Math 3700-3710 or the more theoretical Math 5020-5030. However, you can't get credits for both Math 3700 and Math 5020, or both Math 3710 and Math 5030.
These courses all overlap considerably.
- Analysis Requirement This is satisfied by taking the sequence Math 3600-3610 or the more theoretical Math 5080-509- However, you can't get credits for both Math 3600 and Math 5080, or both Math 3610 and Math 5090.
Note: Majors who begin their mathematics studies with Math 1410-2400 plus a seminar should fulfill at least one of the linear algebra, algebra, and analysis requirements in their sophomore year.
- Mathematics Electives The total number of approved math course units required for a math major is 13. Students should determine how many course units they still need for a math major after completing requirements 1 through 6 above. This will depend on which options have been chosen in completing the requirements. The remaining courses may then be made up from Math 2100 and mathematics courses numbered 3200 or above. One mathematics elective course unit may be taken from the list of approved Cognate Courses given outside the math department. Students who are double majors may take two Cognate courses units.
Students may, for example, take Statistics 4300 (or Systems Engineering 3010 or Econ 103 or ENM 5030), and count such a course as being within the Mathematics Department. Thus by taking one of these courses, one does not lower the number of cognate courses one can take outside the math department, as explained on the page of Cognate Courses.
Example 1: A student is double majoring in math and engineering, did not take a freshman seminar, and completed the Advanced Calculus requirement by taking math 2410. This student thus takes 4 courses related to the Calculus requirements, 4 courses to complete the algebra and analysis requirements, and Math 3140 and Math 4100 for a total of 10 courses. They must take 3 electives to bring their course total up to 13. Because the student is double majoring in math and engineering, two of these electives can be Cognate Courses in other departments. Notice that on the above list of cognate courses, some courses given in other departments are listed as being counted as within the math department as far as the math major and minor are concerned. For example, the student could take Stat 4300 (which is counted as within the math department), use Physics 0150 and Physics 0151 as their cognate courses not counted as within the math department, and then choose two more electives from within the math department to complete their math major requirements.
Example 2: A student is majoring only in math, took a freshman seminar, and completed the Advanced Calculus requirement by taking math 4250. This student thus takes the freshman seminar, 3 calculus courses, math 4100 and 4250 and four algebra and analysis courses in the course of completing the above requirements, for a total of 10 courses. They must take three further electives for a math major. Only one of these can be a Cognate Course, because the student is not a double major.
Students who do not plan graduate study in mathematics or in a highly mathematics-related subject should, as a means of acquiring more background, consider Math 4100, 4200, 4250, and 4300. For glimpses of several beautiful mathematical subjects beyond the basic core, students should consider Math 3500, 5420, 5480, 5490, 5800, 5000, 5300, 4800.
Students who are interested in the physical sciences should consider Physics 0150-0151 or 0170-0171 and the courses beyond. Those interested in the social or biological sciences should consider Math 4300 or Statistics 4300-4310. Those interested in computer science should consider CSE 110, 120-121 and the courses beyond as well as Math 4500, 5700 (previously 473 and 670). For computer programming and numerical methods, students should learn a programming language such as Pascal or C and learn to use symbolic manipulation software such as Mathematica or Maple. They also should consider Math 3200-3210. For discrete methods, in addition to Math 3400 and 3410, 450, 5700 (previously 473), and 5800 (previously 440), students should consider Math 5240-5250 (previously 470) and 5810 (previously 441).
For students who plan to do research in mathematics, or in a highly mathematical subject such as statistics, the considerations which are listed just above still apply. However, since a great deal of further theoretical training is necessary, such students are directed to the basic graduate courses in mathematics: 6000, 6010, 6020, 6030, 6080, and 6090.
All this material must eventually be mastered. It needs to be understood clearly that what is required is a comprehensive grasp of theoretical mathematics. Thus, the student's attention is directed to Method B for obtaining honors in mathematics, and to the joint B.A./M.A. program, pursuing a master's degree at the same time as their undergraduate degree.
The first order of business is to satisfy the first four requirements discussed above. When this has been done, the student usually has sufficient experience and direction to complete the program in consultation with the Major Advisor . It needs to be emphasized strongly, however, that apart from the strict requirements, there are certain other things which all mathematics majors should do. These are:
- Learn to program a computer and learn how to use mathematical symbolic manipulation packages. The latter skill is taught in our Calculus courses.
- Learn statistics. This may be done by taking Math 4300 or Stat 4300 followed by Stat 4310.
- Learn how mathematics is actually used. This can be done by learning something of an applied but highly mathematical field. Operations research, engineering and physics provide examples, but there are many others. (See below.)
- Obtain some job experience. This should be done, if possible, in the summer following the junior year. It should involve some interface between mathematics and the real world.
The importance of the above four recommendations cannot be sufficiently emphasized. Equipped with them, a mathematics major is an attractive candidate for entrance into a great many fields. Without them, job opportunities are limited. These remarks apply to the most theoretical, as well as to the most practical of careers.
To be eligible for honors in mathematics, a student must have an average of at least 3.5 in his/her major and major-related courses. If this condition is satisfied, honors may be obtained by either of the following methods.
Method A. By preparing, through independent study, a body of material approximately equal in amount to a one-semester course and giving a lecture on it as the Honors Committee shall direct.
The area of study chosen should be one that is not normally covered in the department and should involve reading sources outside normal course material. The selected topic may be picked from one field of mathematics or may involve assimilation of topics from different fields. Before beginning the project, the student should ask two members of the faculty, at least one affiliated with the Mathematics Department, to serve as the Honors Committee. The Honors Committee must approve the selected topic and serve as examiners for the lecture (which should be approximately an hour long, seminar-style talk).
Method B. By passing the written Preliminary Examination in undergraduate mathematics. This is required of all incoming Ph.D candidates. Details concerning this examination may be found in the Graduate Admissions Catalogue (also see below).
For further guidance, prospective honors students should consult with their Major Advisor during their junior year. The honors project must be completed by the end of February of the senior year.
Undergraduates who wish to take courses beyond the math major program should consider submatriculation and pursuing a master's degree. The minimum requirements are a A- average in 3600-3610 or 5080-5090 and 3700-3710 or 5020-5030, and permission of the Graduate Chair. Students who plan on a master's degree should submatriculate as early as possible because only courses taken subsequently to this may be counted toward the degree. The degree itself requires the successful completion of eight graduate courses and the written examinations for the Ph.D. The requirements can sometimes be completed by the end of the fourth undergraduate year, but often a fifth year is required.
For more information see the SAS web page Submatriculation and the Math Department Submatriculation page.
- Gifted high school students from the Philadelphia area are encouraged to take courses (usually Math 2400-2410) in the department while they are still in high school. This is done through the Young Scholars Program which is administered by the College of Liberal and Professional Studies.
- High school seniors who wish to major in mathematics and think that they might like to attend the University are invited to visit the mathematics department to meet the faculty and visit classes. They should email ugrad AT math.upenn.edu or call 898-8178 for an appointment.
- For our Math Majors and Minors there is a list of courses often approved as COGNATES for Mathematics Majors (these are courses from other departments often approved for mathematics majors or minor credit). All cognates require the approval of the Undergraduate Chair and must be part of a well-planned selection of electives within the major. The statistics courses enjoy a special status: since they count as being inside the Mathematics Department as far as the major or minor is concerned. Thus, a student who takes one or two of these may count additional outside courses toward the Mathematics Elective requirement. Additional courses may also be approved as cognates upon application to the Undergraduate Chair.
- Undergraduates who plan to teach in secondary schools should refer to the section on the Bachelor of Arts/Master of Science in Education.
- Penn has an active Undergraduate Mathematics Society which conducts seminars, colloquia and other activities for those who wish to encounter Math outside the classroom. Information about Society membership and schedules of its activities can be obtained in the Math Department office or by clicking on the link above.
- Opportunities for summer research exist at many Universities. The Undergraduate Chair is a good source of information about such programs, which are usually announced in October or November.
- National Science Foundation Research Experiences for Undergraduates
- Internships and Summer Employment (maintained by Penn Career Service)
- Penn's Graduate program in Education, including information about obtaining a Masters Degree in Education by submatriculation. |
Algorithms for Some Euler-Type Identities for Multiple Zeta Values
Multiple zeta values are the numbers defined by the convergent series , where , , , are positive integers with . For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is . The well-known result was extended to and by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers and then gave a direct formula for for arbitrary . In this paper we apply a technique introduced by Granville to present an algorithm to calculate and prove that the direct formula can also be deduced from Eisenstein's double product.
The multiple zeta sums, are also called Euler-Zagier sums, where are positive integers with . Clearly, the Riemann zeta function , is the case in (1). The multiple zeta functions have attracted considerable interest in recent years.
For Riemann’s zeta function , Euler proved the following identity: Recently, some identities similar to (3) have also been established. Given two positive integers and (suppose ), define a number by Then, for , the value of is known [1–5].
Following , for , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is ; that is,
In , Gangl et al. proved the following identities:
Recently, using harmonic shuffle relations, Shen and Cai proved the following results in :
In , applying the theory of symmetric functions, Hoffman established the generating function for the numbers . He proved that Based on this generating function, some formulas for for arbitrary are given. For example, Hoffman obtained that where is the th Bernoulli number.
In this paper we use a technique introduced by Granville to present an elementary recursion algorithm to calculate , we also give some direct formula for for arbitrary . Our algorithm may be of some interest if we note that it is obtained through an elementary analytic method and that the statement of the algorithm is fairly simple.
2. Statements of the Theorems
Theorem 1. Let denote a positive integer. Let , be a series of numbers defined by Then, for any two positive integers and with , one has
Theorem 2. Given a positive integer , we have
When is not large, we may use the following recursion algorithm to calculate then use Theorem 1 to get the formula for .
Theorem 3. The coefficients , can be calculated recursively by the following formulas: where , are the numbers defined by
In , Hoffman established an interesting result [6, Lemma 1.3] to obtain his formula (10) for . This lemma might be deduced from the theory of Bessel functions. Using the expressions for the Bessel functions of the first kind with a half integer index, we may deduce from the generating function (13) a direct formula for .
Theorem 4. For , one has
To deduce (17) from (16), we only need to write the expression of , respectively, according to whether is odd or even, and use (if is odd) or (if is even) to replace . In the two cases, we will get the expression (17) for . By Theorem 1, we have which reproduces Hoffman’s formula (10).
3. Proofs of the Theorems
Proof of Theorem 1. The left side of (12) is
The second sum in (19) is the coefficient of in the formal power series
It follows that the coefficient of earlier is
Hence, the sum (19) is
Now, consider the function We partition into two parts. Let Then, we have , , for all, and
Consider the sum (22). For , we treat each sum in (22) with respect to as follows: In the last step, begins with 1 since for .
It follows that the sum (22) becomes that Clearly, the sum in (27) is the coefficient of in the Cauchy product of that is, it is the coefficient of in the power series Therefore, the sum (27) is The proof is completed.
Remark 5. If we take to be a complex variable, then the series is absolutely and uniformly convergent for in any compact set in the complex plane; thus, the function is analytic in the complex plane. Hence, it may be expanded as a Taylor series.
Proof of Theorem 2. First we recall Euler’s classical formula Similar to Euler’s formula, Eisenstein studied a product of two variables and proved that for the following formula holds (see [10, page 17]): Let be temporarily fixed. By (34), for we have Now, let . We get We write . Or equivalently, let . Then, we get
Proof of Theorem 3. Taking logarithms of both sides of (32), we get that By Remark 5, the series may be differentiated term-by-term; hence, we have where we denote The order of the summation can be changed since the series is dominated by for some positive constant . From (39), we get that or Write out the Cauchy product in the right side of (42), then compare the coefficient of on both sides. We get that
Proof of Theorem 4. We now study the the generating function
We may use L’Hospital’s rule to verify that
Now we expand out . We have
By (11) and (13), we have
Consider the function Clearly, the sum in (47) can be rewritten as where means the th derivative of a function with respect to .
We denote . Then, we have and, hence, which implies that Finally, from (47) (49) we get that
We may apply Hoffman’s result [6, Lemma 1.3] to get the direct formula for Here, we use some simple properties of the Bessel functions of the first kind to give its direct expression.
Lemma 6. Let be an integer and let . Then one haswhere denotes the Bessel function of the first kind of index .
The Bessel functions with a half-integer index can be represented by elementary functions. The following lemma is well known.
Lemma 7. Let be an integer, and let . Then, one has
From Lemmas 6 and 7, and (53), we get that This completes the proof of Theorem 4.
The direct formula for can be found from Theorem 4. However, we would like to use Theorem 3 to present some concrete examples to show how to calculate for small . The difficult part of the recursion formula (14) is for to calculate the sum where we denote and .
It follows from that Generally, we can use induction on to prove that if for we have gotten some positive integers such that then the expression for is
Note that if is an even integer, then we have Similarly, if is an odd integer, then we have
From formula (14), we get that
This work is supported by the National Natural Science Foundation of China (1127208).
H. Gangl, M. Kaneko, and D. Zagier, “Double zeta values and modular forms,” in Automorphic Forms and Zeta Functions. In Memory of Tsuneo Arakawa. Proceedings of the Conference, Rikkyo University, Tokyo, Japan, September 2004, pp. 71–106, World Scientific, Hackensack, NJ, USA, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Remmert, Classical Topics in Complex Function Theory, vol. 172 of Graduate Texts in Mathematics, Springer-Verlag, New York, NY, USA, 1998.View at: MathSciNet |
The Magnetic Field of the Hybrid Undulator U27
In order to ensure sufficient overlap of the electrons with the optical beam, the center of the electron trajectory should be at the same position independently of electron energy and of the strength of the magnetic field. The principle of the magnetic structure of such a nearly "passive" undulator is shown in fig. 1. The structure consists of NdFeB permanent magnets and poles of decarborized iron. The field of these arrangement has been investigated by the code RADIA . Corresponding results have been reported . To ensure the correct slope and position of the electron beam at the undulator exit one has to add an additional magnet (m2) and an iron pole (p2) on either side producing appropriate fringe fields.
The undulator was adjusted for a gap g = 12 mm by varying the y-coordinate of p2 and the distance between m2 and p2 (see fig. 1). The resulting displacement and the slope of the trajectory at the exit can be corrected by coils mounted at the entrance sides of both sections and allowing to inject the electrons at a small angle to the z-axis.
Fig. 1: Principle of the magnetic structure of U27. The blue rectangles represent NdFeB permanent magnets, the red ones show poles of decarborized iron. The symbols p2 and m2 denote poles and magnets of reduced thicknesses mounted at the same girder like the rest of the structure.
The undulator field was measured and tuned at HASYLAB (DESY) by means of the Hall probe as shown in fig. 2. After transporting the undulator to Rossendorf control mesurement of the undulator fields have been performed using the pulsed-wire method.
At DESY the probe was mounted on a goniometer with six degrees of freedom for right adjustment within the magnetic structures. The poles were tuned with the aim to get a minimum electron displacement at the exit sides of the modules as well as a peak field roughness lower than 0.4%. As a result of the measurements the fig. 3 shows the field component By(0,0,z) in the middle plane of both undulator units for a gap of g = 12 mm and a distance d = 250 mm between the sections (part a) and the first integral over the measured field (part b). The trajectory of a reference electron with an energy of E = 20 MeV in the wiggle plane is shown in part (c). Only weak magnetic fields of about 2 mT are necessary to keep the electrons within the optical mode. Fig.4 shows the magnetic field distribution By(x,0,z5) perpendicular to the electron beam at different gaps , where z5 is the z coordinate of the 5th pole.
Fig. 2: The Hall probe mounted on a goniometer for field measurements on the axis of one of the undulator sections.
In order to estimate the influence of the remaining field inhomogeneities on the lasing process, we determined for various gap widths the maxima (minima) Bi of the field, their average values Bav and the differences Bi - Bav. For g = 14 mm the results are shown in fig. 5. The standard deviations σB of the Bav contribute to the inhomogeneous line broadening of the emitted light corresponding to the following formula:
Fig. 5: Maxima and minima of the magnetic field By (z) in the middle plane of the whole undulator. For both sections the average values Bav and their standard deviations σ are denounced. The values σB/Bav characterizing the field roughness are given in parenthesis.
The fast Fourier transform obtained from the measured field distribution By(z) is shown in the fig. 6, which indicates for gap widths of g = 8, 12, 16 and 20 mm the absolute values |f(n)| of the contributions of the first, third and fifth harmonics to the magnetic field. The employed FFT - procedure delivers the wave numbers kz as a multiple of the quantity k0:
where L is the length (mm) of the interval used for the analysis. For each gap a window of 1690 points and a length of 885.0 mm was analysed containing 31 full magnetic periods. The quantity k0 has therefore the value k0 = 0.007099 mm-1. Since the momenta A3 in f(n) exhibit always a positive sign, the measured field By(z) is a bit more flat and more broad than a pure sine function. The reason for the appearence of higher harmonics is the width wz = 5 mm of the iron poles in z-direction, influencing the form of the field mostly for lower gap values.
Fig. 6: The Fourier Transform (FFT) of the measured field distribution By(z) for various gap values
If the FEL works with maximum power the energy factor γ of the electrons decreases by dγ = γ/2 Nu (for the first harmonic) along the electron path due to the interaction of the electron beam with the electromagnetic field, where Nu = 68 is the total number of magnetic periods in the two undulator units. The electron energy changes continuously from the undulator entrance to the exit leading to a resonance wavelength λ(z) which depends on the coordinate z along the undulator
Within a certain interval dλ this effect can be compensated by differentially increasing the gap, and hence decreasing the magnetic field along the undulator. To compensate the energy loss dγ by a reduction dB of the magnetic field one has to ensure
from which follows
For hybrid undulators, the Halbach equation allows to estimate the tapering of the gap g needed for the variation dB of the field corresponding to equation (4). Figure 7 shows the effect of field tapering in both sections of the undulator U27, which would be used in a situation typical for high intensity lasing.
For variable gaps the radiation wavelength is changed and consequently the phasing between the two undulator systems has to be changed as well. This has to be done by properly choosing the electron flight path d between the two sections.
The optical phase in an undulator has been analyzed in detail by Walker . It can be expressed by
Here z is the coordinate along the beam axis, λ is the radiation wavelength, γ is the electron kinetic energy in units of its rest mass and Θ is the electrons deflection angle. The terms in the bracket have the following descriptive meaning: the first gives the contribution to the phase, if the electron travels just a distance z in free space, the second term represents the additional contribution due to the magnetic field. The function Θ(z) can be derived from By(z) by
Using the eqs. (5) and (6) the phasing of the two undulator sub-systems can be determined experimentally by measuring By(z) along the axis of the entire undulator.
The phase on the poles in the second section varies with the distance d, which depends on the K-value and has to be chosen appropriately. Moreover, phase matching is obtained periodically after an increase of d by Δd = λu(1+Krms 2). The deviation of the optical phase angle from its nominal value obtained for a perfect undulator is denoted by ΔΨ(z). For a gap g = 17 mm measurements for ΔΨ(z) are shown in fig. 8. The K-value for this case was 0.6688. The nominal phase deviation on the poles is nearly zero, only at the end poles before and after the interspace the phases deviate from zero. The proper choice of the distances d as functions of the K-value as determined experimentally is shown in fig. 9. Six different curves have been measured. They are shifted horizontally by Δd = λu(1+Krms 2(g)). These curves have to be used to choose phasing distances dph in dependence on the gap g.
Fig. 8: Phase differences ΔΨ(z) for a gap g = 17 mm. The K-value is 0.6688. Three phasing distances dph are shown.
Fig. 9: Phasing distances dph (points) between the two sections of U27. The right axis shows the gap widths g, for which the values dph have been found, the left axis shows the corresponding K-values in a linear scale.
P. Elleaume, O. Chubar and J. Chavanne, J. Synchr. Rad. 5 (1998) 481
P. Gippner, W. Seidel and A. Schamlott, Annual Report 1998, FZD-271, p.16
R. P. Walker, Nucl. Instr. Meth. A335 (1993) 328 |
|Part of a series of articles about|
is geometric, because each successive term can be obtained by multiplying the previous term by 1/2.
Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, r and a. The term r is the common ratio, and a is the first term of the series. As an example the geometric series given in the introduction,
may simply be written as
- , with and .
The following table shows several geometric series with different common ratios:
|Common ratio, r||Start term, a||Example series|
|10||4||4 + 40 + 400 + 4000 + 40,000 + ···|
|1/3||9||9 + 3 + 1 + 1/3 + 1/9 + ···|
|1/10||7||7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···|
|1||3||3 + 3 + 3 + 3 + 3 + ···|
|−1/2||1||1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···|
|–1||3||3 − 3 + 3 − 3 + 3 − ···|
The behavior of the terms depends on the common ratio r:
- If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where r is one half, the series has the sum one.
- If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.)
- If r is equal to one, all of the terms of the series are the same. The series diverges.
- If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.
The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely many terms. The sum can be computed using the self-similarity of the series.
Consider the sum of the following geometric series:
This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from the original series s cancels every term in the original but the first,
A similar technique can be used to evaluate any self-similar expression.
For , the sum of the first n terms of a geometric series is
where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:
As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes
When a = 1, this can be simplified to
the left-hand side being a geometric series with common ratio r.
The formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one.
Proof of convergence
Since (1 + r + r2 + ... + rn)(1−r) = 1−rn+1 and rn+1 → 0 for | r | < 1.
Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function,
So S converges to
For , the sum of the first n terms of a geometric series is:
This formula can be derived as follows:
A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
The formula for the sum of a geometric series can be used to convert the decimal to a fraction,
The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:
Note that every series of repeating consecutive decimals can be conveniently simplified with the following:
That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n - 1.
Archimedes' quadrature of the parabola
Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.
Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.
Assuming that the blue triangle has area 1, the total area is an infinite sum:
The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives
This is a geometric series with common ratio 1/4 and the fractional part is equal to
The sum is
For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is
The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is
Thus the Koch snowflake has 8/5 of the area of the base triangle.
The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with .
Book IX, Proposition 35 of Euclid's Elements expresses the partial sum of a geometric series in terms of members of the series. It is equivalent to the modern formula.
For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + ), where is the yearly interest rate.
Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + )2 (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is
which is the infinite series:
This is a geometric series with common ratio 1 / (1 + ). The sum is the first term divided by (one minus the common ratio):
For example, if the yearly interest rate is 10% ( = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.
This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a security.
Geometric power series
The formula for a geometric series
Similarly obtained are:
- Divergent geometric series
- Generalized hypergeometric function
- Geometric progression
- Neumann series
- Ratio test
- Root test
- Series (mathematics)
- Tower of Hanoi
Specific geometric series
- Grandi's series: 1 − 1 + 1 − 1 + ⋯
- 1 + 2 + 4 + 8 + ⋯
- 1 − 2 + 4 − 8 + ⋯
- 1/2 + 1/4 + 1/8 + 1/16 + ⋯
- 1/2 − 1/4 + 1/8 − 1/16 + ⋯
- 1/4 + 1/16 + 1/64 + 1/256 + ⋯
- Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.
- Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278–279, 1985.
- Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.
- Courant, R. and Robbins, H. "The Geometric Progression." §1.2.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13–14, 1996.
- Pappas, T. "Perimeter, Area & the Infinite Series." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 134–135, 1989.
- James Stewart (2002). Calculus, 5th ed., Brooks Cole. ISBN 978-0-534-39339-7
- Larson, Hostetler, and Edwards (2005). Calculus with Analytic Geometry, 8th ed., Houghton Mifflin Company. ISBN 978-0-618-50298-1
- Roger B. Nelsen (1997). Proofs without Words: Exercises in Visual Thinking, The Mathematical Association of America. ISBN 978-0-88385-700-7
- Andrews, George E. (1998). "The geometric series in calculus". The American Mathematical Monthly. Mathematical Association of America. 105 (1): 36–40. doi:10.2307/2589524. JSTOR 2589524.
History and philosophy
- C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0-387-94313-8.
- Swain, Gordon and Thomas Dence (April 1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine. 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.
- Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN 978-0-691-02511-7
- Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0-415-22526-7
- Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN 978-0-393-95733-4
- Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7
- Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0-387-09648-3
- Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press. ISBN 978-0-521-57698-7
- John Rast Hubbard (2000). Schaum's Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0-07-137870-3
- Hazewinkel, Michiel, ed. (2001), "Geometric progression", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Geometric Series". MathWorld.
- Geometric Series at PlanetMath.org.
- Peppard, Kim. "College Algebra Tutorial on Geometric Sequences and Series". West Texas A&M University.
- Casselman, Bill. "A Geometric Interpretation of the Geometric Series" (Applet).
- "Geometric Series" by Michael Schreiber, Wolfram Demonstrations Project, 2007. |
Adv Analytical Chemistry II
Adv Analytical Chemistry II CEM 835
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Spectrochemical Measurements Expressions of Intensity 39 Quantities based on radiometric system not photometric system 0 Basic unit is joule and other SI units sometimes nonSi units for convenience 39 Often de nitions include area volume or solid angle Spectral quantities Bx 7L Partlal quantltles Bk 1 2 k2 Bkdk 1 Total quantities B I Bkdk TABLE 2 1 Radiometric system Quantity Symbols Description Defining equationa Units General Radiant energy Q Energy in the form of radiation J ergs Radiant energy density U Radiant energy per unit U 192 J cm 3 volume 3V Radiant ux or radiant power P Rate of transfer of radiant I 92 W energy at Source Radiant intensity I Radiant power per unit solid 933 W sr 1 angle from a point source an Radiant emittance or radiant M Radiant power per unit area 532 W cm 2 ex1tance 3A Radiant emissivity J Radiant power per unit solid 621 W sr 1 cm 3 angle per unit volume 39 3V Radiance BL Radiant power per unit solid 621 W sr l cmquot2 angle per unit projected area an GAP 621 69 6A cos 0 Receiver Irradiance E Radiant power per unit area E 32 W cm 2 8A I 2 Radlant exposure H Integrated 1rrad1ance H L E dt J cm CEM 835 page 11 Important quantities Radiant flux 1 rate of energy transfer J s391 W Radiant intensity I radiant ux from a point source per unit solid angle QD4n applies to source J s391 sr391 Radiance B radiant intensity I per projected area CD475A applies to source gt depends on angle between detector and radiation propagation direction see Fig 23 Js391sr391cm392 Irradiance E radiant ux CD ontofrom a surface per unit area QDA applies to source or detector Js 1cm392 radiant Exposure H timeintegrated irradiance tAdt J cm392 Fluence often used but meaning is imprecise CEM 835 page 12 Geometric factors Often radiometric quantities include a solid angle or projected area Solid angle l steradian sr is the part of the surface area of a sphere of radius r having an area of r2 2 Asphere 4 7T39 r 4 75 r2 sterad1ans 1n sphere 2 4 71 1257 r 0 x Arc r y r Plane angle One radian 60 Area One steradian Solid angle iIgu b For example intensity is the radiant ux per unit solid angle I 1 4n CEM 835 page 13 How are these quantities related to spectrochemical techniques Emission Spectroscopy Emitted radiation 4 5 E21 hl 21 hcAZI E2 hvz hcKZ E1 hvl hcK1 b 4 5 Thermal electrical A or chemical l I L A energy K2 K1 21 a C 0 Emission refers to thermally excited atoms or molecules ame ICP electrical discharge plasma Interested in number of atoms per unit volume element observed Demission Aji 39hVij 39nj 39V observation volume atoms in excited state j energy per transition Einstein coef cient transition probability j gti CEM 835 page 14 Ilj can be de ned if in thermal equilibrium by Boltzmann expression Demission Aij 39hVij 39nj 39V EjkT gj e oo E kT 39V Z e 1 10 1 fraction of total in state j Z Aij 39hVij 39ntotal The weighting factor gj statistical weight is the number of degenerate states at each energy E g 2 2J 1 J is the total angular momentum LS LSl LS Example g2s2 g2p122 g2p324 Hence radiant ux Qemission can absolutely determine the concentration of atoms in detection volume CEM 835 page 15 Absorption Spectroscopy Incident Transmitted radiation radiation QO P a b E2 hV2 E1 hvl hcAl c Absorbance A given by Beers39 Law related to the measured quantities DO and CD radiant ux by CD A l T l b og ogq 8 c 0 concentration molL39l cell pathlength cm molar absorptiVity L mol391 cm391 lt1 13010 839b39C Luminescence measurements Scattering measurements CEM 835 page 16 Optical Instruments Many spectrochemical instruments share common components 0 a radiation source 0 optics to de ne light paths 0 a sample container 0 a dispersion element 0 detector transducer Speci c names are applied to the various instruments Aperture Photographic film a Spectrochemical or pg fegioimy encoder Exit slit b W Photodetector Dispersion Entrance element Focal slit plane I Q and image transfer system Photodetectors A spectroscope disperses a range of X39s for Visual Viewing A spectrograph disperses a range of X39s onto focal plane for simultaneous measurements by a photographic lm or array detector A monochromator uses entrance slit eXit slit and a dispersion element to separate X39s in space If multiple eXit slits are used the term polychromator is used CEM 835 page 17 A photometer measures intensity but has no provision for 9 scanning X39s can be selected by use of lters A spectrometer includes means of manually or automatically scanning wavelength A spectrophotometer has provision for scanning measurements using two beams of light useful for ratioing incident and transmitted light An interferometer is a nondispersive device that relies on interference to obta1n spectral 1nformatlon A detector is any device whose output is proportional to the intensity of light falling on it A transducer more speci c uses electrical signals CEM 835 page 18 Components of Measurement Instruments Radiation Sources Many radiation sources are based on black body radiation 39 perfect absorber of radiation at all X39s 0 if in thermal equilibrium must also be perfect radiation emitter 2000K E 1750K dWd J m 4 1000 1250K 0 1000 2000 3000 4000 5000 7 visible region 39 391 nm Two obvious points 39 total amount of energy radiated increases rapidly with T U a T4 Stefan39s Law Radiant energy density J cm3 39 position of the maximum spectral radiance kmax blue shifts with increasing T C A 2 max 4965 T where c2 143 8X107 nm K CEM 835 page l9 Energy density U J cm3 is dif cult to measure usually work in radiance B Js1sr1cm2 U c B c B L B V V 47 7 Planck deduced black body equation after consideration of thermodynamics of system with discrete energy levels multiples of hv the beginning of quantum mechanics 2h3 1 B V v C2 ehvkT 1 or in terms of wavelength BKZM 7L5 ehcxkr1 3195 Planck39s radiation law where c12hc2 119x1016 Wnm4cm392sr391 and c2hck same as above 1438x107 nmK CEM 835 page 110 Einstein coef cients Three Einstein coef cients Bij describes probability of absorption from level i gt j Bji describes probability of stimulated emission from level j gti Note These two are simply timereversed processes Aji describes probability of spontaneous emission from level j gti l B A j 8 i i v Absorption Spontaneous Stimulated emission emission The rate of absorption per unit volume s391 cm393 depends on i number of atoms in initial state i ni ii probability of absorption from state i to another state j Bij iii the spectral energy density of incident radiation UV dni B U n dt 1 V 1 absorption removes population from state i so ni decreases CEM 835 page 111 Similarly rate of stimulated emission is The rate of absorption and stimulated emission are the same if there is an equal population in both states Bji 8139 Bij 39gi g is the degeneracy statistical weight Rate of spontaneous emission doesn39t include a UV term E dt If black body is in thermal equilibrium with surroundings rate of absorption and emission must be equal 39Uv ni 39nj Uv nj absorptlon spontaneous stnnulated emission emission U Aji39nj V Bji 39Ili Bji Aji 39nj Alin lt substitutingB g B g ji39 j ij39 i Bjinigigj39nj Aji 39nj CEM 835 page 112 At equilibrium Boltzmann equation can be used to nd nj from ni Aji 39nj Bjini gi gj39nj substituting nj ni eXp hvij kT V Aji 39nj V Bjini gi gj ni eXphVij 1ltT This looks similar in form to Planck39s radiation law B 2hv3 1 v C2 ehvkT1 and gives us the rate of spontaneous emission and absorption UVc 8 h 3 Aji WBij remember1ng BV gjc 4TB Bji39gj Bij39gi CEM 835 page 113 Reflection and Refraction Maxwell39s equations lead to de nition for the velocity of electromagnetic radiation in a vacuum 1 C xSo 39Ho where 80 is the permittivity of freespace 8854x103912 C2N391 m392 uO is the permeability of freespace 4710397 kgmC392 In a medium velocity is reduced VIM The ratio of the velocity in a medium to freespace is refractive indeX gt 100 in a medium 11 varies with wavelength usually increases with frequency called normal dispersion decreases with frequency in region of absorption called anomalous dispersion CEM 835 page 31 7t nm 11 351 1539 458 1525 486 1522 532 1519 644 1515 830 1510 Important Frequency of radiation is xed by source Hence wavelength of radiation in a medium must increase c kz s1ncevot T1 xmedium gt xvacuum When a wave passes from medium with refractive index 111 to medium of refractive index 112 we can write hZ c X nz u 9L1 1121 c 112 CEM 835 page 32 Based on wave representation of electromagnetic radiation and geometry we can quickly deduce the angle of re ection Re ected wavefront Incident wavefront 771 a b 9i 2 93 Law of specular re ectance CEM 835 page 33 The refracted beam does not travel at same velocity as the incident beam v2 v1 411112 first part of the wavefront to strike the interface is retarded preferentially light beam bends towards the interface normal when n2gtm n1 sin61 n2 sin62 Snell39s law of refraction no refraction when 61 00 no transmittance when 61 gt Be critical angle total internal re ection sin 61 sin 62 Snell39s law 111 when sin 62 9O0 61 60 sin 1 111 For airglass 60 z 42 CEM 835 page 34 90 CEM 835 page 35 Fresnel Equations Re ectance losses occur at all at interfaces 0L0 TL pt 1 Conservation Law magnitude increases as the di erence in the refractive indices increases dependent on incidence angle Equation describing the re ectance p00 is the Fresnel equation 1 l sin20i 0 1t2111291 9r 2 Lsin20i 0 1 tan291 9rgti 90 Where 0 i is incidence angle and 01 is refraction angle For the airglass at 589 nm re ectance is about 004 or 4 per interface 10 08 06 Re ectance 04 02 004 CEM 835 page 36 p0 constant for small angles p0 increases rapidly at large angles grazing incidence m Serves several purposes in a spectrometer change the direction of a beam change the polarization of a beam split a beam into two disperse the beam A variety of shapes and materials are available to perform these functions Dispersing prism According to Snell39s Law sinGl sin 62 Snell39s law 111 there will be no dispersion if not is constant dispersion in prism occurs because of the change in refractive index of the prism material as a function of wavelength 0 if prism material exhibits normal dispersion higher frequency shorter wavelength light experiences a higher refractive index than lower frequency longer wavelength light CEM 835 page 37 Light of different wavelengths become divergent and become separated in space angle between incident and refracted beam is called the deviation The variation in deviation with wavelength is called the angular dispersion d6 d6 dn D A dx dn g prism dispersion first term depends on size and shape of the prism and the incidence angle second term prism dispersion depends on the material of the prism and the wavelength 3 glass357 nm 2 194x10 4 nm1 dn 5 1 a glass825 nm 178XlO nm Prisms not often used as dispersion elements because of non constant D A with wavelength produces nonconstant bandwith means range of X39s projected onto eXit slit varies with 9 CEM 835 page 38 Electromagnetic radiation An electromagnetic wave is a transverse wave electric and magnetic elds perpendicular to the propagation direction Plane linearly polarized beam has constant plane containing the electric and magnetic vectors often called unpolarized The timedependent electric eld is E Eosinoa t where E0 is the maximum electric eld strength 0 is the angular frequency 2751 t is time I is the angular phase The angular phase is d0275Xt where X is distance and bo is the phase at x0 275 is number of waves per unit length If two waves maintain the same relative phase difference over i extended period of time ii length they are said to be coherent CEM 835 page 39 Superposition The superposition of two waves states two plane polarized waves can be algebraically summed to produce a resultant wave If waves have same frequency E 2 E1 E2 2 E021 sin03t 11 E022 sin03t 12 Amplitude intensity of wave is E2 E2 2 E1 E22 2 E12 E22 E139E2 E0212 13022 2E0 1 EO Z COS 2 interference term If 1 2 O 27 47 cos0 27 47 l wave amplitude will be reinforced constructive interference If1lt2 7 37 57 cos7 37 57 l wave amplitude will be reduced to zero destructive interference CEM 835 page 310 Interference can result from difference in pathlength If the waves initially start out with same phase the difference in phase 6 due to different paths is 5 1 2 27X1 2nX2 x x 275X1 X2 7 where X1 and X2 are the lengths to the measurement point from source 275 is the number of a complete waves per unit length Thus when 6 O 27 an integral number of wavelengths 275 m 2 n 2 X1 X2 9 5 m 9t 2 construct1ve1nterference 7 when 6 7 37 an integral number of wavelengthsl2 2m 1 5 2 j 2 destruct1ve1nterference Tc CEM 835 page 311 Diffraction Eschellete gratings Parallel grooves etched blazed onto re ective surface asymmetric in profile Groove facet Diffracted ray Incident ray Grating 39 normal N a b Incident light striking long facet is re ected in specular direction With respect to the groove normal light from neighboring grooves travels different distances and so interference occurs in outgoing beam Note angles or and B are de ned With respect to the grating normal not the groove normal Constructive interference occurs When the pathlength difference is an integral number of wavelengths extra pathlength associated With the incident beam is AC AC 2 d sin or extra pathlength associated With the outgoing beam is AD AD 2 d sin B CEM 835 page 312 The total pathlength difference is AC AD AC AD dsinoc sinB m dsin CC sin 3 Grating Formula minimum value of d as M2 because the maximum value of sinoc sinB is 2 The first order m 1 diffraction angle can be calculated for any incidence angle by rearranging the grating formula m s1noc s1nB d m s1nB Y s1n0L where d is found from the groove spacing Important diffraction angle depends on d longer X39s diffracted more than shorter ones 3600 m gt 3500 nm When m0 zero order sinoc sinB or OL B In this case all X39s are diffracted at the same angle If blaze was parallel to the grating plane y 0 the zero order beam would also appear in the specular direction most of the re ected light not dispersed If blaze angle 7 0 specular and zeroorder angles do not correspond and majority of the light is dispersed CEM 835 page 313 Specular re ection Specular re ection Groove normal l3 Incident Incident ray 4 ra Oorder Grating 7 y normal Oorder a b In the special case when incident beam is along the surface normal 0cO and rstorder beam is in specular direction in this case 3 is twice the blaze angle y The wavelength at this angle is called the blaze wavelength m t blaze dsin 0c sin 3 kblaze dSin 3 dsin 2y CEM 835 page 314 Dispersion The angular dispersion D A of the grating can be obtained by differentiating the grating formula with respect to wavelength For constant incidence angle ml dsin 0t sin 3 Grating Formula 1 m DA d E d cosB dsin 0L sin 3 Z d cos 3 sin 0t sin 3 9 cos B sin 0t xed For nearly normal incidence CC is small so 3 is small and so cosB does not change much with k D A does not change much with wavelength much better dispersion element than prism CEM 835 page 315 Monochromators Comprised of 0 dispersive element 0 image transfer system mirrors lenses and adjustable slits an image of the entrance slit is transferred to the eXit slit after dispersion One of the most common arrangements is the CzernyTurner monochromator Entrance Grating SM 81 Collimating mirror M1 Focusing mirror M2 CEM 835 page 316 Wavelength selection Wavelength selection is accomplished by rotating the grating Grating Grating 13 Since angle between the entrance slit grating and exit slit is xed 24 grating formula can be expressed in terms of the grating rotation angle 9 between grating normal and optical axis Sinceoc9and39 mk dsin9 sin9 2dsin 9 coscl the trigonometric identity l2sinABsinAB is sinAcosB Grating formula now in experimental variables 9 the grating rotation angle and I halfangle between the entrance grating and exit and slit CEM 835 page 317 Dispersive characteristics Already mentioned the angular dispersion rate of change of diffraction angle with wavelength for a grating D an ular dis ersion A Cm g P However in monochromator much more interested in dispersion at focal plane eXit slit defined by the linear dispersion D1 gt2 Al K 7 gt 1 x 1 A5 x T Focusing 39 r Ax element 4 TAB gt 2 i l 7 x2 Focal I plane I I r4 f gt dX D1 a lmear d1spers1on units of D1 are mm nm391 or similar For a CzernyTurner arrangement the linear dispersion is D1 f DA where f is the focal length of the focusing exit optic CEM 835 page 318 Sometimes the inverse linear dispersion Rd is used units of 1 nm mm39 or similar d Rd Dfl d 1nverse11near d1spers1on X sin 0t sin 3 A kcos B 1 Rd 2 f D A 9 cosB fsinoc sin 3 Spectral bandpass and the slit function The spectral bandpass nm is the halfwidth of the range of wavelengths passing through the exit slit The geometric spectral bandpass sg Rd W geometric spectral bandpass where Rd is the inverse linear dispersion W is slit width CEM 835 page 319 In a monochromator an image of entrance slit is focused at the eXit slit When input is polychromatic a monochromated version of the image appears at the eXit slit When input is monochromatic image rotating the grating angle 9 Will sweep monochromatic image across the eXit slit W Slit width Fixedpolsition ex1t s it outline I Shtlu ght Moving gt 0 entrance slit image I I i I I l I I I I I I i I I I I I 7 Sg No overlap I I I I I l I l I x I l 39 I I I A 3 sg 25 overlap Z I I I l I I 7 I Direction I I gt of image I 39 travel l A sg 50 overlap I I I l I 7 39 I A III sg 75 overlap l I Z l 7r A 100 overlap I I IA I l i I I I I I I I I l I 39 100 I I I I I I Halfwidth 00 Percentage of image I of ham radiation emerging I Sg from exit slit 39 I 50 l I l l 1 l V o 50 100 200 gto 39 5g 7 0 0 Sg Percentage of image overlap slit function a b CEM 835 page 320 The total intensity t0 measured at the exit slit as image is translated is called the slit function for equal entrance and eXit slits shape is triangular for unequal entrance and eXit slits shape is trapezoidal with a base of s and halfwidth of sg Mathematically the slit function is 9 9 tot 1 x0 sg g x 3 k0 sg s g t0 O elsewhere where 9 is the incident monochromatic wavelength at entrance slit k0 is the wavelength setting of the monochromator the wavelength directed to the center of the eXit slit Resolution Resolution quanti es how well separated two features are at the eXit slit closely related to linear dispersion D1 or angular dispersion DA and physical dimensions of the monochromator through f 0 slit width W CEM 835 page 321 Radiation Sources Continuum sources produce broad featureless range of wavelengths black and gray bodies high pressure arc lamps Line sources produce relatively narrow bands at speci c wavelengths generating structured emission spectrum lasers low pressure arc lamps hollow cathode lamps Line plus continuum sources contain lines superimposed on cont1nuum background medium pressure arc lamps D2 lamp Sources may be continuous or pulsed in time CEM 835 page 21 Continuum sources 0 Continuum sources are preferred for spectroscopy because of their relatively at radiance versus wavelength curves Platinum lead q Glower Re ector H Platinum Platinum wire heater lead L a A 63 U C d Parabolic re ector Anode Window Cathode e a Nernst glower b W lament c D2 lamp d are e are plus re ector CEM 835 page 22 Black body sources Nernst glowers ZrOz YOZ Globars SiC 10001500 K in air 7 lies in IR max 0 relatively fragile 0 low spectral radiance Bi lO394 W39Cm392 nm391 sr391 7500 o Blackbody theoretical at 900 C a Globar 39E 750 P A Nernst glower W a Mantle I 3 739 g 750 3 co I 2 z 75 In 1 4 J 1 l 1 1 1 1 1 1 1 l 1 1 1 A l 1 2 6 10 14 18 22 26 3O 34 38 Wavelength pm a CEM 835 page 23 Heated laments W incandescent lamp QTH 0 20003000 K in evacuated envelope greater radiance UaT4 B7 10392 Wcm39Znm39lsr391 greater UVVis output kmax still in IR QTH heated up to 3600 K wo wg Wg12 gt W12 g WIZ WsIZ Arc sources Hg Xe D2 lamps 0 AC or DC discharge through gas or metal vapor 2070 V 10 mA20 A Ionization necessary for conduction hot cathode thermionic emission cold cathode ignition voltage Nonuniform radiance CEM 835 page 24 50 100 150 150 100 50 Hg are radiance 10 1 I I l lllllll 102 EA uW cm 2 nmquotl ll IIIIIH 10 3 I 39 Irradianee BA from D2 lamp measured by a detector at 25 em CEM 835 page 25 IIHH quotIquot I I Tllllquot I III Bx W cmnzsr391 nlmquot1 9 00 d N d d i y i h h b I I 000I 111111 200300400500600700800 Wavelength nm 39 a high pressure Xe lamp b low pressure Hg lamp Hg arc lamps Hg3P1 gt Hg 1SO hv2537 nm Hg1P1 gt Hg 1SO hv1894 nm 0 if P high gt10 atm pseudocontinuum large current gt5 A many atoms excited I 1 radiance Bx gt10 Wcm39Znm39 sr39 if P low lt1 atm line or line plus continuum small current 1 A 10W radiance Bx ltlO4 WcmZnm1 sr1 selfabsorption at high radiant ux CEM 835 page 26 HIIH W Hg arc lamp selfabsorption CEM 835 page 27 vlam Line sources Generally not much use for molecular spectroscopy useful for luminescence excitation photochemistry eXperiments Where high radiant intensity at one 7 required Arc lamps 0 low pressure lt10 Torr With many different fill vapors Hg Cd Zn Ga In Th and alkali metals 0 excellent wavelength calibration sources Hollow cathode lamps HCL Hollow cathode bk 7 f r Anode Quartz Glass Ne or Ar 0T Pyrex shield at 15 torr window 0 primary line sources in atomic spectroscopy 0 low gas pressure lt10 mtorr linewidths 001 A high currents gtfew mA reduces lifetime and broadens lines 0 single or multielement cathodes 0 moderate radiance Bx 10392 W cm392 nm391sr391 CEM 835 page 28 Electrodeless discharge lamps EDL RF coil Ceramic holder 0 contain a microwave or RFexcited plasma need ignition pulse to start plasma 0 electric eld of RF or microwave drives ions and electrons in plasma no electrodes 0 gas pressures and temperatures relatively low slight pressure broadening linewidths are not as narrow as the HCL lt1A 0 moderate radiance 13 101 W cm392 nm391 sr391 CEM 835 page 29 Lasers intense radiance Bx gt104 Wcm392nm391sr391 nearly monochromatic 00lOl A coherent temporally and spatially directed small divergence pulsed or continuous stable Allow measurements not possible with conventional sources Consider radiation traveling through absorbing medium the change in radiant ux due to absorption is dCD CD ni 6 dz lt change due to absorption length of the medium absorption probability number of molecules in state i radiant ux Similarly for stimulated emission dCD CIgt nj 6 dz lt change due to stimulated emission Total change in ux is amount absorbed minus the amount gained by stimulated emission dd CDcsnj nidz lt total change CEM 835 page 210
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Chiral condensate in the Schwinger model with Matrix Product Operators
Tensor network (TN) methods, in particular the Matrix Product States (MPS) ansatz, have proven to be a useful tool in analyzing the properties of lattice gauge theories. They allow for a very good precision, much better than standard Monte Carlo (MC) techniques for the models that have been studied so far, due to the possibility of reaching much smaller lattice spacings. The real reason for the interest in the TN approach, however, is its ability, shown so far in several condensed matter models, to deal with theories which exhibit the notorious sign problem in MC simulations. This makes it prospective for dealing with the non-zero chemical potential in QCD and other lattice gauge theories, as well as with real-time simulations. In this paper, using matrix product operators, we extend our analysis of the Schwinger model at zero temperature to show the feasibility of this approach also at finite temperature. This is an important step on the way to deal with the sign problem of QCD. We analyze in detail the chiral symmetry breaking in the massless and massive cases and show that the method works very well and gives good control over a broad range of temperatures, essentially from zero to infinite temperature.
Keywords:lattice field theory, Schwinger model, chiral symmetry, non-zero temperature
Investigations of gauge field theories within the Hamiltonian approach have progressed substantially in the last years with the help of tensor network (TN) techniques verstraete08algo ; cirac09rg ; orus2014review . Taking the example of the Schwinger model, numerical calculations have been performed to investigate ground state properties Byrnes:2002nv ; Cichy:2012rw ; Banuls:2013jaa ; Banuls:2013zva ; Rico:2013qya ; Buyens:2015dkc , to demonstrate real-time dynamics Buyens:2013yza ; Buyens:2014pga and to address the phenomenon of string breaking Pichler:2015yqa ; Buyens:2015tea , which has also been explored in non-Abelian models Kuhn:2015zqa . In Refs. Banuls:2015sta ; Saito:2014bda ; Saito:2015ryj , thermal properties of the Schwinger model were studied for massless fermions. From a more conceptual point of view, TN have been developed that incorporate the gauge symmetry by construction, and constitute ground states of gauge invariant lattice models Tagliacozzo:2014bta ; Silvi:2014pta ; haegeman15gauging ; zohar2015peps . Yet a different line of work is the study of potential quantum simulations of these models, using ultracold atoms, see Refs. wiese2013review ; Zohar:2015hwa ; Dalmonte:2016alw for a review. Also in this field, TN techniques can play a determinant role to study the feasibility of the proposals kuehn2014schwinger .
The last numerical developments go beyond standard Markov Chain Monte Carlo (MC-MC) methods. At zero temperature, the Hamiltonian approach allows us to go substantially closer to the continuum limit and reach a much improved accuracy compared to MC-MC. When temperature is switched on, a broad and very large set of non-zero temperature points can be evaluated, ranging from very high to almost zero temperature. In the string breaking calculation, a nice picture of the string breaking phenomenon and the emergence of the hadron states can be demonstrated. Finally, real-time simulations are not even possible in principle with MC-MC methods.
The key to this success is the employment of tensor network states and, in the case of one spatial dimension, as for the Schwinger model, the Matrix Product States (MPS). In this approach, which is closely linked to the Density Matrix Renormalization Group (DMRG) white92dmrg , the problem, which has an exponentially large dimension in terms of the system size, is reduced to an –admittedly– sophisticated variational solution which can be encoded in substantially smaller matrices. The ansatz can represent arbitrary states in the Hilbert space if is large enough (exponential in the system size). Instead in numerical applications, usually an approximation is found to the desired state within the set of MPS with fixed . By varying , an extrapolation of results to can be performed allowing thus to reach the solution of the real system under consideration. A different approach also using tensor network techniques was applied to the Schwinger model with a topological -term in Refs. Shimizu:2014uva ; Shimizu:2014fsa , where the exact partition function on the lattice was expressed as a two dimensional tensor network and approximately contracted using the Tensor Renormalization Group (TRG).
The application of the MPS technique discussed in the present paper is concerned with non-zero temperature properties of the Schwinger model. In Refs. Banuls:2015sta ; Saito:2014bda ; Saito:2015ryj , we have for the first time investigated the thermal evolution of the chiral condensate in the Schwinger model. In the first paper, where we only studied the massless case, we could demonstrate that the MPS technique can be successfully used to compute such a thermal evolution from very high to almost zero temperature. For massless fermions, the results from our MPS calculation could be confronted with the analytical solution of Ref. Sachs:1991en and a very nice agreement was found demonstrating the correctness and the power of the MPS approach.
In the present paper, we will extend our calculations of the thermal evolution of the chiral condensate to the case of non-vanishing fermion masses. Here, no exact results exist anymore, but only approximate solutions are available Hosotani:1998za which can be tested against our results. For our work at zero fermion mass, we also introduced a truncation of the charge sector Banuls:2015sta which was necessary to obtain precise results at high temperature. Here, we will employ this truncation method, too.
It needs to be stressed that the calculations with MPS, as performed here, have a number of systematic uncertainties which are very important to control. This concerns in particular:
an estimate of results for infinite bond dimension; 111For a given system size, , exact results would actually be attained with finite bond dimension, verstraete04dmrg , which is many orders of magnitude larger than the largest one we use in the simulations.
an extrapolation to zero step size in the thermal evolution process;
a study of the truncation in the charge sector of the model;
an infinite volume extrapolation;
and a careful analysis of the continuum limit employing various extrapolation functions with different orders in the lattice spacing.
Controlling these systematic effects renders the calculations with MPS demanding, but it is absolutely necessary to obtain precise and trustworthy results. We have therefore made a significant effort to perform the above extrapolations and we will provide various examples in this paper for the studies of systematic effects carried through here.
2 The Schwinger model and chiral symmetry breaking
The one-flavour Schwinger model schwinger62 , i.e. Quantum Electrodynamics in 1+1 dimensions, is one of the simplest gauge theories and a toy model allowing for studies of new lattice techniques before employing them to real theories of interest, like Quantum Chromodynamics (QCD). Despite its apparent simplicity, it has a non-perturbatively generated mass gap and shares some features with QCD, such as confinement and chiral symmetry breaking, although the mechanism of the latter is different than in QCD – it is not spontaneous, but results from the chiral anomaly.
We start with the Hamiltonian of the Schwinger model in the staggered discretization, derived and discussed in Ref. Banks:1975gq :
where is the site index, , is the lattice spacing, is the coupling, and with denoting the fermion mass and the number of lattice sites. We use open boundary conditions (OBC). The gauge field, , can be integrated out using the Gauss law:
Thus, only at one of the boundaries is an independent parameter and we take , i.e. no background electric field.
We work with the following basis for our numerical computations: Banuls:2013jaa , where is the spin state at site and all the gauge degrees of freedom have been integrated out.
In this paper, we are interested in the chiral symmetry breaking (SB) in the Schwinger model, both at zero and non-zero temperature. The order parameter of SB is the chiral condensate , which can be written in terms of spin operators as . The ground state and thermal expectation values of the chiral condensate diverge logarithmically in the continuum limit for non-zero fermion mass deForcrand98 ; duerr05scaling ; Christian:2005yp . This divergence is present even in the non-interacting case, where the theory is exactly solvable and the Hamiltonian (2) reduces to the XY spin model in a staggered magnetic field. The ground state energy of this model (with OBC) reads: . The ground state expectation value of can then be computed from the derivative :
The free condensate value computed from this formula can be used to subtract the divergence in the interacting case at a finite lattice size , a finite lattice spacing and a given fermion mass . However, one can exactly evaluate the infinite volume limit of the free condensate first, yielding:
where is the complete elliptic integral of the first kind abramowitz . Note that by expanding this expression in the limit , the divergent logarithmic term is indeed seen already in the free case. In this way, we can extrapolate our lattice interacting condensate first to infinite volume limit, , at a finite and a given and then subtract the infinite volume free condensate () given by Eq. (4):
obtaining finally the subtracted condensate , which can then be extrapolated to the continuum limit . Note that a non-zero temperature does not bring any further divergence, hence the above renormalization scheme, subtracting the zero temperature free condensate in the infinite volume limit, can be applied for any . Actually, one can equivalently subtract the free condensate at any finite . This defines an alternative renormalization scheme that we can also implement. Both options would lead to the correct value at , i.e. compatible with the one directly obtained from the ground state calculations, but in order to compare to other results in the literature, we adopt in the following the renormalization scheme for all temperatures.
In the massless case, the temperature dependence of the chiral condensate was computed analytically by Sachs and Wipf Sachs:1991en :
where , is the Euler-Mascheroni constant and is the non-perturbatively generated mass of the lowest lying boson (the vector boson). According to the above formula, chiral symmetry is broken at any finite temperature (zero or non-zero) and it gets restored () only at infinite temperature. There is no phase transition, i.e. chiral symmetry restoration is smooth.
In the massive case, there is no analytical formula describing the temperature dependence of the condensate. However, the massive model was treated by Hosotani and Rodriguez with a generalized Hartree-Fock approach in Ref. Hosotani:1998za , yielding an approximate thermal dependence of . In the following, we will confront our results with ones from this approximation and thus conclude about its validity.
3 Tensor network approach
In this work, we make use of two different applications of tensor network ansatzes. In order to obtain the results at zero temperature, we approximate variationally the ground state of the Schwinger model Hamiltonian (2) on a finite lattice using a MPS. For the temperature dependence, we employ the matrix product operator (MPO) to describe the thermal equilibrium states at finite temperatures.
Although the details of these ansatzes and the basic algorithms involved can be found in the literature, for completeness we compile in this section the fundamental ideas of both approaches, with special emphasis on the particularities associated to the problem at hand.
where is the dimension of the local Hilbert space for each site. For two-level quantum systems, as in the case we are studying, . The state is parametrized by the matrices, , which have dimension , except for the ones at the edges, and , which, for the open boundary conditions we consider, are -dimensional vectors. The parameter is called the bond dimension, and determines the number of variational parameters in the ansatz. The MPS can efficiently approximate ground states of local gapped Hamiltonians in one spatial dimension, and the ansatz lies at the basis of the success of the Density Matrix Renormalization Group (DMRG) method white92dmrg ; schollwoeck11age . In practice, they have been successfully applied to much more general problems, including long range interactions and two dimensional systems.
Different algorithms exist to find an MPS approximation to the ground state of a certain Hamiltonian. We use a variational search verstraete04dmrg ; schollwoeck11age , in which the energy is minimized over the set of MPS with a given bond dimension, , by successively optimizing over one of the tensors, while keeping the rest fixed. The procedure is repeated, while sweeping over all the tensors, until convergence is attained in the value of the energy, to a certain relative precision, , ultimately limited by machine precision. The computational cost of this procedure scales as with the dimensions of the tensors. The effect of running the algorithm with a limited bond dimension is to suffer a truncation error. By running the algorithm with increasing values of , we can estimate the magnitude of this error and extrapolate to the limit, as discussed in detail in Sec. 4.
While any MPS (7) can represent a valid physical state, as far as it is normalized, in order to describe a physical density operator, the MPO needs in addition to be positive. This condition cannot be guaranteed locally for generic tensors . However, it is possible to ensure the positivity of a MPO using the purification ansatz verstraete04mpdo ; delascuevas2013 , in which each tensor of the MPO has the form . This corresponds to a (pure state) MPS ansatz for an extended chain, with one ancillary system per site, such that is the reduced state for the original system, obtained by tracing out the ancillas. It has been shown that thermal equilibrium states of local Hamiltonians can be well approximated by this kind of ansatz hastings06gapped ; molnar15gibbs in arbitrary dimensions.
In the case of finite temperature, a MPO approximation can be constructed for the Gibbs state via imaginary time evolution of the identity operator verstraete04mpdo , , where is the inverse temperature. To achieve this, we apply a second order Suzuki-Trotter expansion trotter59 ; suzuki90 to the exponential, and approximate every step of width by a product of five terms,
where is diagonal in the basis, and the hopping term is split in two sums , with the () term containing the two-body terms that act on each even-odd (odd-even) pair of sites. If each of the exponential terms can be exactly computed, the error of this approximation scales as . The exponentials of and have indeed an exact MPO expression with constant bond dimension . The term contains long range interactions, but its structure allows us to also write it exactly as a MPO, with bond dimension , as detailed in Ref. Banuls:2015sta . The only non-vanishing elements of the tensors specifying the MPO are , for , where for , and . The virtual bond then carries the information about the electric flux on each link, which can assume values . Instead of working with the exact exponential of , which has a bond dimension , we find it convenient, given the large system sizes we want to study, to truncate the dimension of the MPO, by defining a maximum value the virtual bond can attain, . This is equivalent to truncating the physical space to those states where the electric flux on a link cannot exceed and is thus related to approaches where one explicitly truncates the maximum allowed occupation number of the bosonic gauge degrees of freedom Buyens:2013yza .
Starting with the identity operator, , which has a trivial expression as a MPO with bond dimension one, we successively apply steps of the evolution, using the approximation above, and approximate the result by a MPO with the desired maximum bond dimension. This is achieved with the help of a Choi isomorphism choi , , to vectorize the density operators, such that the MPO is transformed in a MPS, with physical dimension per site , on which the evolution steps act linearly. The approximated effect of the evolution is then found by minimizing the Euclidean distance between the original and final MPS. The procedure can be repeated until inverse temperature is reached. Then we construct (up to normalization) such that the purification ansatz is realized and we ensure a positive thermal equilibrium state. The computational cost of this calculation is the same as that of time evolution of a MPS state, with the increased physical dimension, i.e. it scales as .
Using the MPO ansatz with limited bond dimension induces also a truncation error in the case, which is not equivalent to the one described for . First of all, different ansatzes are used for both cases, and while the MPS truncation in the pure state case can be related to the entanglement in the state, the same is not true for the MPO ansatz in the case of mixed states. 222In the case of operators one should instead talk about operator space entanglement entropy, a measure related to truncation error in the MPO that was introduced in Ref. PhysRevA.76.032316 . Moreover, the distinct numerical algorithms used in both cases also mean that errors are introduced in different ways. In the thermal algorithm, each application of one of the exponential factors in (9) potentially increases the bond dimension of the resulting MPO. Hence, after every step, the ansatz needs to be truncated to the maximum desired value of the bond dimension. In practice, this is achieved by minimizing a cost function that corresponds to the Frobenius norm of the difference to the true evolved operator. As in the ground state search, this optimization is done by an alternating least squares (ALS) scheme, in which all tensors but one are fixed, and repeated sweeping is performed over the chain. Also in this case, we use a tolerance parameter, , to decide about the convergence of the iteration, but now the value bounds the relative change in the cost function during the sweeping that follows the application of each single exponential factor. This procedure leads to errors accumulating along the thermal evolution, and while at the state can be exactly written as a MPO with , the largest truncation errors will occur for the lowest temperatures. Thus, recovering zero temperature results from such a procedure is a non-trivial check that the method is working correctly. The calculation, in contrast, does not suffer from this effect, as it directly targets the ground state variationally.
Additionally, the Suzuki-Trotter expansion (9) introduces another systematic error in the thermal evolution, by using a finite step width , which we need to extrapolate to , and another one in the form of the truncation of the physical subspace to a maximum , described above. All these factors need to be taken into account when performing the extrapolations required to extract the continuum values of the observables under study (see Sec. 4 for details).
4.1 Zero temperature
We begin with our results for the ground state chiral condensate for various fermion masses. For the massless case, an analytical result can be obtained, . We are able to reproduce this number with great accuracy and also obtain results in the massive case, where no analytical results exist.
Our numerical procedure consists in computing several sets of data points corresponding to different values of (, , ) and extrapolating in the way described below.
Infinite bond dimension () extrapolation. We use several values of to check the effects from changing the bond dimension. Our final value is taken as the condensate corresponding to the largest computed value of and its error as the difference between the value for and . The lower values of serve to ensure that the two highest bond dimensions are large enough, such that it can be argued that the difference between and is smaller than the one between and , which makes our error estimate valid.
A typical example of such extrapolation is shown in Fig. 1 for and in Fig. 2 for , at . In both cases, we observe very good convergence towards the limit, with the above defined error from this step being of for the former and for the latter. This error is represented by a red band. Note that despite going to , the convergence in bond dimension is so good that actually even with we would already obtain the result with an outstanding precision, of for (i.e. only an order of magnitude worse than with ) or even of for (i.e. the same as with ). The case illustrates that in some cases the convergence in is so good that our uncertainty comes from issues with the numerical precision. The MPS optimization procedure is considered to be converged when the relative change in the ground state energy in subsequent sweeps falls below a certain tolerance parameter, taken to be in our case. Notice, however, that this precision refers to the ground state energy, which typically converges better than other observables, so it will correspond to a somewhat worse precision in the chiral condensate, which we estimate to be in the region. In the case, the variation of values for different becomes smaller than this, which explains the irregular behaviour of the -dependence for this case (left plot of Fig. 2), compared to the apparently regular convergence for the case . We account for this bias (that happens only for our smallest values) in our next step, the infinite volume extrapolation. We emphasize that this is definitely not a drawback of the method, but even better precision could be attained for certain parameter ranges with the same values, by adopting a more demanding convergence criterion. On the other hand, since the ultimate limit of machine precision, which we label by , affects the optimization of individual tensors, so that after one sweep over the whole chain, it may affect the value of the energy in This means that for chains of hundreds of sites, as required for the largest values of we explore, is the best allowed by double precision numerics.
Infinite volume () extrapolation. The results corresponding to our estimates of the limit can then be extrapolated to infinite volume by using a linear fitting ansatz:
where is the infinite- condensate for a fixed fermion mass, volume and lattice spacing. The fitting parameters are (infinite volume condensate at a given lattice spacing and fermion mass) and the mass and lattice spacing-dependent slope of the finite volume correction, .
We show an example of such extrapolation in Fig. 3, again for (left) and (right), at .
We always choose the volumes to be large enough, such that the above linear fitting ansatz yields a good description of data.
We have found that this holds when the volumes used are scaled proportionally to and we take .
Indeed, in all cases where no issues with machine precision are observed, this leads to very good fits.
The resulting error of the fitting coefficient is the propagated error from the -extrapolation.
For very small values of (lower than approx. 30), we need to deal with the numerical precision bias.
The errors from the -extrapolation are in such case underestimated, since they do not take into account the finite numerical precision.
This leads to values of .
However, we know from the analysis for large values of that the linear fitting ansatz (10) yields an excellent description of data, with usually much smaller than 1.
Hence, we account for the bias by inflating the -extrapolation errors to such levels that by construction.
In this way, the final error after the infinite volume extrapolation step is properly rescaled and becomes comparable to the one at larger (e.g. approx. for and for ).
In the end, all our errors of infinite volume condensates, , differ by less than an order of magnitude in the whole considered range of and for all fermion masses.
Continuum limit () extrapolation. Finally, the infinite volume results can be extrapolated to the continuum limit. First, we subtract the infinite volume free condensate according to Eq. (5), obtaining the subtracted condensate . Then, we apply the following fitting ansatz:
with fitting parameters (the continuum condensate for a given fermion mass), , and . This is a fitting ansatz quadratic in the lattice spacing (the role of the lattice spacing is played by ), with logarithmic corrections. The latter appear already in the free theory, where their presence can be shown analytically (see Sec. 2). Note that the final result obtained from this procedure will, to some extent, depend on the fitting range in . To quote final values independent from such choices, we adopt a systematic procedure analogous to the one we used in our spectrum investigation in the Schwinger model, described in detail in the appendix of Ref. Banuls:2013jaa . In short, this consists in performing fits in different possible fitting ranges by varying the minimal and maximal values of entering the fits. The number of fits that we obtain in this way is of and allows us to build a distribution of the continuum values, weighted with of the fits. The final value that we quote is the median of the distribution and the systematic error from the choice of the fitting range comes from the 68.3% confidence interval (such that in the limit of infinite number of fits it corresponds to the width of a resulting Gaussian distribution). This error is then combined in quadrature with our propagated error from - and -extrapolations, which we take as the error of one selected fit, taken to be the one in the interval .
|Our result||Ref. Buyens:2014pga||Exact ()|
|or Ref. Hosotani:1998za ()|
Our continuum limit extrapolations are shown in Fig. 4 for all fermion masses that we considered. We show in these plots the fit from which we estimated our propagated error from earlier extrapolations (), i.e. one of the fits that enter the distribution built to assess our final values and their uncertainties. The final values for each fermion mass are summarized in Tab. 1. We compare to the result of a similar calculation in Ref. Buyens:2014pga and to the exact result in the massless case or the approximation of Ref. Hosotani:1998za . For the former, we observe perfect agreement, which is quite remarkable given the precision of both results being at the level. Similarly good is the agreement with the analytical result at . We will comment more on the agreement with Ref. Hosotani:1998za in the next subsection.
4.2 Thermal evolution
In our previous papers Saito:2014bda ; Saito:2015ryj ; Banuls:2015sta , we showed results for the temperature dependence of the chiral condensate in the massless case. We employed a method without any truncations in the gauge sector and found that it is numerically very demanding to achieve lattice spacings small enough to reliably extrapolate to the continuum at high temperatures. This led us to the method of introducing a finite cut-off, , in the gauge sector and we showed that this method works very well in the massless case, allowing for good precision of results for the whole range of temperatures. In the present paper, we test the method, explained in detail in Sec. 3, in the massive case. Although this method is different from the one used for , the analysis procedure at a given temperature is rather similar to the one described in the previous subsection. We begin by shortly outlying the new parts of the analysis in the thermal case. In the following, we typically express the temperature with its inverse, .
There are two new parameters with respect to computations, apart from the bond dimension, , the system size, , and the inverse coupling, — the parameter describing the cut-off in the gauge sector and the step width, . Thus, our sequence of extrapolations follows the order given below.
Infinite bond dimension () extrapolation.
This extrapolation is done as in the case and we again take the result at our largest as the central value and the difference between this value and one at as the estimate of the uncertainty from the finite bond dimension.
Examples of such extrapolations are shown in Figs. 5 and 6, for and , respectively (both at , , ).
They illustrate a general feature in the -dependence of the chiral condensate — the convergence becomes worse towards the continuum limit.
However, this convergence is in all cases good — the difference between our two largest bond dimensions (140 and 160) is of at and of at .
This difference also depends on the temperature — since lower temperatures are reached by increasing , the error from the finite bond dimension also increases at increasing , approximately linearly.
Note that in the thermal case, the convergence in is somewhat worse than at and we do not observe issues with insufficient machine precision (cf. Sec. 4.1 and the comments about double precision as not enough for certain parameter ranges).
Finally, there is little dependence on the value of , the volume and on the fermion mass.
Zero step width () extrapolation. We denote the results from the previous step as and they differ from the limit by . Hence, we extrapolate to with:
with the fitting parameters and . We always use three values of for each , which allows us to verify that a fitting ansatz linear in is proper. Since we want to access inverse temperatures with a step of , we use values of small enough such that this is possible. Examples are shown in the lower right plots of Figs. 5 and 6, for and , respectively (again at , , ), and three volumes that are later used for infinite volume extrapolation. Since the resulting errors are the propagated errors from the -extrapolation, one again observes similar parameter dependences for the error obtained at this step. We also note that the linear ansatz (12) works very well.
Infinite volume () extrapolation.
The results corresponding to our estimates of the and limits are extrapolated to infinite volume by using the same kind of linear fitting ansatz as in the case, i.e. Eq. (10), and volumes .
An example extrapolation is shown in Fig. 7, for , , five values of the lattice spacing and two temperatures: (left) and (right).
As in the case, we observe that the fitting ansatz gives very good description of our data.
Removing the cut-off ( extrapolation). The physical results have to be independent of the used gauge sector cut-off. We found empirically that for all ranges of our parameters, always yields results compatible with and . Hence, this value of is effectively and no explicit extrapolation is needed (see also Ref. Banuls:2015sta ).
Continuum limit () extrapolation. As our final step, we perform the continuum limit extrapolation of the infinite volume results . Before this is done, we subtract the infinite volume free condensate according to Eq. (5) and obtain the subtracted condensate . We consider the following three fitting ansatzes:
which differ by the order of the polynomial in . We refer to them as linear+log, quadratic+log and cubic+log, respectively. We observe that the discretization effects are very different at different temperatures, in particular these effects become very strong at high temperatures and a polynomial cubic in is needed to obtain a good description of data. We adopt a modified procedure to obtain the systematic error from the choice of the fitting range and the fitting ansatz. The procedure used to analyze the data is inappropriate here, because of the large dependence of the uncertainty from the -extrapolation on the lattice spacing. This uncertainty at a fine lattice spacing () is up to four orders of magnitude larger than the one for our coarsest lattice spacings. Hence, the analogue of the weighted histogram built at is no longer reliable, as it contains fits with very large uncertainties. This does not happen at , where the fine lattice spacings have only slightly larger uncertainties from the and -extrapolations than the coarse lattice spacings. This reflects the difference in strategies used to approximate thermal and ground states as tensor networks. In practice, it translates into a somewhat different manner the truncation errors are accumulated in the thermal evolution with respect to the algorithm. At large , i.e. after several steps of imaginary time evolution, the truncation errors are much larger than in the ground state. As a consequence, the procedure of obtaining the systematic error does not make sense in the case, since only one or two fits dominate the weighted histogram.
For this reason, the procedure to extract the fitting range/ansatz uncertainty is the following. It is performed separately for each temperature at a given fermion mass . We fix the maximum entering each fit to be the one corresponding to the finest lattice spacing. Then, we build all possible fits of Eqs. (13)-(15) changing only the minimal entering (). We take as the central value that corresponds to the smallest uncertainty propagated through , and -extrapolations, but one that satisfies the condition and has all its fitting coefficients statistically significant. We denote it by and its error by . We combine this uncertainty quadratically with the uncertainty from the choice of the fitting interval, , and from the choice of the fitting ansatz, . The former is defined as the difference between and the most outlying (corresponding to the same , i.e. the same functional form of the fitting ansatz) which has still all the fitting coefficients statistically significant. The latter is taken to be the difference between and the most outlying (where , i.e. from another fitting ansatz) which has again statistically significant fitting coefficients.
Below, we illustrate this procedure with a few examples at the fermion mass (Fig. 8). We start with a low temperature, , effectively corresponding to (after a certain -dependent , the continuum result does not change any more — in the case of , zero temperature is reached around ). Here, taking the linear+log fitting ansatz and yields a good fit, with . It can be compared to only two other fits, both of them linear+log, with and . Increasing further or changing the fit form to quadratic+log or cubic+log leads to at least one of the fitting coefficients becoming statistically insignificant. Hence, our final result for this temperature and fermion mass is and is dominated by the uncertainty from the choice of the fitting interval. The error from the choice of the fitting ansatz is zero, since no quadratic+log or cubic+log fit produces a significant result. Since is effectively , this result can be compared to our result at this fermion mass in Tab. 1. We observe full consistency, although the precision of the thermal computation is four orders of magnitude worse than of the ground state one. This is hardly surprising, as thermal evolution is definitely not the best method to investigate ground state properties.
Another example continuum extrapolation is shown for (upper right plot of Fig. 8). In this case, the central value comes from a linear+log fit with and it is compared to the same functional form of the fit with as well as to a quadratic+log fit with . Finally, we get . Towards higher temperatures, cut-off effects become increasingly important, in the sense that one needs higher order polynomials in . For (lower left of Fig. 8), the central value that we take comes from a quadratic+log fit with , compared to and a cubic+log fit with . This leads to . Our final example is (lower right of Fig. 8). Here, the central value comes from a cubic+log fit with , compared to and a quadratic+log fit with . We get . In all these cases, the error is dominated by the uncertainty from the choice of the fitting interval and ansatz. Nevertheless, with the adopted systematic error estimation procedure, one can have these uncertainties reliably under control.
We repeat the analysis steps for all our fermion masses and we summarize the continuum limit results in Fig. 9, where we show results up to ( and ) or (). The most important feature confirming the validity of our results is that we always reproduce the result within our errors — actually the difference between our central values at large enough and the MPS result is much smaller than our errors, suggesting that the error estimation procedure is rather conservative. We also note that our systematic error procedure makes the final errors strongly dependent on temperature — with sometimes irregular jumps of the error caused by some other fitting interval or fitting ansatz entering the procedure at certain values333For example, at all the quadratic+log fits have at least one fitting coefficient statistically insignificant above and at this temperature and higher (smaller ), quadratic+log fits become statistically significant and thus enlarge our error.. Apart from the agreement with the result, we observe that the approach to this result is faster for higher fermion masses — for , is already effectively zero temperature, while for our lowest mass, , we have small changes of the central value even above . Concerning the agreement with the approximation of Ref. Hosotani:1998za (referred to as “Hosotani HF” in the plot), the latter provides good qualitative description of the temperature dependence of the chiral condensate. However, the quantitative agreement is not perfect, with typical deviations of 10-20%. It is known that the approximation becomes exact in the massless limit and indeed, e.g. Hosotani’s result at is relatively closer to the MPS result than the one at . On the other hand, the approximation of Ref. Hosotani:1998za also approaches the analytical result of zero at infinite fermion mass and — hence one also expects an increasing agreement in this regime. Indeed, the relative difference at is the smallest from among all our considered masses. However, when we consider the slope of the -dependence, we clearly observe that the agreement between Hosotani HF and our computation becomes better towards small fermion masses, with both curves being almost parallel for .
5 Summary and prospects
In this paper, we have performed a study of the temperature dependence of the chiral condensate for the one-flavour Schwinger model using a Hamiltonian approach. We emphasize that while for zero temperature we employ a matrix product state (MPS) ansatz, for non-vanishing temperature we use a matrix product operator (MPO) ansatz. In addition, for the non-zero temperature calculation, we have to perform a thermal evolution by starting from a well defined infinite temperature state and evolve the system in incremental inverse temperature steps towards zero temperature using a density operator.
Thus, non-zero temperature calculations within the Hamiltonian approach are rather different from the so far carried out zero temperature ones and hence non-zero temperature computations for gauge theories are novel and need to be tested. While in Ref. Banuls:2015sta we have initiated such non-zero temperature computations for massless fermions, in this paper we went substantially beyond this work by studying the system at various fermion masses. In addition, we employed consistently a truncation of the gauge sector. This allowed us to reach very large system sizes and, keeping the physical extent of the model fixed, very small values of the lattice spacing.
Within our calculation of the chiral condensate, we carried out a substantial and challenging effort to control the systematic effects. To this end, we performed extrapolations to zero thermal evolution step size, infinite bond dimension, infinite volume and zero lattice spacing. In addition, we tested that our cut parameter for the gauge sector truncation has been sufficiently large. The final non-trivial check of the validity of our approach has been to recover the zero temperature result of the chiral condensate after the long thermal evolution performed.
As a result of our work, we could compute the chiral condensate over a broad temperature range from infinite to almost zero temperature with controlled errors. This has been done for zero, light and heavy fermion masses. For zero fermion mass, we found excellent agreement with the analytical results of Ref. Sachs:1991en . Moving to non-zero fermion masses, a comparison to Ref. Hosotani:1998za did not lead to a clear conclusion, see Fig. 9. Although qualitatively the temperature dependence of the chiral condensate shows a comparable behaviour between the analytical result of Ref. Hosotani:1998za and our data, there does not seem to be an agreement on the quantitative level. This is presumably due to the fact that the approximations made in Ref. Hosotani:1998za are too rough to reach a satisfactory quantitative agreement.
We consider the here performed work, besides of the clear interest in its own, as a necessary step towards investigating the Schwinger model when adding a chemical potential. This setup leads to the infamous sign problem and it would be very reassuring to see whether the here used MPS and MPO approaches can lead to a successful application for this very hard problem, which is very difficult, if not impossible to solve by standard Markov chain Monte Carlo methods.
Acknowledgements.We thank J. I. Cirac for discussions. This work was partially funded by the EU through SIQS grant (FP7 600645). K.C. was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse and in part by the Deutsche Forschungsgemeinschaft (DFG), project nr. CI 236/1-1 (Sachbeihilfe). Calculations for this work were performed on the LOEWE-CSC high-performance computer of Johann Wolfgang Goethe-University Frankfurt am Main and in the computing centers of DESY Zeuthen and RZG Garching.
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• Paid search prices are rising.
• SEO isn’t easy anymore
• You can’t buy links anymore
• Your website is only a billboard
• Great ROI on the Marketing Budget
• Customer Service
• Market Research
• Communication/Brand Management
• “Thought Leadership”
• Stay current.
• Reduce email overload.
• Communicate anytime, anywhere.
• Create and contribute to ideas, content, and products.
• Participate in existing and new online communities.
• Listen to others.
• Learn from others.
• Share with others.
• Engage others.
• Build relationships.
• Relate to new and traditional audiences.
• Corporate Transparency
• Build Trust with Customers
• Generate Inbound Links
• Reputation Management
• Promote your blog and website
• Be a part of the conversation that is already going on
• Extend your events, seminars, webcasts, conventions
• Provides a new way to engage and communicate with your customers
• Engagement: Increase loyalty, foster word of mouth
• Research: Identify trends/niches, consultation
• Marketing: Promote business, brand awareness
• SEO: link building, content factors
• PR: Manage reputation, get news out
• Management: Collaboration, knowledge sharing
• Sales: Gain new business, new contact routes
Social Networks in General
1. People visit social networking sites 5 days per week, checking their
accounts 4 times a day
2. 52% of social networkers had friended or become a fan of at least one
3. 64% were neutral or didn’t care about brands on social networks
4. 45% connect only to family and friends and 18% will connect only to people
they’ve met in person
• More than 250 million active users
• More than 120 million users log on to Facebook at least once each day
• More than two-thirds of Facebook users are outside of college
• The fastest growing demographic is those 35 years old and older
• Average user has 120 friends on the site
• More than 5 billion minutes are spent on Facebook each day (worldwide)
• More than 30 million users update their statuses at least once each day
• More than 8 million users become fans of Pages each day
• People demographics on Facebook are more likely to be married (40%),
white (80%) and retired (6%) than users of the other social networks.
• Facebookers have the second-highest average income ($61,000) and an
average of 121 connections.
• 75% say Facebook is their favorite site with 59% saying they’ve increased
their usage over the past 6 months.
• More than 1 billion photos uploaded to the site each month
• More than 10 million videos uploaded each month
• More than 1 billion pieces of content (web links, news stories, blog posts,
notes, photos, etc.) shared each week
• More than 2.5 million events created each month
• More than 45 million active user groups exist on the site
• More than 50 translations available on the site, with more than 40 in
• About 70% of Facebook users are outside the United States
Twitter Users Are Trending Older
• 72.5% of all users joining during the first five months of 2009.
• 85.3% of all Twitter users post less than one update/day
• 21% of users have never posted a Tweet
• 93.6% of users have less than 100 followers, while 92.4% follow less than
• 5% of Twitter users account for 75% of all activity
• New York has the most Twitters users, followed by Los Angeles, Toronto,
San Francisco and Boston; while Detroit was the fast-growing city over the
first five months of 2009
• More than 50% of all updates are published using tools, mobile and Web-
based, other than Twitter.com.
• There are more women on Twitter (53%) than men (47%)
• Of the people who identify themselves as marketers, 15% follow more than
2,000 people. This compares with 0.29% of overall Twitter users who follow
more than 2,000 people.
• Average Twitter user has 28 followers and follows 32 others
Why you need Twitter?
• Fastest Growing Social Networking site - +813% from January 2008 to January 2009
• 44.500.000 users
• Twitter is dominated by newer users - 70% of Twitter users joined in 2008
• An estimated 5-10 thousand new accounts are opened per day
• 35% of Twitter users have 10 or fewer followers
• 9% of Twitter users follow no one at all
• There is a strong correlation between the number of followers you have and the
number of people you follow Level Access to Influencers
• Very Social
• Conversation Monitoring
• Real Time Couponing
• Passive Chatting
• Comcast Cares, Dell Outlet
• Meet Cool Tweeple (not a word)
• Not surprisingly, most users (97%) agree that brands should engage their customers
on Twitter. This is 8 percentage points higher than the fall survey. Clearly Twitter
users want to engage with their brands.
• The majority also have a better impression of brands that use Twitter for customer
• Proper usage of Twitter however, is paramount as 90% of users would frown upon
poor or inappropriate brand use of Twitter.
• The power of a relationship is extremely strong on Twitter. 80% of respondents
would recommend a company based on their presence on Twitter, a huge 20
percentage point increase from the prior survey and 84% of Twitter users will
reward those brands they have key relationships by being more willing to purchase
from them. This was a 5 percentage point increase from the original survey.
• Influencers: More than 80% of respondents have 100+ followers and almost 35% of
respondents have posted more than 1000 Tweets since they signed up for the
• Twitter is a growing microblogging network where people answer the question:
What are you doing now? And write their answer in 140 characters or less.
• Originally designed for text messaging via cellphones (which have a limit of 140
characters for text messages), Twitter has become a whole lot more than the
answer to What are you doing now?
• 9.4 million people belong to Twitter right now. More than 7,500 people join Twitter
• About 500 users have 20,000 or more followers. The average tweeter probably has
500 followers or less. 35% of Twitterati have 10 or less followers; 9% have none.
• Those numbers will change as people begin to realize the true value of Twitter for
helping to create relationships, forge partnerships, and engage in meaningful |
Current Index to Statistics
IEEE Transactions on Reliability
Hypothesis-test for reliability in a stress-strength model, with prior information
Nandi, S. B.
Aich, A. B.
Modified `practical Bayes-estimators'
Forced-outage rates of generating units based on expert evaluation
Noor, S. Fayyaz
McDonald, J. R.
The use of imprecise component reliability distributions in reliability calculations
Roberts, Ian D.
Samuel, Andrew E.
Fuzzy reliability using a discrete stress-strength interference model
Wang, J. D.
Liu, T. S.
Weibull component reliability-prediction in the presence of masked data
Usher, John S.
Bayes estimation of component-reliability from masked system-life data
Lin, Dennis K. J.
Usher, John S.
Guess, Frank M.
Estimating component-defect probability from masked system success/failure data
Flehinger, Betty J.
Conn, Andrew R.
A cautionary tale about Weibull analysis
Mackisack, M. S.
Stillman, R. H.
Characterization of bivariate mean residual-life function
Kulkarni, H. V.
Rattihalli, R. N.
Linear-spline approximation for semi-parametric modeling of failure data with proportional hazards (STMA V38 1917)
Love, C. E.
A reliability model of a system with dependent components (STMA V38 2380)
Robust parameter-estimation using the bootstrap method for the 2-parameter Weibull distribution
Optimum 3-step step-stress tests
Khamis, Imad H.
Higgins, James J.
Nonparametric model for step-stress accelerated life testing
Tyoskin, Oleg I.
Krivolapov, Sergey Y.
Some results on discrete mean residual life
Salvia, Anthony A.
Engineering notion of mean-residual-life and hazard-rate for finite populations with known distributions
Estimating the cumulative downtime distribution of a highly reliable component
Jeske, Daniel R.
A system-based component test plan for a series system, with type-II censoring
Development test programs for 1-shot systems: 2-state reliability and binary development-test results
Vardeman, Stephen B.
Proportional hazards modeling of time-dependent covariates using linear regression: A case study (STMA V38 3217)
Computational algebra applications in reliability theory (STMA V38 3686)
The use of precautionary loss functions in risk analysis (STMA V38 3705)
Norstroem, J. G.
Norstrom, J. G.
Norstrøm, J. G.
Solving ML equations for 2-parameter Poisson-process models for ungrouped software-failure data (Corr: 1997V46 p349 STMA V39 4232))
Knafl, George J.
Using neural networks to predict software faults during testing (STMA V38 3692)
Khoshgoftaar, T. M.
Szabo, R. M.
Generalized linear models in software reliability: Parametric and semi-parametric approaches (STMA V38 3681)
El Aroui, M.-A.
Bayes estimation for the Pareto failure-model using Gibbs sampling
Tiwari, Ram C.
Zalkikar, Jyoti N.
Hierarchical Bayes estimation for the exponential-multinomial model in reliability and competing risks
Papadopoulos, Alex S.
Tiwari, Ram C.
Zalkikar, Jyoti N.
Predictive Bayes design of Scram systems: Related mathematics and philosophical implications
Clarotti, Carlo A.
A Bayes approach to step-stress accelerated life testing
van Dorp, J. Rene
van Dorp, J. René
Mazzuchi, Thomas A.
Fornell, Gordon E.
Pollock, Lee R.
A Bayes approach to step-stress accelerated life testing (STMA V38 3715)
van Dorp, J. R.
Mazzuchi, T. A.
Fornell, G. E.
Pollock, L. R.
A Bayes ranking of survival distributions using accelerated or correlated data
Zimmer, William J.
Deely, John J.
A reliability-growth model in a Bayes-decision framework
How to model reliability-growth when times of design modifications are known
Needed resources for software module test, using the hyper-geometric software reliability growth model (STMA V38 5088)
A conservative theory for long-term reliability-growth prediction (STMA V38 5078)
A survey of discrete reliability-growth models
Comparing the importance of system components by some structural characteristics
Meng, Fan C.
Accelerated life tests for products of unequal size
Bai, D. S.
Yun, H. J.
Accelerated life tests analyzed by a piecewise exponential distribution via generalized linear models
Barbosa, Emanuel P.
Colosimo, Enrico A.
Optimal release policy for hyper-geometric distribution software-reliability growth model (STMA V38 5089)
A Bayes nonparametric framework for software-reliability analysis (STMA V38 5084)
The failure of Bayes system reliability inference based on data with multi-level applicability
Philipson, Lloyd L.
Bayes and classical estimation of environmental factors for the binomial distribution
Elsayed, E. A.
Prediction intervals for Weibull observations, based on early-failure data
Hsieh, H. K.
Confidence interval for the mean of the exponential distribution, based on grouped data
A coherent model for reliability of multiprocessor networks (STMA V38 5079)
Failure-rate functions for doubly-truncated random variables
Ruiz, Jose M.
Analysis of step-stress accelerated-life-test data: A new approach
Tang, L. C.
Sun, Y. S.
Goh, T. N.
Ong, H. L.
A discretizing approach for stress/strength analysis
English, John R.
Landers, Thomas L.
An input-domain based method to estimate software reliability (STMA V38 1002)
De Agostino, E.
Di Marco, G. |
Received 30 May 2016; accepted 19 July 2016; published 22 July 2016
1. Literature review
During the last few years and due to the numerous advances in information systems, the use of means designed for the acquisition of data such as web and mobile applications, social networks, etc. has massively increased. As a result of this “information revolution”, the world of science has been saturated with data of varied origin. It is estimated that 90% of all data have been created in the last two years (2013-2015) . At the IOD (Information On Demand) Conference held in 2011, IBM presented the explosion of data in today’s society as a problem, and put forward how companies are facing the challenge of obtaining relevant and valuable information from this vast amount of data. The amount of data in the world is expected to double every two years, according to the data scientist Mark van Rijmenam, founder of Datafloq, in addition to increase 2.5 exabytes per day .
This enormous amount of information is known as Big Data. The vast majority of these data, which come from astronomy, genomics, telephony, credit card transactions, Internet traffic and web information processing, primarily, are acquired systematically with a certain frequency, being therefore time series - . The tendency to manipulate large quantities of data is due to the need, in many cases, to include the data obtained from the analysis of large databases in new databases, such as business analyses . Besides data manageability, other factors to consider are the speed of analysis/scanning speed, access, search and return of any element. It is important to understand that conventional databases are a significant and relevant part of an analytical solution .
Today, the explosion of data poses a problem given the amount of these increases overwhelmingly; in fact this situation reaches occasionally the point when it is not possible to gain an useful insight from them. Therefore, it is necessary to organise, classify, quantify and of course exploit this information to obtain maximum performance for the benefit of scientific research. In response to this difficulty the concept of Data Mining arises that refers to the non-trivial automated process which identifies valid, previously unknown, potentially useful and funda- mentally understandable patterns in the data.
The literature shows that Data Mining techniques are used to extract information from very diverse back- grounds as the power consumption of a region , modelling and optimisation of wastewater pumping systems and the establishment of the position of wind turbines to obtain the maximum possible wind currents .
A common pattern of all previous studies is the use of time series for the analysis and visualisation of information. A way to perform the processing of time series is through the creation of mathematical models that identify and predict their behaviour. One of these are the ARIMA models , that extract the most relevant data from the dataset identifying the patterns of the series at different levels of the timescale and simplify a large amount of data in a simple equation, hence their utility and application in Data Mining. ARIMA models are within the Data Mining techniques, as these are used in time series, therefore being a very useful tool to extract relevant information from Big Data.
In the field of the analysis and visualisation of data, the development of free software is a good tool for both analytical and visual integration of information. In this section of the processing of data, software for the analysis and visualisation of data allow to work with large volumes of data completed over a period of time . The development of statistical software that allows to work on the analysis of time series further facilitates the implementation of ARIMA models.
The use of free access software as Rstudio, which is an integrated development environment for R, has the advantage of enabling programming statistical packages as required, as well as of applying all kinds of time series analysis, in addition to reducing economic costs in any research project. In the present work a script has been developed in the environment of programming R language that allows the implementation, processing and visualisation of ARIMA models, in order to make it easier for scientists to know about the exploration, exploitation and manipulation of large volumes of univariate data carrying associated timescales. The script development and implementation structure is shown in Figure 1.
this script achieves the implementation of the Box-Jenkins methodology for the development of ARIMA-models; in this way, the researcher is able to decompose the time series and to obtain the most relevant information of the characteristics of the temporal series, showing the extent to which this script helps in the exploration, exploitation and manipulation of data.
2. Information about the ST.R File
This document provides information about what is and how to use the ST.R script.
2.1. What Is ST.R?
ST.R is a code in R language developed for the treatment of time series and the realisation of ARIMA models following the Box-Jenkins methodology . The script is split into two blocks. The first one is a collection of
Figure 1. Structure of development and implementation of the script in R. The different actions to be followed for the implementation of the script are shown. It is a conceptual model of implementation where Excel is used as a possible tool for data management. Source: own elaboration.
commands for the numerical and graphic description of the time series, and the development of the ARIMA models. In the second block the commands of different precision measurements are set up, which allow to compare the forecasts made by the models with the actual data with the aim of selecting the model with the most optimal fit to actual observations .
2.2. How to Use ST.R
In order to successfully run the ST.R script, the necessary libraries are lmtest and tseries. These libraries are available from the repository Comprehensive R Archive Network (CRAN) at http://CRAN.R-project.org/package=OptGS. In this work, the R “stat” package version 3.3.0 was used, using “ARIMA” argument. The fitting methods are described in the R manual .
2.3. ST.R Structure
2) Trend analysis: the existence or non-existence of the trend is studied from the graphical results. A linear trend will be removed with first differences. However, for a nonlinear trend two differences are used. The Dickey-Fuller and KPSS tests are used for the analysis (Figure 4).
3) Homocedasticity analysis: This is done from both a visual and a mathematical perspective. From a visual point of view, it is carried out through the study of the thickness of the series. If this thickness remains constant, with no major irregularities observed, the series will be homocedastic; otherwise, the series will be considered heterocedastic. From a mathemathical, it is carried out with the application of the homoscedasticity Breusch- pagan test (Figure 5).
Figure 2. An R Graphical User Interface (GUI) for step 1. Graphical representation.
Figure 3. An R Graphical User Interface (GUI) for step 1. Graphical representation.
4) Stationarity analysis: As a result of the steps above, when neither seasonal cycle, nor trend, nor a significant thickness alteration of the series are to be perceived, the series is regarded as stationary (Figure 5).
5) Model identification: the most optimal model type is determined from the order of the Autoregressive procedure and moving averages of the constituents, both uniform and seasonal. This choice is made from autocorrelation (FAC) and partial autocorrelation (partial FAC) functions (Figure 5).
6) Estimation of the coefficients of the model: the order of the model having been established, the estimation of its parameters is made. Given it is an iterative calculation process, initial values (pool of models) can be suggested (Figure 5).
Figure 4. An R Graphical User Interface (GUI) for step 2. Trend analysis.
Figure 5. An R Graphical User Interface (GUI) for steps 3 - 7, 10. Homocedasticity analysis; stationarity analysis; model identification; estimation of the coefficients of the model; detailed error analysis; forecast.
7) Detailed error analysis: It is made from the verified differences between values observed empirically and estimated by the model for their final assessment. It is necessary to check an inconsistent regime of them and analyse the existence of significant errors. The Ljung-Box test is applied (Figure 5).
8) Contrast of model validity: the model or models initially selected are quantified and valued using various statistical measures. The measures applied are: R2 (coefficient of determination), % SEP (standard error percentage), E2 (coefficient of efficiency), ARV (average relative variance), AIC (Akaike information criterion), RMSE (root mean square error) and MAE (mean absolute error) (Figure 6).
9) Model selection: based on the results of the previous steps, the model to work on is decided upon (Figure 6).
10) Forecast: the most optimal model will be used as the prediction base tool (Figure 5).
2.4. ARIMA Models
The univariate ARIMA models (p,d,q) try to explain the behaviour of a time series from past observations of the series itself and from past forecast errors. The compact notation of the ARIMA models is as follows:
where p is the number of autoregressive parameters, d is the number of differentiations for the series to be stationary, and q is the number of parameters of moving averages. The Box-Jenkins model (p,q) is represented by the following equation:
The autoregressive part (AR) of the model is, while the part of moving averages of the
Figure 6. An R Graphical User Interface (GUI) for steps 8 and 9. Contrast of model validity; model selection.
from the data, by means of any consistent statistic. The ARIMA models allow fitting the trend plus the stationarity in data. In this case, the model is noted as:
where P is the number of autoregressive parameters in the seasonal part, D is the number of differentiations for the series to be seasonal in the seasonal part, Q is the number of parameters of moving averages in the seasonal part and S is the series frequency.
The Box-Jenkins method provides forecasts without any previous conditions, apart from being parsimonious with regard to coefficients . Once the model has been found, forecasts and comparisons between actual and estimated data for observations from the past can be done immediately .
The identification of the parameters p, q, P, Q and S is done by inspecting the autocorrelation function (ACF) and the partial autocorrelation function (PACF), taking into account differentiation and seasonal differentiation .
To create models, the most suitable values of p, d and q were used, according to the measures of accuracy which are presented in the section of criteria for model selection. The parameters ϕ and θ are set through the use of the function minimisation procedures so that the square sum of residues be minimised.
The time series trend is studied applying the Dickey-Fuller and KPSS tests. The Dickey-Fuller test contrasts the null hypothesis that there is a unit root in the autoregressive polynomial (non-stationary series) against the alternative hypothesis that holds the opposite. The KPSS is another test with the same aim, but not exclusive of autoregressive models, supplementary of the former, which contrasts the null hypothesis that the series is stationary around a deterministic trend against the unit root alternative (non-stationary series). Homoscedasticity is studied through the Breusch-Pagan test , which contrasts the null hypothesis that holds heteroscedasticity exists against its nonexistence.
2.5. Model Selection Criteria
The correlation between the actual and forecast data for a variable (x) is expressed by using the correlation coefficient. The coefficient of determination (R2) describes the proportion of total variation in the actual data, which can be explained by the model. The coefficient of determination shows a range of variation [0-1]. If R2 = 1, it means a perfect linear fit, that is to say the proportion of total variation in the actual data is explained by the model. Instead if R2 = 0, the model does not explain anything of the proportion of total variation in the actual data .
Other selection measures applied in R are the standard error of prediction percentage (% SEP) , the efficiency coefficient (E2) , the average relative variance (ARV) and the Akaike information criterion (AIC) . The first four estimators are unbiased estimators which are used in order to check to what extent the model is able to explain the total variance of the data, while the AIC uses the maximum likelihood function to select the model which best fits data. Moreover, it is advisable to quantify the error in the same units as the studied variable.
These measures, or absolute error measures, include the root mean squared error (RMSE) and the mean absolute error (MAE), both expressed as follows:
where is the variable observed at moment t, is the estimated variable at the same moment t and N is the
total number of observations of the validation set.
The standard error of forecast percentage, % SEP, is defined as:
where is the average of the variable observed of the validation set. The main advantage of %SEP is its non-dimensionality, which allows to compare the forecasts of the different models on the same base.
The efficiency coefficient (E2) and the average relative variance (ARV) are used to verify how the model explains the total variance of data and to represent the proportion of the variation of the data observed considered for the model. E2 and ARV are defined as:
The sensitivity to the atypical values due to squaring the terms of the difference is associated to E2 or to ARV. The Akaike information criterion (AIC) combines the maximum likelihood theory, theoretical information and information entropy , and is defined by the following equation :
where N is the total number of observations of the validation set, k is the number of the parameters of the estimated model, MSE is the mean square error estimated, which is defined by the following equation :
where N is the total number of observations of the validation set, k is the number of parameters of the estimated
Depending on the fit, a model which explains a high variance level (R2, ARV, E2) in the validation period is associated to low absolute error (RMSE, MAE), relative (% SEP) and Akaike (AIC) values. Hence, the hypothesis is validated that when using AIC the best model will be that which presents the lowest value, since its likelihood function will fit the data more accurately .
The nature of information differs now from that of information in the past. Due to the vast amount of measuring devices (sensors, microphones, cameras, medical scanners, images, etc.), the data generated by these elements are the largest of the entire available information spectrum. For this reason, the analysis of the wealth of time series has been carried out in a continuous and frequent way in order to obtain the prediction variables and thus to be able to warn behaviour in the environment these occur.
The analyses of time series take into account the degree of dependence between observations and allow to obtain valid inferences without violating basic assumptions of the statistical model or introducing variations in order to overlook this problem; this way, the model further fits the real behaviour of the series.
Since time series are currently employed in different and various fields of knowledge―telecommunications , fisheries , medicine , etc.―it is important to perform a script that allows to give a global and integrated vision on the treatment of time series grouping all the relevant information with the characteristics of the series and prediction models.
Treatment and analysis of time series using free software such as R presents advantages and disadvantages in comparison with private software. On the one hand R has been used in this work as a free and cross-platformer software, making it easy to work with different operating systems. As it has an open source, it is continuously updated by users, not to mention its great graphical power. On the other hand we are aware that the development of this script in the R programming environment presents a number of drawbacks, such as abundant but unstructured help information or packages and functions that make it difficult to locate specific information in a given search. Error messages do not show clearly where in the development of the script the bug is committed, which creates problems for users with little experience in this programming environment making the initiation tedious. R is a programming language in lines of commands, which does not use menus as other statistical programs (e.g., Statgraphics) interfaces. However this can also be an advantage since R advanced users are able to schedule the treatment and analysis of data, in order to understand the basis of the statistical development and data analysis.
To this aim the ST.R script has been created, whose main objective is the analysis and development of forecasting models for time series. It can be established that time series models allow to estimate the degree of significance of a level change which is operated as a result of the application of a treatment . These models not only allow to obtain statistical inferences on treatment action, but also solve the problem of dependence inherent to this type of designs which use a single subject.
In this work, Excel has been used for the database structure management. We know that this system is not sufficiently solvent to support the current data productions . Although Excel is satisfactory for time series management since this working field is univariate based, Excel has also the advantage of being user friendly and accessible for most users. Then this system is considered an efficient tool when it comes to structuring univariate time series.
In conclusion, the present script aims to be a useful and efficient tool to give a global and integrated vision on the time series treatment through the application of Data Mining based on ARIMA models. Introducing this script has made it possible to group all the most relevant information related to the series and prediction models characteristics in order to be able to optimise decision-making in research, in the sense of obtaining more robust and reliable results to support the study.
We thank Sonia Páez-Mejías for the edition of the manuscript in English. We also wish to acknowledge Miguel Ángel Rozalén Soriano for the constructive comments and suggestions about Big Data and Data Mining. This study has been submitted to the V International Symposium on Marine Sciences (July, 2016). The authors are grateful to anonymous referees for their helpful comments and CACYTMAR (Centro Andaluz de Ciencia y Tecnología Marinas) for funding support.
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What are the foundations of mathematics? Early answers to this question were closely related to geometry, and historically, the philosophy of mathematics and the mathematics of geometry maintained a unique connection for more than two thousand years. During this period absolute certainty reigned, and here we shall survey major developments in the evolution of geometry and metamathematics in relation to certitude. We will begin with the origins of the belief in mathematical certainty in Classical Greece, then survey its connection to science through to the seventeenth-century. In closing, we will examine the decline of certainty in the early nineteenth-century, when the discovery of non-Euclidean geometry forced uncertainty on to mathematics and philosophy.
Perhaps the first inquiry in to mathematical foundations was by the Greek philosopher Thales (c. 624 - 547 BCE). Thales saw that in counting and measuring, the practices of unconnected regions coincided, and the practices of one region applied to others. This coincidence enabled different groups to make calculations in the same way, for example when working with physical spaces that approximated elementary mathematical shapes, such as rectangular grain fields. Observing that geographically diverse peoples treated numbers and numeric operations similarly, Thales asked: why?
The practices Thales observed had developed independently, but appeared to share the same general form, and to be generally applicable and accurate, and this was a remarkable fact when compared to the non-generality of other regional practices, for example in politics and religion. In attempting to account for his observations, Thales approached his explanation empirically and universally, and his mode of explanation differed dramatically from the prevalent mode of explanation, which was pre-deductive (and which we refer to as pre-deductive precisely because of the power and prevalence of deduction, after Thales).
Pre-deductive discourse, as seen for example in the religious texts of Thales' era, presented claims in a de facto manner, and presented idealized assertions and idealized consequences, while Thales attempted to arrive at conclusions about observations, and also inquired about the very basis of his observations. Thales was therefore grasping towards a new mode of discourse that we might describe as proto-deductive.
Owing to the nature of his investigations, Thales introduced the term "geometry," meaning "earth measurement," in reference to land plotting and similar activities. The term "mathematics" meaning "knowledge," was introduced after Thales by his mathematical successors, the Pythagoreans. With respect to metamathematics, the origin of these terms is important, being an indicator of the reason geometry and mathematics came to be well-defined fields of inquiry. Geometry arose to organize regionally diverse but conceptually united practices, and approached the real world in terms of magnitudes, and elementary operations that related those magnitudes; and mathematics arose to treat of magnitudes and operations more generally.
Enthralled by the incredible utility and uniformity of mathematics, the Pythagoreans developed a mystical belief system based on the idea that mathematical associations were the framework within which the physical world unfolded. In their framework the concept of number was central, and the Pythagoreans equated math and numbers with metaphysical genesis, as can be seen from one of their oaths; "Bless us, divine number, thou who generates gods and men!"
The Pythagoreans made a number of discoveries that correlated nature closely with mathematics, such as the discovery that musical harmonies may be represented in terms of whole number ratios. This provided fodder for the idea that mathematics was not merely the prism through which nature could be understood, but that nature was mathematics; that "all things are numbers." This metamathematical idea led the Pythagoreans to categorize nature hierarchically, such that math was the source of the universe, and expressed itself in terms of the discrete and the continuous, where the discrete gave rise to the absolute (arithmetic) and the relative (music), and the continuous gave rise to the static (geometry) and the moving (astronomy). Mathematics was the fountainhead, prior to both "gods and men," and generated and organized all of nature; an important claim, because it made mathematics more basic than gods, and was therefore connected to Thales' reasoning process, in that both reassessed religious thinking.
In sum, Thales considered the practices of mathematics generally, and approached math in a way that prefigured deduction, and the Pythagoreans took the universality of mathematics to indicate that the universe was identifiable with mathematics. Thus, mathematical practices had directly spurred metaphysical reflections, and those reflections yielded metamathematical conclusions that led to realignments in existing philosophies. Although claims that appealed to God in pre-deductive modes of explanation still dominated, by the era of the Pythagoreans they were increasingly challenged by mathematical considerations.
Like the Pythagoreans, the Greek philosopher Plato (c. 424 - 347 BCE) believed mathematics was fundamental to being, however, unlike the Pythagoreans, Plato did not believe a hierarchy of categories such as the discrete and continuous captured the foundations of mathematics. For Plato, mathematics existed in the eternal world of Forms, while humans lived in the temporal world, in an ever-changing process of becoming. The Forms effected the universe, and the universe's physical forms were constantly undergoing change, and because of this the real world presented only a shadow of the Forms to humans, meaning humans had limited access to the perfect Forms of mathematics. Mathematics did underpin nature, but natural sensations presented nature and math to humans incompletely.
Because mathematical Forms existed independently of human experience and could not be properly perceived via the senses, Plato eschewed the incompleteness of sensation, turned inwards, and concluded true knowledge of the Forms was to be achieved through cogitation. Because mathematics transcended human experience, it was a natural truth that could be established by transcendent thought. Thus, Plato accepted the Pythagorean belief in mathematics as a basic reality that exists independently of humans, and combined it with Thales' concern for understanding the connections between ideas in a universally consistent manner.
Responding to Plato's metamathematical deliberations, his student Aristotle (384-322 BCE) took up the project of formalizing Thales' reasoning procedure, and elaborated on the relationship between claims and conclusions, and denied that mathematical truth corresponded to the contemplation of ideal mathematical Forms. For Aristotle, Forms inhered within physical existence, and the foundation of mathematics was forms inhering in the world. True mathematics were indeed arrived at by reasoning, however reasoning was to be based on observations of the Forms in nature, rather than arguing from purely intellective premises about the Forms. Physical experience was the foundation for arriving at accurate mathematics: observing the world, analyzing those observations generally, and categorizing those analyses produced truth. Only thus could humans draw objective and accurate conclusions about the mathematical Forms.
Building on the work of Thales, the Pythagoreans, Plato, and Aristotle (and others), the Greek expositor Euclid (c. 300 BCE) set forth in his Elements a series of mathematical proofs using the recently developed logico-deductive format, beginning with mathematical axioms and postulates, combining these with mathematical rules, and setting out the conclusions that followed from these combinations.
In the Elements, Euclid exhibited the mathematics of his era, which were primarily concerned with geometrical results, by taking mathematical truths that were seemingly self-evident, and using precise, repeatable procedures, that any reader could reapply to develop the exact same theorems. Metamathematically, the Elements is important philosophically and historically, because if its reader accepted the mathematical axioms and operations as defined within -- as they apparently had to -- they were also forced to accept its conclusions. For this reason, the Elements possessed a finished quality; there was no room for further development of the theorems laid out, because none found a reason to disagree with them. Hence, in a sense, the Elements completed the project Thales' started, in its development and presentation of an apparently universally applicable and accurate mathematics.
Mathematics, then, was not seen like other subjects such as politics and religion, which permitted contention and ceaseless disputation and were therefore a collection of claims that were in at least some degree vague or indefinite. It seemed that in mathematics, one observed reality as it was, by universally proving the validity of a theorem. All observers could reproduce a theorem, and thus be certain they shared in the knowable reality of that theorem in exactly the same way as all other observers.
Therefore, as the end of Classical Greek civilization approached, mathematics was regarded as a domain that advanced certain knowledge, because of the metamathematical belief that math's foundations were perfectly natural, and that math's theorems were equivalent to natural relations, as revealed through systematic observation and testable manipulation.
The enduring power of this metamathematical certitude was captured in the results of the Greek mathematician Archimedes (c. 287 - 212 BCE), who combined physical motion with mathematics in such an innovative and lasting manner that many regard his proper intellectual successor to be Isaac Newton (1642 - 1727 CE). Addressing the ancient problem of squaring the circle, Archimedes provided an extraordinary geometric solution that synthesized circular and linear motions. Although these motions were acceptable in Euclidean geometry their synthesis was unprecedented, and though Archimedes' results were not strictly Euclidean, they were rigorous and had all the certainty of a Euclidean result.
This was of singular importance in the history of metamathematics, for after Euclid and Archimedes, the development of geometry, and advances in the investigation of metamathematical certainty languished, for nearly two millennia. Looking forward, we find it was not until the seventeenth-century that new and significant progress occurred in the study of geometry; and, pursuant to the progress of geometry, it was only in the eighteenth-century that significant progress occurred in the study of the foundations of mathematics.
With respect to geometry, the objective of Galileo Galilei (1564 - 1642 CE) was to apprehend the algebra of objects moving in space. In Particular, Galileo's goal was to determine which properties of natural objects and motion could be measured and related to each other mathematically. Accordingly, he came to focus on physical features such as weight, velocity, acceleration, and force. Investigating the foundations of mathematics was not one of Galileo's direct concerns, as he noted in his Discourses and Mathematical Demonstrations Concerning Two New Sciences (1638); "The cause of the acceleration of the motion of falling bodies is not a necessary part of the investigation."
Nonetheless, though Galileo aimed at practical explanations and not foundational ones, he did comment on natural philosophers that developed systems based on mere argumentation, rather than systems based on physical experimentation. Importantly, though Galileo was catholic, and his metamathematics reflected his metaphysics -- God was the basis of existence, and therefore math -- Galileo felt God had no immediate place in physical explanations of the world, because "the universe ... is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures." Proportionately, nature was revealed to humanity by direct study of the world, rather than otherworldly speculation.
This practical bent was shaped by Galileo's metamathematical belief that there was a fundamental difference between idealizing on the one hand, and measuring and then idealizing on the other. In terms of historical continuity, the importance of Galileo was that he took up the methods of Aristotle and Euclid, and picked up the physically oriented studies of Archimedes, in order to develop mathematical equations that correlated natural properties to natural regularities.
In connection to foundations, René Descartes (1596 - 1650 CE) agreed that God was the source of reality and the designer of mathematics, and that God was the reason humanity was able to perceive truths about reality. For Descartes, the fact that God had designed reality mathematically was evident in the patterns we observed, and, as a perfect being, God presented patterns to humans only if they represented truth, and therefore we could be sure of our observations.
Like Plato, Descartes posited a world of perfection that was partially accessible via the senses, and like Aristotle and Galileo, Descartes believed sense datum should be analyzed to arrive at true mathematical theorems. Combining natural patterns with intellective analysis, Descartes associated the properties of lines and points with the symbolic mode of representation, and revolutionized the study of nature by introducing the concepts of variable magnitude and coordinate geometry -- the latter having also been developed by Pierre de Fermat (c. 1607 - 1665 CE), independently of Descartes.
Using Euclidean theorems as a basis, coordinate geometry correlated geometric properties to general algebraic statements that related those properties, and defined curves using symbolic relations. Like the equations of Galileo, coordinate geometry tied physical phenomena to quantitative relations, and, when taken altogether, the works of Galileo, Descartes, and Fermat redefined both the purpose and content of natural philosophy, by grounding it in mathematics. This was a new science imbued with a new type of certainty, based on the authority of God through the certainty of his mathematics.
Adopting both the foundations and practices of the new science, Isaac Newton (1642 - 1727 CE) also maintained that God was the foundation of the universe, and therefore mathematics. In contrast to Galileo and Descartes however, Newton's religion was primary, and was a personal motivation for his mathematical work.
Like Galileo and Descartes, Newton regarded his mathematical intuitions and discoveries as confirmation of his religious ideals, and like Galileo, Newton's emphasis was practical. Building on coordinate geometry, Galileo's studies of motion, and Descartes' conception of variable magnitude, Newton developed the calculus, which approached a curve as a flowing quantity that moved across time, thus defining a close relationship between time and motion. The calculus was a sort of procedural algebra that could be used to manage and understand relations between changing variables, per real world examples such as planetary orbits. For Newton, the harmony of his algebraic mechanics with real world mechanics demonstrated that the universe proceeded along its course mathematically, and the calculus was a testament to its supernatural designer.
Motivated by religion and drawing religious conclusions from his science, Newton's mentality was reminiscent of the Pythagoreans, and his esoteric declarations and studies mark him as somewhat of a mathematical mystic. This fact is easily understandable, in reference to the historical milieu he lived in, but salient metamathematically, because for Newton, Galileo, Descartes, Fermat, and a preponderance of their contemporaries, there was an essential accord between the qualities of God and the quantitative relations of mathematics.
Considering the transformation of natural philosophy from the period beginning immediately before Galileo, and ending with Newton, we observe that science underwent a mathematical reformulation. Before Galileo, natural philosophers concerned themselves with testing ideas against other ideas. By the time of Newton, scientific investigations were concerned with scrutinizing experience, and collating results mathematically. This was crucial in the history of metamathematics, because with the advent of Galileo's equations of motion, Descartes' and Fermat's coordinate geometry, the calculus, and Newtonian mechanics, the goal of science became aligned with the early mathematical goal of defining axioms that were self-evident. Much like Euclid's Elements, if one accepted the physical axioms and postulates of science as well as the rules and equations that related them -- as they apparently had to -- they were also forced to accept the conclusions of science. Unlike the controversies permitted by natural philosophy before Galileo, the experiments and conclusions of science were now repeatable and testable, and there was an air of inevitability and certainty about the new science, because it presented a universally applicable physics based on a universally applicable mathematics. With respect to its algebraic and geometric foundations, there appeared to be no room for disagreement, whether mathematical or metamathematical, because through science mathematics clearly represented nature.
The new science (specifically the calculus), was in fact attacked, on religious grounds, by the influential philosopher George Berkeley (1685 - 1753 CE), the Bishop of Cloyne, in Ireland. However, Berkeley's attack yielded no immediate metamathematical consequences, and this is relevant because the incredible practical utility of algebra and geometry in science continued to be interpreted as proof positive of the correctness of mathematics, and its foundation, God.
The next major development that concerned the relationship between geometry and the foundations of mathematics was the philosophy of Immanuel Kant (1724 - 1804 CE), whose epistemology maintained the content of mathematics, but radically altered its foundations. For Kant, the essence of mathematics was not simply nature as it is, because nature as it is, is unknowable for humans. Human minds possess an architecture that systematizes observations and perceptions by its own internal rules, rather than apprehending the foundations of the universe, and we can never know a thing in itself, independent of our mental architecture. That architecture is natural, but it is does not capture nature, and the well-ordered certainty of math and mathematical science arises from the prescripts of the mind, which include a non-empirical form of knowledge about temporality and spatiality, which we express in the form of our self-evident axioms of mathematics. Geometry and therefore mathematical science were not valid because they were built on proper observation and reflection, but because they rested atop valid spatio-temporal intuitions.
Here, Kant vouchsafed the soundness of Euclidean geometry in a new way, and united his philosophy of mind with Euclid's axioms, postulates, and theorems. Not long after Kant passed away however, this aspect of Kantian philosophy and the long-standing certainty of Euclidean geometry were invalidated by the discovery of non-Euclidean geometries, when it was realized the Euclidean system was not the one system, but only one system among many.
In the first half of the nineteenth-century, János Bolyai (1802 - 1860 CE) and Nikolay Lobachevsky (1792 - 1856 CE) independently demonstrated geometries that were consistent, and did not respect Euclid's fifth postulate;
If a straight line incident to two straight lines has interior angles on the same side of less than two right angles, then the extension of these two lines meets on that side where the angles are less than two right angles.
Contrary to the fifth postulate, Bolyai's and Lobachevsky's geometries permitted the construction of multiple "parallel" lines for any given line through a given point. This can be seen, for example, by considering a plane in the shape of a circle, thus enabling one to draw an arc line across the diameter of the circle, and then selecting a point inside the circle that is not on the diameter line, such that numerous lines pass through that point, on angles such that these lines never meet the diameter, because all lines are terminated by the boundary of the circle.
The existence and features of non-Euclidean geometries completely undermined metamathematical certainty, and foisted uncertainty on all scientific and metaphysical suppositions that rested on mathematics. This sparked vigorous attempts to retrieve certainty, including many non-geometric programs such as logicism and formalism, all aimed at rigorously explicating and certifying the foundations of mathematics. Ultimately however, the long-term result of these efforts was only to further separate mathematics from certainty in unexpected ways, and this gave rise to the post-modern perception of mathematics as rooted in reality and internally cohesive, but not certain in any absolute physical or metaphysical sense.
Reflecting on the rise and fall of certainty in geometry and metamathematics from Thales to Lobachevsky, we see that when mathematics first arose it was taken straightforwardly, as a practical device that solved problems in the real world. In prehistory and Classical history, mathematics was approached as a device that simply was and simply worked, much like a door or field plough. When Thales took up mathematics however, he latched on the fact that mathematics was not quite like other devices, and he observed its physical manifestations, and speculated on it supra-physically. This mode of speculation was instrumental in generating Classical Greek metaphysics, and culminated in the logico-deductive method, and the incredibly powerful Euclidean system.
The Euclidean system reigned with certainty for millennia, and though mathematics continued to evolve, and explanations for its certainty changed, the fact of certainty remained. Attempts to explain the basis and correctness of mathematics ranged from Forms and God, to nature and mental architectonics, but even though metamathematical claims varied, mathematical claims did not. Whatever its metamathematics, mathematics itself was absolutely accurate.
The discovery of non-Euclidean geometries instantly destroyed the possibility of absolute mathematical certainty, and this is an extraordinary fact, because for millennia brilliant mathematicians were exactly wrong in their metamathematical certitude. Looking back to the end of certainty, it appears certainty was as much a goal as a hypothesized feature of mathematics; that mathematicians undertook mathematics because they wanted to work with something that was guaranteed.
At a fundamental level, the rise and subsequent fall of mathematical certainty was central to the philosophical and scientific recognition of human fallibility. Today it is believed that nature exists, but because of the peculiarities of our experience of it, there always remains the possibility that our metamathematical and metaphysical claims are inaccurate and perhaps entirely false. Thus, the end of mathematical certainty has given rise to a new kind of certainty, that regardless of its foundations, mathematics remains the most powerful tool humans possess for mediating between themselves and nature, and that the development of mathematics enables us to expose falsities -- such as the absolute certainty of mathematics -- and thus allows us to work towards the refinement and extension of better justified, if not certain beliefs.
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By Peter Petersen
Meant for a three hundred and sixty five days direction, this article serves as a unmarried resource, introducing readers to the real concepts and theorems, whereas additionally containing sufficient history on complex subject matters to entice these scholars wishing to specialise in Riemannian geometry. this is often one of many few Works to mix either the geometric components of Riemannian geometry and the analytic points of the idea. The e-book will attract a readership that experience a uncomplicated wisdom of normal manifold concept, together with tensors, varieties, and Lie groups.
Important revisions to the 3rd variation include:
a vast addition of exact and enriching routines scattered during the text;
inclusion of an elevated variety of coordinate calculations of connection and curvature;
addition of basic formulation for curvature on Lie teams and submersions;
integration of variational calculus into the textual content taking into consideration an early remedy of the field theorem utilizing an explanation via Berger;
incorporation of numerous fresh effects approximately manifolds with optimistic curvature;
presentation of a brand new simplifying method of the Bochner strategy for tensors with program to certain topological amounts with common decrease curvature bounds.
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Extra info for Riemannian Geometry (3rd Edition) (Graduate Texts in Mathematics, Volume 171)
11. More generally, the map I S1 S1 ! I S1 t; ei Â1 ; ei Â2 7! t/dÂ22 and the target has the rotationally symmetric metric dr2 C . 12. z; w/ D . z; w/). The quotient map I S2nC1 S1 ! I S2nC1 S1 =S1 can be made into a Riemannian submersion by choosing an appropriate metric on the quotient space. To find this metric, we split the canonical metric ds22nC1 D h C g; where h corresponds to the metric along the Hopf fiber and g is the orthogonal component. In other words, if pr W Tp S2nC1 ! t/d 2 : Observe that S2nC1 S1 =S1 D S2nC1 and that the S1 only collapses the Hopf fiber while leaving the orthogonal component to the Hopf fiber unchanged.
5. R/ revolution by revolving t 7! R sin R3 can be thought of as a surface of t R ; 0; cos t R around the z-axis. The metric looks like dt2 C R2 sin2 t R d 2 : Note that R sin Rt ! t as R ! 1, so very large spheres look like Euclidean space. 7 by observing that it comes from the induced metric in R2;1 after having rotated the curve t 7! R sinh t R ; 0; cosh around the z-axis. t/d 2 of rotationally symmetric metrics. 1= k/. t/ D 0. In the revolution case, the profile curve clearly needs to have a horizontal tangent in order to look smooth.
T/ to the integral R is part of the great circle. t//. dt (5) Show that there is no Riemannian immersion from an open subset U Rn into Sn . Hint: Any such map would map small equilateral triangles to triangles on Sn whose side lengths and angles are the same. Show that this is impossible by showing that the spherical triangles have sides that are part of great circles and that when such triangles are equilateral the angles are always > 3 . 21. Let H n Rn;1 be hyperbolic space: p; q 2 H n ; and v 2 Tp H n a unit vector.
Riemannian Geometry (3rd Edition) (Graduate Texts in Mathematics, Volume 171) by Peter Petersen |
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A change in demand on the other hand, is causedby other variables such as a change in tastes, income orcompetition from related goods. Have you ever observed why the inessential things like diamonds, platinum, gold are very expensive, whereas necessities like food, clothes, water are inexpensive? The Law of Demand The law of demand states that, if all other factors remain equal, the higher the price of a good, the less people will demand that good. The amount of a good that buyers purchase at a higher price is less because as the price of a good goes up, so does the opportunity cost of buying that good. How severely is the change in the quantity demanded impacted by a change in the price? Thus you get two benefits : Added security as well as aproof. Conclusion Demand is inversely related to price, i.
So if the Price of complements goes up then the demand for the good goes down thus shifting the graph to the left. If, on the other hand, there is a change in any other factor except the price of the commodity under consideration the demand curve will shift to a new position. In such a case, it is incorrect to say increase or decrease in demand rather it is increase or decrease in the quantity demanded. On the other hand, quantity demanded is a particular point on the demand curve. The quantity demanded lies in the demand curve and can be determined by just assuming a point and calculating its intercepts, on the price and quantity planes respectively. These movements are sometimes described as extensions or contractions of demand. Only the combination of the willingness and the affordability will be considered as a demand.
Conversely, if a person talks about expansion or contraction of demand, he refers to the change in quantity demanded. As against this, a shift in the demand curve represents a change in the demand for the commodity. It refers to a particular point on the curve. On the contrary, a shift in demand curve occurs due to the changes in the determinants other than price i. For example, when technology advances, or the cost of production decreases, supply increases. A shift in the opposite direction would imply a decrease in demand. Relationship Between Decrease in Demand and Decrease in Quantity Demanded Understanding the difference between a change in demand and change in quantity demanded is a key concept in economics.
Quantity supplied increases in the above case as the equilibrium point shifts along the supply curve from point A to point B. Reasons Factors other than price Price Measurement of change Shift in demand curve Movement along demand curve Consequences of change in actual price No change in demand. A change in demand is the sum of all the changes in quantities demanded that consumers can buy at a specified price level. Also, when there is a change in the determinants of demand ie. Increse in quantity demanded:: Movement up the demand curve.
A change in Demand is affected by either a change in productivity or a change in the price of a certain product. Hence, more quantity of a good is demanded at low prices, while when the prices are high, the demand tends to decrease. A change in quantity demanded is represented as a movement along a demand curve. Change in income, change in number of consumers, taste and preferences, price of related goods, and future expectations all cause shifts in demand curve. Achange in quantity continues to move along the same demand curve,whereas a change in demand shifts it either to the left or right ofthe original line. However, the following day a report is published that finds pesticides used on bananas can cause lasting health problems. Recently, he has increased his sales of luxury products, and his manager considers promoting him to sales manager in the store.
However, if the average income of doctors goes up, the demand for tractors would not change. Summary Definition Define Change in Demand: A change in demand is an economic term that describes when the entire demand curve shifts upward or downward because the market changes the quantity it demanded. From the business point of view, demand can indicate the possible sales that take place. On a national level, if consumer income decreases, the demand for goods and services will decrease, thereby shifting the demand curve downwards. In economics, demand is defined as the quantity of a product or service, that a consumer is ready to buy at various prices, over a period. This means that even at the current price, that person is willing to buy more video games due to the increase in income.
The price elasticity of demand is a measure of the responsiveness of quantity demanded to a change in price. Change in the position of the curve. Quantity demanded is represented on the graph by moving up and down on the curve, rather than side-to side. A rightward shift in the demand curve shows an increase in the demand, whereas a leftward shift indicates a decrease in demand. A decrease in demand results from the presence of a factor that shifts the demand curve to the left such as a damaging study or introduction of a competing product. Going back to the video game example if the price of video games drops the quantity demanded for those video games is going to increase. The following graph illustrates an increase in supply and an increase in quantity demanded.
It is the actual amount of goods desired at a certain price. In economics, demand is defined as the quantity of a good or service consumers are willing and able to buy at a range of prices. The law of demand states that as the price of a good or service increases ceteris paribus , the quantity demanded will decrease and vice versa. Determinant Price Non-price Indicates Change in Quantity Demanded Change in Demand Result Demand Curve will move upward or downward. Therefore change in factors other than price. To understand the difference more clearly, we need to study the difference between demand and quantity demanded. Thus less is q 0 instead of q1 demanded at a fixed price po A change in quantity demanded for the commodity resulting from a change in its own price will lead to a movement along the curve itself this indicates either a contraction or an extension of demand.
When you look at these two statements together, it may appear confusing and contradictory. Quantity-demanded shifts can go either up or down based on the changes in the marketplace relating to prices and consumer demand. The payment will be made through an account of the payee. . If demand is elastic, there are alternatives readily available in the market. Therefore, demand and quantity demanded are two different things.
If price falls there is a downward movement to the right. Many variables can change the demand for a product. Whenever there is a shift in the demand curve, there is a shift in the equilibrium point also. Generally, when demand rises, supply increases and when demand falls, supply is decreased. A decrease in income would contract his spending, allowing for a limited quantity of goods. Substitutes- A good where in can be used in place of another. The figure given below represents the shift in demand curve due to various factors such as income, taste or preferences, the price of complementary or substitute goods etc. |
Sketcher of various interrelated fourfolds.
July 17, 2007.Recentest (mildly) significant change: January 2, 2009 (third such change since August 6, 2007). This post is much less speculative in style than the others on this blog. But my other blog "feels" filled up, I can't quite say why. Still, maybe I'll eventually move this post to there.
We tend to consider the logical quantity of the term and not only that of the proposition, especially when a logical quantity such as the singular gets involved. Yet tradition has kept the spotlight on propositions (or sentences, etc.) because of the interest in valid argumentation involving them. That seems to be why logical quantity from the term's viewpoint has lain largely unexplored by philosophy. Philosophy hasn't stopped and smelt the roses long enough to see what vistas might spread thence. Given a term "H" predicated (truly or purportively truly) of something (call it "x"), the question of its logical quantity then depends on quantification over the rest of the universe of discourse: Is there something which isn't that thing x and of which the term "H" is also true? -- and -- Is there something which isn't that thing x and of which the term "H" is instead false? The twin questions stand mutually independent and resolve into four answers, conjoinable in four ways (notwithstanding issues of term purport which multiply relevant options). For the polyadic case, incorporate criteria requiring one-to-one correspondences as needed and slackening as needed to compensate for sequence variety. None of the four conjunctions enframes a blind or almost blind window as long as we class the singular and the singulars-in-a-polyad together in logical quantity, just as we class both the monadic general and the polyadic general as general. One such conjunction, the monadic-or-polyadic singular-cum-universal, is a logical quantity corresponding to a gamut, a total population and its parameters, a universe of discourse, etc. The eventual result of a systematic approach to logical quantity from the term's viewpoint is a surrounding scene of various categories of the 'essences' -- attributes/modifications, modes of attributability, and forms of mathematical correspondence -- whereto nonsingular terms are often allied, 'essences' categorially as different each from the others as they are from the scene-completing object -- this man, this horse, etc. -- of a typical concrete singular term.
The "problem of universals" is a philosophical perennial. Now, before one does a metatheory about, say, the theory of geology, one needs first to do theory of geology. And, before that, one needs to do physical geography. The "geography" of logical quantity (singular, general, universal, etc.) seems to have lain largely unexplored by philosophers. Aristotle and C.S. Peirce are exceptions.
On July 17, 2007, I searched on Google for the two phrases problem-of-universals logical-quantity. Only two results came up, both mine -- the first version of this post and a similar thing which I posted to peirce-l some weeks ago. I searched for problem-of-universals logical-quantification and found few results, half of them mine. (My own earlier post on the topic at The Tetrast doesn't come up, and of course the problem of universals isn't always called that by name, but it still seems fair to take the paucity of Google results as significant). The lack of an adequate systematic terminology is another sign of how little attention philosophers have given to the topic of logical quantity, despite their long interest in the problem of universals.
The problem of universals gets its standard name from the noun "universal" in the sense in which one finds it used in translations of Aristotle -- that which is true of more than one object, a sense for which the word "general" is now sometimes employed as a noun in philosophical discussion and is in any case usually so employed here.
Singular and general in standard 1st-order logic
Now, in the standard terminology of first-order logic, a "general term" is a term which does not purport as to logical quantity (or has only a "default" purport to the existential particular affirmative when the term is true of something). If the monadic general term were to purport, when true of an object at all, to denote more than one object, then a proposition claiming in effect that the term were uniquely true of some given object would be formally false. Instead such a proposition is merely contingent. In other words, a so-called general term in standard first-order logic is vague in logical quantity and is 'general' from a kind of second-order viewpoint -- one might call it "general" across various possible logical quantities. On the other hand, a "singular term" in standard first-order logic is a term (and indeed a subject term rather than a predicate term) which does purport as to logical quantity, and purports to singularity, so that a proposition which claims in effect that a monadic singular term corresponds to two different objects is formally false. I am speaking of constant a.k.a. definite terms such as "blue" and "Jack." Constancy versus variability is a similar yet distinct issue or dimension which complicates an elementary discussion.
Generality more generally
In speaking philosophically of generality, not adhering to the linguistic habits of standard first-order logic, we may mean neither vagueness in logical quantity nor a purportive (or still some other de jure) generality; instead we may mean a de facto generality, for instance that of a monadic term like "blue" which happens to be true of more than one object. In speaking of singularity we may likewise mean a de facto singularity. I'm not sure what there is for all this, except to get used to the distinction between purportive and other sorts of de jure, and de facto. It seems difficult to limit one's discussion to examples of just one kind or just the other. The distinction does not seem so hard and fast to intuition. "Blue" -- as term or as idea or as quality -- is the kind of thing which one would not expect to be true of just one single object.
In order to distinguish the sense of "general" as that which corresponds to more than one object (in the monadic case), I will speak of the coaliant general. (I could just call it the "coaliant" per se but I wish it remembered that I'm speaking of a kind of general. I coin it from co- + aliud + -ant.) The coaliant general corresponds, purportively, etc., or de facto, to something but not to that thing alone but also to something else. In the polyadic case, consider it to correspond to polyads whose intersections lack objects from each polyad. (As for re-orderings or re-sequencings of the same polyad, they are another issue which complicates an elementary discussion.)
A bustling floor under generality
Since one thinks in terms of greater and lesser generality, there arises an imagery of limits. Such imagery is itself limited in usefulness but inevitable in its way.
Now, the coaliant general (monadic or polyadic) encounters something like a limit, closure, or bound, at the "low" end, in the singular or singulars in a polyad. A polyadic version of a singular is not strictly to be called "singular" in that it is not monadic, and "plural" already has specialized meanings in logic. One might say only loosely that it is a polyadic singular. The word "singular" isn't quite right for a logical quantity definable by its opposition to the general -- the mind places "singular" opposite not only to "general" but also to "plural" and thus also to "polyadic." In order to unglue term adicity a.k.a. term valence from logical quantity and instead to treat all logical quantities on the same plane, I'll call any monadic-or-polyadic singular transingular. The coaliant general encounters an excluded or external limit, at the "low" end, in the transingular. A transingular term can be a subject but also can be a predicate or other things.
A bustling ceiling into generality
The coaliant general, if it has an upper limit in some sort of "most general," will include it in a way that it does not include the transingular, since the coaliant general is definable as the determinately non-transingular. What would non-arbitrary utmost generals be? They would be something like the Scholastic transcendentals (unity, truth, goodness) which are true of each and every thing automatically, in sheer virtue of the thing's existing at all -- the given thing is one thing, a true thing, and a good thing, at least in respect of its existence if not of its character. That seems to make of the utmost general a rather narrow window, while other logical quantities at the same level of analysis are rich and, in their way, panoramic. Is the world's symmetry really that deeply broken? A systematic understanding of logical quantity does not foster a view of the world as arranged mainly into genus-species type relations, strict inclusions, etc., with one or a few utmost generals monotonous at the top. Confronted with the Scholastic transcendentals one, true, good, one may ask, what about two things? Aren't any two things two in sheer virtue of their being things xy such that ~(x=y)? Now, if one views collections in such a way as to see othernesses and unities among selected parts as definitive attributes of the whole, then, since obviously not every such collection consists of exactly two things or of exactly one thing or of etc., in that sense such numberish predicates are not utmost generals. However, any object (in a large enough universe) will fairly belong among polyadized objects whereof "two" is true collectively. Keeping this in mind, we have a notion of universality reached by utmost generality, universality which can be extended to sequence schemata, etc., and which seems, as a "window," practical and cornucopious like the singular or transingular. If we "arbitrarily" declare a given predicate term universal, equivalent to a predicate like in "Tx" or in "Hxyz v ~Hxyz", it can be refined by formal schemata. (Via a richer formalism such as set theory or the like, mathematics can treat these universals as more or less general and even unique properties of various sets or the like, and mathematics can re-generate the world's wild variegation, while building imaginative, metamorphosic bridges of equivalences across the greatest disparities of outward appearance.) The point is that the 'accidents' or 'modifications' of the objects xyz in the above example don't matter. All that can matter is their othernesses and unities, relationships defined within the formalism (of first-order logic with equality a.k.a. ...with identity). On the other hand, with things like "blue," we're getting into modifications of objects. Such terms or ideas or qualities as "blue" and "Jack" befit (at least in a realistic universe where not everything is blue or Jack) that which I will call the special, or contraliant special to ensure clarity as to just what sense of the polysemic word "special" I mean. ("Contraliant" from contra- + aliud + -ant). The contraliant special term is (or purports to be) true of something (or things in a polyad) but decidedly not of everything.
Yet the universal can be either transingular (as in the case of a total population, its parameters, etc.) or (coaliant) general (or indeterminate about that alternative in the case of a term's de jure applicability). So the universal is better pictured as a ceiling into generality than as a ceiling in generality.
The universal supplies the upper limit of the coaliant general, and is a kind of extreme to which the coaliant general reaches, like a line segment which includes its endpoint adjoinment with something else (a universal may be general or instead transingular). In the other direction, generality's "line segment" includes everything till the transingular but not the transingular itself, like when a mathematician replaces an endpoint with a little bubble. A coaliant general is either universal or contraliant special (or indeterminate about that alternative in the case of a term's purport, its de jure applicability, or the like).
- The (coaliant) general has two limits -- an excluded limit, the transingular, and a partly included limit, the universal (a universal is not necessarily general).
- The (contraliant) special has two limits -- a partly included limit, the transingular (a transingular is not necessarily special), and an excluded limit, the universal.
- Should the general-cum-special be considered a fully included "limit" of both the (coaliant) general and the (contraliant) special? Here we seem to approach a limit to the usefulness of the imagery of limits.
A transingular may be universal too. If the transingular is a total population, a universe, a gamut, then it is also universal, at least in the relevant universe of discourse. When it is not the universe, the transingular is (contraliant) special. (In the case of term purport, the transingular may be indeterminate about that alternative.)
Universals & universes
Basically one ends up with two kinds of (coaliant) general and two kinds of universal. Now, in the universe of a plinker's distinct notes cdefgab, that gamut is the universe. It is both unique and universal. In its universe of discourse there's no polyad that contains notes uncontained in the gamut. "The gamut" is true of cdefgab and there's nothing else of which "the gamut" is true. In that sense it is not general. Yet it is universal, it is the universe and, in that sense, it is not (contraliant) special. A gamut, a universe of discourse, a total population is a transingular universal. Also universal is a monadic or polyadic term which does not exhaust the universe's population in a single predication yet which, like "one," is true of each object distributively or which, like "two," is such that every object is among some objects whereof the term is true collectively. Such a universal is also general, since there is more than one instantiation of it in its universe. One the other hand, "THE one" and "THE two," etc., are not general, insofar as they are true of the one object in a one-object universe, the two objects in a two-object universe, etc., respectively.
So we have two kinds of universal, one a transingular and the other a (coaliant) general. A universal which does not exhaust its universe in a single predication is (coaliant) general, not transingular, and is closer to the kind of thing which one usually has in mind with the word "universal," something like a rule, with more instances than the given one, indeed sometimes infinitely more, as with the "miraculous jar" of positive integers.
To be in the world
A transingular which does not exhaust its universe in a single predication is much closer (than the universal transingular) to the sort of thing which one usually has in mind with the word "singular," a singular or singulars-in-polyad among still more singulars in a larger world. Such a transingular is not its universe, it is not universal. It is (contraliant) special.
So a transingular may be universal or special. Likewise, a general may be universal or special. Just because a term is general, having more than one instantiation, doesn't mean that every object is covered one way or another in its instantiations. For instance, "blue" is, eclectically, true of some things and false of the others. So now we have four comparatively simple logical quantities -- universal, (coaliant) general, (contraliant) special, and transingular -- and four conjunctions nameless except for such improvised unwieldy names as "universal-cum-general," "universal-cum-transingular," "special-cum-general," and "special-cum-transingular."
To be systematic
Any pair of statements are TT, FF, FT, or TF. We define logical binary compounds in that way. Formal logic wouldn't even think of not systematizing the four mutually exclusive and collectively exhaustive cases -- the four conjunctions based on truth conditions. And we get "and," "neither-nor," "no, but," and "and not."
In the same inevitable way, any term true of something is, de facto: -- (1) universal & (coaliant) general -- or (2) universal & transingular -- or (3) (contraliant) special & (coaliant) general -- or (4) (contraliant) special & transingular. All that's being done is to answer two mutually independent logical-quantity questions, which bring us --
To the heart of it
In the monadic case, the two logical-quantity questions are:
"Given that there's a thing (call it 'x') which is H, is there a thing (call it 'y') which isn't that thing x and which also is H?" If yes, then "H" is (coaliant) general. If no, then "H" is transingular.
"Given that there's a thing (call it 'x') which is H, is there a thing (call it 'y') which isn't that thing x and which is not H?" If yes, then "H" is (contraliant) special. If no, then "H" is universal.
The mutual independence of the twin questions needs to be appreciated; they result in four possible conjunctions. The result is not simply two separate extremes of universal and singular with the somewhat-general somewhat-special as a third, in between. The habitual swerve of thinking of the singular only in monadic terms even while thinking of all three of its kindred logical quantities (special, general, and universal) in both monadic and polyadic terms, leads to thinking incorrectly of the universal singular as a trivial combination (if one notices it at all), a nearly blind window, confined to a one-object universe. In fact the window's vista is quite populous. A grand boat gets missed there, that of a logical quantity corresponding to a gamut, a total population and its parameters, etc., along with a whole class of research, research starting from given parameters of a total population, universe of discourse, etc., to draw deductive conclusions.
(There are even more than four options for term purport, de jure applicability, or the like, 16 including the formally false option, mostly since indeterminateness becomes an option in various alternatives. Such options for de jure applicability seem to become 2^16=65,536 if we admit options for objective indeterminateness and an option for objective inconsistency.)
Now, in a large enough universe, the general-cum-special will be mostly vague in range. In the monadic case it could be true of just two things or it could be true of all but one thing or it could be anywhere in between. It is so much like logic's "general term" as to be barely distinguishable except under certain near-the-limit conditions. For similar reasons, one might question at least the utility of some of the other combinations. One might say, instead of column A, why not column B?:
|General-cum-special||Logic's "general," logical-quantitatively indeterminate like the predicate term letters in logical schemata.|
|Transingular-cum-special||"Just plain" Transingular (be it universal or (contraliant) special).|
|General-cum-universal||"Just plain" Universal (be it transingular or (coaliant) general).|
|Transingular-cum-universal||Transingular-cum-universal (a universe, total population, gamut).|
Now, if we're defining kinds of terms by purportive logical quantity for the purpose of a formalism or grammar, then Column B seems the more convenient way to go. However, Column A is logically "nicer" and more consistent in its criteria; its four logical quantities are on a par with each other. In any case Column A girt by the simple logical quantities as shown in A.1 is the completed relevant picture (almost completed -- one could also devise terms for the diagonals). And if one is interested in logical quantities as characterizing typical mental perspectives distinguishing classes of research, Column A is the way to go, and even a pair of terms for A.1's diagonals would be useful. Now, I speak of the perspective as represented by the given subject matter, not the object(ive) or goal which, for instance in the special sciences, may include finding generals true of multitudes of singular objects and events.
|Perspective in typical |
|Class of research:||Typical inferential character |
|Transingular-cum-special.||The special sciences a.k.a. idioscopy. Human/social, biological, material, physical.||Surmise (ampliative-cum-precisive).|
|General-cum-special.||Sciences of positive phenomena in general, rather than of special classes.|
Philosophy, cybernetic theory*, statistics, and inverse-optimization theory.
|Strictly ampliative induction.**|
|Transingular-cum-universal.||Deductive math theories of logic, information***, probability, and optimization.||Strict (precisive) deduction.|
|General-cum-universal.||'Pure' mathematics. Ordering, calculation, enumeration/measure, graphing/topology.||"Reversible" deduction.****|
|Kinds of |
or in related posts.
don't formally imply
|Strictly ampliative |
don't formally imply
** That's notwithstanding the internal properties of the 'domain-independent' deductive formalisms with which these fields sometimes occupy themselves.
*** Deductive mathematical theory of information considerably overlaps into 'pure' math, abstract algebra in particular, because of the pure-mathematically deep treatment of laws of information, laws which also turned out to be equivalent to some principles of group theory.
**** In mathematical induction, the minimal case and the heredity, conjoined, are equivalent to the conclusion, given the well-orderedness of the relevant set. The proof of the minimal case or of the heredity is sometimes not reversibly deductive, especially when inequalities or greater-than or less-than statements get involved. More generally, pure maths are rife with inference through equivalences and equipollencies.
Update August 6, 2007: Am I analytic?
Thank you to Enigmania for including me in the 51st Philosophers' Carnival. In answer to his implied question: Well, I don't take the analytic linguistic turn, and I went through a Merleau-Ponty phase, but I like C.S. Peirce more and don't regard science as sinister to some great extent that would distinguish science from the humanities. Indeed, as "Enigman" says, my stuff "seems to be more analytic" than Continental, "but who can say?" and this is also partly because I'm an insufficiently disciplined amateur, not a professional philosopher. If wishes were horses, and so forth. To date, I've engaged in discussion mainly with Peirceans (at peirce-l), which has been good for me and, I hope, not bad for them. I've read some of the important early papers in analytic philosophy and some books by Quine, but I haven't engaged in discussions with analytic philosophers, so I've lacked the benefit of criticism from them. I don't know how to rectify that but, if I'm lucky, the Philosophers' Carnival will help.
• I regard philosophy's best bet to be to define itself (A) as having, as its subject matter, positive phenomena in general in their inferential issues, and (B) as properly tending to draw, as its conclusions, inductive generalizations to or toward totalities -- all in all, sort of like statistical theory, but tackling the inductive inverse of the problem of deductive theory of logic rather than of probability, and thus lacking the quantitative-measurement emphasis and having multiplicity of levels, reflexivity, and so on, pursuing problems of estimating, interpolating, extrapolating the logical structure of a universe rather than the parameters of a total population, and rising to consider general processes of experience, mind, heart, society, etc., and complex inference processes including all mathematical and scientific research, to say the least. (Note: The kinship between statistical theory and philosophy isn't very close -- they're still far apart like, say, matter science and human/social studies.)
• I certainly don't oppose deductive formalisms (not to mention deductive arguments) in philosophy, any more than a statistician opposes probability formalisms. Statistics' normal curve of distribution is a way of looking at Pascal's Triangle extended indefinitely. A piece of logical formalism transits the heart of the ideas in this post.
• Still, recognition of its underlying kinship with inductive, totality-targeting fields like statistical theory could help philosophy manage and temper its own aspirations to a "God's eye view" (pace Rorty, who, complaining of its aspirations, essentially gave up on philosophy), help philosophy reduce attendant hyperbole and disillusionment, and help it be more pragmatic about vagueness, discriminate in hyperbolic doubt, fallibilistic, etc., without tending to substitute some idea of utility (not to mention power) in place of the idea of truth be it ever so slippery. My 2¢ worth. End of August 6, 2007 update (Edited, January 2, 2009).
A few informal assertions about the problem of universals.
Areas of research can be ordered according to their appeals to principles of how we know things (ordo cognoscendi, the order of learning or familiarity) and, in pretty much reverse order, to principles (entities, laws, etc.) whereby we explain things (ordo essendi, the order of being). The order of being is often preferred in the special sciences (physics first, etc.), while the order of learning and of the verificatory bases on which we know things is sometimes preferred in maths (where such preference tends to put logic and order theory first). Maybe those researches which I call "sequenced in the order of being" you would call "sequenced in the order of abstractness." Still could well be the same ordering. I'm not saying that the ontological questions are unimportant, to the intellectual climate, the human spirit, and the ultimate bearings which people take in their decisions. But for my part I generally take their involvement in questions of math and science classification as an intrusion signifying that the classification is either deficient in firm and fertile constraints or just plain nebulous. And, if people argue over whether some sciences should be ordered by increasing concreteness or increasing abstractness, and if it's essentially the same ordering forwards versus backwards,
On various topics I prefer compatibility with a range of ontological viewpoints, but I do I have my own ontological views. Generally, when people deny the reality or ontological legitimacy of generals in any usual sense, I don't know what to think but that they regard Scholastic Realism as "secretly" believing that generalities like redness and threeness exist like lamps and chairs. As if we might expect to hear a news bulletin, "Blueness, as such, has been finally been found, orbiting a house in New Orleans." Now, if "blue" is not itself a real individual object like a blue thing, still the real individual object is really blue. So blue has really-ness. But that extrapolates to coming up with syntactically complicated words for variations of "real" and you know that sooner or later we'll find some general word for them all. I foreshorten the process and take that word to be the word "real" itself and will merrily consider in what senses and what universes Santa Claus, Planet Pluto, and Cthulhu are real. Sure, some things are "realer" than others. Indeed even with reality we can admit graduality, etc., if we don't try to live always in the armor of a flat first-order logical universe, as interesting a challenge as that can sometimes be, and as needful as it may be for those whose sense of reality is unfortunately shaky. Coarse is what it is, like that browser Safari which should instead be called Tour by Tank. Anyway, Peirce's definition of the real as that which is what it is, and indeed in some sense persists, independently of that which you or I or any finite community thinks of it and which would be discovered by research adequately prolonged, suffices for a definition of "real" which takes things like blue in and is a critically unfolded version of the common-sense interpretation of the word "real." Now, if somebody, Quine or Stuart Rankin or whoever wants to come along and define "real" as "singular object" or as "Scottish" or as whatever, they can do that, but only the Peircean kind of definition has earned the force and feeling of the everyday word "real" which everybody in the discussion prizes. I certainly don't know what would be a "naturalistic solution" to generals and mathematicals and I see no germane practical significance in the idea.
The transingular subject is a this, or a this, this, that, yon,, etc., and, as a more or less haecceitous rest point or useful stopper to analysis, is also a hook or polyad of hooks on which, to borrow Peirce's phrase, to hang the hat of a predicate, it is a point of general indetermination and freedom regarding how the predicate relates to components or sections or durations (and so forth) of the singular subject(s). For instance, it is left to the definition, context, etc., of the predicate "blue" whether "something blue" means something entirely blue or mostly blue, etc.; one is not automatically forced to quantify over parts or stuff of the described subject. Many a natural thing, through such characteristics as forcefulness, endurance, vigor, and firmness/integrity, lends itself to treatment as a singular. As Peirce argued persistently, some things impose themselves on us, whether we like it or not. The haecceitous thing may come crashing in through a hundred windows. And things could not be alike in their bare singularness -- they could not all be singulars -- but for generality. And the general would not be general but for ranging over more than one thing.
The singular seems just as mysterious as the general to me, and neither one of them makes sense without the other. I can't see anything in the limitation of the real to the singular but a kind of fetish arising from the fight against the unmoored generalities so involved with causing chaos and destruction to people and society.
To go on being systematic
Also, to be concerned with the singular and the general and not also with the universal and the special seems unsystematic, unthoroughgoing, and illogical to me. The possibilities of a term's being true or false of objects besides that of which the term is predicated in the given instance don't play such favorites.
There's plenty in all that to examine philosophically. As the transingular-cum-special term lends itself to use as a subject term, and as the general-cum-special term lends itself to use as a predicate term, so a transingular-cum-universal term lends itself to adaptation as a predicate-formative functor such as "with a probability of 75%," and a general-cum-universal term lends itself to adaptation as a subject-formative functor such as "double of". There is a parallelism which runs among logical quantity, grammatical form, and philosophical category such as substance, attribute/modification, mode of attributability (modalities and "indeed," "not," "if," "novelly," "probably," "feasibly," "optimally," etc.), and correspondences/variances (such as "another than," the combinatory "Inv," "double of," "product of," "antiderivative of," etc.) The parallelisms, as non-binding affinities, seem to help empower thought.
|Logical Quantity:||Grammatical Form:||Philosophical Category:|
|Transingular-cum-universal.||Predicate-formative functor.||Mode of attributability.|
|General-cum-universal.||Subject-formative functor.||Mathematical correspondence/variance.|
Whatever one thinks of the problem of universals, still for inquiry on the problem of universals to get off on the right foot, it's a good idea to develop more than a nodding, dozing acquaintance with logical quantity. For really what there is is not simply a problem of universals but instead, from the start, a systematic complex of issues of the (comparatively) simple logical quantities and their conjunctions.
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When working with and reporting results about data, always remember what the units are. Created by Sal Khan.ShareTweetEmailEstimating a population proportionConfidence interval exampleMargin of error 1Margin of error 2Next tutorialEstimating a population meanTagsConfidence intervalsConfidence interval exampleMargin of error 2Up NextMargin of error 2 Υπενθύμιση αργότερα Margin of error = Critical value x Standard deviation of the statistic Margin of error = Critical value x Standard error of the statistic If you know the standard deviation of Refer to the above table for the appropriate z*-value. http://facetimeforandroidd.com/margin-of/margin-or-error-equation.php
Check out the grade-increasing book that's recommended reading at Oxford University! If we multiply this result by the FPCF, we get MOE with FPCF = sqrt[(2401-865)/(2401-1)]*(0.033321) = sqrt[1536/2400]*(0.033321) = (0.8)(0.033321) = 0.026657 So these survey results have a maximum margin of error On this site, we use z-scores when the population standard deviation is known and the sample size is large. The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x
Required fields are marked *Comment Name * Email * Website Find an article Search Feel like "cheating" at Statistics? z*-Values for Selected (Percentage) Confidence Levels Percentage Confidence z*-Value 80 1.28 90 1.645 95 1.96 98 2.33 99 2.58 Note that these values are taken from the standard normal (Z-) distribution. In general, for small sample sizes (under 30) or when you don't know the population standard deviation, use a t-score.
The sample proportion is the number in the sample with the characteristic of interest, divided by n. Note: The larger the sample size, the more closely the t distribution looks like the normal distribution. Take the square root of the calculated value. Margin Of Error Formula Algebra 2 Of these three the 95% level is used most frequently.If we subtract the level of confidence from one, then we will obtain the value of alpha, written as α, needed for
Easy! How To Find Margin Of Error On Ti 84 T-Score vs. Emerson © 2010
In other words, if you have a sample percentage of 5%, you must use 0.05 in the formula, not 5. Sampling Error Calculator How to Find the Critical Value The critical value is a factor used to compute the margin of error. Multiply the sample proportion by Divide the result by n. If you perform 100 surveys with the same sample size drawn from the same poplulation, then 95% of the time you can expect the margin of error to fall within the
The number of Americans in the sample who said they approve of the president was found to be 520. This means that if you perform the same survey 100 more times, then 95% of the time the number of people who like chocolate more than vanilla should be between 44.9% Margin Of Error Excel The number of standard errors you have to add or subtract to get the MOE depends on how confident you want to be in your results (this is called your confidence Margin Of Error Calculator Without Population Size Get the best of About Education in your inbox.
Most surveys you come across are based on hundreds or even thousands of people, so meeting these two conditions is usually a piece of cake (unless the sample proportion is very Margin Of Error Definition How to Calculate a Z Score 4. You need to make sure that is at least 10.
For example, if your CV is 1.95 and your SE is 0.019, then: 1.95 * 0.019 = 0.03705 Sample question: 900 students were surveyed and had an average GPA of 2.7 Todd Grande 7.419 προβολές 7:12 How to calculate Confidence Intervals and Margin of Error - Διάρκεια: 6:44. Pets Relationships Society Sports Technology Travel How to Compute the Margin of Error Margin of Error Calculator Enter the sample size n. Margin Of Error Sample Size Discrete vs.
To express the critical value as a t statistic, follow these steps. For some margin of error formulas, you do not need to know the value of N. 95% Confidence Interval Margin of Error If you have a sample that is drawn from But if the original population is badly skewed, has multiple peaks, and/or has outliers, researchers like the sample size to be even larger. http://facetimeforandroidd.com/margin-of/margin-of-error-equation-stats.php Otherwise, calculate the standard error (see: What is the Standard Error?).
In each formula, the sample size is denoted by n, the proportion of people responding a certain way is p, and the size of the total population is N. Thank you,,for signing up! Next, we find the standard error of the mean, using the following equation: SEx = s / sqrt( n ) = 0.4 / sqrt( 900 ) = 0.4 / 30 = The choice of t statistic versus z-score does not make much practical difference when the sample size is very large.
Share Pin Tweet Submit Stumble Post Share By Courtney Taylor Statistics Expert By Courtney Taylor Many times political polls and other applications of statistics state their results with a margin of This chart can be expanded to other confidence percentages as well. Statistics and probability Confidence intervals (one sample)Estimating a population proportionConfidence interval exampleMargin of error 1Margin of error 2Next tutorialEstimating a population meanCurrent time:0:00Total duration:15:020 energy pointsStatistics and probability|Confidence intervals (one sample)|Estimating It is not uncommon to see that an opinion poll states that there is support for an issue or candidate at a certain percentage of respondents, plus and minus a certain
Expected Value 9. from a poll or survey). Rett McBride 7.293 προβολές 5:31 How to calculate margin of error and standard deviation - Διάρκεια: 6:42. Previously, we described how to compute the standard deviation and standard error.
Statistics Statistics Help and Tutorials Statistics Formulas Probability Help & Tutorials Practice Problems Lesson Plans Classroom Activities Applications of Statistics Books, Software & Resources Careers Notable Statisticians Mathematical Statistics About Education In other words, if you have a sample percentage of 5%, you must use 0.05 in the formula, not 5. What's the margin of error? (Assume you want a 95% level of confidence.) It's calculated this way: So to report these results, you say that based on the sample of 50 Learn more You're viewing YouTube in Greek.
Most surveys you come across are based on hundreds or even thousands of people, so meeting these two conditions is usually a piece of cake (unless the sample proportion is very Rumsey When you report the results of a statistical survey, you need to include the margin of error. Red River College Wise Guys 77.998 προβολές 8:46 Why are degrees of freedom (n-1) used in Variance and Standard Deviation - Διάρκεια: 7:05. A t*-value is one that comes from a t-distribution with n - 1 degrees of freedom.
However, when the total population for a survey is much smaller, or the sample size is more than 5% of the total population, you should multiply the margin of error by The presence of the square root in the formula means that quadrupling the sample size will only half the margin of error.A Few ExamplesTo make sense of the formula, let’s look Hence this chart can be expanded to other confidence percentages as well. If the confidence level is 95%, the z*-value is 1.96. |
Instead of Zernike polynomials, ellipse Gaussian model is proposed to represent localized wave-front deformation in researching pointing and tracking errors in inter-satellite laser communication links, which can simplify the calculation. It is shown that both pointing and tracking errors depend on the center deepness h, the radiuses a and b, and the distance d of the Gaussian distortion and change regularly as they increase. The maximum peak values of pointing and tracking errors always appear around h=0.2λ. The influence of localized deformation is up to 0.7µrad for pointing error, and 0.5µrad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the machining precision of optical devices, which should be more greater than 0.2λ, is proposed. The principle of choosing the optical devices with localized deformation is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the results can be used in the design of inter-satellite lasercom systems.
©2008 Optical Society of America
Comparing to microwave communications, inter-satellite laser communication (lasercom) has many advantages, such as smaller size and weight of the terminal, less power consumption, greater immunity to interference, larger data rate, and denser satellite orbit population, consequently it provides an attractive alternate to microwave communications for both commercial and military applications. [1–5] Inter-satellite lasercom relates to laser beam transmission which has recently been extensively studied. [6, 7] Due to the small beam divergence and the ultra-long distance of the communication links, wave-front aberrations strongly affect the spatial pointing and tracking of laser beams.
There are two major reasons which cause the wave-front aberrations. The first reason is the space environment which includes space radiation, contamination, and especially temperature variation. Temperature variation causes local changes in the optical properties of the devices, such as variation of the reflective index, variation of the curvature of the lens surface, variation of the thickness of the lens, and variation in the gap between lenses. The second reason is the processing technic. It is difficult for the optical devices, especially for that with large aperture, to be processed to the precision of 0.01λ and remain unchanged for long time, consequently localized distortions is almost inevitable. Both of the two reasons are equivalent to the deformation of the optical devices. When the beam transmits the optical devices with deformation, its wave-front will change locally.
Toyoshima et al. have studied mutual alignment errors in circle region using Zernike polynomials expressing wave-front aberrations. Furthermore, Sun et al. developed the research to annular region. Due to the orthogonality of Zernike polynomials, almost all the wave-front aberrations in the optical system can be represented by them. [12–14] However, it generally needs too many items of Zernike polynomials to express localized distortion, which complicates the calculation. To simplify the analysis, we proposed ellipse Gaussian model to represent localized deformation, which is proved simple in the calculation by comparison with Zernike polynomials. Based on ellipse Gaussian model, the effects of localized wave-front deformation on pointing and tracking errors are researched. The purpose of the research is to estimate how much the influence of localized wave-front deformation on pointing and tracking errors, and try to provide the evidence of processing precision for the optical devices used in lasercom.
This paper has the following outline. In Section 2 the ellipse Gaussian Model is introduced to describe local distortion. In Section 3 pointing and tracking errors are defined. Section 4 is devoted to numerical analysis. Section 5 summarizes our results.
2. Ellipse Gaussian model
Due to the limitation of processing technic and the effects of space environment, the localized distortion is extremely likely to appear in satellite optical system, especially in the primary mirror of transmitter antenna due to the large aperture. To simplify the analysis, we propose ellipse Gaussian model to express them, which is shown in Fig. 1 and can be written as
where A is the center value of the ellipse Gaussian function (the center deepness h=A(1-1/e)), a and b are the radiuses of the localized distortion, (x 0,y 0) is the coordinate of the center, d is the distance from (0,0) to (x 0,y 0), which can be represented as
Assuming that there is localized deformation in the primary mirror of reflection-style antenna, when the beam is reflected by it, localized wave-front deformation is generated. The forming process is shown in Fig. 2. The wave-front deformation can be written as
where ψ denotes the center amplitude of ellipse Gaussian function, which is considered to be 4Aπ/λ. Equation (3) is composed of two parts, Φ1 and Φ2. Φ1 is ellipse Gaussian function, and Φ2 is a constant.
The optical field of the beam reflected by mirrors can be shown in the form
where H(x,y) is the optical field before the optical device, exp(jΦ) is called aberration term caused by the localized distortion.
Root mean square (rms) is a conventional factor to evaluate the degree of wave-front aberrations, for the ellipse Gaussian function Φ1, which can be expressed as
where S denotes the deformation area which is an ellipse with major axis radius a and minor axis radius b. From Eq. (5) we can find that rms proportionably depends on the center deepness h, but has no relation to the radiuses a and b.
3. Pointing and tracking errors
Mutual alignment errors are defined in Ref. 10 as the angle between the transmitting and receiving optical axes. We consider that, in fact, mutual alignment errors include two parts: pointing and tracking errors. They are described in the following subsections.
3.1. Pointing error
Pointing error is defined as the angle between the transmitting optical axes with and without wave-front aberrations. Transmitting optical axis is determined by the direction with the peak intensity at a far-field. The definition of the coordinate systems is shown in Fig. 3, which is similar to that in Ref. 10. The transmitter beam is Gaussian beam with localized wave-front aberrations, which can be written as
where C is a constant, F 0 is the radius of curvature at the transmitter, M 1(x 0,y 0) is transmitter aperture function which is determined by the transmitter antenna with primary mirror radius R 1 and secondary mirror radius R 2, ω 0 is waist radius of the Gaussian beam. The intensity distribution Ire(x,y) in the receiver plane is obtained as the following
where λ is the wavelength, zf is the distance of the two communication terminals. For transmitter beam free of aberrations, the peak intensity is at the origin. And for the beam with aberrations, it is at the position of Ire(x,y) |max=Ire(xmax,ymax). In this case, pointing error θP can be written in the form
3.2. Tracking error
Tracking error is defined as the angle between the receiving optical axes with and without wave-front deformation. Receiving optical axis is obtained by the gravity center of the received optical power on an optical tracking sensor. Owing to the long distance between the two communication terminals, the received wave can be considered as plane wave. When the plane wave passes through the optical terminal which is equivalent to a lens with focal length f, it is focused on the focal plane, and the intensity is given by
where B is a constant, M 2(x,y) is receiver aperture function which is determined by the receiver antenna, r 1 and r 2 are the primary mirror radius and secondary mirror radius, Φ(x,y) is wave-front deformation in receiver plane. Similarly, when there is no aberrations in the optical systems, the gravity center of the received optical power in the focus plane is at the origin. However, normally the center of gravity is at (X,Y) when aberrations exist in the optical systems. By definition tracking error θT can be written as
where X and Y are given by the following equations
Similar to pointing error θP, tracking error θT also depends on the following parameters: the center deepness h, the radiuses a and b, and the distance d.
where H(x) denotes Gaussian beam for pointing error, or plane beam for tracking error. F is the distance between two satellites for pointing error, or the focal length of receiver optical system for tracking error. And D=2R 1 for pointing error, or D=2r 1 for tracking error. Substituting Eq. (3) into Eq. (14), we can obtain the following equation
Equation (15) shows that the optical field consists of three parts. By definition the first and the second parts don’t cause pointing and tracking errors which are mainly influenced by the third part. Therefore, to simply the analysis, we can only consider the third part which is shown as
From Eq. (16), we can find that pointing and tracking errors are mainly determined by the aberration term exp(jΦ1). It is known that exp(jΦ1)=exp[j(Φ1+2 π)], namely the aberration term is a periodic function whose period is 2π. Therefore, pointing and tracking errors would vary periodically with the change of Φ1. When Φ1=0, pointing and tracking errors are zeros. We know that the wave-front difference for Φ1=0 and Φ1=(2n-1)π is the maximum. Therefore, the peaks of pointing and tracking errors would appear around Φ1=(2n-1)π (n is positive integer). Due to Φ1 being a function of x and y, the peaks should be around rms=(2n-1)π. Furthermore, from the integral region we can conclude that it is the localized deformation, not the whole aperture, which determines the pointing and tracking errors, consequently rms should obtained from localized deformation area (See Eq. (5)). Related to Eq. (6), we can conclude that pointing and tracking errors would change periodically as the center deepnees h increases. Though the radiuses a and b don’t contribute to the rms of Φ1 according to Eq. (6), it determines the aberration area. When a rises, the value of h(u) increases, namely the influence of wave-front deformation increases too. According to the definitions of pointing and tracking errors, they would increase as the distortion becomes wide.
4. Numerical results and analysis
To show the advantages of ellipse Gaussian model, the comparison between ellipse Gaussian function and Zernike polynomials to represent the localized deformation is addressed in Figs. 4 and 5. For the localized deformation which is expressed accurately by ellipse Gaussian function, we represent it using Zernike polynomials with different terms. The term numbers are N=20, 40 and 60, respectively. The results are in the Fig. 4. When N=20 the result of Zernike polynomials is very poor, and when N=40 the result becomes better. When N=60 the result is close to that of Gaussian function. The results show that it does need many terms for Zernike polynomials to express the localized deformation with less error, which will complicate the calculation. Fig. 5 gives the results of Zernike polynomials with N=40 for different a/D. As can be seen that the result is better for large value of a/D than for small value of a/D. In a word, by comparison with Zernike polynomials, ellipse Gaussian model can really simplify the calculation due to its simple expression, especially for small value of a/D.
Based on ellipse Gaussian model, the numerical results of the effects of localized wave-front deformation on pointing and tracking errors are given in Figs. 6 and 7. In the calculation process, the parameters are D=2R 1=2r 1=250 mm, R 2=r 2=40 mm, λ=800 nm, ω 0=125 mm, and f=1000 mm. The distance of the two satellites is taken to be zf=50,000 km. Fig. 6 shows how pointing and tracking errors vary with the center deepness h, the radiuses a and b, and the distance d. In calculation we only consider the condition that the Gaussian deformation is totally in the aperture of the antenna, and the center of Gaussian distortion is in x axis. As can be seen from Fig. 6, pointing and tracking errors do not monotonically rise with h increasing as generally expected, but fluctuates like damped oscillation. On the other hand, pointing and tracking errors monotonically increases as a rises. In other words, the wider localized distortion, the stronger influence on pointing and tracking errors. With the distance d increasing, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases secondly. The difference is considered that the beam contributing to tracking error is plane beam, while that contributing to pointing error is Gaussian beam whose intensity decreases with d increasing. Fig. 7 shows clearly the fluctuation of pointing and tracking errors with h and rms rising. The peak appears around h=0.2λ (rms=π), h=0.75λ (rms=3π), h=1.25λ (rms=5π), et. al.. The fluctuation period for rms value is 2π. The results show that to reduce the impact of localized deformation on pointing and tracking errors, the center deepness h should be more less than 0.2λ, namely the machining accuracy of the optical devices should be more greater than 0.2λ. Moreover, the influence of localized deformation is up to 0.7µrad for pointing error, and 0.5µrad for tracking error.
The comparison of pointing and tracking errors for localized deformation expressed by ellipse Gaussian function and Zernike polynomials are shown in Fig. 8. As can be seen that pointing and tracking errors due to wave-front aberrations described by Zernike polynomials, are gradually close to that expressed by Gaussian function with N increasing. Figs. 10(a) and 10(d) show that Zernike results are better for small value of h than for large value h. The reason is that, for small value of h, the localized deformation plays an important role, and the effect of Zernike error is comparatively weak. With h rising, the influence of the localized deformation reduces, then the impact of Zernike error gradually increases. Furthermore, as shown in Figs. 8(b) and 8(e), Zernike results are obviously worse for small value a than for large value a. The reason is that Zernike error is large for small value a/D than large value a/D, which is shown in Fig. 5.
From above numerical analysis, we can conclude that ellipse Gaussian model is an effective method for the localized distortion, especially for that with small values of a/D. To weaken the effect of localized deformation on pointing and tracking errors, processing precision of optical devices should be more than 0.2λ. If we have to use the optical devices with localized deformation, we may select them according to the following principles: (1) The deepness h is more less than 0.2λ; (2) The radiuses a and b are small; (3) The center position (x 0,y 0) is near by the center of the optical device. In addition, if we know the localized deformation before laser beam transmitting/receiving, we can adjust pointing direction to compensate the pointing and tracking errors caused by localized aberrations.
To research localized deformation on pointing and tracking errors in inter-satellite lasercom, ellipse Gaussian model is proposed, which can simplify the calculation especially for small value of a/D by comparison with Zernike polynomials. It is found that pointing and tracking errors due to localized deformation are mainly determined by the center deepness h, the radiuses a and b, and the distance d. With the increasing of the deepness h, both of pointing and tracking errors fluctuate like damped oscillation with peak values around h=0.2λ (rms=π), h=0.75λ (rms=3π), h=1.25λ (rms=5π), et al.. The wider the localized deformation is, the more for the influence on pointing and tracking errors being. With the distance d rising, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases monotonically. The effects of localized deformation is up to 0.7urad for pointing error, and 0.5urad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the processing accuracy of optical devices should be more greater than 0.2λ. The principle of choosing the optical devices with localized distortion is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the conclusion can be used in the design of inter-satellite lasercom systems.
References and links
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2. B. Laurent and G. Planche, “SILEX overview after flight terminals campaign, in Free-Space Laser Communication Technologies IX, G. S. Mecherle, ed.,” Proc. SPIE 2990, 10–22 (1997). [CrossRef]
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4. M. Renard, P. Dobie, J. Gollier, T. Heinrichs, P. Woszczyk, and A. Sobeczko, “Optical telecommunication performance of the qualification model SILEX beacon, in Free-Space Laser Communication Technologies VII, G. S. Mecherle, ed.,” Proc. SPIE 2381, 289–300 (1995). [CrossRef]
5. K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies VIII, G. S. Mecherle, ed.,” Proc. SPIE 2699, 114–120 (1996). [CrossRef]
8. Brian R. Strickland, Michael J. Lavan, Eric Woodbridge, and Victor Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt. 38, 424–431 (1999). [CrossRef]
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11. J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt. 44, 4953–4958 (2005). [CrossRef] [PubMed]
14. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981). [CrossRef]
15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).
16. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).
17. M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987). |
2 edition of Transition flow ion transport via integral Boltzmann equation. found in the catalog.
Transition flow ion transport via integral Boltzmann equation.
Thomas Edward Darcie
Written in English
|The Physical Object|
|Pagination||222, [20, 6] leaves.|
|Number of Pages||222|
Grad’s assumption allows to split the collision operator in a gain and a loss part, Q(f, g) = Q+(f, g) − Q−(f, g) = Gain - Loss The loss operator Q−(f, g) = f R(g), with R(g), called the collision frequency, given by NOTE: The loss bilinear form is local in f and a weighted convolution in g. while the gain is a bilinear form with a weighted symmetric convolution structureFile Size: 5MB. Modeling of Flow Transition Using an Intermittency Transport Equation Y. B. Suzen and P. G. Huang I)epartmenl Mechanical Engineering l'niversitv of I,exington. Kentucky Abstract A new transport equation for illtermittencv factor is l)VOl)osed 1o model transitional Size: 1MB.
Here E l () and are the kinetic energy and the velocity of electrons in the valley l, and U() is the potential moderate electric fields, when the electron energy spectrum may be assumed to be parabolic, we will use the simple relation E l =P 2 /2 m l with the electron effective mass m l in the valley the electron transport under high fields and high electron energies, we. Boltzmann's Transport Equation With his ``Kinetic Theory of Gases'' Boltzmann undertook to explain the properties of dilute gases by analysing the elementary collision processes between pairs of molecules. The evolution of the distribution density in space,, is described by Boltzmann's transport equation. A thorough treatment of this.
The Linear Boltzmann Equation 1. Introduction One must distinguish between the “linear Boltzmann equation” and the “linearized Boltzmann equation.” The former has no self interaction, just scattering with the medium, whereas the latter is the linearization of the fully nonlinear Boltzmann equation. We will deal with the linear equationFile Size: KB. A similar approach can be used to calculate the contribution of electron–electron collisions to , considering the transformation in a reference frame that moves with the electron current ().In any case, the contribution of the flow to the collision integrals S 0 and S 1 is a second order correction.. To couple electron kinetics with a fluid dynamic model it is necessary to determine the.
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The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up.
4 CHAPTER 1. BOLTZMANN TRANSPORT functions2 k(r) = exp(ik r)= p V, as well as the result X k2 ^ A(k) = V Z ^ d3k (2ˇ)3 A(k) () for smooth functions A(k).
Note the factor of V 1 in front of the integral in eqn. What this tells us is that for a bounded localized potential U(r), the contribution to theFile Size: 2MB.
Transition flow ion transport via integral Boltzmann equation. Title. Transition flow ion transport via integral Boltzmann equation. Author. Darcie, T.E. Institution. University of Toronto Institute for Aerospace Studies. Date. To reference this document use: Author: T.E. Darcie. The Boltzmann transport equation and the diffusion equation Sergio Fantini’s group, Department of Biomedical Engineering, Tufts University Modeling light propagation in scattering media with transport theory The Boltzmann transport equation (BTE) is a balance relationship that describes the flow of particles in scattering and absorbing Size: KB.
where I[f] is defined to be the collision integral and is a functional of the distribution function. We can now arrive at the final form of the Boltzmann transport equation ∂f(r,p,t) ∂t + p m ∇ rf(r,p,t)−∇ rV ext(r,t)∇ pf(r,p,t) = I[f].
(12) The Collision Integral Our main problem. • Transition Flow regime can use approximations of Boltzmann Equation to solve • Knudsen Number provides indication of range of Equation validations Kn (Knudsen Number) = λ /L λ flow molecular mean free path length. L distance between boundaries.
Laminar N-S Equation. Transition between. and. Full Boltzmann. f is a distribution function ofFile Size: 1MB. 12 January Volumenumber 8 PHYSICS LETTERS A APPLICATION OF THE BOLTZMANN TRANSPORT EQUATION TO CALCULATIONS OF FLUX AND RANGE DISTRIBUTIONS OF ENERGETIC IONS Gyula BARDOS Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Moscow, USSR Received 19 June ; revised manuscript received 5 November ; Cited by: 9.
Explanation of the various gain and loss terms in the Boltzmann transport equation, which is the starting point for modeling how light propagates in. For transition and turbulent flow, use Figure (the f in this figure only applies for fully turbulent flow corresponding to the flat portions of the curves in Figure ) with Figureand Figures a and b as appropriate.
Friction factor in long steel pipes handling wet (saturated with water vapor) gases such as hydrogen, carbon monoxide, carbon dioxide, nitrogen, oxygen, and. The structure and the contents of the present book have some com mon features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical by: Physics for Solid State Applications Lecture Introduction to Boltzmann Transport • Non-equilibrium Occupancy Functions • Boltzmann Transport Equation • Relaxation Time Approximation Overview • Example: Low-field Transport in a Resistor Outline Ap Scattering Rate Calculations Overview Step 1: Determine Scattering File Size: KB.
Ludwig Boltzmann, Transport Equation and the Second law 3 influential and vociferous of the German-speaking physics community - the so-called energeticists, led by Ernst Mach ( - ) and Wilhelm Ostwald ( - ) did not approve of this. For them, energy was the only fundamental physical entity.
They dismissed with contempt any. Moment Methods for Solving the Boltzmann Equation. Transition flow ion transport: Experimental critical comments are introduced concerning the treatment of path-integral methods in a well Author: Larry Viehland.
An Introduction to the Boltzmann Equation and Transport Processes in Gases the basic principles of this theory within an elementary framework and from a more rigorous approach based on the Boltzmann equation. The subjects are presented in a self-contained manner such that the readers can understand and learn some methods used in the kinetic.
Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework. Philipp Neumann, Till Rohrmann. Faculty of Informatics, TU München, Munich, Germany. Email: [email protected] Received ; revised J ; accepted J ABSTRACT. We present simulation results of flows in the finite File Size: 2MB.
B BOLTZMANN TRANSPORT EQUATION In analogy to the diffusion-induced changes, we can argue that particles at time t = 0 with momentum k - k 6t will have momentum k at time 6t and which leads to the equation = -k- ’ afk dk vxB h dk B Scattering-Induced Evolution of fk(T) We will assume that the scattering processes are local and instantaneous and change.
Mass Flow Rate. Mass per unit time [MT-1] Flux. Mass flow rate through unit area [ML-2 T-1] 3. The Transport Equation. transport equation. Advective flux.
Dispersive flux. Equation 26 advection J J dispersion t x C + ion 2 z 2 2 2 y 2 x. RADIATIVE TRANSPORT 1. The Boltzmann equation The Boltzmann equation accounts for changes in the phase space number density. Physically, such The change in the number of particles of interest in the element due to their motion or flow is the integral representing the sum over initial states or other groups.
Again for simplicity File Size: KB. The Boltzmann Transport Equation The Boltzmann equation describes the time evolution of the electron distri-bution function f(r,k,t). Its physical interpretation is that f(r,k,t)drdk is the number of electrons (wavepackets) at point r with wavenumber k in the phase space volume drdk.
If integrated in all space over k, we would get theFile Size: KB. Students learn to solve the Boltzmann equation in the classical limit under relaxation time approximation in this lecture.
Students also learn to derive the Fourier law, Newton shear law, and the electron transport process with the Ohm's Law. Transport properties - Boltzmann equation goal: calculation of conductivity Boltzmann transport theory: distribution function number of particles in infinitesimal phase space volume around evolution from Boltzmann equation collision integral for static potential.
Transport properties - Boltzmann equation ion lattice density fluctuation File Size: 1MB.M. Bahrami Fluid Mechanics (S 09) Integral Relations for CV 8 The angular momentum A control volume analysis can be applied to the angular momentum, by letting B equal to angular‐ momentum vector H.
If O is the point about which moments are desired, the angular moment about O isFile Size: KB.Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$.
Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$.Cited by: 2. |
The collected works of F. W. Lawvere
Please inform us of any inaccuracies or missing works.
| Title |Source| Year | |:------|:-----|:----:| |The Language of Algebra: Supplement|TEMAC programmed learning materials|1961| |The Category of Probabilistic Mappings – With Applications to Stochastic Processes, Statistics, and Pattern Recognition|Unpublished; seminar handout notes|1962| |Functorial Semantics of Algebraic Theories|Original unpublished; Ph.D. thesis, Columbia University, 1963. See extended TAC reprint 2004|1963| |Functorial Semantics of Algebraic Theories (short notice)|Proceedings of the National Academy of Science 50, No. 5 (November 1963), 869-872|1963| |Functorial Automata Theory (abstract)|AMS Notices 603-151, vol 10 (1963), 477-478|1963| |The group ring of a small category (abstract)|Notices Amer. Math. Soc., 10, 280 (1963); Errata, Notices Amer. Math. Soc., 10, 516|1963| |An Elementary Theory of the Category of Sets (cf. 2005 long version with commentary)|Proceedings of the National Academy of Science 52, No. 6 (December 1964), 1506-1511|1964| |Algebraic Theories, Algebraic Categories, and Algebraic Functors|Theory of Models; North-Holland, Amsterdam (1965), 413-418|1965| |Functorial Semantics of Elementary Theories (abstract)|J. Symb. Logic, 31, 294-295, 1966|1966| |The Category of Categories as a Foundation for Mathematics|La Jolla Conference on Categorical Algebra, Springer-Verlag (1966), 1-20|1966| |Theories as Categories and the Completeness Theorem (abstract)|Journal of Symbolic Logic, 32:562|1967| |Some Algebraic Problems in the Context of Functorial Semantic of Algebraic Theories|Springer Lecture Notes in Mathematics No. 61 (Reports of the Midwest Category Seminar II), Springer-Verlag (1968), 41-61|1968| |Ordinal Sums and Equational Doctrines|Springer Lecture Notes in Mathematics No. 80, Springer-Verlag, 141-155.|1969| |Diagonal Arguments and Cartesian Closed Categories|Springer Lecture Notes in Mathematics No. 92, Springer-Verlag (1969), 134-145.|1969| |Adjointness in Foundations|Dialectica 23 (1969), 281-296|1969| |Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor|Proceedings of the American Mathematical Society Symposium on Pure Mathematics XVII (1970), 1-14|1970| |Quantifiers and Sheaves|Proceedings of the International Congress on Mathematics, Nice 1970, Gauthier-Villars (1971) 329-334|1970| |Introduction to "Toposes, Algebraic Geometry and Logic"|Springer Lecture Notes in Mathematics No. 274, New York: Springer, pp. 1–12|1971| |Theory of Categories over a Base Topos|Lectures given at Università di Perugia|1972| |Metric Spaces, Generalized Logic, and Closed Categories|Originally published in Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135-166; Republished in Reprints in TAC, No.1 2002 pp. 1-37|1973| |Logic of Topoi: inside and outside|Lecture Notes, Université de Montréal|1974| |Introduction to "Model Theory & Topoi"|Springer Lecture Notes in Mathematics No. 445, Springer-Verlag, pp. 3-14|1975| |Variable Sets Etendu and Variable Structure in Topoi|Notes by Steven Landsburg of Lectures and Conversations, Spring 1975, University of Chicago|1975| |Continuously Variable Sets – Algebraic Geometry=Geometric Logic|Studies in Logic and the Foundations of Mathematics, Volume 80, pp. 135-156|1975| |Variable Quantities and Variable Structures in Topoi|Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), pp. 101–131|1976| |Categorical Dynamics|Proceedings of Aarhus May 1978 Open House on Topos Theoretic Methods in Geometry|1978| |Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body|Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume: 21, Issue: 4, pp. 377-392|1980| |On C-∞ functions|Preprint, State Univ. of New York, Buffalo|1981| |Thermodynamics of deformations of continuous bodies, non homogeneous, with memory, far from equilibrium|Handwritten notes from a seminar held in Trieste|1982| |Introduction to "Categories in Continuum Physics"|Springer Lecture Notes in Mathematics No. 1174, Springer-Verlag (1986)|1982| |Measures on toposes|Proceedings of Aarhus Workshop on Category Theoretic Methods in Geometry|1983| |Functorial Remarks on the General Concept of Chaos|IMA Research Report #87, University of Minnesota (1986)|1984| |State Categories Closed Categories and the Existence of Semi-continuous Entropy Functions|IMA Research Report #86, University of Minnesota|1984| |State Categories and Response Functors|Preprint|1986| |Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs|Revista Colombiana de Matemáticas XX (1986), pp. 179-185|1986| |Taking Categories Seriously|Revista Colombiana de Matemáticas XX (1986), pp. 147-178|1986| |Some "New" Mathematics Arising From the Study of Grassmann 1844|Unpublished manuscript|1987| |Concepts and Problems in Mathematical Toposes (abstract)|CMS Winter Meeting, December 1988, Toronto, Program p. 31, Special Session on Category Theory|1988| |Fractional Exponents in Cartesian Closed Categories|Unpublished manuscript|1988| |Möbius algebra of a category|Handwritten Notes by S. Schanuel at the Sydney Combinatorics Seminar organized by Don Taylor|1988| |Toposes generated by codiscrete objects in combinatorial topology and functional analysis|TAC Reprint (2021) of notes for Colloquium lectures given at North Ryde, New South Wales, Australia 1988-89|1989| |On the Complete Lattice of Essential Localizations (with G.M. Kelly)|Bull. Société Mathematique de Belgique, XLI, 289-319|1988| |Intrinsic boundary in certain mathematical toposes exemplify logical operators not passively preserved by substitution|Preprint, Univ. of Buffalo|1989| |Display of Graphics and their Applications Exemplifed by 2 Categories and the Hegelian Taco|Proceedings of the First International Conference on Algebraic Methodology and Software Technology, The University of Iowa, 51-75|1989| |Qualitative Distinctions Between Some Toposes of Generalized Graphs|Proceedings of AMS Boulder 1987 Symposium on Category Theory and Computer Science, Contemporary Mathematics, 261-299|1989| |Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes|Category Theory, Proceedings Como 1990, A. Carboni, M. C. Pedicchio, G. Rosolini (Eds). Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) pp. 279-281|1991| |More on Graphic Toposes|Proceedings of the 1989 Bangor Category Theory Meeting, Cahiers de Topologie et Géométrie Différentielle Catégorique XXXII - 1 (1991), pp. 5-10|1991| |Some Thoughts on the Future of Category Theory|Category Theory, Proceedings Como 1990. A. Carboni, M. C. Pedicchio, G. Rosolini (Eds). Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) pp. 1-13|1991| |Categories of Space and Quantity|The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, International Symposium on Structures in Mathematical Theories (1990), San Sebastian, Spain; DeGruyter, Berlin (1992), pp. 14-30.|1992| |Cohesive Toposes and Cantor's Lauter Einsen|Philosophia Mathematica, The Canadian Society for History and Philosophy of Mathematics, Series III, Vol. 2 (1994), pp. 5-15.|1994| |Tools for the Advancement of Objective Logic Closed Categories and Toposes|The Logical Foundations of Cognition, J. Macnamara, G. E. Reyes (Eds). Oxford University Press (1994), pp. 43-56|1994| |Adjoints in and among Bicategories|Logic & Algebra, Proceedings of the 1994 Siena Conference in Memory of Roberto Magari. Lecture Notes in Pure and Applied Algebra 180: pp. 181-189, Ed. Ursini/Aglianò, Marcel Dekker, Inc. Basel, New York|1996| |Grassmann's Dialectics and Category Theory|Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar pp. 255-264, Boston Studies in the Philosophy of Science Series (BSPS, volume 187)|1996| |Unity and Identity of Opposites in Calculus and Physics|Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures, 4: pp. 167-174 Kluwer Academic Publishers|1996| |Algebra Step by Step|Buffalo Workshop Press, ISBN: 0963180525|1997| |Linearization Revisited (abstract)|IIIrd Joint Meeting AMS-SMM, Special Session on Rings and Category Theory, Oaxaca Mexico, December 1997, Program p. 59|1997| |Toposes of Laws of Motion|Transcript from Video, Montreal September 27, 1997|1997| |Volterra's Functionals and Covariant Cohesion of Space|Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, 64, R. Betti, F. W. Lawvere (Eds.), (2000), pp. 201-214|1997| |Everyday physics of extended bodies or why functionals need analyzing (abstract, cf. paper 2017)|Public Lecture, CMS Summer 1998 Meeting, University of New Brunswick, Saint John, June 13-15, (received May 1998)|1998| |Outline of Synthetic Differential Geometry|Buffalo Geometry Seminar notes, February 1998|1998| |Are Homotopy Types the Same As Infinitesimal Skeleta? (abstract)|CMS Summer 1998 Meeting|1998| |Categorical Analyses of the Whole/Part Relation (abstract)|Mitteleuropaeisches Kulturinstitut, Bolzano, Italy|1998| |Categorie e Spazio: Un Profilo|Lettera matematica PRISTEM 31, Springer, Italy, (1999), 35-50. [Reprinted in La Mathematica a cura di Bartocci, Claudio, Giulio Einaudi editore (2010) vol. 4, 107-135.|1999| |Kinship and Mathematical Categories|Language, Logic, and Conceptual Representation, P. Bloom, R. Jackendoff, and K. Wynn (Eds), MIT Press, (1999), pp. 411-425|1999| |Comments on the Development of Topos Theory|Development of Mathematics 1950-2000, J.-P. Pier (Ed) Birkhäuser Verlag, Basel, (2000), pp. 715-734|2000| |The Role of Cartesian Closed Categories in Foundations (Interview with Felice Cardone)||2000| |Explicit foundational concepts in the teaching of mathematics|Matematica e filosofia: il problema dei fondamenti oggi PRISTEM/Storia 14-15 - Translation: Filsofia, scienza e bioetica nel dibattito contemporaneo a cura di Minazzi, Fabio, Instituto Poligrafico e Zecca dello Stato, Roma|2001| |Categorical Algebra for Continuum Micro Physics|Journal of Pure and Applied algebra 175, (2002), pp. 267-287|2001| |On the Duality Between Varietes and Algebraic Theories (with J. Adámek & J. Rosický)|Algebra Universalis, (2003), pp. 35-49|2003| |How Algebraic is Algebra? (with J. Adámek & J. Rosický)|Theory and Applications of Categories , Vol. 11, 2003, No. 11, pp. 252-282|2001| |Linearization of Graphic Toposes via Coxeter Groups|Journal of Pure and Applied Algebra, vol. 168, (2002), pp. 425-436|2002| |Foundations and Applications – Axiomatization and Education|The Bulletin of Symbolic Logic, vol. 9, No. 2, (2003), pp. 213-224|2003| |Continuous Categories Revisited (with J. Adámek & J. Rosický)|Theory and Applications of Categories , Vol. 11, 2003, No. 11, pp 252-282|2003| |Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories|Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121. Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, pp. 41-61|2004| |Functorial Concepts of Complexity for Finite Automata|Theory and Applications of Categories, Vol. 13, 2004, No. 10, pp. 164-168|2004| |Left and Right Adjoint Operations on Spaces and Data Types|For Dana Scott's Seventieth Birthday, Copenhagen 2002, Theoretical Computer Science, Elsevier, vol 316/1-3, (2004) pp. 105-111|2004| |An Elementary Theory of the Category of Sets (long version) with commentary|Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1-35. Expanded version of Proceedings of the National Academy of Science of the U.S.A 52, 1506-1511, with commentary by Colin McLarty and the author.|2005| |Grassmann Book Reviews|Historia Mathematica, vol. 32, (2005), pp. 101-106|2005| |John Isbell's Adequate Subcategories|Topological Commentary, Vol. 11 #1|2006| |Axiomatic Cohesion|Theory and Applications of Categories, online publication, Special volume from the CT2006 Conference at Whitepoint Nova Scotia, vol 19, (2007), 41-49|2007| |Cohesive toposes: combinatorial and infinitesimal cases (video) (lecture notes)|Lectures in Como (Italy), January 10, 2008|2008| |Core Varieties Extensivity and Rig Geometry|Theory and Applications of Categories, Vol. 20, 2008, No. 14, pp 497-503.|2008| |Interview with Maria Manuel Clementino and Jorge Picado|Bulletin of the International Center for Mathematics (part 1, December 2007, part 2, June 2008)|2008| |Foreword To "Algebraic Theories"|Cambridge Tracts in Mathematics 184, J. Adámek, J. Rosický & E. M. Vitale (2012)|2009| |Open Problems in Topos Theory|88th Peripatetic Seminar on Sheaves and Logic, For Martin Hyland and Peter Johnstone in honor of their sixtieth birthdays (updated July 2016)|2009| |The Hopf Algebra of Möbius Intervals (with M. Menni)|Theory and Applications of Categories, online publication, vol. 24, (2010), 221-265|2010| |Categorical Dynamics (abstract)|Talk at International Category Theory Conference, Vancouver, July 2011|2011| |Euler's Continuum Functorially Vindicated|Logic, Mathematics, Philosophy: Vintage Enthusiasms, Essays in Honour of John L. Bell, D. DeVidi et al. (Eds), Western Ontario Series in Philosophy of Science 75, (2011)|2011| |What Is A Space? (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |The Dialectic of Continuous and Discrete in the History of the Struggle for a Usable Guide to Mathematical Thought (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |Categorical Dynamics Revisited: Category Theory and the Representation of Physical Quantities (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |What are Foundations of Geometry and Algebra? (abstract) (video) (transcript)|Fifty Years of Functorial Semantics Conference|2013| |Internal Choice Holds in the Discrete Part of any Cohesive Topos Satisfying Stable Connected Codiscreteness (with M. Menni)|Theory and Applications of Categories, online publication, vol. 30, (2015), No. 26, pp 909-932|2015| |Alexander Grothendieck and the Concept of Space|Invited address at CT 2015 Aveiro, Portugal|2015| |Birkhoff's Theorem from a Geometric Perspective: A Simple Example|Categories and General Algebraic Structures with Apllications Vol. 4, no. 1, (2016), pp. 1-7|2016| |Everyday physics of extended bodies or why functionals need analyzing|Categories and General Algebraic Structures with Applications, Volume 6 2017, pp. 9-19|2017| |
1 Operational Amplifiers Table of contents 1. Design 1.1. The Differential Amplifier 1.2. Level Shifter 1.3. Power Amplifier 2. Characteristics 3. The Opamp without NFB 4. Linear Amplifiers 4.1. The Non-Inverting Amplifier 4.2. The Voltage Follower 4.3. The Inverting Amplifier 5. Frequency Characteristics 5.1. Band width 5.2. Slew Rate 6. Applications 6.1. Non-Inverting Amplifier 6.2. Inverting Amplifier 6.3. With push-pull output 6.4. Summing Amplifier 6.5. Logarithmizing Amplifier 6.6. Signal Rectification 6.7. Voltage Regulator 6.8. Comparator 6.9. Schmitt Trigger Astable Multivibrator Phase Shifter
2 Operational Amplifiers The theory of electrical signal processing requires amplifiers to perform, with electrical signals, mathematical operations such as addition, subtraction, multiplication, division, differentiation, integration, etc. These amplifiers must fulfil the following requirements: Differential inputs D.C. amplification Very high voltage gain Very high input resistance Very low output resistance They are then called "operational amplifiers" (opamps) because they are able to perform mathematical operations. With opamps, even analog computers are constructed which surpass any digital computer when high speed of signal processing is required. The first opamps were built using discreet transistors, but it a was difficult and expensive process because of temperature drift problems. The big breakthrough came with integrated circuits. Having all circuit elements on one monolithic silicon chip solved most of the temperature drift problems and allowed for cheap mass production. Today we have to consider the opamp as a circuit element. We will study its characteristics but not dwell on how it works internally. 1. Design The basic form of an opamp is a high gain dc-amplifier with a differential input port and a single output port. A differential input has two terminals, which are both independent of ground or common. The signal between these two terminals is the input signal, which will be amplified. The terminals are called non-inverting input and inverting input. The two inputs can be used in three different ways: 1. Non-Inverting Amplifier: The input signal is applied between the non-inverting input and ground. The inverting input is connected to ground. The output signal will be in phase with the input signal 2. Inverting Amplifier: The input signal is applied between the inverting input and ground. The non-inverting input is connected to ground. The output signal will be 180 out of phase with the input signal. 3. Differential Amplifier: Two input signals are each connected to the non-inverting and the
3 inverting input, using both common as second terminal. The output signal will be the amplified difference between the two. U o = (U i+ - U i- ) g Fig The three basic ways of applying input signals to the opamp. When there is no voltage difference between the input terminals, the output voltage should be 0. The internal circuit of opamps consists basically of three main parts: 1.1. The Differential Amplifier: A differential amplifier stage consists of two transistors in common emitter configuration which are supplied with a common emitter current.
4 Fig The basic design of a differential amplifier stage. As long as there is no voltage difference between the two bases of the transistors, the two transistors will draw the same collector currents and a certain voltage will appear at the output. If the base of T 1 becomes more positive than of T 2, T 1 will draw more current, the voltage across R C1 will increase. As the total current is constant, the current through T 2 will decrease by the same amount. The voltage across R C2 will decrease and the output voltage becomes more positive. So the base of T 1 is the non-inverting input. If the base of T 2 becomes more positive than that of T 1, T 2 will draw more current. The voltage across R C2 increases and the output voltage becomes more negative. Thus the base of T 2 is the inverting input. If the voltages at the bases of T 1 and T 2 are varied by the same amount, the current distribution between the two transistors does not change and no voltage results at the output. This case is called common mode and should not produce an output signal. The general requirements for the differential amplifier: high differential mode gain low common mode gain high input impedance
5 low base currents temperature stability Some opamps use FET as input transistors to achieve extremely high input resistances Level Shifter The level shifter fulfils two main tasks: it provides most of the voltage amplification of the opamp; it provides dc-matching between differential amplifier and the output to obtain zero output voltage for zero input (offset voltage). The level shifter consists mainly of a number of dc-coupled transistor stages which are arranged and biased in such a way that zero offset voltage with a high temperature stability is achieved. Requirements to the level shifter: low distortion wide frequency range 1.3. Power Amplifier The final stage of an opamp is in most cases a complementary push-pull amplifier. It has to provide the required output current at a low output resistance. Requirements: symmetrical output swing from +U b to -U b low output impedance short-circuit protection low distortions
6 Fig An example of the circuit of a simple integrated opamp. The circuit symbol for an opamp is a triangle pointing towards the output. The input terminals are drawn to the vertical left side. Any further auxiliary terminals such as supply voltages or offset adjustment are drawn at the top and bottom slopes of the triangle. Fig The circuit symbol for a general opamp.
7 2. Characteristics Voltage gain An ideal opamp should have an open loop voltage gain g (without NFB) which is infinite. Practical opamps may have values from 60dB to 120d, which equals 10 3 to In general, all practical opamps have sufficient gain for most requirements. Input resistance An ideal opamp should have an input resistance R i which is infinite. Practical opamps may have values from 10k to 1M Input up to 1G can be reached for opamps with MOSFET. The input resistance of opamps will further be increased by NFB, so that the achieved values will satisfy most practical requirements. Output Resistance An ideal opamp should have an output resistance R o of zero. Practical opamps may have values from 50 to 500 These values are not made lower in order to achieve short circuit protection of the output. The output resistance will be reduced by NFB, so that the achieved values will satisfy most practical requirements. Supply Voltage In general, opamps require two symmetrical (equal but of opposite polarity) supply voltages +U b and -U b in respect to ground. These voltages must be large enough in order to properly bias all internal transistors. On the other hand, they may not exceed a specific maximum value. Practical supply voltages range from ±3V to ±30V. A common value is ±15V. Some opamps are also designed to be operated on one supply voltage only. This requires a special design for the input and output stage. Either supply terminal may then be connected to ground. Output voltage Swing The maximum output signal U sat (saturation voltage of the output stage) will depend on the supply voltage. It is obvious that the output voltages cannot be higher than the supply voltages. As the output of the amplifier will always require a certain voltage drop, the maximum output voltage swing will be 1V to 3V lower than the supply voltage, depending on the type of opamp.
8 Fig The relationship between supply voltage and maximum output voltage swing. The maximum output voltage will depend on the supply voltage. The higher the supply voltage, the more output amplitude can be achieved. As for opamps operated on one supply voltage only, the amplitude of the output signal can only be less than half of the supply voltage. Input Offset Voltage The output voltage of an opamp should be zero, if the input voltage is zero (input terminals shorted). In practice, there will always be some asymmetry in the differential amplifier. This voltage is then amplified through all stages and, depending on the gain, there might be a high voltage at the output of the opamp.
9 Fig The output voltage which is measured at the output of an opamp with shorted input terminals is the internal offset voltage U iofs multiplied by the gain g. This voltage could be compensated by feeding a dc-voltage to the input which opposes the internal offset. This voltage is equal to the input offset voltage Uofs. This process is called offset compensation or offset null-balance. It is required for most cases of dc-amplifiers (e.g. measuring amplifiers). Fig If the input offset voltage U iofs is fed into the inverting input terminal, the output voltage can be set to zero.
10 In order to keep the input terminals free for the signal, some opamps provide separate terminals for offset adjustment. These offset adjustment terminals must be used according to the specifications of the data sheets. Fig Example of the offset compensation using the separate terminals of an opamp (741). Input Bias Current The input terminals of opamps can be considered as base terminals of the transistors of a differential amplifier stage. In order to operate the transistors in the active region, they require a certain bias current I ib. For opamps with bipolar input this will be in the range of some na or µa. Although these currents are very small, they may produce a voltage drop across any resistance in series with the input. This is then a voltage difference at the input which again produces an offset at the output. If the two resistors are equal, the voltage drops will be equal and there will be no voltage difference at the input.
11 Fig The input bias current I ib of the input transistors will produce a voltage drop across any resistor connected in series to the input. Making both resistors equal will cancel out the two voltages U R1 and U R2. Care is therefore often taken that both inputs of the opamp have an equivalent resistance to ground to avoid offset due to bias current. Input offset Current The bias current of the two transistors may not be equal, so even if both inputs have equal resistors in series, there might be an offset voltage. In practice, this effect cannot be distinguished from the effect of the input offset voltage, so they will be compensated together.
12 3. The Opamp without NFB Let us look at how the opamp can amplify signal. We will assume that the opamp has an open loop gain of g = 6000 = 76dB. This means an input voltage of 1mV will produce an output voltage of 6V. Fig Opamp as amplifier with its transfer characteristic. Input voltages of more than 2mV will drive the output to saturation. In practice, it will be found that an amplifier with such a large dc-gain will not work properly because the offset voltage drift will not allow a stable working point. An opamp without NFB can not be used as linear amplifier. The opamp in this "pure" form is only used as COMPARATOR. The comparator compares two input signals and provides a digital (high/low) output signal, depending on which of the two is larger. U o = +U sat (approx. +U b ) if U i+ > U i-
13 U o = -U sat (approx. -U b ) if U i+ < U i- Fig The opamp as comparator. The output signal is either +U sat or -U sat, depending on which of the two input voltages is larger. Normally one of the two input voltages is used as a reference or threshold for the other. If the reference voltage is connected to the inverting terminal, we will get a non-inverting comparator. If the reference voltage is connected to the non-inverting input, we will have an inverting comparator.
14 4. Linear Amplifiers Opamps can only be used as linear amplifiers with external negative feedback. The NFB is achieved by a voltage divider circuit which feeds back a fraction of the output signal to the inverting input. As opamps have a very high open loop gain, very strong NFB can be provided. This makes strong use of all of the advantages of NFB such as: - reduction of distortion, - favourable input and output resistances, - stable working parameters. Depending on how (in which form) the NFB is achieved and how the signal is fed to the input, different types of amplifiers with different characteristics are created The Non-Inverting Amplifier The non-inverting amplifier feeds the input signal to the non-inverting input. The NFBsignal is derived from a voltage divider from the output signal and is fed to the inverting input. Fig The basic configuration of the non-inverting amplifier. The properties of this amplifier are controlled entirely by the NFB voltage divider (see chapter on NFB): Close Loop Voltage Gain
15 This formula is correct if g' << g (g' is much smaller than g) Input Resistance The input resistance is increased by the degree of reduction of gain. This factor will in practice be at least 10 or 100, so the input resistance of this amplifier will be very high (>1M ) in all cases. Output Resistance The output resistance will be reduced by the same factor by which the input resistance is increased. In practice, this leads to very low values (<1 ). Summary of properties of the non-inverting opamp: the signals at input and output are in phase, the closed loop gain g' depends on the external elements R 1 and R 2 only, the input resistance is very high, the output resistance is very low. The non-inverting amplifier is used for audio amplification and as a measuring amplifier. The NFB tends to eliminate all kinds of negative influences which appear between the input and output of the amplifier. It can be used to reduce the influence of any other circuit elements which are used in conjunction with opamps. Any resistance which is in series with the output of the amplifier will increase the output resistance. The effect of this resistance can be reduced if the resistor is taken into the NFB-loop.
16 Fig A resistance in series with the output of an amplifier. a.) If the resistance in series with the output is outside of the NFB-loop, the resistance adds fully to the output resistance. b.) If the resistance in series with the output is within the NFB-loop, the resistance is eliminated by the NFB. If more output current is required, a push-pull stage can be connected to the output of the opamp. A push-pull stage can produce distortions, mainly cross-over distortions. Taking the push-pull stage into the NFB-loop will strongly reduce the distortions. Fig A push-pull state may be used to boost the output current of the opamp. a.) If the push-pull stage is outside of the NFB-loop, the distortions of this stage appear at the output. b.) If the push-pull stage is within the NFB-loop, the distortions of this stage are reduced by the NFB The Voltage Follower The smallest gain to be achieved with a non-inverting amplifier is one. This is achieved if the entire output signal is fed back to the input. Considering the formulas above, this means that R 1 = 0 and R 2 = (infinit).
17 Fig When all the output voltage is fed back to the input, the non-inverting amplifier becomes a voltage follower with unity gain. The gain of this amplifier is one and so the output voltage is identical to the input voltage. Because of this, the circuit is called UNITY GAIN AMPLIFIER or VOLTAGE FOLLOWER. Important characteristics of this amplifier: Gain: g' = 1 Input Resistance: R i ' = R i * g Output Resistance: R o ' = R o /g Summary of important properties: the signals at input and output are in phase, the closed loop gain g' is one the input resistance is extremely high, the output resistance is extremely low. Voltage followers are used as impedance converters in audio amplifiers and measuring amplifiers The Inverting Amplifier Inverting amplifiers feed the input signal and the NFB-signal into the inverting input. The non-inverting input is connected to ground. The output signal is shifted 180 in phase to the input signal.
18 Fig The basic configuration of the inverting amplifier. The function of the inverting amplifier can be explained by taking two points into consideration: 1. The input voltage of the opamp U i will be negligible compared to the input voltage of the amplifier U i ', or even compared to the output voltage U o. The inverting input of the opamp therefore has approximately the same voltage as the non-inverting terminal, which is connected to ground. This point of the circuit is therefore called VIRTUAL GROUND. From the point of view of the signal, this point has the same properties as the ground point of the circuit. 2. The input current to the opamp I i- is approximately zero. The sum of the currents I R1 and I R2 must therefore sum up to 0. The inverting input is therefore also called the SUMMING POINT. The main characteristics can be derived from these considerations: Closed loop gain: The resistor R1 and R2 are virtually connected to ground at the inverting input. The currents through the resistors R1 and R2 are equal. This requires that the input and output voltage have the same ratio as the resistors R1 and R2. This formula is correct if g' Input Resistance
19 The input resistance is only the resistor R2, because it is connected between input and virtual ground. Output Resistance The output resistance will be reduced by the same factor as the gain. In practice, this leads to very low values (<1 ). Summary of properties of the inverting opamp: the signals at input and output are 180 out of phase, the closed loop gain g' is set by the ratio of R 1 to R 2 the input resistance is set by R 2 the output resistance is very low. the inverting input of the opamp can be considered as virtual ground. If bias current compensation is required, a compensation resistor Rcomp can be used to offset current compensation. It should be selected so that the resistance in series with both inputs is approximately equal. Therefore: R comp = R 1 //R 2 (R 1 parallel with R 2 ) Fig The inverting amplifier with compensation resistor for the bias current.
20 5. Frequency Characteristics Opamps have a frequency range which starts at 0Hz (d.c.). At the upper end, the frequency range is limited by the BAND WIDTH and by the SLEW RATE. Both have the effect of limiting the upper operational frequencies, but have different physical causes and must be considered separately Band width Opamps without NFB have only a relatively small frequency range. Some types only have an upper frequency limit (-3dB) of a few Hz or a few hundred Hz. The gain decreases with increasing frequency due to the low-pass behaviour of the internal transistor amplifier stages. Furthermore, the opamp will have several internal transistor stages in series, each forming a low-pass with its own critical frequency. Fig The different amplifier stages of an opamp eacg form a low-pass, which is connected in series. The gain decreases after the first critical frequency with a slope of 20 db/decade, after the second critical frequency with a slope of 40 db/decade, etc. Each low pass will also produce a certain phase shift of up to 90 per low-pass. With increasing frequency, a growing phase shift will occur between input and output. The so- called "Bode-plot" shows the relations:
21 Fig Example of the Bode plot of an opamp (TAA 861). The critical frequency of the open loop gain (g=85db) is about 10 Hz. Over 1kHz the gain drops with 40dB/decade due to a second internal low pass. At 5kHz the phase shift between differential input and output is more than 180. The limited band width makes this device unsuitable for audio applications, but introducting NFB, the band width can be increased. Assume for the TAA 861 the gain is set by NFB to 40dB (100). Thus below 1kHz, the open loop gain will be higher than the closed loop gain, and the gain will be defined entirely by the NFB. Above 1kHz the open loop gain will be less than the desired closed loop gain, and the gain will be equal to the closed loop gain.
22 Fig The frequency response of the same opamp with the gain set to 40dB by NFB. The upper critical frequency has been improved to 1kHz. The band width of this amplifier could be increased to approximately 30kHz. Then the open loop gain becomes 1. But at higher frequencies only little gain is achieved. (In fact, the TAA 861 is not a suitable opamp for audio circuits!) The lower the chosen gain, the higher the band width. As the opamp without NFB is not used as a linear amplifier, the band width of the open loop gain plays no practical role and is thus not mentioned in the data sheets. Instead, the UNITY-GAIN BAND WIDTH is given. This is the band width of the opamp with a closed loop gain of 1. Some examples of unity-gain band width of practical opamps: - type TAA 861: 30kHz - type 741: 300kHz - type 081: 3MHz A problem arises from the phase shift inside the opamp which increases with frequency. The NFB-signal is supplied with a nominal phase shift of 180 to the input signal (anti-phase). Additional internal phase shifts will turn the negative feed back into a positive feed back. If the gain is then still larger than 1 (0 db), this will cause oscillation of the amplifier (instability). In the case of the TAA 861: the lowest gain for stable conditions is 25 db. In practice, a phase security margin of 60 is respected. This determines the lowest possible gain to 48 db and the upper critical frequency to 900 Hz. For an uncompensated opamp the danger of instability increases with increasing NFB.
23 To allow higher band widths at smaller gains - particularly for voltage followers (g' = 0 db) - opamps are provided with terminals for EXTERNAL FREQUENCY COMPENSATION by means of R and C components. The required circuit elements and their wiring depends on the type of opamp and has to be determined from the data sheets. In general, frequency compensation is achieved by a low pass function, reducing the first open loop corner frequency and providing a gain decrease of 20 db/decade down to unity gain. Sufficient phase margin is achieved, though band width and slew rate are reduced compared to uncompensated operation. Several opamps provide internal frequency compensation (e.g. 741-types) and secure stable conditions for all gains. Fig Frequency compensation of TAA 861 with C k according to the data sheets. (This Op Amp is an open-collector device and requires the load-resistor to be connected to +U b ) Slew Rate If a step function (pulse) is applied to the input of an opamp, the output signal will not respond immediately. This is due to internal capacitances which cannot be charged instantaneously. The output will respond with a slope function, representing the highest speed in voltage change. This is called the slew rate (or slewing rate). It is given in volts per microseconds (V/µs).
24 Fig When a step function is applied to the input of an opamp, the output will respond with its maximum possible voltage rise, called the slew rate. (The gain of this opamp is set to 2.) In addition, when a sine wave is applied to the opamp, the output is only able to follow with its maximum slew rate. For a sine wave, the highest voltage change occurs during zero crossing and is related to frequency and magnitude. Sine waves follow the function:
25 Fig The maximum slope of a sine function occurs at the zero crossing. The slope depends on the amplitude and on the frequency. If the voltage continues to rise with the zero-slope of the sine function, it will reach U max at: The maximum slope can therefore be expressed in terms of the amplitude and the frequency of the sine function: This means for a given slew rate: the higher the output voltage, the smaller the maximum frequency, resp. band width; and vice versa: the larger the required band width, the smaller the maximum amplitude. The slew rate relates the maximum amplitude and the maximum frequency of the output signal. The slew rate cannot be influenced by NFB. Examples of the slew rate of some practical opamps: - type 741: O.3V/µs - type 081: 13V/µs
26 6. Applications This chapter sums up some of the most important opamp applications and gives their main characteristics and design rules Non-Inverting Amplifier (very high) (very low)
27 6.2. Inverting Amplifier (very low) 6.3. With push-pull output The complementary push-pull stage boosts the output current. If it is included in the NFB-loop, the take-over distortions are compensated.
28 6.4. Summing Amplifier The input signals U 1, U 2, etc. are added up and amplified. As the summing point is the virtual ground ( Zero-Ohms-Circuit), the inputs are fully decoupled from each other Logarithmizing Amplifier
29 A non-linear NFB-circuit will result in an non-linear characteristic of the amplifier. The exponential U-I-characteristic of the diode produces a logarithmic U in -U out - relationship. (U T is the inherent temperature voltage of the diode which, for silicon diodes, is approx. 40mV at 25 C. I o is the minority current of the diode at 0V, which is appr. 10nA at 25 C) 6.6. Signal Rectification The threshold voltage of rectifier diodes produce incorrect indications when small signal voltages have to be rectified for indication. Putting the rectifier into the NFBloop of an opamp will produce a linear indication of the meter. It is a disadvantage of this circuit that the meter cannot be grounded on one side.
30 6.7. Voltage Regulator The opamp is used as an error amplifier, comparing the reference voltage with the actual output voltage. Depending on how much output current is required, several current amplifier transistor stages are required Comparator
31 The comparator is an analog-digital converter. The output signal is high or low, depending on whether the input voltage is higher or lower than the reference voltage. If the reference voltage is applied to the non-inverting input, it will be an inverting comparator Schmitt Trigger The Schmitt Trigger can be considered a comparator with hysteresis. By applying positive feedback, the output is always saturated. The threshold voltages for changing the output from positive to negative is different from the voltage which will change it from negative to positive.
32 6.10. Astable Multivibrator This circuit produces a symmetrical square wave at the output of the opamp. The amplitude is given by the saturation voltage of the opamp. The steepness of the flanks is limited by the slew rate.
33 6.11. Phase Shifter This circuit provides a frequency depending phase shift between the input and output signal, but has a linear amplitude response. It is therefore also called an ALL PASS FILTER. The phase shift will vary between 0 and 180. The gain is defined by the negative feedback of R 1 and R 2. Normally, the gain is set to 1 (R 1 =R 2 ). |
On Y. Nievergelt’s inversion formula for the Radon transform
We generalize Y. Nievergelt’s inversion method for the Radon transform on lines in the 2-plane to the -plane Radon transform of continuous and functions on for all .
Key words and phrases:The -plane Radon transform, Nievergelt’s inversion method, the convolution-backprojection method
2000 Mathematics Subject Classification:Primary 42C40; Secondary 44A12
Inversion formulas for Radon transforms of different kinds are of great importance in mathematics and its applications; see, e.g., [2, 4, 7, 8, 9, 14, 16, 17, 20, 24], and references therein. Since many of them are pretty involved, especially for new-comers in the area, or applicable under essential restrictions, every “elementary” inversion method deserves special consideration. In 1986 Yves Nievergelt came up with intriguing paper , entitled “Elementary inversion of Radon’s transform”. His result can be stated as follows.
. Any continuous compactly supported function on the -plane can be reconstracted from the Radon transform over lines in this plane by the formula
where the double integral on the right-hand side equals the average of over the disc of radius centered at .
1. What is the basic idea of the Nievergelt’s method from the point of view of modern developments?
2. Is this method applicable in the same elementary form to -plane Radon transforms on for all and arbitrary continuous or functions, satisfying minimal assumptions at infinity?
In the present article we answer these questions and indicate possible generalizations.
Notation and main results. Let and be the affine Grassmann manifold of all non-oriented -planes in , and the ordinary Grassmann manifold of -dimensional subspaces of , respectively. Given , each vector can be written as where and , being the orthogonal complement to in . Each -plane is parameterized by the pair where and . The manifold will be endowed with the product measure , where is the -invariant measure on of total mass , and denotes the usual volume element on . We write for the space of continuous functions on vanishing at infinity; denotes the area of the unit sphere in .
The -plane transform of a function on is a function on defined by
This expression is finite for all if is continuous and decays like with . Moreover [20, 22, 24], if , , then is finite for almost all planes . The above-mentioned bounds for and are best possible.
Following , we define the wavelet-like transform
where denotes the Euclidean distance between the point and the -plane .
[19, Th. 3.1] Let , and let be a radial function on , which has an integrable decreasing radial majorant. If is a solution of the Abel type integral equation
The limit in (1.7) is understood in the -norm and in the almost everywhere sense. If for some , then (1.7) holds uniformly on .
This theorem is a core of the convolution-backprojection method for the -plane Radon transform, and the most difficult task is to choose relatively simple functions and satisfying (1.5). The crux is that the left-hand side of (1.5) has, in general, a bad behavior when . Hence, to achieve integrability of , the solution must be sign-changing.
Our first observation is that the essence of Y. Nievergelt’s Theorem 1.1 can be presented in the language of Theorem 1.2 as follows.
For the convenience of presentation, we will keep to the following convention.
Of course, Theorem 1.2 deals with essentially more general classes of functions than Theorem 1.1, however, the main focus of our article is different: we want to find auxiliary functions and , having possibly simple analytic expression.
(i) The Nievergelt’s method is applicable to the -ray transform (the case ) in any dimension. Namely, if is chosen according to (1.8), then (1.5) has a solution
and inversion formula (1.7) holds.
(ii) If , then Nievergelt’s method is inapplicable.
An integral in (1.11) can be expressed through the hypergeometric function and explicitly evaluated in some particular cases; see Section 2.2. For instance, if , then (1.11) is the Nievergelt’s function (1.9).
To include all , we modify the Nievergelt’s method by choosing in a different way as follows.
Theorem B. Let
. Then the corresponding function has a decreasing radial majorant in and (1.5) has the following solution:
(i) In the case
(ii) In the case
In both cases the inversion result in Theorem 1.2 is valid.
Theorems A and B are proved in Sections 2 and 3, respectively.
. The convolution-backprojection method is well-developed in the general context of totally geodesic Radon transforms on spaces of constant curvature. Apart of , the latter include the -dimensional unit sphere and the hyperbolic space ; see [1, 18, 19]. As above, the key role in this theory belongs to a certain Abel type integral equation and the relevant sign-changing solution . Moreover, passage to the limit in (1.7) as , can be replaced by integration in from to against the dilation-invariant measure . This leads to inversion formulas, which resemble the classical Calderón’s identity for continuous wavelet transforms . The corresponding wavelet function is determined as a solution of a similar Abel type integral equation; see [1, 18, 19] for details. In all these cases analogues of Theorems A and B can be obtained. We leave this exercise to the interested reader.
. Unlike the classical -plane transforms on , the corresponding transforms on matrix spaces [5, 11, 12, 13] are much less investigated. To the best of our knowledge, no pointwise inversion formulas (i.e., those, that do not contain operations in the sense of distributions) are available for these transforms if the latter are applied to arbitrary continuous or functions . One of the reasons of our interest in Nievergelt’s idea is that it might be applicable to the matrix case. Moreover, as in , it may pave the way to implementation of wavelet-like transforms in the corresponding reconstruction formulas. We plan to study these questions in our forthcoming publication.
2. The case
2.1. Proof of Theorem A
We will be dealing with Riemann-Liouville fractional integrals
Changing variables, we transform the basic integral equation
(cf. (1.5)) to the form
Suppose that is defined by (1.8). If , then, by homogeneity,
being the constant from (2.2). Hence, for , we necessarily have
If , this equation has no solution , because, otherwise, we get
This proves the second statement in Theorem A.
Consider the case . If , then . If , then, setting , from (2.6) we get
or (set )
Thus, if , then, necessarily,
One can readily see that function (2.7) is locally integrable on . Let us prove that it satisfies for all . It suffices to show that when . We have , where
Both integrals can be expressed in terms of hypergeometric functions. For , owing to 3.197 (3) and 9.131 (1) from , we obtain
For , changing the order of integration and using [6, 3.238 (3)], we have
To complete the proof, we recall that , which gives
This coincides with (1.11).
Let us give some examples of functions defined by (1.11) in the case . By [6, 3.197(3)],
Keeping in mind that , and using formulas 156, 203, and 211 from [15, 7.3.2], we obtain:
3. The general case
As in Section 2.1, our main concern is integral equation (2.2), which is equivalent to
We want to find relatively simple functions and , which are admissible in the basic Theorem 1.2 and such that the corresponding functions and obey (3.1). It is convenient to consider the cases of even and odd separately.
3.1. The case of even
Let . We choose
The corresponding function obviously has a decreasing radial majorant in . By (3.2), equation (3.1) is equivalent to
The th derivative
is integrable on . The fact that (3.5) satisfies (3.4) can be easily checked using integration by parts. Thus, the pair of functions and , defined by
and (3.3), falls into the scope of Theorem 1.2 and the “even part” of Theorem B is proved.
3.2. The case of odd
Let . We define by (3.3), as above. Then, instead of (3.4), we have
This gives , where
(use [6, 3.238(3)]). Let us show that satisfies (3.7). Integrating by parts, we have , where
This gives where (use [6, 3.238(3)] again)
as desired. Thus, functions and , defined by
and (3.3), obey Theorem 1.2. This completes the proof of Theorem B.
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Symmetries for Galileons and DBI scalars on curved space
We introduce a general class of four-dimensional effective field theories which include curved space Galileons and DBI theories possessing nonlinear shift-like symmetries. These effective theories arise from purely gravitational actions for 3-branes probing higher dimensional spaces. In the simplest case of a Minkowski brane embedded in a higher dimensional Minkowski background, the resulting four-dimensional effective field theory is the Galileon one, with its associated Galilean symmetry and second order equations. However, much more general structures are possible. We construct the general theory and explicitly derive the examples obtained from embedding maximally symmetric branes in maximally symmetric ambient spaces. Among these are Galileons and DBI theories with second order equations that live on de Sitter or anti-de Sitter space, and yet retain the same number of symmetries as their flat space counterparts, symmetries which are highly non-trivial from the d point of view. These theories have a rich structure, containing potentials for the scalar fields, with masses protected by the symmetries. These models may prove relevant to the cosmology of both the early and late universe.
- I Introduction and Summary
- II General brane actions and symmetries
- III Actions with second order equations of motion
IV Maximally Symmetric Examples
- IV.1 A Minkowski brane in a Minkowski bulk: in – DBI Galileons
- IV.2 A Minkowski brane in an anti-de Sitter bulk: in – Conformal Galileons
- IV.3 A de Sitter brane in a Minkowski bulk: in
- IV.4 A de Sitter brane in a de Sitter bulk: in
- IV.5 A de Sitter brane in an anti-de Sitter bulk: in
- IV.6 An anti-de Sitter brane in an anti-de Sitter bulk: in
- V Small field limits: the analogues of Galileons
- VI Conclusions
- A Some useful expressions
I Introduction and Summary
The possibility that the universe may contain large, and possibly infinite, spatial dimensions beyond the three we commonly perceive has opened up entirely new avenues to address fundamental questions posed by particle physics and by cosmology. The precise manner in which the dynamics of the higher-dimensional space manifests itself in the four dimensional world depends on the geometry and topology of the extra-dimensional manifold, and the matter content and action chosen. At low enough energies, the relevant physics is then captured by a four-dimensional effective field theory with properties inherited from the specific higher-dimensional model under consideration. The simplest example of this is the Kaluza-Klein tower – the hierarchy of higher mass states that accompany zero mass particles when compactifying a five-dimensional theory on a circle. There are, however, much more exotic possibilities. Many of these describe viable higher-dimensional theories, while others are merely mathematical tools with which to construct interesting physical four-dimensional effective field theories.
A particularly interesting and well studied example of a higher-dimensional model is the Dvali-Gabadadze-Poratti (DGP) model Dvali:2000hr , for which the ambient space is a flat -dimensional spacetime in which a Minkowski -brane floats, subject to an action consisting merely of two separate Einstein Hilbert terms – one in d, and the other only on the brane, constructed from the induced metric there. In an appropriate limit, the resulting four-dimensional effective field theory describes gravity plus a scalar degree of freedom parametrizing the bending of the brane in the extra dimension Luty:2003vm ; Nicolis:2004qq . The specific form of the four dimensional action for the scalar inherits a symmetry from a combination of five dimensional Poincaré invariance and brane reparametrization invariance. In the small field limit this symmetry takes a rather simple form and has been called the Galilean symmetry, with the associated scalar becoming the Galileon Nicolis:2008in .
Abstracting from DGP, a four dimensional field theory with this Galilean symmetry is interesting in its own right. It turns out that there are a finite number of terms, the Galileon terms, that have fewer numbers of derivatives per field than the infinity of competing terms with the same symmetries. These terms have the surprising property that, despite the presence of higher derivatives in the actions, the equations of motion are second order, so that no extra degrees of freedom are propagated around any background. Much has been revealed about the Galileon terms, including such useful properties as a non-renormalization theorem Luty:2003vm ; Hinterbichler:2010xn ; Burrage:2010cu , and applications in cosmology Agarwal:2011mg ; Burrage:2010cu ; Creminelli:2010ba ; Creminelli:2010qf ; DeFelice:2010as ; Deffayet:2010qz ; Kobayashi:2011pc ; Mota:2010bs ; Wyman:2011mp . The Galileons have been covariantized Deffayet:2009mn ; Deffayet:2009wt ; Deffayet:2011gz , extended to p-forms Deffayet:2010zh , and supersymmetrized Khoury:2011da . Further, it was recently shown that the general structure of Galileon field theories can be extended to multiple fields, finding their origins in braneworld constructions with more than one codimension Hinterbichler:2010xn ; Padilla:2010de ; Padilla:2010ir ; Padilla:2010tj ; Zhou:2010di . If some of the resulting symmetries of the four dimensional effective field theory are broken, then they are related to low energy descriptions of cascading gravity models in which a sequence of higher dimensional branes are embedded within one another deRham:2007rw ; deRham:2007xp ; Agarwal:2011mg ; Agarwal:2009gy .
If our universe really is a brane world, then theories of this sort are generic, since they share, in a certain limit, the symmetries of the Dirac-Born-Infeld (DBI) action. The DBI action encodes the lowest order dynamics of a brane embedded in higher dimensions, and provides an important arena within which to study inflation Silverstein:2003hf ; Alishahiha:2004eh , late-time cosmic acceleration Ahn:2009xd , tunneling Brown:2007zzh , and exotic topological defects Andrews:2010eh ; Babichev:2006cy ; Sarangi:2007mj ; Bazeia:2007df ; Babichev:2008qv . The Galileon terms can be thought of as a subset of the higher order terms expected to be present in any effective field theory of the brane, and which will be suppressed by powers of some cutoff scale. The Galileons are a special subset in the class of all possible higher order terms because they contain fewer derivatives per field than competing terms with the same symmetries, and because they yield second order equations. Crucially, there can exist regimes in which only a finite number of Galileon terms are important, and the infinity of other possible terms within the effective field theory are not (see section II of Hinterbichler:2010xn , as well as Nicolis:2004qq ; Endlich:2010zj , for more on this and for examples of such regimes.) This fact, coupled with a non-renormalization theorem for Galileons and the fact that there are a finite number of such terms, holds out the hope of computing non-linear facts about the world which are exact quantum mechanically. Finally, it should be remembered that even if our universe is not a brane world, the same conclusions follow if one postulates the existence of symmetries of the same form as those of a brane world.
In this paper, we construct a general class of four-dimensional effective field theories by writing an action on a 3-brane probing a higher dimensional bulk, of which the Galileon theory and DBI scalars are special cases. This extends the construction of deRham:2010eu to its most general form. We observe that the symmetries inherited by scalar fields in the d theory are determined by isometries of the bulk metric, and are present if and only if the bulk has isometries. The precise manner in which the symmetries are realized is determined by the choice of gauge, or foliation, against which brane fluctuations are measured. We derive in general the symmetries of these effective field theories, and classify the examples that result when embedding a maximally symmetric brane in a maximally symmetric background. This approach yields a set of new Galileon-like theories which live on d curved space but retain the same number of non-linear shift-like symmetries as the flat-space Galileons or DBI theories.
These theories have their own unique properties. For example, in curved space the field acquires a potential which is fixed by the symmetries – something that is not allowed for the flat space Galileons. In particular, the scalars acquire a mass of order the inverse radius of the background, and the value of the mass is fixed by the nonlinear symmetries. Although not addressed in detail here, allowing for de Sitter solutions on the brane opens up the possibility of adapting these new effective field theories to cosmological applications such as inflation or late time cosmic acceleration in such a way that their symmetries ensure technical naturalness.
The paper is structured as follows. In the next section we discuss general brane actions and symmetries, and the ways in which these symmetries may be inherited by a four-dimensional effective field theory. In section III we then consider constructing actions with second order equations and explicitly derive all possible terms in such theories. We then provide six separate examples, exhausting all the maximally symmetric possibilities: a d Minkowski brane embedded in a Minkowski bulk; a d Minkowski brane embedded in ; a d de Sitter brane embedded in a Minkowski bulk; a d de Sitter brane embedded in ; a d de Sitter brane embedded in ; and a d Anti-de Sitter brane embedded in . In each case, we describe the resulting d effective field theories and comment on their structure. In section V we take the small field limits to obtain Galileon-like theories, discuss their stability, and compare and contrast these theories with the special case of the original Galileon, before concluding.
Conventions and notation:
We use the mostly plus metric signature convention. The 3-brane worldvolume coordinates are , , bulk coordinates are , . Occasionally we use 6-dimensional cartesian coordinates , , for constructing five dimensional and as embeddings. Tensors are symmetrized and anti-symmetrized with unit weight, i.e , .
When writing actions for a scalar field in curved space with metric and covariant derivative , we use the notation for the matrix of second derivatives . For traces of powers of we write , e.g. , , where all indices are raised with respect to . We also define the contractions of powers of with using the notation , e.g. , , where again all indices are raised with respect to .
Ii General brane actions and symmetries
We begin with a completely general case - the theory of a dynamical 3-brane moving in a fixed but arbitrary (4+1)-dimensional background. The dynamical variables are the brane embedding , five functions of the world-volume coordinates .
The bulk has a fixed background metric . From this and the , we may construct the induced metric and the extrinsic curvature , via
Here are the tangent vectors to the brane, and is the normal vector, defined uniquely (up to a sign) by the properties that it is orthogonal to the tangent vectors , and normalized to unity . (Note that the extrinsic curvature can be written , demonstrating that it depends only on quantities defined directly on the brane and their tangential derivatives.)
We require the world-volume action to be gauge invariant under reparametrizations of the brane,
where is the gauge parameter. This requires that the action be written as a diffeomorphism scalar, , of and as well as the covariant derivative and curvature constructed from ,
This action will have global symmetries only if the bulk metric has Killing symmetries. If the bulk metric has a Killing vector , i.e. a vector satisfying the Killing equation
then the action will have the following global symmetry under which the shift,
We are interested in creating non-gauge theories with global symmetries from the transverse fluctuations of the brane, so we now fix all the gauge symmetry of the action. We accomplish this by first choosing a foliation of the bulk by time-like slices. We then choose bulk coordinates such that the foliation is given by the surfaces . The remaining coordinates can be chosen arbitrarily and parametrize the leaves of the foliation. The gauge we choose is
In this gauge, the world-volume coordinates of the brane are fixed to the bulk coordinates of the foliation. We call the remaining unfixed coordinate , which measures the transverse position of the brane relative to the foliation (see Figure 1). This completely fixes the gauge freedom. The resulting gauge fixed action is then an action solely for ,
Global symmetries are physical symmetries that cannot be altered by the unphysical act of gauge fixing. Thus, if the original action (4) possesses a global symmetry (6), generated by a Killing vector , then the gauge fixed action (8) must also have this symmetry. However, the form of the symmetry will be different because the gauge choice will not generally be preserved by the global symmetry. The change induced by is
To re-fix the gauge to (7), it is necessary to simultaneously perform a compensating gauge transformation with gauge parameter
The combined symmetry acting on ,
is then a symmetry of the gauge fixed action (8).
ii.1 A special case
We now specialize to a case which includes all the maximally symmetric examples of interest to us in this paper. This is the case where the foliation is Gaussian normal with respect to the metric , and the extrinsic curvature on each of the leaves of the foliation is proportional to the induced metric. With these restrictions, the metric takes the form
where denotes the Gaussian normal transverse coordinate, and is an arbitrary brane metric. Recall that in the physical gauge (7), the transverse coordinate of the brane is set equal to the scalar field, .
Working in the gauge (7), the induced metric is
Defining the quantity
the square root of the determinant and the inverse metric may then be expressed as
The tangent vectors are
To find the normal vector we solve the two equations
Using the non-vanishing Christoffel symbols , , , the extrinsic curvature is then
Note that when the d coordinates have dimensions of length, has mass dimension and is dimensionless.
The algebra of Killing vectors of contains a natural subalgebra consisting of the Killing vectors for which . This is the subalgebra of Killing vectors that are parallel to the foliation of constant surfaces, and it generates the subgroup of isometries which preserve the foliation. We choose a basis of this subalgebra and index the basis elements by ,
where we have written for the components, indicating that these components are independent of . To see that this is the case, note that, for those vectors with , the Killing equations (5) tell us that is independent of . Furthermore, the Killing equations tell us that is a Killing vector of .
We now extend our basis of this subalgebra to a basis of the algebra of all Killing vectors by appending a suitably chosen set of linearly independent Killing vectors with non-vanishing . We index these with , so that is a basis of the full algebra of Killing vectors. From the component of Killing’s equation, we see that must be independent of , so we may write .
A general global symmetry transformation thus reads
From this, we see that the symmetries are linearly realized, whereas the are realized nonlinearly. Thus, the algebra of all Killing vectors is spontaneously broken to the subalgebra of Killing vectors preserving the foliation.
ii.2 Maximally symmetric cases
In this paper, we will focus on the case in which the 5d background metric has 15 global symmetries, the maximal number. Thus, the bulk is either d anti-de Sitter space with isometry algebra , 5d de-Sitter space with isometry algebra , or flat 5d Minkowski space with isometry algebra the five dimensional Poincare algebra . In addition, we focus on the case where the brane metric , and hence the extrinsic curvature, are maximally symmetric, so that the unbroken subalgebra has the maximal number of generators, 10. This means that the leaves of the foliation are either d anti-de Sitter space with isometry algebra , 4d de-Sitter space with isometry algebra , or flat 4d Minkowski space with isometry algebra the four dimensional Poincare algebra . In fact, there are only 6 such possible foliations of d maximally symmetric spaces by d maximally symmetric time-like slices, such that the metric takes the form (12). Flat can be foliated by flat slices or by slices; can be foliated by flat slices, slices, or slices; and can only be foliated by slices. Each of these 6 foliations, through the construction leading to (8), will generate a class of theories living on an , or background and having 15 global symmetries broken to the 10 isometries of the brane. These possibilities are summarized in Figure 2.
It should be noted that the missing squares in Figure 2 may be filled in if we are willing to consider a bulk which has more than one time direction111We thank Sergei Dubovsky for pointing this out.. For example, it is possible to embed into a five-dimensional Minkowski space with two times (indeed, this is the standard way of constructing spaces). From the point of view that the bulk is physical, and hence should be thought of as dynamical, these possibilities may be unacceptable on physical grounds. However, if one thinks of the bulk as merely a mathematical device for constructing novel four-dimensional effective theories, then there is nothing a priori to rule out these possibilities. In this paper, we focus on those cases in which the bulk has only one time dimension. The construction in the other cases will, however, follow the same pattern.
Finally, note that the only invariant data that go into constructing a brane theory are the background metric and the action. Theories with the same background metric and the same action are isomorphic, regardless of the choice of foliation (which is merely a choice of gauge). For example, given the same action among the theories listed in Figure 2, the three that have an background, namely the conformal DBI Galileons, the DBI Galileons, and the type III DBI Galileons, are really the same theory. They are related by choosing a different foliation (gauge), shuffling the background configuration into the background metric.
Iii Actions with second order equations of motion
Up until now we have discussed the degrees of freedom and their symmetries, but it is the choice of action that defines the dynamics. A general choice for the function in (8) will lead to scalar field equations for which are higher than second order in derivatives. When this is the case, the scalar will generally propagate extra degrees of freedom which are ghost-like Ostrogradski ; deUrries:1998bi . The presence of such ghosts signifies that either the theory is unstable, or the cutoff must be lowered so as to exclude the ghosts. Neither of these options is particularly attractive, and so it is desirable to avoid ghosts altogether. It is the Galileon terms which are special because they lead to equations of at most second order. Furthermore, as mentioned in the introduction, there can exist regimes in which the Galileon terms dominate over all others, so we will be interested only in these terms.
A key insight of de Rham and Tolley deRham:2010eu is that there are a finite number of actions of the type (8), the Lovelock terms and their boundary terms, that do in fact lead to second order equations for and become the Galileon terms. The possible extensions of Einstein gravity which remain second order are given by Lovelock terms Lovelock:1971yv . These terms are specific combinations of powers of the Riemann tensor which are topological (i.e. total derivatives) in some specific home dimension, but in lower dimensions have the property that equations of motions derived from them are second order. (For a short summary of some properties of these terms, see Appendix B of Hinterbichler:2010xn .) The Lovelock terms come with boundary terms. It is well known that, when a brane is present, bulk gravity described by the Einstein-Hilbert Lagrangian should be supplemented by the Gibbons-Hawking-York boundary term Gibbons:1976ue ; York:1972sj
Similarly, Lovelock gravity in the bulk must be supplemented by brane terms which depend on the intrinsic and extrinsic curvature of the brane (the so-called Myers terms Myers:1987yn ; Miskovic:2007mg ), which are needed in order to make the variational problem for the brane/bulk system well posed Dyer:2008hb . Of course we are not considering bulk gravity to be dynamical, but the point here is that these boundary terms also yield second order equations of motion for in the construction leading to (8).
The prescription of deRham:2010eu is then as follows: on the 4-dimensional brane, we may add the first two Lovelock terms, namely the cosmological constant term and the Einstein-Hilbert term . (The higher Lovelock terms are total derivatives in 4-dimensions.) We may also add the boundary term corresponding to a bulk Einstein-Hilbert term, , and the boundary term corresponding to the Gauss-Bonnet Lovelock invariant in the bulk. The zero order cosmological constant Lovelock term in the bulk has no boundary term (although as we will see, we may construct a fifth term, the tadpole term, from it) and the higher order bulk Lovelock terms vanish identically. Therefore, in total, for a 3-brane there are four possible terms (five including the tadpole) which lead to second order equations. These are the terms we focus on.
iii.1 The tadpole term
As mentioned, there is one term that contains no derivatives of and is not of the form (8). This Lagrangian is called the tadpole term, denoted by . The value of the tadpole action is the proper 5-volume between some surface and the position of the brane,
Note that .
Under a general nonlinear symmetry of the type (24), its change is
Using the Killing equation (5), it is straightforward to check directly that a general variation of the right-hand side vanishes, demonstrating that the change in the tadpole term under the symmetry transformation is a total derivative. Thus the tadpole term has the same symmetries as the other terms.
iii.2 Explicit expressions for the terms
Including the tadpole term there are thus five terms that lead to second order equations for ,
where the explicit form of the Gauss-Bonnet boundary term is
Indices are raised and traces are taken with . At this stage, each of these terms would appear in a general Lagrangian with an arbitrary coefficient. As we will see later, requiring stability will, however, force certain choices on us in specific examples.
En route to presenting specific examples of our new theories, we now evaluate these terms on the special case metric (12). We make use of formulae catalogued in Appendix A. Our strategy is to collect coefficients of , , and , eliminate everywhere in favor of , and then to group like terms by powers of . A lengthy calculation yields
The quantities and are various contractions of derivatives of the field, and the notation is explained in the conventions at the end of Section I. In these expressions, all curvatures are those of the metric , and all derivatives are covariant derivatives with respect to . We point out that no integrations by parts have been performed in obtaining these expressions.
The equations of motion derived from any of these five terms will contain no more than two derivatives on each field, ensuring that no extra degrees of freedom propagate around any background. After suitable integrations by parts, these actions should therefore conform to the general structure presented in Deffayet:2011gz for actions of a single scalar with second order equations (see also the Euler hierarchy constructions Fairlie:1991qe ; Fairlie:1992nb ; Fairlie:1992yy ; Fairlie:2011md ). In the above construction, however, we can immediately identify the nonlinear symmetries by reading them off from the isometries of the bulk.
Finally, we note that by keeping the metric in (12) arbitrary rather than fixing it to the foliation, we can automatically obtain the covariantizaton of these various Galileon actions, including the non-minimal curvature terms required to keep the equations of motion second order, the same terms obtained by purely 4-d methods in Deffayet:2009mn ; Deffayet:2009wt ; Deffayet:2011gz . Of course, this in general ruins the symmetries we are interested in considering. But from this point of view, we can see exactly when such symmetries will be present. The symmetries will only be present if the which is used to covariantly couple is such that the full metric (12) has isometries.
Iv Maximally Symmetric Examples
We now proceed to construct explicitly the maximally symmetric examples catalogued in Section II.2 and Figure 2. The construction starts by finding coordinates which are adapted to the desired foliation, so that the metric in the bulk takes the form (12), allowing us to read off the function . Plugging into (LABEL:generalterms) then gives us the explicit Lagrangians. To find the form of the global symmetries, we must write the explicit Killing vectors in the bulk, and identify those which are parallel and not parallel to the foliation. We may then read off the symmetries from (24).
The construction for each case is similar, and some of the results are related by analytic continuation, but there are enough differences in the forms of the embeddings and the Killing vectors that we thought it worthwhile to display each case explicitly. The reader interested only in a given case may skip directly to it.
iv.1 A Minkowski brane in a Minkowski bulk: in – DBI Galileons
Choosing cartesian coordinates on , the foliation of by is simply given by slices, and the metric takes the form
Comparing this to (12), we obtain
and the terms (LABEL:generalterms) become (again, without integration by parts)
iv.1.1 Killing vectors and symmetries
The Killing vectors of d Minkowski space are the 10 boosts , and the 5 translations . The 6 boosts and the 4 translations are parallel to the foliation and form the unbroken symmetries of . The 5 broken generators are
Using the relation from (24), we obtain the transformation rules
under which the terms (LABEL:DBIGalileonterms) are each invariant up to a total derivative. The symmetry breaking pattern is
iv.2 A Minkowski brane in an anti-de Sitter bulk: in – Conformal Galileons
In this section, indices run over six values and are cartesian coordinates in an ambient d two-time Minkowski space with metric , which we call .
Five dimensional anti-de Sitter space (more precisely, a quotient thereof) can be described as the subset of points in the hyperbola of one sheet satisfying
with the radius of curvature of , and where the metric is induced from the flat metric on . This space is not simply connected, but its universal cover is . The scalar curvature and cosmological constant are given by
We use Poincare coordinates on which cover the region ,
where , and is the Minkowski 4-metric. The coordinates and all take the range . Lines of constant foliate the Poincare patch of with Minkowski time-like slices, given by intersecting the planes with the hyperbola.
The induced metric is
Comparing this with (12) we obtain
and the terms (LABEL:generalterms) become (without integration by parts)
These are the conformal DBI Galileons, first written down in deRham:2010eu .
iv.2.1 Killing vectors and symmetries
The 15 Lorentz generators of ; (here are the coordinate basis vectors in the ambient space , and indices are lowered with the flat metric ) are all tangent to the hyperboloid, and become the 15 isometries of the isometry algebra of . Of these, 10 have no components and are parallel to the foliation. These form the unbroken isometry algebra of the slices.
First we have
which taken together are the 6 Lorentz transformations of the .
For the remaining 4, we focus on
which may be grouped as
If we now take the following linear combinations, |
Source: Deep Learning on Medium
Talented Mr. 1X1: Comprehensive look at 1X1 Convolution in Deep Learning
With startling success of AlexNet in 2012, the Convolutional Neural Net (CNN) revolution has begun! The CNN based frameworks in Deep Learning like GoogleNet, ResNet and several variations of these have shown spectacular results in the object detection and semantic segmentation in computer vision.
When you start to look at most of the successful modern CNN architectures, like GoogleNet, ResNet and SqueezeNet you will come across 1X1 Convolution layer playing a major role. At first glance, it seems to be pointless to employ a single digit to convolve with the input image (After all wider filters like 3X3, 5X5 can work on a patch of image as opposed to a single pixel in this case). However, 1X1 convolution has proven to be extremely useful tool and employed correctly, will be instrumental in creating wonderfully deep architectures.
In this article we will have a detailed look at 1X1 Convolutions
First a quick recap of Convolutions in Deep Learning. There are many good blogs and articles that intuitively explain what convolutions are and different types of convolutions (few of them are listed in the reference). While we will not delve deep into the convolutions in this article, understanding couple of key points will make it easier to get what 1X1 convolution is doing and most importantly How & Why it is doing it.
Quick Recap: Convolution in Deep Learning
As mentioned, this article will not provide a complete treatment of theory and practice of Convolution. However, we will recap key principles of Convolution in deep learning. This will come in handy when we examine 1X1 Convolution in depth.
Simply put, Convolutions is an element wise multiplication and summation of the input and kernel/filter elements. Now the data points to remember
1. Input matrix can and, in most cases, will have more than one channel. This is sometimes referred to as depth
a. Example: 64X64 pixel RGB input from an image will have 3 channels so the input is 64X64X3
2. The filter has the same depth as input except in some special cases (example 3D Convolutions to reconstruct medical images). This specific point, for some unknown reason, is not explicitly mentioned in most of the literature, causing some misunderstanding (Especially for someone new to convolutions, Deep learning etc)
a. Example: filter of 3X3 will have 3 channels as well, hence the filter should be represented as 3X3X3
3. Third and critical point, the output of Convolution step will have the depth equal to number of filters we choose.
a. Example: Output of Convolution step of the 3D input (64X64X3) and the filter we chose (3X3X3) will have the depth of 1 (Because we have only one filter)
The Convolution step on the 3D input 64X64X3 with filter size of 3X3X3 will have the filter ‘sliding’ along the width and height of the input.
So, when we convolve the 3D filter with the 3D image, the operation moves the filter on the input in 2 directions (Along the width and height) and we do the element wise multiplication and addition at each position to end up with an output with a depth of 1.
Armed with this, we are ready to dive into the 1X1 convolution
1X1 Convolution — What is it?
Introduced first in a paper by Min Lin et all in their Network In Network, the 1X1 Convolution layer was used for ‘Cross Channel Down sampling’ or Cross Channel Pooling. In other words, 1X1 Conv was used to reduce the number of channels while introducing non-linearity.
In 1X1 Convolution simply means the filter is of size 1X1 (Yes — that means a single number as opposed to matrix like, say 3X3 filter). This 1X1 filter will convolve over the ENTIRE input image pixel by pixel.
Staying with our example input of 64X64X3, if we choose a 1X1 filter (which would be 1X1X3), then the output will have the same Height and Weight as input but only one channel — 64X64X1
Now consider inputs with large number of channels — 192 for example. If we want to reduce the depth and but keep the Height X Width of the feature maps (Receptive field) the same, then we can choose 1X1 filters (remember Number of filters = Output Channels) to achieve this effect. This effect of cross channel down-sampling is called ‘Dimensionality reduction’.
Now why would we want to something like that? For that we delve into usage of 1X1 Convolution
Usage 1: Dimensionality Reduction/Augmentation
Winner of ILSVRC (ImageNet Large Scale Visual Recognition Competition) 2014, GoogleNet, used 1X1 convolution layer for dimension reduction “to compute reductions before the expensive 3×3 and 5×5 convolutions”
Let us look at an example to understand how reducing dimension will reduce computational load. Suppose we need to convolve 28 X 28 X 192 input feature maps with 5 X 5 X 32 filters. This will result in 120.422 Million operations
Let us do some math with the same input feature maps but with 1X1 Conv layer before the 5 X 5 conv layer
By adding 1X1 Conv layer before the 5X5 Conv, while keeping the height and width of the feature map, we have reduced the number of operations by a factor of 10. This will reduce the computational needs and in turn will end up being more efficient.
GoogleNet paper describes the module as “Inception Module” (Get it — DiCaprio’s “We need to go DEEPER” in the movie Inception)
Usage 2: Building DEEPER Network (“Bottle-Neck” Layer)
2015 ILSVRC Classification winner, ResNet, had least error rate and swept aside the competition by using very deep network using ‘Residual connections’ and ‘Bottle-neck Layer’.
In their paper, He et all explains (page 6) how a bottle neck layer designed using a sequence of 3 convolutional layers with filters the size of 1X1, 3X3, followed by 1X1 respectively to reduce and restore dimension. The down-sampling of the input happens in 1X1 layer thus funneling a smaller feature vectors (reduced number of parameters) for the 3X3 conv to work on. Immediately after that 1X1 layer restores the dimensions to match input dimension so identity shortcuts can be directly used. For details on identity shortcuts and skip connection, please see some of the Reviews on ResNet (Or you can wait for my future work!)
Usage 3: Smaller yet Accurate Model (“FIRE-MODULE” Layer)
While Deep CNN Models have great accuracy, they have staggering number of parameters to deal with which increases the training time and most importantly need enterprise level computing power. Iandola et all proposed a CNN Model called SqueezeNet that retains AlexNet level accuracy while 50X times smaller in terms of parameters.
Smaller models have number of advantages, especially on use-cases that require edge computing capabilities like autonomous driving. Iandola et all achieved this by stacking a bunch of “Fire Modules” which comprise of
1. Squeeze Layer which has only 1X1 Conv filters
2. This feeds an Expansion layer which has mix of 1X1 and 3X3 filters
3. The number of filters in Squeeze Layer are set to be less than number of 1X1 filters + Number of 3X3 in Expand Layer
By now it is obvious what the 1X1 Conv filters in Squeeze Layer do — they reduce the number of parameters by ‘down-sampling’ the input channels before they are fed into the Expand layer.
The Expansion Layer has mix of 1X1 and 3X3 filters. The 1X1 filters, as you know, performs cross channel pooling — Combines channels, but cannot detect spatial structures (by virtue of working on individual pixels as opposed to a patch of input like larger filters). The 3X3 Convolution detects spatial structures. By combining these 2 different sized filters, the model becomes more expressive while operating on lesser parameters. Appropriate use of padding makes the output of 1X1 and 3X3 convolutions the same size so these can be stacked.
In this article we reviewed high level Convolution mechanism and threw ourselves into the deep end with 1X1 Convolution to understand the underpinnings, where they are effectively used and to what end.
To recap, 1X1 Convolution is effectively used for
1. Dimensionality Reduction/Augmentation
2. Reduce computational load by reducing parameter map
3. Add additional non-linearity to the network
4. Create deeper network through “Bottle-Neck” layer
5. Create smaller CNN network which retains higher degree of accuracy
1. Andrew Ng’s Video on 1X1 Convolution
4. Network in Network — Min Lin et All
5. Going Deeper with Convolutions — Christian Szegedy et All
6. Deep Residual Learning for Image Recognition — Kaiming He et All
7. SqueezeNet — Forest Iandola et All
8. CNN Architecture — Lecture 9 (Stanford) : Fei-Fei Lin et All |
. Please provide any two values below and click the Calculate button to get the third value. In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol % or simply as percent or pct. For example, 35% is equivalent to the decimal 0.35, or the fraction Answer:75Step-by-step explanation:the three digit number is = 90×100/30=300so 25% of the number is= 300×25/100=7 30% of a number is 90, then what is 50% of the same number?a, 250b, 50c,350d, 150 Get the answers you need, now! siddiquaasma605 siddiquaasma605 05.04.2021 Math Secondary School answered 30% of a number is 90, then what is 50% of the same number? a, 250 b, 50 c,350 d, 150 30% Percent Calculator. Use this calculator to find percentages. Just type in any box and the result will be calculated automatically. Calculator 1: Calculate the percentage of a number. For example: 30% of 25 = 7.5. Calculator 2: Calculate a percentage based on 2 numbers. For example: 7.5/25 = 30% Finally, we have found the value of Y which is 27 and that is our answer. If you want to use a calculator to know what is 30 percent of 90, simply enter 30 ÷ 100 × 90 and you will get your answer which is 27 You may also be interested in: What percent of 90 is 2
What is 30 less than 90? We are looking for a new number which is 30 less than 90. We will get the new number by subtracting 30 from 90. We write it down as: 90-30=60. And finally the solution for: What number is 30 less than 90? is 60 There is a method, where you find a small percentage, then multiply. Below you can find 10%, then multiply by 7 to get 70% of the original number (100%-30%=70% of. Finally, we have found the value of Y which is 90 and that is our answer. You can easily calculate 27 is 30 percent of what number by using any regular calculator, simply enter 27 × 100 ÷ 30 and you will get your answer which is 90 You may also be interested in: What is 30 percent of 90 Type 3: The number 32 is 8% of what number? As usual, this problem requires to steps: Step 1: Write 8% as a decimal number: 8% = 0.08. Step 2: Divide 32 and 0.08: 32 ÷ 0.08 = 400. Type 4: percentage increase: What is the percentage increase from 20 to 90? In this type of problem we use formul
30 is 30% of 100. Steps to solve 30 is 30 percent of what number? We have, 30% × x = 30; or, 30 / 100 × x = 30 Multiplying both sides by 100 and dividing both sides by 30, we have x = 30 × 100 / 30 x = 100. If you are using a calculator, simply enter 30×100÷30, which will give you the answer $90 / 3 = $30 each: 20% of $90 is $18. Now find 20% of $30 and add that as the tip. The total bill amounts to: 20% of $30 is $6. $90 + $18 = $108: Now, add that $6 to $30. This amounts to $36. Now, split the total bill in thirds: $108 / 3 = $36: Total share: Each of you would pay $36. Total share: Each of you would pay $36 Answer provided by our tutors. let 'x' represent the original number, then: x * 0.3 = 12. x = 12 / 0.3 = 40. 12 is 30% of the number 40 Calculator 1: Calculate the percentage of a number. For example: 90% of 30 = 27. Calculator 2: Calculate a percentage based on 2 numbers. For example: 27/30 = 90%. How much is 90% of 30? What is 90% of 30 and other numbers? 90% of 30.00 = 27.0000. 90% of 30.25 = 27.2250. 90% of 30.50 = 27.4500
60. One of the ways to solve this is by proportion. A/B (A:B) = C/D (C:D) A - the number (18) B - the base number (unknown) C - Percentage (30%) D - Total Percentage. seventy-eight is 15% of what number so there's some unknown number out there and if we take 15% of that number we will get 78 so let's just call that unknown number X and we know that if we take 15% of X so if we take 15% of X so multiply X by 15% we will get we will get 78 and now we just literally have to solve for x now 15% mathematically you can deal directly with percentages but it's much. The number is 400 Let x be what number. We can then write this problem as: 30% of x = 120 or 30/100 x = 120 Solveing for x while keeping the equation balanced gives: 100/30 30/100 x = 120 100/30 x = 12000/30 x = 40 The strategy here is to see how many times the percent number (in this case, 25) goes into 100, and then count by that number until we reach 100-the whole thing. Here, we're told that 25% of a number is 5. So, to find 100% of the number, we count by 25s up to 100: 25, 50, 75, 100. 100% is 20 30 percent of 91 is the same as 30 per hundred of 91. We can therefore make the following equation: 30/100 = X/91 To solve the equation above for X, you first switch the sides to get the X on the left side, then you multiply each side by 91, and then finally divide the numerator by the denominator on the right side to get the answer
10/10 = 1. When we put that together, we can see that our complete answer is: 27. /. 1. The complete and simplified answer to the question what is 3/10 of 90 is: 27. Hopefully this tutorial has helped you to understand how to find the fraction of any whole number A% of B is C as in: 10% of 90 is 9 where A=10, B=90, C=9 The percentage formula is: A/100 x B = C as in: 10/100 x 90 = 9 Rearranging: A 100 = C B as in: 10 100 = 9 90 The percentage formula is sometimes expressed as 30% of 90=27 what is the base, rate the element and the number expressed in % and number found by multplying the base by the rate - 941602 marrienereyes marrienereyes 26.09.2017 Math Elementary School answered • expert verifie
As of 2019, startup failure rates are around 90%. 21.5% of startups fail in the first year, 30% in the second year, 50% in the fifth year, and 70% in their 10th year To find 30 percent of a number, multiply that number by 0.30. For mental arithmetic it is sometimes easier to divide the number by 100 (to calculate 1%) then multiplying by 30 (for 30%)
Systemic overfishing is only made worse by illegal catches and trade. In fact, some of the worst ocean impacts are caused by pervasive illegal fishing, which is estimated at up to 30% of catch or more for high-value species. Experts estimate illegal, unreported, and unregulated (IUU) fishing nets criminals up to $36.4 billion each year Although Bob's model assumed 90 women instead of 30 had cancer in total, it predicted Breast Cancer correctly 22.2% of the time as opposed to Hawkins Model with Precision of 0. Also, out of the 30 women that actually has Breast Cancer, Bob's Model was able to correctly recall that someone has Breast Cancer 67% of the time as opposed to. The markup is 30/100 = 30%. The MARGIN, however, is 30/130 = 23%. This is because selling the item for $130 results in a $30 profit, and 30/130 means that 23% of the money the store took in was profit. We say their margin was 23%. In fact, a 30% markup will always result in a 23% profit margin
If you want to calculate a percentage of a number in Excel, simply multiply the percentage value by the number that you want the percentage of. For example, if you want to calculate 20% of 500, multiply 20% by 500. I.e. type the following formula into any Excel cell: =20%*500. - which gives the result 100. Note that the % operator tells Excel. Colder air cannot handle as much moisture as warmer air. Temperature in relation to humidity is important, especially as we spend 90% of our time indoors. Consider for example a winters day. The outdoor air could have a 100% relative humidity at 41°F, and therefore contain 0.2 grams of water
• About 90% of children who are victims of sexual abuse know their abuser.12,13 Only 10% of sexually abused children are abused by a stranger.12 • Approximately 30% of children who are sexually abused are abused by family members.12,13 • The younger the victim, the more likely it is that the abuser is a family member. Of those molesting a. If the total is t, and the number items under consideration is n, then p= (n/t)x100. Using algebra to solve the formula for t, the number the question requires, gives the following expression: t= (n/p)x100. If 8 is equal to 80 percent of the total, inserting the numbers into the equation results in provides the t= (8/80)x100 so t=10
Use this calculator to find percentages. Just type in any box and the result will be calculated automatically. Calculator 1: Calculate the percentage of a number. For example: 30% of 300 = 90. Calculator 2: Calculate a percentage based on 2 numbers. For example: 90/300 = 30%. How much is 30% of 300 How to calculate 30% off 19 dollars or pounds. In calculating 30% of a number, sales tax, credit cards cash back bonus, interest, discounts, interest per annum, dollars, pounds, coupons,30% off, 30% of price or something, we use the formula above to find the answer. The equation for the calculation is very simple and direct If 10% of a number is 7, what is 80% of the number? Solution Note that 80% of something is 8 times 10% of the same thing. Hence if 10% of a number is 7 then 80% of the same number is given by 8 × 7 = 56 Which is the greatest? 90% of 10 6% of 1000 5% of 1400 3% of 250
32% of 100 is 32. 87.9% of 100 is 87.9. 416% of 100 is 416. For as we saw in Lesson 4, percent is an abbreviation for the Latin per centum, which means for each 100. (Per means for each.)A percent is a number of hundredths.. Example 1. A store paid $100 for a jacket. It then raised the selling price by 28% As of 1998, an estimated 17.7 million American women had been victims of attempted or completed rape. 5. Young women are especially at risk. 82% of all juvenile victims are female. 90% of adult rape victims are female. 6. Females ages 16-19 are 4 times more likely than the general population to be victims of rape, attempted rape, or sexual. 5 is 5% of 100. 12 is 12% of 100. 250 is 250% of 100. For, a percent is a number of hundredths.5 is 5 hundredths -- 5% -- of 100. That is the ratio of 5 to 100. For a percent expresses a ratio, a relationship, between two numbers FICO Scores are calculated using many different pieces of credit data in your credit report. This data is grouped into five categories: payment history (35%), amounts owed (30%), length of credit history (15%), new credit (10%) and credit mix (10%). Your FICO Scores consider both positive and negative information in your credit report 30% 0f 90% is what number? See tutors like this. Suppose you have a quantity x. 30% of 90% of x is (0.30)(0.90)x = 0.27x = 27% of x . Upvote.
You have a 30% off coupon. The cost of the item you want to buy is $249.99. How much money will you save by using the coupon? 30% of 249.99 = ? Entering these values into the percentage calculator will give you the answer of: 74.997 After rounding to two decimal places, you will save $75.00 Click to show this example in the calculator above Scientific Notation is simply a number format that includes a multiplication of 10 to the power of either a negative number, for small numbers, or to the power of a positive number, for larger numbers. This method reduces the amount of digits and especially zeros needed to write in representing a number Find percentage. Calculation of percentage is an interesting part in the world of mathematics and obvious in every math classes. The percentage converter helps you with percent increase, decrease, differences, calculation and to figure out percentage. Get the help you need on finding the percentage here 1% of 90 → 0.01 × 90. 60% of $700 → 0.6 × $700. This gives us another way to calculate the percentage of a number (or percentage of some quantity): To calculate a percentage of some number, change the percentage into a decimal, and the word of into multiplication. Example 1. Find 70% of 80 Enter the original price into our percent off calculator. For example, a TV set might originally set you back $5000. Determine the percentage discount - in our example store, everything is 75% off. The sum that stays in your pocket - your savings - is simply these two values multiplied by each other: 75% * $5000 = 0.75 * $5000 = $3750
I magine 100 people are ill with Covid-19. 90% efficacy means if only they'd had the vaccine, on average only 10 would have got ill. Vaccine efficacy is the relative reduction in the risk. By 2025, millennials will make up the majority of the workforce (75%). There are 56 million millennials in the US workforce. 21% of millennial workers have switched jobs in the last 12 months. 73% of millennials put in more than 40 hours of work per week. Millennial turnover costs the US economy $30.5 billion per year Recycling codes are used to identify the material from which an item is made, to facilitate easier recycling or other reprocessing. The presence on an item of a recycling code, a chasing arrows logo, or a resin code, is not an automatic indicator that a material is recyclable; it is an explanation of what the item is made of. Codes have been developed for batteries, biomatter/organic material. The numbers represent the percentages of importance that varying communication channels have. The belief is that 55% of communication is body language, 38% is the tone of voice, and 7% is the. *Assuming someone with poor credit (620-639) gets a 30-year fixed-rate loan at 4.03% APR compared to someone with excellent credit (760+) getting a 2.441% APR. Interest for the borrower with poor credit would total $217,478. Interest for the borrower with excellent credit would total $123,425
One hundred percent of a number is just the number itself. Two hundred percent of a number is twice that number. 100% of 50 -> 50 200% of 50 -> 2 x 50 = 100. Let's find 30 percent of 400: First change 30% to a decimal by moving the decimal point 2 places to the left. 30% = 0.30. Then multiply. 0.30 x 400 = 120. 30% of 400 is 120 Title: Percentage Worksheet Author: Maria Miller Subject: Percentage of number worksheet Keywords: Percentage, number, worksheet Created Date: 7/18/2021 4:05:23 A 30% of annual giving occurs in December. 10% of annual giving occurs on the last 3 days of the year. 77% believe everyone can make a difference by supporting causes. 4.5 is the average number of charities each person supports. 64% of donations are made by women. 69% of the population gives 74. 11 is 25% of what number? 75. 37 is 4% of what number? 76. 90 is 80% of what number? 77. 8 is 2% of what number? 78. On a 120-question test, a student got 84 correct answers. What percent of the problems did the student work correctly? 79. An engineering student answered 81 questions correctly on a 90-question test. Wha
According to data from the Bureau of Labor Statistics, as reported by Fundera, approximately 20 percent of small businesses fail within the first year. By the end of the second year, 30 percent of. What is eGFR? eGFR - Estimated glomerular filtration rate is the best test to measure your level of kidney function and determine your stage of kidney disease. Your doctor can calculate it from the results of your blood creatinine test, your age, body size and gender. Your GFR tells your doctor your stage of kidney disease and helps the doctor plan your treatment This does not mean: There's a 30% chance it will rain and a 70% chance it won't. Three out of 10 times when the weather is similar, it will rain. Precipitation will fall 30% of the day (or night) Thirty percent of the forecast area will experience rain, snow, or storms. Rather, the correct interpretation would be: there is a 30% chance that 0. 2) Solution: Box 1: Enter your answer as an equation. Example: y=3x^2+1, 2+x+y=3. Be sure your variables match those in the question. x 136 = 38 100 or 38 100 = x 136 x 136 = 38 100 or 38 100 = x 136. Box 2: Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4
Instagram content statistics. The Instagram algorithm has tripped up marketers in the past and that trend looks to continue in 2021. Recent Instagram statistics tell us that organic engagement has fallen from 2020, which was at 1.60%. For reference, research from RivalIQ puts the average engagement rate at 1.22% Based on data collected by CDC from states and territories for year 2017: Over 98% of U.S. newborns were screened for hearing loss. About 6,500 U.S. infants born in 2017 were identified early with a permanent hearing loss. The prevalence of hearing loss in 2017 was 1.7 per 1,000 babies screened for hearing loss Q5. At IIM Bangalore, 60% of the students are boys and the rest are girls. Further 15% of the boys and 7.5% of the girls are getting a fee waiver. If the number of those getting a fee waiver is 90, find the total number of students getting 50% concession if it is given that 50% of those not getting a fee waiver are eligible to get half fee.
Younger People Are at the Highest Risk of Sexual Violence. Ages 12-34 are the highest risk years for rape and sexual assault. 3 Those age 65 and older are 92% less likely than 12-24 year olds to be a victim of rape or sexual assault, and 83% less likely than 25-49 year olds. 4 Read more statistics about about child sexual abuse You can put this solution on YOUR website! When 80% of a number is added to the number, the result is 252. What is the number? 0.8x + x = 252 1.8x=252 x=140 Cheers, Stan H WATCH: Why elephant numbers have fallen so low. Between 2006-2015, around 111,000 African elephants were lost from the wild, mainly due to poaching, and between 2007-2014 30% of Africa's savannah. If your school does not list your percentile, it is easy to figure out. Divide your class rank by the number of students in your grade, multiply by 100, then subtract that number from 100. For example, if there are 600 students in your grade and you are ranked 120th, then you are in the 80th percentile because (120/600)*100=20, and 100-20=80
I have a column that lists bunch of numbers. How can I select the average of top 30% of the values in one column: 'Values' 10 9 8 7 6 5 4 3 2 1 so, the top 30% is '10, 9, 8' and the average is (10+9+8)/3 = If two numbers are respectively 30% and 40% more than a third number, what percent is the first of the second? A. 92 6 7 %. B Honestly, I'd have estimated this in the 20-30 percent range, so it surprised me to see that, from Jumpshot's data, all Google properties earned only 11.8% of clicks from distinct searches (only 8.4% across all searches). That's still significant, of course, and certainly bigger than it was 5 years ago, but given that we know Google's search. Assignment #2 Q1 (Weight 30%): Write a python program that: . Reads the number of rows and columns from the user. • Reads the different elements of the matrix from the user. • Calls a function to displays the matrix. • Calls a function to return the sum of all elements in the matrix. The quality of the code is important
A number decreased by 30% gives 84. The number is (a) 90 (b) 110 (c) 120 (d) 135. Answer: (c) 120 Let. 2.5 million, nearly 90%, are treated and released from an emergency department. TBI is a contributing factor to a third (30%) of all injury-related deaths in the United States. 1. Every day, 153 people in the United States die from injuries that include TBI. 1. Most TBIs that occur each year are mild, commonly called concussions. 2 A number multiplied by 6 and then reduced by 3 gives 69, the number is_____ 3). The number of boys is 3/2 the number . Algebra 1. Two fewer than a number doubled is the same as the number decreased by 38. Find the number. If n is the number, which equation could be used to solve for the number A growing number of consumers (37%, up from 30% in 2017), are willing to pay a fee for access to enhanced loyalty program benefits 95% of loyalty program members want to engage with brands through a mix of new, emerging, and growing tech, including augmented reality, virtual reality, card-on-file and more ( Bond 50 stats that show the importance of online reviews. 1. 92% of consumers now read online reviews vs. 88% in 2014 tweet. 2. 40% of consumers form an opinion by reading just one to three reviews vs. 29% in 2014 tweet. 3.Star rating is the number one factor used by consumers to judge a business tweet. 4. 44% say a review must be written within one month to be relevant.This highlights the.
The share of U.S. children living with an unmarried parent has more than doubled since 1968, jumping from 13% to 32% in 2017. That trend has been accompanied by a drop in the share of children living with two married parents, down from 85% in 1968 to 65%. Some 3% of children are not living with any parents, according to a new Pew Research. According to estimates from Scandinavian research centre Sintef, 90% of all the data the human race has ever produced has been generated in the past two years. That explosion is due to the rise of. Washington, DC, August 29, 2019 - A recent Ipsos poll reveals that more Americans have tattoos today than in early 2012.Three in ten (30%) of Americans have at least one tattoo, an increase from 21% in 2012. The vast majority of those with at least one tattoo (92%) say they are happy with it, and forty-six percent of respondents have had at least one tattoo for more than ten years The proportion of American adults with high-speed broadband service at home increased rapidly between 2000 and 2010. In recent years, however, broadband adoption growth has been much more sporadic. Today, roughly three-quarters of American adults have broadband internet service at home. Chart. Data |
In the beginning of 20th century Sommerfeld introduced closed integral solution for the problem of diffraction by a half-plane [Sommerfeld1954]. It was done in a very elegant way with the help of reflection method. Namely, he reduced the half-plane problem to the problem of plane wave propagation on two-sheeted surface. Then, using plane wave decomposition integral he solved the problem. This integral with particular contour of integration was named after Sommerfeld. Later Sommerfeld integral approach was applied to a number of problems such as problem of diffraction by a strip [Shanin2003a], by a wedge [Babich2008] and some others [Luneburg1997, Hannay2003].
Nowadays with the growth of computational power problems on discrete grids draw more attention. Recently, several discrete diffraction problems were solved rigorously using Wiener-Hopf approach [Sharma2015a, Sharma2015b, Sharma2015c]. In the current work we want to apply Sommerfeld integral approach to some of them. We show that in the discrete case Sommerfeld integral is essentially an integral on torus from algebraic function (elliptic integral) and we derive such integrals for the following problems:
The problem for Green’s function on a plane,
The problem of diffraction by a half-plane,
The problem of diffraction by a right-angled wedge.
2 Discrete Green’s function on a plane
2.1 Problem formulation
Consider the Green’s function for a simplest stencil discrete 2D Helmholtz equation. Namely, let function , , obey the equation
The wavenumber parameter is close to positive real, but has a small positive imaginary part mimicking attenuation in the medium. The radiation condition imposed on is that it should decay exponentially as .
2.2 Preliminary step. Reducing the number of computations of the integral
Our aim is to tabulate function for some set of values . It is clear that
Thus, one should tabulate only for non-negative .
Let it be necessary to tabulate all with
A naive approach requires computations of the integral. However, here we show that one can compute integrals. Namely, we will compute the integrals only for , and all other values find by using “cheap” recursive relations.
Compute the values of row by row. Each row is a set of values with , , , i. e. the rows are in fact diagonals.
Let all values with be already computed, and it is necessary to compute the values with . Find by integration. Then use (1) rewritten as a recursive relation:
Note that all values in the right have the sum of indices , thus they are computed on the previous steps. The left-hand side is a recursive relation for .
2.3 Double integral representation
The result is the following representation of the field :
Introduce the variables
Also introduce the function
The integral (5) can be rewritten as
where contour is the unit circle in the -plane passed in the positive direction anti-clockwise.
2.4 Single integral representation
The integral (9) can be taken with respect to one of the variables by the residue integration. There are four cases, possibly intersecting.
Consider the integral (9). Fix and study the integral with respect to . The -plane is shown in Fig. 1. One can see that there are four possible singular points in this plane. Two of them are the roots of the dispersion equation
considered with respect to . The roots are
Beside (11), there maybe singularities at two other points: and (note that is a certain point of ). However, the presence of singularities at these points depends on the value of . For example, if then the integrand is regular at .
Thus, the only singularity of the integrand inside the circle is . Apply the residue method. The result is
Thus, (13) can be rewritten as
The same analysis can be made for the singular points in the -plane for a fixed . This anayisis shows that there may exist a singularity at , but the behavior at is regular. This means that the integrand has no branching at , and the integrand decays not slower than . For such an integrand one can apply the residue theorem to the domain . The result is
This case can be considered similarly to . The representation for the field is
The field is
2.5 A recursive relation for
Let be and . Rewrite the representation (15) in the form
Using the proof of Legendre’s theorem for the Abelian integrals [Bateman1955], derive a recursive formula for . Introduce the constants as follows:
Using these constants one can write
Then note that
Substituting this identity into (20) and taking into account that contour of integration is closed, get
2.6 Field representation by integration on a manifold. Plane wave decomposition
Consider and being complex variables. Let be , , where is a compactified complex plane, that is a Riemann sphere.
Each point thus belongs to . Let us describe the set of points such that equation (10) is valid. Obviously, this is an analytic manifold of complex dimension 1 or of real dimension 2. This manifold will be referred to as .
Consider defined by (12). Now consider it as a double-valued function, thanks to the presence of the square root in it. Let us study the Riemann surface of this function. Topologically, there is no difference between and the Riemann surface of .
Function has four branch points. They are the points where the argument of the square root in (12) is equal to zero, i. e.
The values , , , possess the following property that can be checked directly. For
Exactly two of these branch points are located inside the circle . One can check that the branch points are the points at which .
The scheme of the Riemann surface is shown in Fig. 2. The branch points are connected by cuts shown by bold curves. For definiteness, the the branch cuts are conducted along the lines at which
The sides of the cuts labeled by equal Roman number should be connected with each other.
Topologically, is a torus (i. e. it has genus equal to 1). This can be eacily understood, since is obtained by taking two spheres, making two cuts, and connecting their shores.
One of the sheets drawn in Fig. 2 is called physical, and the other is unphysical (the naming is meaningless) . The physical sheet is the one on which for . Respectively, on the unphysical sheet for . Note that and cannot be equal to 1 on , since is not real.
We find useful to mark four “infinity points” belonging to :
Note that belongs to . Points Inf 1 and Inf 4 belong to the physical sheet, while points Inf 2 and Inf 3 belong to the unphysical sheet.
The notations of infinity points and branch points on are shown in Fig. 3.
The statement that is an analytic manifold means that in each (small enough) proximity of any point of one can introduce a complex local variable , such that all transhormation matrices between the neighboring local variables are biholomorphic. It is clear that such local variables can be: for all points except four branch points and two infinities Inf 3 and Inf 4; for the branch points; for the infinities Inf 3 and Inf 4.
An analytic 1-form can be defined in the manifold [Gurvitz1968] by introducing a formal expression , where is a local variable (discussed above) in some proximity, and is an analytic function in this proximity. In neighboring proximities the representations can be different (say, and ), but they should match in an obvious way:
The 1-form can be analytic/meromorphic if the functions are analytic/holomorphic. In the same sense the form can have zero or a pole of some order.
Analyticity of a 1-form is an important property since one
One can see that the form
is analytic everywhere on . Let us prove this. The statement is trivial everywhere except the branch points and the infinities. Consider the infinities. At the points Inf 1 and Inf 2 it is easy to show that as , and the denominator is non-zero. At the points Inf 3 and Inf 4 one can show that as , thus . A change to the variable shows that the form is regular.
Finally, consider the branch points (25)–(28). As it has been mentioned, one can take as a local variable at these points. An important observation is that due to the theorem about an implicit function,
everywhere on . Thus,
At the branch points the denominator of the right-hand side of (32) is not zero, so the form is regular.
The representation (15) can be considered as a contour integral of the form
along some contour drawn directly on . The contour is, indeed, shown in Fig. 2. This statement is quite trivial.
What is less trivial, is that three other representations, (16), (17), (18) can be represented as the contour integrals of the same form on , but taken along some other contours. Namely, the contours of integration for the representations (16), (17), (18), are shown in Fig. 4. They are denoted by , , , respectively. The contour for (15) is denoted by for uniformity.
Note that the form is, generally, not analytic on . Depending on and , it can have poles at the infinity points. The list of conditions of regularity for the infinity points is as follows:
The domains of regularities at infinities are shown in Fig. 5.
Note that the contours , , , can be deformed into each other. As we mentioned, is a torus. Topologically, the relative positions of the contours and the infinity points are shown in Fig. 6
One can see that carrying the contours in the direction labeled by the red arrows corresponds to moving the observation point in the -plane in the clockwork direction. The representations are converted into each other, and every time there is a region where at least two representations are valid simultaneously.
2.7 Sommerfeld integral for Green’s function problem
Sommerfeld integral for this problem is formally a plane wave integral (34) with contour of integration that does not cross the line of propagating waves (locus of points ). After a simple analysis of one can obtain the result shown in figure 7.
One can notice that this curve is topologically equivalent to the one of the canonical sections of . To prove it let us show that domains and are simply connected. For simplicity consider the case (obviously, the topology of domains should be the same for any ). In this case domain covers physical sheet and domain covers unphysical sheet. Thus, resulting domains are linearly connected, and any closed contour lying in any of domains can be collapsed through infinity point.
Contour for Sommerfeld integral consists of two closed non-trivial contours lying at different sides of curve of propagating waves.An example of such a contour is shown in figure 8.
In figure 9 we plot these contours on torus .
Finally, Sommerfeld integral takes form:
Obviously, contour is equivalent to the since there is no poles lying on the curve of propagating waves. Nevertheless, representation (36) seems to be more convenient when the problem for an incident plane wave is considered.
3 Diffraction by a Dirichlet half-plane
3.1 Problem formulation
Let the discrete Helmholtz equation
be satisfied everywhere except line . On this line the following boundary condition should be satisfied:
where is an incident plane wave:
Here an angle of incidence. In order to satisfy the discrete Helmholtz equation, incident wave should satisfy the dispersion equation:
where we introduced a notation:
Introduce total field as a sum of incident and scattered field:
Also the scattered field should satisfy the radiation condition.
3.2 Formulation on a branched surface
Consider a branched surface of continuous variables . For this, parametrize the points by the relations
Thus, the points become defined on a surface with two sheets reminding the Riemann surface of the function .
Define an integer lattice on the branched surface. There are two points having coordinates for any pair except . Denote these points by , where as an index labelling the sheet somehow (say, by separating the surface into sheets by making an appropriate cut). The pair will be called an affix of the point.
We assume that there is a wave field defined on the points of the branched surface.
Each point except has exactly four stencil neighbors having affixes , , , . We say that equation (37) is valid on the branched discrete plane at some point if it is valid for the value of at and at four its neighbours.
3.3 Sommerfeld integral for half-plane problem
First let us first construct Sommerefeld integral for a plane wave on a plane. We search for
where is some algebraic function on the torus . It is well known from the theory of elliptic functions [Bateman1955] that non-trivial function on the torus should have at least two poles of the first order or one pole of the second order. For definiteness, let us suppose that function has one simple pole corresponding to the incident wave , and the other simple pole corresponding to an arbitrary point . It can be checked directly that such function has the following form
Here constants and satisfy the following system of linear equations:
Let us choose constant in a way that in point the residue of integral (44) will be equal to . We obtain
Let us construct a plane wave solution on a branched surface. Following Sommerfeld ideas we need to be build a function that covers torus twice and has a unity pole corresponding to a plane wave. There are several obvious candidates that cover twice, such as
Then, multiplying with function (45) we obtain the function with desired properties. Thus on a branched surface Sommerfeld integral has form:
To choose which function should be used one need to check the validity of radiation conditions. It can be showed after simple computation that only the integral with
Finally, solution for the half-plane problem can be obtained using reflection principle:
3.4 Wiener-Hopf solution
Let us find the solution of the half-plane diffraction problem using the Wiener-Hopf approach. First, let us symmetrize the problem. Namely, represent the incident field (39) as a sum:
Then, study the equation (37) separately for the symmetrical and anti-symmetrical part of the field. One can check directly that anti-symmetrical problem is trivial, i.e.:
Thus the solution of symmetrical problem coincide with the solution of (37), i.e.
Without loss of generality we can suppose that .
Introduce direct and inverse bilateral -transform as follows:
where is a unit circle passing in a counterclockwise direction. To obtain functional equation let us apply -transform to boundary condition (38). We have
where is a unilateral -transform of :
and is some unknown function analytical inside the unit circle. Function is analytical outside the unit circle [Sharma2015b]. Equation (60) cannot have unique solution, since it also involve unknown function that is analytical in some ring. To introduce a second functional equation let us study a combination
One can check directly that
where were introduced in (29), and
The equation can be easily factorized. The solution is as follows:
The scattered field is given by the following integral
4 Diffracton by a right-angled wedge
4.1 Problem formulation
On the boundary of this domain the following conditions should be satisfied:
where is an incident plane wave (39). Also the scattered field should satisfy the radiation condition.
Using reflection method this problem can be reduced to the problem of wave propagation on three-sheeted surface [Sommerfeld1954]. It can be checked directly that total field on three sheeted surface is related to the total field of original problem by the following formula:
4.2 Sommerfeld integral on three-sheeted surface
We will search integral in the form (44) as in previous sections. Here we need to construct function covering torus three times and having a pole corresponding to the incident wave with zero residue. Unfortunately, there are no obvious candidates like it is for two-branched surface. Let us study from the topological point of view. The Riemann diagram of torus is shown in figure 11.
It can be noticed that function which covers three times should have Riemann diagram that is shown in figure 12.
It can be easily proved that function having Riemann surface has the following structure:
where , some rational functions. Thus, using (71) we can build Sommerfeld integral for three-sheeted surface with two unknown rational functions.
One can construct (71) by studying polynomial
Namely, suppose that the roots of this polynomial define function , and this function has Riemann surface . Thus, there are exactly four points in which has exactly three roots of order two, i.e. it can be represented as:
where are unknown parameters. These parameters should be determined from the following system of equations:
Equating coefficients at the same powers of we will obtain system of equation for unknown parameters , . Solving this system one can obtain exact expression for function (71).
In this paper we applied Sommerfeld integral approach to several diffraction problem for discrete Helmholtz equation (1). We showed that the field is represented as integral on a manifold. This manifold is torus, and corresponding integrals are Abelian integrals. For point source problem we proposed recursive procedure of field calculations which reduces integral computation to integral computation. For half-plane problem we constructed solution using Sommerfeld integral and showed it is equivalent to the Wiener—Hopf Solution. For the problem of diffraction by a right-angled wedge we showed that the problem can be reduced to the solution of nonlinear equation.
Appendix A. Abelian integrals
Indeed, are Abelian differentials on . The form is an Abelian differential of the first kind, while all other are Abelian differentials of the third kind.
Moreover, since is a torus, the Abelian integrals in this case are elliptic functions.
The classical framework of study of the Abelian integrals is as follows. The surface is cut by several cuts (by two cuts in our case) such that the surface becomes mapped onto a polygon with an edge. These cuts are and (the latter is shown in Fig. 13).
are cyclic periods of an Abelian differential .
Note that is a solution of the problem based on equation (1) but with another radiation condition (one should take with a negative imaginary part).
All theorems related to Abelian and elliptic integrals can be applied to and . This properties can be found in [Bateman1955].
Appendix B. Sommerfeld integral as an integral on the dispersion manifold
Let us build an analogy between Sommerefeld integral for continuous problem and Somerfeld integral for discrete problem.
let the Helmholtz equation
be satisfied everywhere except half-plane
where Dirichlet boundary condition is satisfied:
Here is an incident wave:
Plane wave should satisfy the following dispersion equation:
Also, radiation and Meixner conditions should be satisfied. Let us introduce a plane wave decomposition. Following the idea of field representation by integration on a manifold one should first study (79). This manifold is a Riemann sphere with two punctured points. In this points field has exponential growth. It is more natural to study this manifold as a tube (see figure 14).
Also, equation (79) can be written in a parametrical form:
Thus, this tube can be mapped to the strip:
Thus, the polar coordinates
should be introduced and the plane wave decomposition should be some integral on . It was shown by Sommerfeld that it has the following form:
Contours of integration are showed in the figure 15.
On -branch surface function should be periodical with respect to with period , and should have a pole with unity residue corresponding to the incident wave. For a plane wave on a plane we have:
For a plane wave on 2-sheeted surface we have:
So, the following analogies between discrete and continuous solutions can be seen:
The field is represented as an integral on some manifold defined by dispersion equation. In the discrete case this manifold is torus, and in contentious case it is tube.
There are two contours of integration in Sommerefeld integral. Both contours do not cross the line of propagating waves (the line of real wavenumbers).
To obtain the solution for half-plane problem one need to construct a function that covers the manifold twice.
Appenidx C. Sliding plane wave decomposition
For the discrete problem we can mimic this sliding. Namely, consider torus . There are 4 contours along which the integration can be held, and 4 possible “infinites” at which there can be singularities. They are shown in Fig. 4. The contours are , , , and .
A simpler scheme of the same surface with contours and infinities is shown in Fig. 6. A torus is shown as a torus in a usual sense. The relative position of the contours and the infinity points is drawn.
Consider an integral
Let be some 1-form on having poles only at the four infinities. Let the order of the poles there is for some integer . According to the consideration made above, the form is regular at the infinites under the following conditions:
Take the observation point such that . Move this point about the origin in the direction of the red arrow in the figure. One can see that one can slide the contour sequentially in the order
(In fact, here we have in mind that, for example, in order to slide contour to we should ensure that the form is analytic at the point Inf 1.)
Corresponding contours provide a necessary decay of the solution in corresponding sectors of the plane . |
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It was submitted to the free digital textbook initiative in california and will remain unchanged for at least two years. With few exceptions i will follow the notation in the book. The book is in use at whitman college and is occasionally updated to correct errors and add new material. University of kentucky elementary calculus and its. Catalog description math 143 calculus iii 4 units ge area b1 prerequisite.
These notes are largely based on the optional text calculus by elliot gootman. Indeterminate forms and some theoretical tools about them. The proofs of most of the major results are either exercises or. Calculus early transcendentals an open text base text revision history current revision. A quick and dirty introduction to exterior calculus 45 4. It also does some mathematics providing entertaining takes on the standard concepts of the first year of calculus, providing often. Myers florida international university, miami florida state university, tallahassee new college of florida, sarasota university of central florida, orlando.
I am a sophomore at penn state and i am taking calculus for the second. Together these form the integers or \whole numbers. Calculus i or needing a refresher in some of the early topics in calculus. A problemtext in advanced calculus portland state university. Gootmans text is very readable and has many worked out examples, and often provides more detail than the lecture notes available here.
Refresherbefore embarking upon this calculus revision course. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an algebra or trig class or contained in other sections of the notes. Differential calculus we call the gradient at a point the derivative, which can be written in the following ways. It was written exactly for people like you, who are taking calculus and struggling with it.
Infinite sequences and series, vector algebra, curves. I recommend the paperback calculus by my friend and colleague, elliot gootman. Partial derivatives, multiple integrals, introduction to vector analysis. The complete textbook is also available as a single file. About the cover the maglev magnetic levitation train uses electromagnetic.
Math 142 with a grade of c or better or consent of instructor. There is online information on the following courses. The latter case occurs for example when computing the derivative. This book is a revised and expanded version of the lecture notes for basic calculus and other similar courses o ered by the department of mathematics, university of hong kong, from the. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. Not to be copied, used, distributed or revised without. It is made up of two interconnected topics, differential calculus and integral calculus. All new content text and images is released under the same license as noted above. A kform on rn is a function that takes kvectors in rnand returns a number v 1v k, such that is multilinear and antisymmetric as a function of the vectors. Calculus online textbook chapter 1 mit opencourseware. It is a form of mathematics which was developed from algebra and geometry.
Publication date 192122 topics calculus, integral publisher london, macmillan collection. In the pointslope form we can use any point the graph passes through. The videos, which include reallife examples to illustrate the concepts, are ideal for high school students, college students. An alternative is to add to the calculus the following axiom scheme x. Since the course is an experimental one and the notes written.
For implementations of the calculus the machine has to deal with. Elliot gootman, for agreeing to be on my committee and. Calculus i lhospitals rule and indeterminate forms. You may email me, or use the web form for feedback on the web pages for the course. Here are a set of practice problems for my calculus iii notes. We prefer our version of the theory in which the identi cations are made on syntactic level. Textbook calculus online textbook mit opencourseware. I read this book because id forgotten a lot of the calculus i learned in college. This book is meant to be an accessible introduction to the main ideas, methods and applications of first year calculus. What is the largest possible product you can form from two. If you are viewing the pdf version of this document as opposed to viewing it on the web this document contains only the problems.
We shall cover the material in the first ten 10 chapters of this book, as well as appendix b. Integral calculus with applications to the life sciences. Peterson department of biological sciences department of mathematical sciences clemson university email. Calculus this is the free digital calculus text by david r. Applications and integration poli 270 mathematical and statistical foundations sebastian m. Just find the derivative, which we do using first principles. The author even often says look in your courses calculus book or ask your instructor for more information or a proof, etc. This course will cover the topics from the first ten chapters and supplement of the course text.
These identi cations are done in our mind and not on paper. Erdman portland state university version august 1, 20 c 2010 john m. As the title of the present document, problemtext in advanced calculus, is intended to suggest, it is as much an extended problem set as a textbook. The book is published by barrons, and it will be the primary text for the course. How to use this booklet you are advised to work through each section in this booklet in order. How to ace calculus takes a tongueincheek approach and includes the lowdown on nonmathematical topics such as choosing and dealing with your instructor, asking questions, preparing for and taking exams. University of kentucky elementary calculus and its 110 chapter3. The book can be purchased from the bookstores or online.
Math 221 1st semester calculus lecture notes version 2. Math 221 first semester calculus fall 2009 typeset. In this section we will revisit indeterminate forms and limits and take a look at lhospitals rule. Abstracts should be submitted on special forms which are available in many. Hildebrand advanced calculus for applications prenticehall inc. It was developed in the 17th century to study four major classes of scienti. My book is designed to be an accessible, userfriendly introduction to the main ideas, techniques and applications of first year calculus. These all mean the same thing, so dont panic if youre asked to find the of a function. In this section we will revisit indeterminate forms and limits and take a look at l hospitals rule. The subject of calculus on time scales is a young one being first introduced by. Catalog description math 241 calculus iv 4 units prerequisite. Its a great selfteaching tool with exercises at the end of each chapter. You may need to revise some topics by looking at an aslevel or alevel textbook which contains information about di.
Saiegh department of political science university california, san diego october 7 2010 sebastian m. In fact, i have often loaned my copy of schaums outline of calculus to students. A beginning getting ready for models and analyzing models the seadragons were intrigued by calculus and ocked to the teacher. Students soludons manualcontains solutions to approximately onethird of the. It contains explanations, in straightforward and simple language, of the essential concepts of beginning. Mit professor gilbert strang has created a series of videos to show ways in which calculus is important in our lives.
The notes were written by sigurd angenent, starting. Integral calculus with applications to the life sciences leah edelsteinkeshet mathematics department, university of british columbia, vancouver february 26, 2014 course notes for mathematics 103 c leah keshet. It does not read like a textbook and if youre a math geek, gootmans writing style is actually interesting and engaging not dry like a lot of books out there. Version2017 revisiona extensiveedits, additions, and revisions have been completed by the editorial team at lyryx learning.1169 775 1092 1429 688 432 503 1588 452 1482 1379 1413 783 1044 1318 130 223 297 1250 1025 1439 1083 1623 524 96 42 1223 1129 260 540 890 1244 24 350 621 |
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
Using the statements, can you work out how many of each type of rabbit there are in these pens?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?
Can you use this information to work out Charlie's house number?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
Can you make square numbers by adding two prime numbers together?
Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
Can you work out some different ways to balance this equation?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
Have a go at balancing this equation. Can you find different ways of doing it?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This dice train has been made using specific rules. How many different trains can you make?
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?
The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different? |
kg or lb The SI base unit for mass is the kilogram. 2.5 kg to lbs. One kilogram equals 2.20462262 pounds, to convert 20.5 kg to pounds we have to multiply the amount of kg by 2.20462262 to obtain amount in pounds. 5 kg is equal to 11 pounds See also the following table for related convertions 1 kg = 2.2 pounds 2 kg = 4.4 pounds 3 kg = 6.6 pounds 4 kg = 8.8 pounds 5 kg = 11 pounds 6 kg = 13.2 pounds 7 kg = 15.4 pounds 8 kg = 17.6 pounds 9 kg = 19.8 pounds 10 kg = 22 pounds 11 kg = 24.2 pounds 12 kg = 26.4 pounds To convert 7.5 kg to us lbs you need a formula. To use this calculator, simply type the value in any box at left or at right. Kilograms. We will show you two versions of a formula. 5. kg * 2.2046 lbs. Method 1 How to convert 1.5 kg to pounds To calculate a value in kg to the corresponding value in pounds, just multiply the quantity in kg by 2.2046226218488 (the conversion factor).. 0.5 kg to lb conversion. The avoirdupois pound is equivalent to 16 avoirdupois ounces. 1 kg = 11.02311311. lbs It couldn’t be easier to use. Convert g, lbs, ozs, kg, stone, tons. In most cases, all you need to do to convert is to multiply the number of kilograms by 2.2 to get the number of pounds. Hence, the final answer is 5 kg = 11.023113109 lbs. Let’s start with the first one: Number of kilograms * 2.20462262 = the 16.534669650 outcome in pounds The definition of the kilogram changed in 2019. 245 5 kg to lbs kilograms kilograms stones and pounds chart 21 5 kilograms in pounds how many 1 5 kilograms to pounds converter lbs to kg conversion conversions fast method to convert kg pounds. How to convert. For example, convert 5 kg to lbs. 1 pound (lb) is equal to 0.45359237 kilograms (kg). How many pounds in 2.5 Kilograms? A quick online weight calculator to convert Kilograms(kg) to Pounds(lb). The kilogram, or kilogramme, is the base unit of weight in the Metric system. The Kg to Pounds Conversion Formula to convert 66.5 kg to lbs To know how many pounds in a kilogram, you can use the following formula to convert kg to lbs : X(lb) = Y(kg) / 0.45359237 How to convert 66.5 kg to lbs? To convert kilograms to pounds, multiply the kilogram value by 2.2046226218. Since we know that 1 kg = 2.2046226218 lbs therefore 5 kg = (5 X 2.2046226218) lbs. It converts kilo to pounds or vice versa with a metric conversion table. Kilograms to Pounds Converter. The online Kilograms to Pounds converter is used to convert the weight from kilos to pounds . 5kg=11.0231131 lb Algebraic Steps / Dimensional Analysis Formula. 5 kg to grams: 5 kg to lbs: 5 kg to oz: 5 kg to tons: 5 kg to stone: How much is 5 kilograms in ounces? m (kg) = m (lb) × 0.45359237. Kg to Lbs converter. 2.5 kilograms or 2500 grams equals 5.51 pounds. kg to pounds kg to lb + oz. For example, to calculate how many pounds is 2 kilograms, multiply 2 by 2.2046226218, that makes 4.4092 lbs is 2 kg. Plus learn how to convert Kg to Lb What is 5 kg in pounds, ounces, grams, stone, tons, etc? This is a very easy to use kilograms to pounds converter.First of all just type the kilograms (kg) value in the text field of the conversion form to start converting kg to lbs, then select the decimals value and finally hit convert button if auto calculation didn't work.Pounds value will be converted automatically as you type.. 1 kg = 2.2046226218 lbs 1 lbs = 0.45359237 kg. Definition: A pound (symbol: lb) is a unit of mass used in the imperial and US customary systems of measurement. Pound. The simplest way to find how many pounds is 8.5 kg is to divide the kilogram value by 0.45359237. How to convert 8.5 kg to lbs? In general, it can be said that there are 2.2046 pounds per kilogram. 1 kilogram (kg) = 1 liter (l). Kg to lbs is a kilogram (kg) to Pounds (lbs) weight Converter. 20.5 kg are equal to 20.5 x 2.20462262 = 45.194764 pounds. Definition of kilogram. Enter 100 kg here, and you will get the conversion of 100 kilos in pounds easily. Kilograms and pounds are both units used to measure weight. 5.5 kg are equal to 5.5 x 2.20462262 = 12.125424 pounds. How to convert Pounds to Kilograms. Note that rounding errors may occur, so always check the results. Converting from kilograms to pounds is a common task in the realms of math and engineering, but, luckily, it's an easy one. 0.5 Kilograms = 1.1023113 Pounds (rounded to 8 digits) Display result as. 1.5kg to lbs. Liter (l) is a unit of Volume used in Metric system. Convert 20.5 kg to pounds. 5 kg = 176.36981 ounces. How many grams in 5 kilograms? swap units ↺ Amount. Converting 5 kg to lb is easy. The kilogram (kg) is the SI unit of mass. Pounds : The pound or pound-mass (abbreviations: lb, lbm, lbm, ℔) is a unit of mass with several definitions. To. Example. It is the approximate weight of a cube of water 10 centimeters on a side. Convert 8.5 KG to LBS. Now you learned how many 7.5 kg to lbs and how many kilograms 7.5 pound, so it is time to move on to the 7.5 kg to lbs formula.. 7.5 kg to pounds. Task: Convert 15 kilograms to pounds (show work) Formula: kg ÷ 0.45359237 = lb Calculations: 15 kg ÷ 0.45359237 = 33.06933933 lb Result: 15 kg is equal to 33.06933933 lb Conversion Table For quick reference purposes, below is a conversion table that you can use to convert from kg to lb. Keep reading to learn more about each unit of measure. 1 lb = 0.45359237 kg. Nowadays, the most common is the international avoirdupois pound which is legally defined as exactly 0.45359237 kilograms. The mass m in kilograms (kg) is equal to the mass m in pounds (lb) times 0.45359237:. 100 kg = 220.46226 lbs. 5 kilograms equal 11.0231131092 pounds (5kg = 11.0231131092lbs). 5 kg = 5000 grams. A pound is equal to 16 … Simply use our calculator above, or apply the formula to change the length 5 kg to lbs. Convert 5 lb to kilograms: The kilogram (kg) is the SI unit of mass. 5 kg = (5 × 2.204623) = 11.023113 lb . It is unlikely you will just need to convert 8.5 kg to lbs (pounds). The international avoirdupois pound (the common pound used today) is defined as exactly 0.45359237 kilograms. Definition of kilogram. From. 1 kilogram is equal to 2.204622621849 pounds or lbs. Here is the formula: How much does 0.5 kilograms weigh in pounds? 1 kilogram is equal to 2.2046226218488 lb. 1 Kilogram (kg) is equal to 2.2046226218 pounds (lbs). Above is the actual conversion rate of Kilograms to Pounds and vice versa. If M (kg) represents mass in kilograms and M (lb) represents mass in pounds, then the formula for converting kg to lbs is: M (lb) = 2.204622621849 × M (kg) If you need a quick way to find out what something weighs in stones and pounds instead of kilograms, all you need to do is keep our easy-to-use guide handy and you’ll never need to start working out tricky kilos to stone conversions again! The time will come when you will need to convert 5, 15, 25 kg and so on to pounds (lbs), so knowing the process helps. 1.5 Kilograms to Pounds shows you how many pounds are equal to 1.5 … 5 Kg Into Pounds Saturday, 28 November 2020. Kilograms. It accepts fractional values. How to convert kilograms to pounds? Kilogram (kg) is a unit of Weight used in Metric system. Or at right the common pound used today ) is defined as exactly 0.45359237 kilograms ( kg =... Most common is the SI unit of weight in the imperial and us customary systems of measurement divide! 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Centimeters on a side here, and you will just need to convert the weight kilos... 0.5 kilograms = 1.1023113 pounds ( lb ) times 0.45359237:: pounds ( lbs.. To convert 8.5 kg to lbs occur, so always check the results pounds easily of 100 kilos pounds... In Metric system in general 5kg to lbs it can be said that there are 2.2046 pounds per kilogram pounds lb... Know that 1 kg = 11.02311311. lbs kg or lb the SI unit weight... At left or at right = m ( kg ) is a unit of in... Quick online weight calculator to convert kilograms to pounds, multiply 2 by 2.2046226218, makes! Is equal to 2.2046226218 pounds ( lbs ) on any conversion a quick online weight calculator convert! |
Technician b says that the total resistance is 18 ohms. The patch clamp technique is a laboratory technique in electrophysiology used to study ionic currents in individual isolated living cells, tissue sections, or patches of cell membrane. A combination of a current source in series with a resistor and a voltage source behaves just like the current source alone. In a series circuit, the current is only able to flow through a single path. Series circuits part 3 series voltage sources circuits. In standard wholecell voltage clamp, the goal is simple.
However, when used by themselves, such techniques are not well suited to the task of mapping lowdensity channel distributions. This video provides a lesson on combining independent current and voltage sources, which i did not cover explicitly during our lectures. A patch clamp recording of current reveals transitions between two conductance states of a single ion channel. We describe here a new voltage clamp method the whole cell loose patch wclp method that combines wholecell recording through a tightseal pipette with focal extracellular stimulation through a looseseal pipette. Voltage source andor inductor loop two capacitors in series, without a series resistance, might confuse a simulator. How do resistance and capacitance determine the electrical properties of the cell to. Mathematically, current and voltage sources can be converted to each other using thevenins theorem and nortons theorem. You can also verify this by kvl around the outer loop. This device connects to a host computer through a usb 2. These things are bad, but what is worse is when it changes over time. Recall that in resonance, the voltage across the reactive elements is q times larger than the voltage on the load. Series resistance compensation for wholecell patchclamp.
In practice, this ideal form of the voltage clamp cannot be implemented because of the series resistance r s of the electrode that connects to the cell. To find the output voltages for circuits b and c, you use voltage divider techniques. The experimental artefact components of the model include. Im confused as to how the ideal sources interact with the resistor and each other.
Hopefully by now you should have some idea of how electrical voltage, current and resistance are closely related together. Resistors in a series with current source and resistor in. In csevc, the same electrode is used simultaneously for voltage recording and for current passing. The unit of voltage is the volt which is a measure of electric potential energy per unit of charge. The output impedance is defined as this modeled andor real impedance in series with an ideal voltage source. Parasitic series and shunt resistances in a solar cell circuit. Series resistance does not affect the solar cell at opencircuit voltage since the overall current flow through the solar cell, and therefore through the series resistance is zero.
Apr 25, 2008 the voltage v m at all times is exactly clamped to the battery voltage v cmd. Impact of both series and shunt resistance pveducation. Apr 10, 2020 the study of ohc nlc by admittance techniques in whole cell voltage clamp is compromised by contributions from stray capacitance, membrane conductances, and electrode series resistance r s, the. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Conversely, the perfect constant voltage source has zero resistance and adding parall. Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals ab by an equivalent combination of a voltage source v th in a series connection with a resistance r th. Along with voltage and current, resistance is one of the three basic units in electricity. Patch clamp technique method electrophysiology technique. In the experiments of parallel and series connection, electrodes were. In the case of a current source in series with a resisitor, can this be transformed to just a current source with 0 resistance, or is the resistance infinity in the norton equivalent. And using the ohms law, we conclude that no current passes through.
Seriesconnected flexible biobatteries for higher voltage. Axon axopatch 200b microelectrode amplifier molecular devices. Assume we want to apply a voltage across the cell membrane by injecting. However, near the opencircuit voltage, the iv curve is strongly affected by the series resistance. Circuit theory current source, voltage source, resistor in. Series aiding voltage sources are sources that are connected so that current in both sources flows in the same direction. Many electrical circuits have more than one voltage source and these sources may be series aiding or series opposing.
Complex nonlinear capacitance in outer hair cell macro. Sources and elimination of interference in patch clamp electrophysiological. A novel voltage clamp technique for mapping ionic currents. Why does a resistance become redundant, when in parallel with. The voltage across is and therefore, the voltage drop on will be zero. As explored below, the glowing filament in an incandescent light bulb allows us to view resistance in action. The ideal current source will produce any voltage across itself to maintain a 2ma output but im unsure how this affects the voltage drop across the resistor or the current needed by the ideal voltage source to maintain 24v. Using kirchhoffs laws to solve circiut with two power supplies task number.
Oct 03, 20 shows how to calculate the voltage, resistance and current in an electric circuit containing resistors in series. The current of is the same as before because its voltage is not changed. Since the tails of the arrows conververge at a common node, lets consider the junction of the current source and resistor to be ground. To model this as a current source you cant put a resistor in series because the current source can still generate 1 amp and the open circuit voltage would be infinite. As originally stated in terms of dc resistive circuits only, thevenins theorem aka helmholtzthevenin theorem holds that. Continuous single electrode voltageclamp csevc molecular. Axon axopatch 200b microelectrode amplifier key features ultra lownoise current and voltage patch clamp amplifier integrates with any data acquisition system optimized for wholecell and singlechannel recordings three recording modes from subpa to hundreds of na currents. Relationship between voltage current and resistance. The battery shown here is an ideal voltage source which delivers 1.
Technician a says that the source voltage is 12 volts. Series current and voltage source all about circuits. Series resistance compensation for wholecell patch clamp studies using a membrane state estimator adam j. Abstract wholecell patch clamp techniques are widely used to measure membrane currents from isolated cells. The relationship between voltage, current and resistance forms the basis of ohms law. In this learning activity youll explore the effect of connecting voltage sources in series to increase voltage applied to a load. Analyze circuits with two independent sources using. How can i calculate the voltage error in a wholecell. Voltage, current, and resistance flashcards quizlet.
Sep 16, 2014 if the circuitry in the box is represented by its thevenin equivalent. Ideal voltage source explained learning about electronics. Set up the circuit connections by referring to the figure below. The graphs will display the output voltage and voltage across the capacitor over time. In wholecell recordings an important problem is the series access resistance. Electronics for electrophysiologists optical imaging and.
In essence, that is enough to perform the impedance transformation. Electronics internal resistance of a voltage source. The perfect constant current source has infinite resistance and adding series load resistance to the circuit has no effect on the magnitude of the current. First connect the source voltage from the output terminals of the interface across the series combination of the 100 22 resistor and 100 uf capacitor using terminals a red and b black. An ideal voltage source is a voltage source that supplies constant voltage to a circuit despite the current which the circuit draws. By kirchhoffs current law, the current i equals the current that flows through the passive cell membrane, which is the sum of the current flowing through the cell resistance and the current flowing through the cell. As we have already shared ohms law p,i,v,r calculator in which you can also calculate three phase current. It should be noted that older patchclamp amplifiers implement a differ ent circuit that is. May 19, 2018 this electronics video tutorial provides a basic introduction into voltage, current, and resistance. If you look at it here we have a power supply and here we have current flowing, negative to positive current is flowing in this direction and the other voltage. Resistors is electric circuits 2 of 16 voltage, resistance.
A straightforward method of estimating the series resistance. Using kirchhoffs laws to solve circiut with two power supplies. Continuous single electrode voltageclamp csevc is an electrophysiological patchclamping method to pass a membrane voltage into a cell and measure the change in current as the voltage steps. The current source is going to supply the resistor with 14 volts so 2 ma goes through it. The problem with series resistance in this case is that the voltagedrop across this. Internal resistance of a voltage source theory example internal resistance of a voltage source any device which produces a voltage output has a limit to the current it can provide. The circuit diagram for three resistors in parallel, connected to a voltage source looks like the following. These may help you to relatively stable the series resistance during the voltage clamp recordings. So, if the original voltage source was 10 volts and had a 10 ohm resistor in series, the equivalent current source would be 1 amp in parallel with 10 ohms. Understanding the cell as an electrical circuit scientifica. That is, you use the idea that a circuit with a voltage source connected in series with resistors divides its source voltage proportionally according to the ratio of a resistor value to the total resistance. Automotive electronics flash cards flashcards quizlet. Current source in series with resistor physics forums. The patch clamp amplifier thus must function as a currenttovoltage converter to allow this.
While suitable for a broad range of ionic currents, the series resistance rs of the. Jun 08, 2019 for understanding the ideal voltage source, we can take an example of a circuit shown above. Aug 27, 2016 mam, sorry, i think your understanding of networks is going little bit, in wrong direction. The challenge in the case of the voltage clamp is the previously discussed inaccuracy in measuring membrane voltage. The intan clamp system allows users to perform single amplifier or multiamplifier patch clamp electrophysiology experiments with small, affordable hardware and free, open source software. Ok, we have a current source and a voltage source fighting each other. This means that despite the resistance which a load may be in a circuit, the source will still provide constant and steady voltage. The patch clamp technique is a refinement of the voltage clamp. To combine the effect of both series and shunt resistances, the expression for ff sh, derived above, can be used, with ff 0 replaced by ff s 1. Put the alligator clips on the ends of the voltage sensor.
Ideal voltage and current sources in series stack exchange. I compensate my series resistance 7580%, i have sodium currents that range from 0. Can someone advise on series resistance in current clamp. Internal resistance of a voltage source theory example internal resistance of a voltage source any device which produces a voltage output has a limit to the. In voltage clamp, series resistance prevents your amplifier from charging the membrane capacitor, and in current clamp, series resistance stops your cell from being able to charge the capacitance of your pipette. This value is used by the amplifier to generate the expected voltage drop across the resistance and to correct the.1288 1128 1377 1421 563 689 459 415 355 663 1238 361 1279 422 344 1528 163 956 197 366 66 1285 436 1484 1473 1176 686 566 |
Iterative methods for AQA A-level Maths
This page covers the following topics:
1. Iterative formulas
2. Iterative Bisection
3. Linear Interpolation
4. Cobweb diagrams
5. Staircase diagrams
6. Iterative methods problems in context
Suppose we have an equation such as x² – 3x + 2 = 0. One approach of solving this is to rearrange the equation so that x is the subject of the formula, giving x = √(3x – 2). A value of x which satisfies this is also a solution of our original equation. We can find this value of x by considering this as an iterative formula xₙ₊₁ = √(3xₙ – 2). We then pick some starting value x₀ and put it in the right hand side of our equation to give some new x₁. We then iteratively put the x₁ in the right side again to get x₂, and so on. The values we get from the equation should converge on an approximate solution to our original equation, but this can depend on the function and what starting x₀ we use.
Suppose we are trying to find the root of some equation f(x). We can use the change of signs method and find 2 points a and b so that f(a) < 0 and f(b) > 0, and therefore we know the root will lie somewhere in between. We then consider the midpoint m of these 2 points (dashed line on graph). If f(m) < 0, then we now know the root is between m and b, and otherwise it's between a and m. Iterative bisection just involves repeadedly doing this process, finding a new smaller intervals until it converges to an approximate solution. After a certain number of iterations one obviously needs to stop, and then we just give the interval's midpoint as the approximate root. It doesn't strictly need to find a root either, we could check for values f(a) < 1 and f(b) > 1 when trying to solve f(x) = 1.
Consider this graph of f(x), where we know f(x) = 0 between some f(a) < 0 and f(b) > 0. Linear Interpolation is similar to iterative bisection, but instead of finding the midpoint of a and b to get our new interval, we consider the straight line from f(a) to f(b). The x–intercept (c) of this straight line can be easy calculated as c = ( a|f(b)| + b |f(a)| ) / ( |f(a)| + |f(b)| ), and this will be closer to the root of f(x) (note that the '| |' means the absolute, positive value). Then just as we did for iterative bisection, we identify the smaller interval which must contain the root and iteratively repeat the process, converging onto the root.
Iterative methods can be used to find the roots of f(x) = 0 after rearranging it to the form x = g(x) and using the iterative formula x_(n + 1) = g(x_n). One method of doing this is with successive iterations which alternate between being below and above the root. If the iterations converge, a cobweb diagram is formed. The interval in which a root lies can be found by plugging values for x in the formula and seeing where the value of f(x) changes sign.
One iterative method is one in which the iterations get progressively closer to the root from the same direction. When this process is plotted, a diagram called a staircase diagram is formed.
Iterative methods can be used to model situations and find their solutions.
A ball is dropped from a vertical distance of 15 m. The motion of the ball is modelled by the function f(x) = −2x² + 15, x > 0, where x is the horizontal distance travelled by the ball. Show that the horiznotal distance is between 2 m and 3m.
f(2) = −2(2)² + 15 = 7.
f(3) = −2(3)² + 15 = −3.
Since there is a change in sign, the horizontal distance travelled by the ball is between 2 m and 3 m.
Show that there is a root of f(x) = 2x + tan(x) + 1 at around x = –0.3 using iterative bisection between x = 0 and x = 0.5.
As iterative bisection is repeatedly performed, one should see the interval becoming smaller and smaller around the point x = –0.3.
Find the root of f(x) = x³ + √(x) – 3/2, using linear interpolation (to 1 decimal place).
x = 0.8 (rounded to 1 decimal place). To start, one needs to spot an interval where the function has a change of sign. It should be pretty easy to spot f(0) = –3/2, and f(1) = 1/2. Now one starts the linear interpolation and repeatedly narrows down the interval. Once we reach an interval anywhere within the range x = 0.75 to 0.84, then we know the root in this range would always round to x = 0.8.
Find x² + 5x + 6, using linear interpolation and knowing a root exists in the interval between x = – 3.5 and x = –2.5.
f(–3.5) = 0.75, and f(–2.5) = –0.25. The line intersects the x–axis at x = (–3.5*|f(–2.5)| + (–2.5)*|f(–3.5)|) / (|f(–3.5)| + |f(–2.5)|) = (–3.5*(0.25) + (–2.5*0.75)) / (0.75 + 0.25) = –2.75. We have f(–2.75) = –0.1875. The new interval must be between f(–3.5) and f(–2.75) since there is a change of sign. One then repeats the process, finding the new straight line's intercept with the x–axis. One progressively gets closer to the root of x = –3.
Find a root of f(x) = sin(x) + x + 2 using linear interpolation, knowing the root is between x = –1 and x = –2.
The actual root is x = –1.106… Using linear interpolation should converge to this root, but you obviously stop the iterative process at some point on an approximate root.
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Unformatted text preview: UCSD Physics 10 Special Relativity
Einstein messes with space and time UCSD Physics 10 How Fast Are You Moving Right Now? 0 m/s relative to your chair 400 m/s relative to earth center (rotation) 30,000 m/s relative to the sun (orbit) 220,000 m/s relative to the galaxy center (orbit) 370,000 m/s relative to the CMB cosmic wallpaper Relative to What?? This is part of the gist of special relativity it's the exploration of the physics of relative motion only relative velocities matter: no absolute frame very relevant comparative velocity is c = 300,000,000 m/s UCSD Physics 10 A world without ether For most of the 19th century, physicists thought that space was permeated by "luminiferous ether" this was thought to be necessary for light to propagate Michelson and Morley performed an experiment to measure earth's velocity through this substance first result in 1887 Michelson was first American to win Nobel Prize in physics Found that light waves don't bunch up in direction of earth motion shocked the physics world: no ether!! speed of light is not measured relative to fixed medium unlike sound waves, water waves, etc. UCSD Physics 10 Speed of light is constant: so what? Einstein pondered: what would be the consequences of a constant speed of light independent of state of motion (if at const. velocity) any observer traveling at constant velocity will see light behave "normally," and always at the same speed Mathematical consequences are very clear forced to give up Newtonian view of space and time as completely separate concepts provides rules to compute observable comparisons between observers with relative velocity thus "relativity": means relative state of motion UCSD Physics 10 Simultaneity is relative, not absolute
Observer riding in spaceship at constant velocity sees a flash of light situated in the center of the ship's chamber hit both ends at the same time But to a stationary observer (or any observer in relative motion), the condition that light travels each way at the same speed in their own frame means that the events will not be simultaneous. In the case pictured, the stationary observer sees the flash hit the back of the ship before the front UCSD Physics 10 One person's space is another's time If simultaneity is broken, no one can agree on a universal time that suits all the relative state of motion is important Because the speed of light is constant (and finite) for all observers, space and time are unavoidably mixed we've seen an aspect of this in that looking into the distance is the same as looking back in time Imagine a spaceship flying by with a strobe flashing once per second (as timed by the occupant) the occupant sees the strobe as stationary you see flashes in different positions, and disagree on the timing between flashes: space and time are mixed see description of light clock in text Space and time mixing promotes unified view of spacetime "events" are described by three spatial coordinates plus a time UCSD Physics 10 The Lorentz Transformation There is a prescription for transforming between observers in relative motion
ct' = (ct - vx/c); x' = (x - vt); y' = y; z' = z "primed" coordinates belong to observer moving at speed v along the x direction (relative to unprimed) note mixing of x and t into x' and t' time and space being nixed up multiplying t by c to put on same footing as x now it's a distance, with units of meters the (gamma) factor is a function of velocity: UCSD Physics 10 The gamma factor Gamma ( ) is a measure of how whacked-out relativistic you are When v = 0, = 1.0 and things are normal At v = 0.6c, = 1.25 a little whacky At v = 0.8c, = 1.67 getting to be funky As vc, UCSD Physics 10 What does do? Time dilation: clocks on a moving platform appear to tick slower by the factor at 0.6c, = 1.25, so moving clock seems to tick off 48 seconds per minute standing on platform, you see the clocks on a fast-moving train tick slowly: people age more slowly, though to them, all is normal Length contraction: moving objects appear to be "compressed" along the direction of travel by the factor at 0.6c, = 1.25, so fast meter stick will measure 0.8 m to stationary observer standing on a platform, you see a shorter train slip past, though the occupants see their train as normal length UCSD Physics 10 Why don't we see relativity every day? We're soooo slow (relative to c), that length contraction and time dilation don't amount to much 30 m/s freeway speed has v/c = 10-7 = 1.000000000000005 30,000 m/s earth around sun has v/c = 10-4 = 1.000000005 but precise measurements see this clearly UCSD Physics 10 Velocity Addition Also falling out of the requirement that the speed of light is constant for all observers is a new rule for adding velocities Galilean addition had that someone traveling at v1 throwing a ball forward at v2 would make the ball go at v1+v2 In relativity, reduces to Galilean addition for small velocities can never get more than c if v1 and v2 are both c if either v1 OR v2 is c, then vrel = c: light always goes at c UCSD Physics 10 Classic Paradoxes The twin paradox: one twin (age 30) sets off in rocket at high speed, returns to earth after long trip if v = 0.6c, 30 years will pass on earth while only 24 will pass in high speed rocket twin returns at age 54 to find sibling at 60 years old why not the other way around? Pole-vaulter into barn high-speed runner with 12 meter pole runs into 10 meter barn; barn door closes, and encompasses length-contracted 9.6 m pole (at 0.6c) but runner sees barn shrunken to 8 m, and is holding 12 m pole! can the barn door close before the pole crashes through the back? resolution in lack of simultaneity: "before" is nuanced UCSD Physics 10 If I'm in a car, traveling at the speed of light... If I turn on my headlights, do they work? Answer: of course--to you, all is normal you are in an un-accelerated (inertial) frame of reference all things operate normally in your frame To the "stationary" outsider, your lights look weird but then again, so do you (because you're going so fast) in fact, at the speed of light, all forward signals you send arrive at the same time you do And the outside, "stationary" world looks weird to you But I must inquire: how did you manage to get all the way up to the speed of light?! UCSD Physics 10 What would I experience at light speed? It is impossible to get a massive thing to travel truly at the speed of light energy required is mc2, where as vc so requires infinite energy to get all the way to c But if you are a massless photon... to the outside, your clock is stopped so you arrive at your destination in the same instant you leave your source (by your clock) across the universe in a perceived instant makes sense, if to you the outside world's clock has stopped: you see no "ticks" happen before you hit UCSD Physics 10 E = mc2 as a consequence of relativity Express 4-vector as (ct, x, y, z) describes an "event": time and place time coordinate plus three spatial coordinates factor of c in time dimension puts time on same footing as space (same units) We're always traveling through time our 4-velocity is (c, 0, 0, 0), when sitting still moving at speed of light through time dimension stationary 4-momentum is p = mv(mc, 0, 0, 0) for a moving particle, p = ( mc, px, py, pz) where px, etc. are the standard momenta in the x, y, and z directions the time-component times another factor of c is interpreted as energy conservation of 4-momentum gets energy and momentum conservation in one shot UCSD Physics 10 E = mc2, continued can be approximated as = 1 + v2/c2 + ...(small stuff at low velocities) so that the time component of the 4-momentum c is:
m c2 = mc2 + mv2 + ... the second part of which is the familiar kinetic energy Interpretation is that total energy, E = m c2 mc2 part is ever-present, and is called "rest mass energy" kinetic part adds to total energy if in motion since sticks to m in 4-momentum, can interpret this to mean mass is effectively increased by motion: m m gets harder and harder to accelerate as speed approaches c UCSD Physics 10 Experimental Confirmation We see time dilation in particle lifetimes in accelerators, particles live longer at high speed their clocks are running slowly as seen by us seen daily in particle accelerators worldwide cosmic rays make muons in the upper atmosphere these muons only live for about 2 microseconds if not experiencing time dilation, they would decay before reaching the ground, but they do reach the ground in abundance We see length contraction of the lunar orbit squished a bit in the direction of the earth's travel around the sun E = mc2 extensively confirmed nuclear power/bombs sun's energy conversion mechanism bread-and-butter of particle accelerators UCSD Physics 10 References Relativity Visualized by Lewis Carroll Epstein http://www.anu.edu.au/physics/Searle/ movie Assignments Q/O #3 due today by midnight Partial read of Chapters 9 & 10 (pages on assignment page) Read Chapters 35 & 36 on relativity HW5: 9.R.13, 9.E.9, 9.E.14, 9.E.43, 9.P.7, 10.E.16, 35.R.27, 35.E.6, 35.E.19, 35.E.20, 35.E.37, 35.P.3, 35.P.10, 36.R.7, 36.E.2, 36.E.6 ...
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Presentation on theme: "Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic,"— Presentation transcript:
Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic, Diatomic, Mass Defect, 2D Lattices) Strain (compliance, reduced notation, tensors) Harmonic Oscillator (Destruction/Creation, Hamiltonian & Number Operators, Expectation Values) Energy Density and Heat Capacity (phonons, electrons and photons) Quasiparticle Interactions (e-e, e-phonon, e-photon, defect interations) Electrical and Thermal Conductivity
Lattice Vibrations Longitudinal Waves Transverse Waves When a wave propagates along one direction, 1D problem. Use harmonic oscillator approx., meaning amplitude vibration small. The vibrations take the form of collective modes which propagate. Phonons are quanta of lattice vibrations.
The force on the n th atom; The force to the right; The force to the left; The total force = Force to the right – Force to the left aa U n-1 U n U n+1 Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies Monatomic Linear Chain Thus, Newton’s equation for the n th atom is
Brillouin Zones of the Reciprocal Lattice 1st Brillouin Zone (BZ=WS) 2nd Brillouin Zone 3rd Brillouin Zone Each BZ contains identical information about the lattice 2 /a Reciprocal Space Lattice: There is no point in saying that 2 adjacent atoms are out of phase by more than (e.g., 1.2 =-0.8 ) Modes outside first Brillouin zone can be mapped to first BZ
Diatomic Chain(2 atoms in primitive basis) 2 different types of atoms of masses m1 and m2 are connected by identical springs U n-2 U n-1 U n U n+1 U n+2 K KK K m1 m2 m a) b) (n-2) (n-1) (n) (n+1) (n+2) a Since a is the repeat distance, the nearest neighbors separations is a/2 Two equations of motion must be written; One for mass m1, and One for mass m2.
As there are two values of ω for each value of k, the dispersion relation is said to have two branches Upper branch is due to the positive sign of the root. Negative sign: k for small k. Dispersion- free propagation of sound waves Optical Branch Acoustical Branch This result remains valid for a chain containing an arbitrary number of atoms per unit cell. 0л/a2л/a–л–л/a k A B C A when the two atoms oscillate in antiphase At C, M oscillates and m is at rest. At B, m oscillates and M is at rest.
Number and Type of Branches Every crystal has 3 acoustic branches, 1 longitudinal and 2 transverse Every additional atom in the primitive basis contributes 3 further optical branches (again 2 transverse and 1 longitudinal)
2D Lattice K U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Write down the equation(s) of motion What if I asked you to include second nearest neighbors with a different spring constant?
2D Lattice C U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Similar to the electronic bands on the test, plot w vs k for the and directions. Identify the values of at k=0 and at the edges.
Specific Heat or Heat Capacity The heat energy required to raise the temperature a certain amount The thermal energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Classical Picture of Heat Capacity When the solid is heated, the atoms vibrate around their sites like a set of harmonic oscillators. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3kT, and consequently the energy per mole is = Dulong-Petit law: states that specific heat of any solid is independent of temperature and the same result (3R~6cal/K-mole) for all materials!
Average energy of a harmonic oscillator and hence of a lattice mode at temperature T Energy of oscillator The probability of the oscillator being in this level as given by the Boltzman factor Thermal Energy & Heat Capacity Einstein Model
Mean energy of a harmonic oscillator Low Temperature Limit Zero Point Energy exponential term gets bigger High Temperature Limit is independent of frequency of oscillation. This is a classical limit because the energy steps are now small compared with thermal/vibrational energy <<
Heat Capacity C (Einstein) Heat capacity found by differentiating average phonon energy where T(K) Area = The difference between classical and Einstein models comes from zero point energy. Points:Experiment Curve: Einstein Prediction The Einstein model near T= 0 did not agree with experiment, but was better than classical model. Taking into account the distribution of vibration frequencies in a solid this discrepancy can be accounted for.
1.Approx. dispersion relation of any branch by a linear extrapolation 2.Ensure correct number of modes by imposing a cut-off frequency, above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that: Debye approximation to the dispersion Debye approximation has two main steps Einstein approximation to the dispersion
Density of states (DOS) per unit frequency range g( ) The number of modes/states with frequencies and +d will be g( )d . # modes with wavenumber from k to k+dk= for 1D monoatomic lattice
The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus Mean energy of a harmonic oscillator for 1D It would be better to find 3D DOS in order to compare the results with experiment. Debye Model adjusts Einstein Model
3D Example: The number of allowed states per unit energy range for free electron? Each k state represents two possible electron states, one for spin up, the other is spin down.
L L L Octant of the crystal: k x,k y,k z (all have positive values) The number of standing waves;
The Heat Capacity of a Cold Fermi Gas (Metal) Close to E F, we can ignore the variation in the density of states: g( ) g(E F ). By heating up a metal (k B T << E F ), we take a group of electrons at the energy - (with respect to E F ), and “lift them up” to . The number of electrons in this group g(E F )f( )d and each electron increased its energy by 2 : The small heat capacity of metals is a direct consequence of the Pauli principle. Most of the electrons cannot change their energy. kBTkBT
Bam! Random Collisions On average, I go about seconds between collisions with phonons and impurities electron phonon Otherwise metals would have infinite conductivity Electrons colliding with phonons (T > 0) Electrons colliding with impurities imp is independent of T The thermal vibration of the lattice (phonons) will prevent the atoms from ever all being on their correct sites at the same time. The presence of impurity atoms and other point defects will upset the lattice periodicity
Fermi’s Golden Rule Transition rate: Quantum levels of the non-perturbed system Perturbation is applied Transition is induced (E) is the ‘density of states available at energy E’. See Fermi‘s Golden Rule paper in Additional Material on the course homepage
Absorption When the ground state finds itself in the presence of a photon of the appropriate frequency, the perturbing field can induce the necessary oscillations, causing the mix to occur. This leads to the promotion of the system to the upper energy state and the annihilation of the photon. This process is stimulated absorption (or simply absorption). Einstein pointed out that the Fermi Golden Rule correctly describes the absorption process. - degeneracy of state f
Quantum Oscillator Atoms still have energy at T=0. What is for the ground state of the quantum harmonic oscillator? (1D Case) For 3D quantum oscillator, the result is multiplied by 3: ⇒
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical particles which have no classical analogue. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many different areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs
Phonon spectroscopy = Constraints: Conservation laws of MomentumEnergy Conditions for: elastic scatteringin In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector.
x=(a-b)/2 or The cubic axes are equivalent, so the diagonal components for normal and shear distortions must be equal. And cubic is not elastically isotropic because a deformation along a cubic axis differs from the stress arising from a deformation along the diagonal. e.g., vs. Zener Anisotropy Ratio: |
Anya Hageman and Pauline Galoustian
In our previous two chapters we explored how the age structure of the population affects the economy. Now we focus on how the rate of population growth affects the economy.
The model of economic growth by Robert Solow (1956) is very well-known, simple, and easy to manipulate, so we’ll have a look at it and see what it predicts about the economic consequences of population growth. Its message will not be about efficiency, because efficiency is held constant in the Solow model. It will not be about hours worked or about the fraction of the population that works, because in the Solow model, labour is measured as the number of people in the population: everyone is assumed to work full-time.
The Solow model’s message will be about the capital-labour ratio, K/L, and the importance of accumulating capital to keep up with the number of workers.
The Solow model uses the aggregate production function Y = A F(K,L)
Y = aggregate output of the economy. There’s just one thing produced.
A = efficiency. This is held constant.
F(K,L) = the production function. The production function exhibits constant returns to scale; that is to say, if you double K and double L, F(K,L) doubles in size.
K = physical capital. This time we are not using K+ as we did in Chapter 16. K+ represents a number of different kinds of capital, and different kinds of capital may present mathematical complications. For example, if we include human capital, we might reasonably expect that human capital accumulation would affect efficiency, A, and we’d need an equation to show how that happens. If we included non-renewable natural resource capital, we’d need an equation to show how the resource stock is being depleted.
L = labour. This is identical to the number of people in the population. It grows at rate n.
Why is efficiency A held constant? We might think of including an equation that shows how efficiency grows, either exogenously (automatically) or endogenously (in response to something in the model, such as population growth). However, if A is allowed to grow, then output Y exhibits increasing returns to scale. Doubling K and L while increasing A would result in more than double the output. If increasing returns to scale were in place, output Y and consumption could grow forever in the presence of population growth. Sustainability would be too easy, at least mathematically.
Because efficiency growth such as technological change makes sustainability so easily to achieve, it’s more interesting to hold technological change constant and see what happens when technology and other forms of efficiency don’t improve.
We think of efficiency A as possibly changing depending on the age structure of the workforce, but in the Solow Model there is no change to the age structure.
This model is almost as simple as the Malthusian model. One difference is that in Solow’s model the population growth rate never changes. A second difference is that in Solow’s model, there is not only labour, but also physical capital. People can either eat the output Y or save some of it. The saved Y is invested and is transformed into capital. Think of Y as corn, which you can either eat or save for planting next spring.
The production function F(K,L) can be any positive monotonic function of K and L so long as
- Both K and L are essential to production. If one of them is equal to zero, then output must be equal to zero.
- There is some degree of substitutability between K and L.
- F(K,L) demonstrates diminishing marginal returns to K and to L.
- F(K,L) exhibits constant returns to scale. As noted above, if you double the inputs, you double the output. Similarly, if you divide the inputs by some number, you divide the output by that same number. This makes it possible easily to express the production function in terms of output per worker, dividing by L.
Assuming constant returns to scale, Solow divides everything by the number of workers so that there is one production process y=f(k) which uses capital-per-worker k to produce output-per-worker y. Capital-per-worker is denoted by lowercase k and is called the capital-to-labour ratio or capital:labour ratio.
Figure 18-1 shows output-per-worker as a function of the capital:labour ratio. This function demonstrates diminishing marginal returns to k.
Figure 18-2 shows the same function multiplied by fraction s. s is the savings rate.
In the Solow model, we produce y, which represents output per worker or output per person (since all people work). Fraction (1-s) of this is consumed, and fraction s is saved/invested into the stock of physical capital, K. Every year, K increases by sY. It can be shown (if interested, see the end of this chapter) that the amount of savings needed to keep the capital:labour ratio steady must satisfy Equation 18-1.
The amount saved per worker must equal the population growth rate/labour force growth rate multiplied by the capital:labour ratio.
The original Solow model also included another term, called d. d is the rate at which physical capital depreciates, by rusting away or becoming obsolete. If you include d in the model, Equation 18-1 becomes sy=(n+d)k. We show that at the end of this chapter.
According to Equation 18-1, sustainability requires that a population must save enough to offset something, something to do with population growth. Let’s have a look at Figure 18-3 to learn more.
The blue line above, sy, represents the left-hand side of the Solow condition (Equation 18-1), while the yellow line, nk, represents the right-hand side. Where these lines intersect is the level of k, where the Solow condition is satisfied.
The lines intersect at a particular level of k, called the steady state capital:labour ratio, or k*.
Wherever the blue savings line is higher than the yellow population line, sy > nk, and the capital:labour ratio rises. As it rises, we move rightward along the horizontal axis until we get to k*.
Wherever the blue savings line is lower than the yellow population line, sy < nk, and the capital:labour ratio falls. We call this “capital shallowing“. As k falls, we move leftward along the horizontal axis until we get to k*.
So whatever capital:labour ratio k we start out with, we tend to reach k*. k* is an equilibrium.
If we start out with a k that is higher than k*, k falls. That’s because k is so high that diminishing returns are kicking in, and the extra output we get from our investment is not enough to prevent capital shallowing. If we start off with a low k, we will find that our savings and investments are so productive that capital accumulates faster than population.
This is exciting! Whatever savings rate we choose, we can achieve:
Even though the labour force in the Solow model is constantly growing at rate n, the amount each worker/person consumes will never fall. Consumption per person never falls = that’s what theoretical macroeconomists call sustainability. Like a lot in economics, the conception of sustainability is very anthropocentric.
Population growth will cause capital shallowing, but we can save and reverse that capital shallowing. By means of saving, the capital stock can grow as quickly as the population.
If the population growth rate were to rise for some reason, our yellow population line would become steeper. It would intersect the blue savings line at a lower k*. This would means lower output per worker and lower consumption per worker. However, the output and consumption per worker are still constant every year. They are still sustainable. Consumption may be reduced because population growth has accelerated, but it is still sustainable, unless it falls below the critical threshold needed to support human life.
If the efficiency level A were to rise for some reason, again our blue savings line would pivot up and intersect the yellow population line at a higher steady state k*. Output per worker would rise, and so would consumption per worker, since output has increased and the savings rate has not changed.
If the savings rate were to increase for some reason, our blue savings line would pivot up and intersect the yellow population line at a higher steady state k*. This would result in a higher steady state level of output per person; however, because a higher fraction of that output is being saved, it’s not clear whether consumption per person would actually increase.
We can solve mathematically for the savings rate that would result in the highest level of consumption per person. That savings rate is the one that achieves the “golden rule” k*. The golden rule k* is found where the slope of the per-worker production function (y=Y/L=Af(k*)) is equal to n.
The Solow model tells us that, if we save, we can achieve sustainability of consumption despite population growth, unless the rate of population growth is so very high that k* and the resulting consumption (=(1-s)y) are too low to support life. In the real world, things are more complicated. The Solow model ignores the natural environment. It ignores the fact that not everyone works. And it assumes that the rate of population growth doesn’t affect how much people save. In the real world there is environmental degradation, there is dependency, and there is the likelihood that dependency reduces workers’ savings.
When the rate of population growth is increasing, it is likely that the young dependency ratio (YDR) and total dependency ratio (TDR) are increasing. It is likely that savings will fall as parents devote resources to caring for the young. Kelley (1998) calls this the youth dependency effect. It is also likely that governments will spend on education and health care for the young, diverting money away from investment in various forms of physical and knowledge capital. Most of the spending on children will not improve the productivity of the current working generation. Kelley calls this the investment-diversion effect.
A. C. Kelley (1998) reviewed the many journal articles on population and economic growth available at that time. He concluded that,ceteris paribus, the rate of population growth likely reduces the standard of living through the youth dependency effect, the investment-diversion effect, and capital shallowing. However, there is no clear empirical relationship between the population growth rate and per capita output. Many other important factors also influence per capita output, factors like the economy’s overall size, its civil and political institutions, its educational achievement, and its openness to trade (Bloom 2003).
If we observe an apparent negative correlation between income growth and population growth, we have to remember that causation can flow both ways in a negative feedback loop. Low economic growth could mean lower rates of female education, low rates of female employment, higher rates of infant mortality, and a smaller social safety net, all of which tend to increase fertility.
The graph below shows GDP per capita growth, adjusted for inflation, between 1960 and 2000, for 98 different countries, as a function of the average population growth rate for each country between 1960 and 2000. A simple best fit line has been generated. Variation in population growth rates “explains” 25% of the variation in GDP per capita growth, or vice versa. 75% of the variation remains unexplained.
The Solow model paints a scenario where saving prevents consumption per worker from falling as the population grows. Could that work if there are natural resources needed for production?
The Solow model will continue to generate sustainable output and consumption per worker when the production function includes a renewable resource. The renewable resource must be harvested sustainably. No more of the resource each year can be harvested than can grow back in one year. In fact, we should harvest less than that, because the stock must continue to grow as long as population grows.
When we add a non-renewable resource, such as oil, coal, or lithium to the production function, the Solow model can no longer generate sustainable output or sustainable consumption in the presence of population growth, at least not as long as population growth is geometric or exponential.
In the Solow model, savings can compensate for capital shallowing and capital depreciation, or savings can compensate for non-renewable resource depletion, but not both.
If there were no population growth (i.e. n=0) and no depreciation of capital (i.e. d=0), then consumption could be sustained despite the depletion of non-renewables like oil. Solow (1974) and Hartwick (1977) showed that IF physical capital can substitute to some degree for non-renewables like oil, and IF enough physical capital were accumulated to make up for the diminishing stock, then consumption could be sustained indefinitely.
Hartwick derived the formula for the precise amount of savings needed to make up for declining non-renewable resource stocks. The amount needed is equal to the amount of non-renewable resource extracted multiplied by rent (price minus marginal cost) on the marginal ton. This amount is known as Total Hotelling Rent.
Hartwick’s Rule tells us to invest non-renewable resource rents in other forms of capital. This will keep our output and consumption steady, so long as the savings rate is not affected and dependency is not an issue. If there is geometric or exponential population growth, sustainability is not achievable; however, if there is arithmetic or quasi-arithmetic population growth, sustainability can be achieved by investing even more than Total Hotelling Rent.
In real life, our population has grown and our capital stock has more than kept up, because of technological improvements, other efficiency improvements, and the colonization of new lands and peoples. Neither Malthus nor Solow nor Hartwick include technological change in their models. That is because the introduction of technical change into a mathematical model will too easily generate sustainability.
Hartwick’s Rule depends on the production function being the kind where the inputs are multiplied together to yield the output. This means that it is always possible to make up for a shrinking amount of one input by using more of another input. In Hartwick’s model, an expanding stock of K makes up for a diminishing stock of non-renewable resource. R. Herman Daly (1990), one of the founders of Ecological Economics, has pointed out that there may be critical thresholds below which all the physical capital in the world cannot make up for the loss of natural or environmental capital.
Ecological Economics was established as a discipline in 1990 by economists who were concerned that traditional economics does not adequately consider the economy’s size and the population’s size relative to the carrying capacity of the environment. We will discuss population size in our next Chapter.
We can take Hartwick’s Rule as suggestive rather than definitive. It recommends something that common sense immediately recognizes: do not allow the stock of your capital to diminish. Invest the profits you earn from nonrenewable resources. Save for the day when your resources run out.
Many nations have created sovereign wealth funds to invest the tax revenue that their governments collect from the oil and gas industry. Alaska, Kuwait, and Norway have such sovereign wealth funds. Alberta contributed to its Sovereign Heritage Savings Trust Fund between 1976-1988, and 2005-8, but otherwise the revenue has been used by the government or distributed among Alberta’s residents. Chile and Venezuela use resource tax revenues to help with government spending needs.
Genuine Savings, also known as Adjusted Net Savings, is an estimate of whether a nation’s capital stock (including physical, human, natural, and environmental capital) is really growing or not. If genuine savings is positive, then the nation is wisely building up its capital stock. If genuine savings is negative, then the nation is dissipating its capital.
Here is the calculation:If the rate of genuine savings is positive, the nation is accumulating capital. The question then is whether the rate of capital accumulation is high enough to match population growth.
In Table 18-1 we see estimates of Genuine Savings (as a rate) for several countries, computed by the World Bank.
Compiled by Pauline Galoustian. Sources: World Bank (data.worldbank.org) /United Nations Population Division/Eurostat: Demographic Statistics/United Nations Statistical Division/Secretariat of the Pacific Community (CC BY 4.0)
How can we tell whether the Genuine Savings of a country is enough to keep its consumption sustainable? If the population is not growing, any genuine savings above zero indicates an increase in the productive capacity of the economy and an improvement in its ability to provide consumption sustainably into the future. If the population is growing geometrically or exponentially (as is usually the case), then Genuine Savings needs to be impossibly high unless technical change and efficiency improvements are occurring (as is usually the case). In the situation of population growth with technical change, we don’t have a simple equation to calculate how high a nation’s Genuine Savings needs to be.
The World Bank (2011), in its Appendix E, tried to estimate that anyway. Their calculations for the year 2005 purported to show that Canada’s genuine savings that year had been sufficient to cover its population growth. They also estimated that the United States had needed to save an additional 2% of gross national income in 2005 to keep its capital-per-person (they did not calculate it per worker) intact.
In our next chapter, we’ll study the effects on the economy of the absolute size of the population.
The end-of-chapter questions follow the Appendix below.
Let K(t) be the capital stock at time t. s is the savings rate. d is the rate at which capital breaks down or becomes obsolete: the depreciation rate. We will set d = 0 for simplicity. L(t), the labour force at time t, is growing every year at rate n.
The production function is:
A(t) is efficiency or technology at time t and we just hold it constant, meaning:
Dividing by L(t), we write y(t) = Af (1, k(t)) where y is output per worker and k is capital per worker.
We can ignore the 1 and write:
The following equation shows how the capital stock grows from year to year:
Translation: capital next year = capital this year + amount saved minus capital lost to decay and obsolescence. Rearranging:
What we’re going to do now is divide everything by K(t).
Using the fact that,
the equation becomes:
The left-hand side is “percentage change in K”. Now the percentage change in little k is equal (by definition) to the percentage change in K minus the percentage change in L. The percentage change in L is the population growth rate, since in this model, everyone is in the labour force. So let’s replace the left-hand side of our Solow equation with the percentage change in k plus n, the population growth rate.
Now multiply both sides by k(t) and we have our final version:
This is Equation 18.1 It tells us that, for capital-per-worker to be constant over time – i.e. for the left hand side of this equation to be equal to zero, – savings per worker must equal the population growth rate multiplied by capital per worker.
1. What assumptions about the economy does the Solow model make?
2. In the Solow model, what are three ways that n, the rate of growth of population, can affect the steady-state capital:labour ratio, k*?
3. What is Hartwick’s Rule and how has it been criticized?
4. If in 2008, Country X sells 100,000,000 barrels of oil, and if the marginal cost of this oil is $92 per barrel, and if the price of this oil is $100 per barrel, what is Total Hotelling Rent for Country X in 2008?
5. Country Y in 2008 has:
- investment in physical capital=15.9 percent of GNI (gross national income)
- current spending on education = 5 percent of GNI
- depreciation of physical capital = 11.5 percent of GNI
- Total Hotelling Rent of 0
- over-harvesting renewable resources valued at 0.1 percent of GNI
- estimated damages from pollution valued at 0.3 percent of GNI
a) What is genuine savings for Country Y?
b) What can you tell me about Country Y?
- The intuition is that we keep raising k* until the net benefit of doing so is zero. The net benefit of raising k* is the resulting increase in output per worker minus the extra n units of output required to prevent capital shallowing. (Van Gaasbeck (2022)). ↵
- In quasi-arithmetic growth, N(t) = a + b(t) ↵ |
This guide covers Series RL Circuit Analysis, its Phasor Diagram, Power & Impedance Triangle, and several solved examples.
In a purely resistive AC circuit, any inductive effects are considered negligible. Similarly, in a purely inductive AC circuit, any resistive effects are considered extremely small, and as a result they are omitted from any calculations. In many AC circuits, however, the load is actually a combination of both resistance and inductance. That is, the circuit can no longer be treated as either purely resistive or as purely inductive.
The combination of a resistor and inductor connected in series to an AC source is called a series RL circuit. Figure 1 shows a resistor and a pure or ideal inductor connected in series with an AC voltage source.
The current flow in the circuit causes voltage drops to be produced across the inductor and the resistor. These voltages are proportional to the current in the circuit and the individual resistance and inductive reactance values.
As in any series circuit the current will be the same value throughout the circuit. The resistor voltage (ER) and the inductor voltage (EL) expressed in terms of Ohm’s law are
Figure 1 Series RL circuit diagram
The total opposition to current flow in any AC circuit is called impedance. In a series RL circuit, this total opposition is due to a combination of both resistance (R) and inductive reactance (XL). The symbol for impedance is Z, and like resistance and reactance, it too is measured in ohms.
From Ohm’s law, the impedance of a circuit will be equal to the total supply voltage (ET) divided by the circuit current:
It was previously shown that the current flowing through a pure resistance was in phase with the voltage across the resistance and that the current through a pure inductance lagged the voltage across the inductance by 90 degrees. For this reason, in the series RL circuit the two voltage drops will not be directly additive but will be a vector sum.
The relationship between the current and voltages in a series RL circuit is shown in the vector (phasor) diagram of Figure 2 and can be summarized as follows:
- The reference vector is labeled I and represents the current in the circuit, which is common to all circuit elements.
- Since the voltage across the resistor is in phase with the current flowing through it, the voltage vector ER, it is drawn superimposed on the current vector.
- The inductor voltage EL leads the current by 90 degrees and is drawn leading the current vector by 90 degrees.
- The total supply voltage (ET) is the vector sum of the resistor and inductor voltages:
- The phase shift between the applied voltage and current is between 0 and 90 degrees.
- As the frequency increases, the inductive reactance (XL) increases, which causes the phase angle, or shift between the applied voltage and current, to increase.
Figure 2 Series RL circuit vector (phasor) diagram.
Due to the phase shift created by the inductor, the impedance of a series RL circuit cannot be found by simply adding the resistance and inductive reactance values. The total impedance of a series RL circuit, similar to its total voltage, is the vector sum of the resistance and inductive reactance.
The impedance triangle for a series RL circuit is shown in Figure 3. Note that the impedance triangle is geometrically similar to the circuit vector diagram and will have the same phase angle theta (θ). The reason for this is that the voltage drops for the resistor and the inductor are a result of the current flow in the circuit and their respective opposition. Equations used to solve the impedance triangle include:
Figure 3 Series RL circuit Impedance triangle.
Impedance Calculation in Series RL Circuit Example 1
Problem: An AC series RL circuit is made up of a resistor that has a resistance value of 150 Ω and an inductor that has an inductive reactance value of 100 Ω. Calculate the impedance and the phase angle theta (θ) of the circuit.
Once the impedance of a circuit is found it is possible to find the current by using Ohm’s law and substituting Z for R as follows:
Since the current is the same throughout the series circuit, the individual voltage drops across the inductor and resistor can be calculated by applying Ohm’s law as follows:
RL Series Circuit Calculations Example 2
Problem: For the series RL circuit shown in Figure 4:
- Calculate the value of the current flow.
- Calculate the value of the voltage drop across the resistor.
- Calculate the value of the voltage drop across the inductor.
- Calculate the circuit phase angle based on the voltage drops across the resistor and inductor.
- Express all voltages in polar notation.
- Use a calculator to convert all voltages to rectangular notation.
Figure 4 RL series circuit for Example 2.
The various power components associated with the series RL circuit are shown in Figure 5 and can be identified as follows:
- True power is measured in watts (W) and is the power drawn by the resistive component of the circuit. For a pure resistor the voltage and current are in phase, and power dissipated as heat is calculated by multiplying voltage by current (W=ER× IR).
- Reactive power is measured in volt-amperes reactive (VARs). Reactive power is the power continually stored and discharged by the magnetic field of the inductive load. For purely inductive loads, the voltage and current are 90 degrees out of phase, and true power in watts is zero. The inductive reactive power is calculated by multiplying the inductor voltage by its current (VARs=EL×IL)
- Apparent power is measured in volt-amperes (VA) and is the combination of the reactive and true power. For a series RL circuit the phase shift between the applied voltage and current is between 0 and 90 degrees. The apparent power or volt-amps is calculated by multiplying the applied voltage by the current flow (VA=ET×IT).
Figure 5 Power components associated with the RL series circuit.
The power triangle of Figure 6 shows the relationship between the various power components of a series RL circuit. In this triangle:
- The length of the hypotenuse of a right-angle triangle represents the apparent power.
- The angle theta (θ) is used to represent the phase difference.
- The side adjacent to theta (θ) represents the true power.
- The side opposite theta (θ) represents the reactive power.
- The power triangle is geometrically similar to the impedance triangle and the series RL circuit vector diagram.
Figure 6 Series RL circuit power triangle.
Power Calculations in RL Series Circuit Example 3
Problem: For the series RL circuit shown in Figure 7, determine:
- True power.
- Inductive reactive power.
- Apparent power.
Figure 7 RL series circuit for Example 3.
The power factor (PF) for any AC circuit is the ratio of the true power (also called real power) to the apparent power:
Power factor is a measure of how effectively equipment converts electric current to useful power output, such as heat, light, or mechanical motion. The power factor for a RL circuit is the ratio of the actual power dissipation to apparent power and can be summarized as follows:
- The power factor ranges from 0 to 1 and is sometimes expressed as a percentage.
- A 0 percent PF indicates a purely reactive load, while 100 percent PF indicates a purely resistive load.
- For circuits containing both resistance and inductive reactance, the power factor is said to be lagging (current lags) in some value between 0 and 1.
- The greater the power factor, the more resistive the circuit; the lower power factor, the more reactive the circuit.
- Circuit power factor is an indication of the portion of volt-amperes that are actually true power; a high PF indicates a high percentage of the total power is true power.
For many practical applications, the power factor of a circuit is determined by metering total circuit voltage, current, and power, as illustrated in the circuit of Figure 8. The power factor can then be determined by dividing the reading of the wattmeter by the product of the voltmeter and ammeter readings as follows:
Figure 8 Determining RL circuit power factor.
The power factor is not an angular measure but a numerical ratio with a value between 0 and 1. As the phase angle between the source voltage and current increases, the power factor decreases, indicating an increasingly reactive circuit. Any of the following equations can be used to calculate the power factor of a series RL circuit:
RL Series Circuit Example 4
Problem: For the series RL circuit shown in Figure 9, determine:
- Inductive reactance (XL).
- Impedance (Z).
- Current (I).
- Voltage drop across the resistor (ER) and inductor (EL).
- The angle theta (θ) and power factor (PF) for the circuit.
- True power (W), reactive power (VARs), apparent power (VA).
Figure 9 RL Circuit for Example 4.
- Step 1. Make a table and record all known values.
Step 2. Calculate XL and enter the value in the table.
Step 3. Calculate Z and enter the value in the table.
Step 4. Calculate IT, IR, and IL and enter the values in the table.
Step 5. Calculate ER and EL and enter the values in the table.
Step 6. Calculate the angle θ and PF for the circuit and enter the values in the table.
Step 7. Calculate the W, VARs, and VA for the circuit and enter the values in the table.
A real inductor has resistance due to the wire. It is impossible to have a pure inductance because all coils, relays, or solenoids will have a certain amount of resistance, no matter how small, associated with the coils turns of wire being used. This being the case, we can consider our simple coil as being a resistance in series with a pure inductance.
- Define the term impedance as it applies to AC circuits.
- What symbol is use to represent impedance?
- A circuit consists of a resistance of 20 Ω and an inductive reactance of 40 Ω connected in series and supplied from a 240-volt, 60-Hz source. Determine:
- The circuit impedance.
- Amount of current flow.
- The phase angle theta (θ) of the circuit.
- For the series RL circuit vector (phasor) diagram shown in Figure 10, determine the value of the voltage drop across the inductor.
Figure 10 Vector for review question 4.
5. The known quantities in a given series RL circuit are as follows: Resistance equals 8 Ω, inductive reactance equals 39 Ω, current equals 3 A, and the applied voltage is 120 volts, 60 Hz. Determine the following unknown quantities:
- Voltage across the resistor.
- Voltage across the inductor.
- Angle by which the applied voltage leads the current.
6. A wattmeter connected to a 240-volt, 60-Hz series RL circuit indicates a reading of 691 watts. A clamp-on ammeter used to measure current flow indicates a current of 4.8 A. Determine the:
- True power.
- Apparent power.
- Reactive power.
- Circuit power factor.
7. For the series RL circuit shown in Figure 11, determine:
- Apparent power.
- True power.
- Reactive power.
- Circuit power factor.
Figure 11 RL Series Circuit for review question 7.
8. Complete a table for all given and unknown quantities for the series RL circuit shown in Figure 12.
Figure 12 Circuit for review question 8.
9. The frequency to an RL series circuit is decreased. What effect will this have on the phase angle between the applied voltage and current? Why?
Review Questions – Answers
- The total opposition offered to the current flow in the AC circuit.
- (a) 44.7 Ω (b) 5.4 A (c) 63.4°
- 352 V
- (a) 40 Ω (b) 24 V (c) 117 V (d) 78.4 °
- (a) 691 Watts (b) 1152 VA (c) 921.6 VAR (d) 60%
- (a) 18.1 kVA, (b) 17 k Watts, (c) 6,380 VAR, (d) 94%
|E||I||R /XL /Z||W/ VA /VARs||PF|
|R||155 V||3.1 A||50 Ω||480.5 W||0|
|L||155 V||3.1 A||50 Ω||480.5 VARs||90|
|Total||220 V||3.1 A||70.7 Ω||682 VA||45||71 %|
9. The inductive reactance (XL) deceases causing the phase angle between the applied voltage and current to decrease. |
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Home / physics / modern physics problems and solutions pdf Modern Physics Problems And Solutions Pdf 29/08/2021 Modern Physics ProblemsTwo locations of modern physics are addressed via example troubles on this page. Special Relativity troubles ask you to relate the observations of two observers measuring the exact same thing. In Quantum Mechanics troubles, you might look at wave or pwrite-up behavior of light and subatomic pshort articles. As always, fundamental definitions difficulties are found via various other Definitions examples. How to Solve Special Relativity Problems1.Identify the ProblemAny problem that asks you to relate the monitorings made by two observers measuring the exact same point is a relativity trouble. Keep in mind that sometimes one of the observers is implied—for instance, if you are asked for length of a somepoint “in the structure of the electron” there is an implied observer in the very same recommendation frame as the electron.You watching: Modern physics problems and solutions pdf2. Draw a PictureIn Special Relativity problems, you relate the monitorings made by 2 observers in different referral frames measuring the exact same thing. Therefore, valuable photos are images of the activity being described via the 2 observers and the experiment (the thing that they are measuring) clearly indicated. The family member velocity of the two observers need to also be shown. If the observers are measuring size or a time interval, one of the frames is the appropriate framework and have to be identified as such. If the observers are measuring velocity, your mathematical solution will be simpler if you choose the ‘ frame to be that of the observer whose velocity measurement you already understand.3. Select the RelationTright here are three crucial relationships for Special Relativity. The correct relation to understand also a case is based upon what the observers are measuring. If the 2 observers are measuring the length of something, their dimensions are related by When two observers time the very same event, their measurements are associated byWhen two observers meacertain the velocity of a 3rd object, their measurements are associated by Note that the symbol v in all three equations refers to the loved one velocity in between the two observers (reference frames.) In the instance of the 3rd equation, make sure to asauthorize v the authorize to correctly reexisting the direction of the ‘ observer’s motion according to the other observer.4. Solve the ProblemIf your picture clearly reflects your choice of frames, the experiment, and the family member motion in between frames, fixing the problem is merely a issue of filling values from the image into the equation and doing algebra.Keep in mind that for the velocity equation you have to take into consideration the authorize of u, u’, and also v. Sign is provided by the direction of the velocity. Note that v is the velocity of the observer in the ‘ framework as measured by the other observer.See more: Fossil Black Mechanical Watch Es For Sale, Fossil Automatic, Mechanical & Skeleton Watches 5. Understand the ResultsOnce you have completed the problem, look at it aget. Does your answer make sense? Did it give the habits you intuitively expected to find? (Remember, you suppose objects relocating loved one to you to be shorter, and occasions in frames relocating family member to you to take longer. Relative velocities need to never before be higher than c, and also objects approaching each other should have a greater family member velocity than either of their speeds family member to some other allude.) If your numerical answers carry out not make feeling, the many prevalent errors are incorrect alternative for appropriate frame or incorrect sign of v in the velocity equation. Can you currently do measures that led to you problems earlier? Can you explain in words what is happening? Help! I can’t uncover an example that looks favor the problem I have to work!Yes, my difficulty is absolutely a Modern Physics problem.In that instance, think even more generally around what makes a helpful instance. Remember, you were given your assignment to practice the problem solving strategy, not because the answers to your problems are specifically amazing. An example in which you merely substitute your numbers for those in the difficulty will certainly offer you exercise entering numbers on your calculator however will teach you nopoint about physics, and as soon as you take your exam eextremely difficulty on it will feel brand-new and also different to you. So think around your example as support for helping you to exercise the problem resolving method.In certain, if two troubles attend to the very same key physics they will certainly be addressed in the exact same method regardmuch less of what quantity you are asked to find. If two observers meacertain the size of some object, for example, the difficulty is approached in an the same way regardmuch less of whether you desire to understand the size measured by observer 1, the size measured by observer 2, or the loved one velocity between them.That sassist, tbelow are many type of different locations of contemporary physics and many kind of subtopics within those areas. You will certainly not uncover examples of topics such as cosmology or product scientific research on this web page. |
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
How many trains can you make which are the same length as Matt's, using rods that are identical?
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
An investigation that gives you the opportunity to make and justify predictions.
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
Can you find the chosen number from the grid using the clues?
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
Can you work out some different ways to balance this equation?
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?
Have a go at balancing this equation. Can you find different ways of doing it?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?
The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.
Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?
Number problems at primary level that may require resilience.
Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my square?
How will you work out which numbers have been used to create this multiplication square?
If you count from 1 to 20 and clap more loudly on the numbers in the two times table, as well as saying those numbers loudly, which numbers will be loud?
Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
I throw three dice and get 5, 3 and 2. Add the scores on the three dice. What do you get? Now multiply the scores. What do you notice?
If there is a ring of six chairs and thirty children must either sit on a chair or stand behind one, how many children will be behind each chair?
Follow the clues to find the mystery number.
You can make a calculator count for you by any number you choose. You can count by ones to reach 24. You can count by twos to reach 24. What else can you count by to reach 24?
Find the squares that Froggie skips onto to get to the pumpkin patch. She starts on 3 and finishes on 30, but she lands only on a square that has a number 3 more than the square she skips from.
Can you sort numbers into sets? Can you give each set a name?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
Can you place the numbers from 1 to 10 in the grid?
Can you help the children in Mrs Trimmer's class make different shapes out of a loop of string?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
This activity focuses on doubling multiples of five.
How many different sets of numbers with at least four members can you find in the numbers in this box?
Can you make square numbers by adding two prime numbers together?
Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?
Number problems at primary level to work on with others.
How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?
If you have only four weights, where could you place them in order to balance this equaliser? |
Figure 5: A comparison of the amount of accumulated dissolved solids
between a tank that receives 10-percent daily water changes versus one
that receives 70-percent weekly water changes.
10% Daily - 70% Weekly
to endure a third of the pollution the fish in the other tank endure
every single day.
It would take a series of complex and carefully structured
experiments to determine empirically whether fish are more harmed
by a constant level of pollution or by varying pollution levels.
Intuitively, it seems that overall exposure is more significant. After
all, what would be your choice if you could choose between living
in a town with a consistently high level of air pollution and smog
and living in a town where once a week pollution and smog reach
that same high level, but then it drops to one-third that level and
gradually rises back to the maximum over the rest of the week?
Even the deadliest poisons are relatively harmless at a certain
low concentration. As the concentration increases, so does the
11 It is therefore reasonable to conclude that a regimen
which provides pollution concentration levels that are frequently
lower than those with another regimen will be less harmful.
There is nothing to suggest that living things can get used to
poisons in their environment, making a steady concentration of
a toxin preferable to a cyclical concentration that often is below
the steady one. A large weekly change (as against smaller daily
changes) exposes your fish to a considerable amount of time at
lower pollution concentrations and minimizes the time they spend
at the highest concentration.
What about the argument that large water changes
stress fish by changing water chemistry (other than what
pertains to pollution) too quickly? There does seem to
be some evidence that a massive difference between the
parameters of the tank water and of the replacement
water can shock fish. The underlying assumption here is
that metabolic processes and evaporation change the
chemistry of the water so that over time it becomes vastly
different from that of your tap water. A simple solution is
not to let that happen!
Why wait until water quality degrades to do something
about it? If you change water often enough and in
sufficient volumes, there will be no major changes in its
chemistry, and the new water won’t stress your fish at all.
Often the claim is made that small daily changes are preferable to large weekly ones because they provide a consistently higher water quality. This claim is demonstrably false. A regimen of 10 percent daily changes does produce a consistent water quality, but it is a much poorer water quality than a 70 percent weekly change produces after every change and for several days thereafter.
We’ve seen how contrary to much advice large weekly changes
provide a better environment for your fish than do small daily
changes when the total amount of water changed is the same (here
10 percent daily and 70 percent weekly). In Part 2 we will challenge
common wisdom about the benefits of different total percentages
of water change and reveal more startling discoveries of this
1. Removing 20 percent of the water removes 20 percent of the total
dissolved pollution, which total is made up of 80 percent of the first
week’s leftover pollution and 100 percent of the second’s additional
pollution, leaving 0.80( 80) = 64 percent of first week and 0.80( 100)
= 80 percent of the second week. Alternatively, you could say that
of the 180 percent of one week’s pollution still in the tank before
the second water change, 144 percent, or 0.80(180), remains in the
aquarium after the second water change.
2. Appropriately pronounced pee-yoo, pyoo, or even poo.
3. If the biofilter is immature or inadequate, then ammonia and/or
nitrite will also be part of the total dissolved pollution.
4. Or, at the end of the first day there are x pollution units dissolved in
the water. At the end of the second day there are 2x, the third 3x, etc.
If on the seventh day (accumulation 7x) half the water is removed
and replaced, half of the pollution ( 3.5x) is also removed, leaving
3.5x in the tank.
5. Calculations for Figure 1: The data for this graph are generated by
the simple equation y = x + 10 for x > 0. Or using a spreadsheet, the
formula for B1, starting with A1 at a value of 0, is =SUM(A1, 10), and
continuing on across.
6. Evaporation increases the concentration even further, of course,
but to keep things simple we will assume a very tight-fitting
lid and minimal evaporation. In fact, replacing evaporation with
tap water actually increases the concentration of some dissolved
substances—further motivation to change water rather than to top
7. Calculations for Figure 2: With A1 at 0, the formula for B1 is
=SUM(A1, 10), continuing for six cells. The value for G1 is 0, with
the cycle repeating every seven days.
8. Calculations for Figure 3: A1= 10, B1 =(SUM(A1, 10)) *0.9, continuing
9. The exact value at which the graph reaches asymptote depends on
how many decimal places you carry in your calculations. The more
places you use, the more water changes it will take to reach a steady
value for the remaining pollution at a given rate of water changing.
In fact, if you use infinite decimal places, the graph will never reach
asymptote. After each change there will be an infinitesimally greater
amount of pollution remaining. However, the difference between 1.0
x 10-15 pu and 2.0 x 10-15 pu can hardly be of consequence. In fact,
carrying any decimal places at all is certainly inaccurate, since you
will not be changing precisely the same amount of water each time.
Change a few more drops than last time, and the remaining dissolved
pollution will actually be less than it was after the last change. This
is for the most part way beyond our ability to measure and certainly
a fish’s ability to perceive. Buckets, not pipettes!
10. Calculations for Figure 4: The formula for B1 is =SUM(A1, 10),
continuing for six cells. The value for G1 is =(SUM(F1, 10))*0.3, with
the cycle repeating every 7 days.
11. Toxicity is typically defined in terms of the LC50, the concentration
at which 50 percent of victims exposed die. D
Tropical Fish Hobbyist www.tfhmagazine.com |
|Ch 01: Units & Vectors||2hrs & 22mins||0% complete||WorksheetStart|
|Ch 02: 1D Motion (Kinematics)||3hrs & 11mins||0% complete||WorksheetStart|
|Ch 03: 2D Motion (Projectile Motion)||3hrs & 8mins||0% complete||WorksheetStart|
|Ch 04: Intro to Forces (Dynamics)||3hrs & 42mins||0% complete||WorksheetStart|
|Ch 05: Friction, Inclines, Systems||4hrs & 32mins||0% complete||WorksheetStart|
|Ch 06: Centripetal Forces & Gravitation||3hrs & 51mins||0% complete||WorksheetStart|
|Ch 07: Work & Energy||3hrs & 55mins||0% complete||WorksheetStart|
|Ch 08: Conservation of Energy||6hrs & 54mins||0% complete||WorksheetStart|
|Ch 09: Momentum & Impulse||5hrs & 35mins||0% complete||WorksheetStart|
|Ch 10: Rotational Kinematics||3hrs & 4mins||0% complete||WorksheetStart|
|Ch 11: Rotational Inertia & Energy||7hrs & 7mins||0% complete||WorksheetStart|
|Ch 12: Torque & Rotational Dynamics||2hrs & 9mins||0% complete||WorksheetStart|
|Ch 13: Rotational Equilibrium||4hrs & 10mins||0% complete||WorksheetStart|
|Ch 14: Angular Momentum||3hrs & 6mins||0% complete||WorksheetStart|
|Ch 15: Periodic Motion (NEW)||2hrs & 17mins||0% complete||WorksheetStart|
|Ch 15: Periodic Motion (Oscillations)||3hrs & 16mins||0% complete||WorksheetStart|
|Ch 16: Waves & Sound||3hrs & 25mins||0% complete||WorksheetStart|
|Ch 17: Fluid Mechanics||4hrs & 39mins||0% complete||WorksheetStart|
|Ch 18: Heat and Temperature||4hrs & 9mins||0% complete||WorksheetStart|
|Ch 19: Kinetic Theory of Ideal Gasses||1hr & 40mins||0% complete||WorksheetStart|
|Ch 20: The First Law of Thermodynamics||1hr & 49mins||0% complete||WorksheetStart|
|Ch 21: The Second Law of Thermodynamics||4hrs & 56mins||0% complete||WorksheetStart|
|Ch 22: Electric Force & Field; Gauss' Law||3hrs & 32mins||0% complete||WorksheetStart|
|Ch 23: Electric Potential||1hr & 55mins||0% complete||WorksheetStart|
|Ch 24: Capacitors & Dielectrics||2hrs & 2mins||0% complete||WorksheetStart|
|Ch 25: Resistors & DC Circuits||3hrs & 20mins||0% complete||WorksheetStart|
|Ch 26: Magnetic Fields and Forces||2hrs & 25mins||0% complete||WorksheetStart|
|Ch 27: Sources of Magnetic Field||2hrs & 30mins||0% complete||WorksheetStart|
|Ch 28: Induction and Inductance||3hrs & 38mins||0% complete||WorksheetStart|
|Ch 29: Alternating Current||2hrs & 37mins||0% complete||WorksheetStart|
|Ch 30: Electromagnetic Waves||1hr & 12mins||0% complete||WorksheetStart|
|Ch 31: Geometric Optics||3hrs||0% complete||WorksheetStart|
|Ch 32: Wave Optics||1hr & 15mins||0% complete||WorksheetStart|
|Ch 34: Special Relativity||2hrs & 10mins||0% complete||WorksheetStart|
|Ch 35: Particle-Wave Duality||Not available yet|
|Ch 36: Atomic Structure||Not available yet|
|Ch 37: Nuclear Physics||Not available yet|
|Ch 38: Quantum Mechanics||Not available yet|
|Intro to Moment of Inertia||30 mins||0 completed|
|Moment of Inertia via Integration||19 mins||0 completed|
|More Conservation of Energy Problems||55 mins||0 completed|
|Moment of Inertia of Systems||23 mins||0 completed|
|Conservation of Energy in Rolling Motion||45 mins||0 completed|
|Moment of Inertia & Mass Distribution||10 mins||0 completed|
|Intro to Rotational Kinetic Energy||17 mins||0 completed|
|Energy of Rolling Motion||18 mins||0 completed|
|Types of Motion & Energy||24 mins||0 completed|
|Parallel Axis Theorem||14 mins||0 completed|
|Conservation of Energy with Rotation||36 mins||0 completed|
|Torque with Kinematic Equations||59 mins||0 completed|
|Rotational Dynamics with Two Motions||51 mins||0 completed|
|Rotational Dynamics of Rolling Motion||27 mins||0 completed|
Concept #1: Intro to Moment of Inertia
Hey guys so in this video we're going to talk about the moment of inertia which is really just the rotational equivalent of mass, so in linear motion you have mass in rotational motion you have moment of inertia, you can think of moment of inertia as rotational mass or rotational inertia, the name is tricky moment of inertia sort it's sort of a name from Engineering but don't get confused its just rotational mass let's check it out. So, you might remember that when we solve the motion problems which were those problems we solved with the 3 to 4 motion equations or kinematic equations those equations didn't have a mass in it, right? So they did not depend on mass but once you moved into energy problems or force problems dynamics problems mass was important so here is kinetic energy is 1/2MV squared, gravitational potential energy has a mass, force F=MA that has a mass as well so before we can talk about energy in rotation in forces in rotation we have to talk about mass in rotation because it's a little different, again linear motion mass and rotational motion we have this new thing that we're going to talk about, I want to quickly remind you that mass is the amount of resistance to linear acceleration, linear acceleration is A, mass is resistance to A and we call that inertia when you resist acceleration you're resisting change in velocity resisting change is inertia, inertia means you want things to keep going the way they're gone you want to resist change, I can show real quick how this looks like sum of all force = MA, I can rewrite this as A equals sum of all forces over M, so notice that the greater my mass the smaller my A, right? So, the more mass I have the more resistance and remember resistance is inertia the more mass the more inertia so we say that mass is the quantity of inertia this is old stuff, mass is the quantity of inertia. Well it's going to the same thing in rotation, in rotation the only difference is that in rotation the amount of resistance depends on mass and depends on something else, in linear motion depends only on mass but in rotation it depends on mass and it also depends on distance to axis so if an object spins at a distance of 10 it's going to have a different resistance than if it spins at a distance of 20 from its axis of rotation, OK? So, if you go like this you have less inertia than if you're going like this, OK? So, this combination of mass and distance to axis is what we call moment of inertia and it's the amount of rotational inertia that you have, moment of inertia takes the letter I you can think of this as just inertia, right? And it's the rotational equivalent of mass, cool? And again, you can think of it as rotational inertia or rotational mass, cool? So, depending on the kind of problem you have if you have motion, linear motion or any kind of linear problem you use mass if you have any kind of rotational problem you're going to use rotational mass which is called moment of inertia, cool. There's two types of objects and two types of problems you can see, you can have point masses again a point mass is a tiny little objects that goes around the circle of radius R and we're going to say that the mass itself has no radius, OK? So remember this distance here is little r, little r is distance big R is the radius so it's a tiny object that doesn't really have a radius it doesn't have a volume, usually you hear something like a small objects and then the other type is when we have a shape or a rigid body these are like a cylinder with a radius or something so the radius here is not 0 and the reason I put shape is because it's usually going to the problems usually that tell you what kind of shape this is, it's a solid cylinder if it says it's a solid cylinder you know it's one of these guys and not a point mass, cool? Now if you have a point mass, the moment of inertia is given by an equation which MR squared and M is the mass of that object and R again is the distance to the axis of rotation, OK? Distance to axis, alright? And if it's a rigid body, I will be given by a table look up what I mean by that? Well your textbook has a table of moments of inertia and it's going to say for a solid cylinder the moment of inertia for example is let me write it here for a solid cylinder the moment of inertia is 1/2 MR squared so this will be given to you most professors don't require you to memorize this they'll give this to you in some way, alright? So, look through your book find a table it's got some pretty shapes, something like this I pulled this from Wikipedia it shows you the shape and it shows where it's rotating, OK? I want to point out that if you spin here if you spin this object at the end of the object so imagine that you're spinning sort of like here, right? At this edge right here so it's doing this it's different than if you're spinning it in the middle like this, notice how these equations are different this is 1/3ML squared and this is 1/12ML squared it's because for moment of inertia where you spin matters, OK? One last point here is that the most moments of inertia will follow this general form here it will be some fraction like 1/2 or 2/3 or 2/5 or whatever MR squared, in this case this is a thin rod so what matters in the rod is not the R not the radius of the rod because it's very thin so it's small it's negligible but it's the length of the rod but even then you see that it's instead of MR squared it's ML squared so you should expect to see something like that, OK? So, we're going to do this quick example here so you see how this works and then we'll do a practice problem.
It's says system is made of two small masses the one of the left right here it's this guy has a mass of M let's call it M left= 3 and then mass on the right M right=4 and they are attached to the ends of a 2 meter long thin rod so it's this guy right here, I'm going to write it like this length = 2 meters that is massless so it's a thin rod that has no mass at all and we want to calculate the moment of inertia of the system if it spins about a perpendicular axis through the center of the rod there's a lot of words here and you're going to get used to this but I'm going to start slowly here, OK? So, I want to in the moment of inertia I=? of the system let me put a little system here moment of inertia of the whole system if it spins so it's spinning about a perpendicular axis you're going to see this all the time perpendicular axis, perpendicular means 90 degrees, Perp=90 degrees this is a remember the symbol for perpendicular, cool? So, what does it mean it's a perpendicular axis? Well here's the rod, right? Perpendicular axis means it's making 90 degrees with the rod so it looks like this, cool? Like that, now this is also perpendicular because it's also making 90 degrees so sometimes it's hard to tell which one so you have to be careful so it says perpendicular axis through the center of the rod so it means it's perpendicular makes 90 degrees and it goes like this, you can have a perpendicular axis or you can have a parallel axis, parallel axis would look like this it would go with the rod but then the rod is just spinning around itself and that doesn't do anything, OK? So, the axis will be like this through the middle which means the rod is spinning around itself like this, OK? So, it's a very visual chapter a very visual topic so I'm going to draw it like this and the idea is that this guy is spinning around itself like this, OK? The moment of inertia of a system is the sum of the individual moments of inertia, OK? So we have 3 objects but the rod has no mass and look at the moment of inertia, moment of inertia is either 1/2MR squared or if you are shape it looks like this, both of these guys have masses what it means is that if both of the them require mass so if you have no mass you have no moment of inertia, moment of inertia is rotational mass if you have no regular you have no rotational mass either so what that means is that we're only going to really have two guys, I left + I right, OK? So now we're going to expand this, what goes here and what goes here so what goes here and what goes here? Now you have to decide or you have to figure out are these masses a tiny point mass or are they sort of a bigger shape and here it says two small masses, small is a hint that these are point mass PM points masses which means the equation is an MR squared, OK? So, there's two clues here one it says small that's a dead giveaway and the other one isn't it didn't indicate a shape so if I tell you a small solid cylinder it's still a rigid body because I said that it was a solid cylinder I gave you the shape here I tell you it's small and I don't give you a shape it's a point mass, OK? So, what we're going to write here is M of the left R left squared because that's the equation MR squared, same thing here MR squared but this is for the right, OK? So, the masses are a 3 and a 4, 3 and a 4 so I like to set up.... Whoops This is a 2 right there I like to set up this way what I've done is I've written the mass here, mass here, this mass this mass and I've left the space for us to plug in the Rs, OK? I left a space for us to plug in the R this is where you have to slow down make sure you find the right number, R is the distance between the object and its axis of rotation so it's not the two R is this so this is R for the right ball or right object and this is R for the left object the distances are one for both because it sits right down the middle so it's going to be 1 and 1, OK? 1 squared is just one so the answer here is simply 7, now let's talk about units because we haven't done that yet if you look at I=MR squared which is the I for a point mass the units are going to be kilograms because of the mass and meter squared because of the distance squared, distances is in meters so it's going to be 7 kilograms meter squared, alright? That's it for this one I want to point out that we actually didn't use this table, right? We had a rod here we didn't use this table because this rod didn't have a mass, if it had a mass you would have used this equation right here, OK? You would have used this equation here and I'm going to write here but no mass sad face so we didn't get to use it but you would have used this one because it's spinning around its middle point, cool? That's it for this one let's keep going.
Practice: A system is made of two small masses (MLEFT = 3 kg, MRIGHT = 4 kg) attached to the ends of a 5 kg, 2-m long thin rod, as shown. Calculate the moment of inertia of the system if it spins about a perpendicular axis through the mass on the left.
Example #1: Moment of inertia of Earth
Hey guys so as you know the Earth is rotating therefore it has a moment of inertia and if we make some assumptions about the shape of the Earth we can actually calculate the moment of inertia of the earth let's check it out, so it says here Earth has a mass and radius given by these big numbers and they also tell you that the radial distance between the Earth and the sun is this, what I mean by a radial distance is that if the Sun is here the Earth spins around the sun and this distance here little r is 1.5x 10(11) squeeze it in there, that's what I mean by that and then I gave you the mass of the Earth and the radius of the Earth as well I want to know the moment of inertia of the earth as it spins around itself and as it spins around the Sun, as you know the Earth has 2 motions and we can calculate a moment of inertia about or relative to those two motions or for those two motions, remember moment of inertia depend on the axis of rotation that's why these numbers will be different so if you want a moment of inertia of the Earth around itself you would have to treat the earth as an object with a significant size you can't treat it as a tiny object so what we do here is we're going to treat the earth as a solid sphere So the Earth is a big ball that spins around itself. Now technically it's at an angle like that but it doesn't really matter you can just do this, OK? So, it's spinning around itself and your book would show you that solids sphere have a moment of inertia given by this equation right here so when I tell you solid sphere I'm indirectly telling you hey use this equation for I, OK? So, for A part we're going to do part A is over here, we're going to say I= 2/5 MR squared and all we're going to do is plug in the numbers here so M is the mass of the earth which is 5.97x10(24) and R is the radius of the earth which is this and not the radial distance, it's the earth going around itself so it's the radius of the actual object or the sphere, 6.37x10(6) squared, OK? If you look at this number I got a 24 and then I got a 6 squared so you should imagine that this is going to be a gigantic number and it is I multiplied everything I get 9.7x10(37) kilograms meter square, the Earth has a lot of inertia and what that means is that we're being incredibly hard to make the earth stop spinning, OK? Now if you were to google this number you would see that it's actually a little bit off the actual moment of inertia is a little bit off and that's because the earth is not a perfect sphere it's got different layers it's not even a sphere so but this number is a pretty good approximation.
For Part B we want to find out what is the moment of inertia of the earth as it spins around the sun, now in this case relative to the Sun the Earth is tiny so we're going to treat it as a point mass which is crazy the earth is huge thing and you're going to just treat it as a little point mass of negligible radius and that's because relative to the Sun the Earth is negligible in size, OK? So, I'm going to put the earth here as a tiny M earth and it's going around the sun and the distance here the radial distance which is little r, big R is radius of an object and little r is distance to the center is 1.5x10(11) meters in this case we're going to use instead of 2/5 MR we're going to use MR squared because the Earth is being treated as a point mass here, M is the mass of the object itself, right? It's the object that spinning it's not Sun so it's going to be 5.97x10(24) same thing but r is going to be the distance to the center which is 1.5 so 1.5x10(11) squared I got a 24, I got an 11 squared this going to be again a gigantic number, 1.34x10(47), this number is like a billion times bigger than the other number, right? So as hard as it would be to get the earth to stop spinning it would be way harder, right? It would be 10 to 10 times harder to make the earth stops going around the sun, and that's it so that's very typical classic problem hopefully it makes sense you should try this out on your own and let's keep going.
Practice: A solid disc 4 m in diameter has a moment of inertia equal to 30 kg m2 about an axis through the disc, perpendicular to its face. The disc spins at a constant 120 RPM. Calculate the mass of the disc.
Example #2: Inertia of planet of known density
Calculation error. Should be 5.49x1043 kgm2.
Hey guys, so here's another pretty straightforward moment of inertia question the only difference here is that we're going to have density to deal with so let's do this real quick. So, it says we have a planet that is nearly spherical, nearly spherical means that it's a sphere, OK? So you can basically ignore the word nearly It means we're going to approximate it as a sphere with nearly continuous mass distribution, again you can ignore the word nearly and assume that it has continuous mass distribution, continuous mass distribution we'll talk more about this later but it basically means that a sphere has the mass evenly distributed throughout the sphere so you sometimes you see drawn like something like this, this is a solid sphere and this is opposed to a hollow sphere which is a sphere that has nothing inside, it's not continuously or evenly distributed all the mass is concentrated on the edges this is not what we have here this is what we have here the reason why that's important is because you're going to get a different I equation a different moment of inertia equation depending on what kind of sphere you have and the moment of inertia equation for this guy here is 2/5MR squared, the question didn't give you this but you would look this up in your book or in a test he would have to give you this somehow unless your professor requires you to memorize these and then you have to do that but most of them don't, alright? So that's the equation you're supposed to use I give you the radius right here radius is 8x10(7) meters and I give you the density, density you can use D but the official variable if you will is row, right? It's 10000 Kilograms per cubic meter, I want to remind you that if you have a volume the density of a volume is mass over volume and you could have seen this from the units kilograms cubic meter, cube right so it's a volume it's a three dimensional object, and that's it that's all we're given I also give you here the equation for volume of a sphere, volume right here of a sphere, now if you were looking for I and if you start plugging stuff in here you would realize you don't have M but you have R, OK? So we don't have M we got to figure this out and if you look around you, you realize well we have another piece of information that has some connection to M so maybe I can use this to solve for M and that's exactly what we're supposed to do so I want to find M I have 10000 but I don't have a V but once again I have another piece of information here that allows me to find the V, V= the volume of a sphere= pie r cubed and I know, I know r so I can get V I'm going to V so I'm going to be able to get M I'm going to know M o I'm going to be able to get I, OK? That's how it's going to flow. Alright so what I'm going to do is right here I'm going to solve for M in other words I'm going to move V over here so M=10000V and V is according to this equation right here 4/3 pie r cubed so I'm going to get this whole thing which is M and I'm going to stick it in here, OK? So, 2/5 x 10000 x 4/3 pie r cubed, this is just M and then I also have the R squared here, tons of Rs, OK?
And if you multiply this whole crap you're going to end up with let's see this is 18...I'm sorry this is 8 so this is going to be 80000/15 so it's 80000 pie, did I get that right? Yep, divided by 15 times r to the fifth, these two guys combined R to the fifth so it's going to be 8x10(7) this thing to the fifth, OK? So, you should expect a pretty big number and I get 3.15x10(45). Now how exactly do you arrive at this number doesn't really matter it's I so it's kilograms meter square you could have you know gotten a number here plugged it in here it really doesn't matter as long as you're right here it's a bunch of multiplication, cool? So that's it for this let me know if you have any questions.
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Leibniz is generally recognized as a master of notation, but is not so easy to substantiate this impression without getting lost in the labyrinthine technicalities of the development of the differential and integral calculus. Joseph Hoffmann and A. Rupert Hall, supplied a good basis for a convincing account of the story. In spite of our shared misgivings early on, we saw that the story of the priority conflict between Newton and Leibniz, though forbidding both in terms of the quantity of ink spilled on it, the animosities it once engendered, and its apparent triviality, actually deserves some attention for the light it casts, if approached judiciously, on scientific matters. Fortunately both Hoffmann and Hall, though not quite taking the same side, approach it in this spirit. As Hall points out, prior to his own work it had not actually been dealt with as a subject in its own right; we noticed that Hofmann's intention to set out his own views in a monograph on the subject was thwarted by his death in an accident.
Early on you make the following claim about Leibniz: "And yet although he became acquainted quite late in his life with the mathematical achievements of his generation, it will always be his innovations in this field that put him to the forefront of the enlightened thinkers of his era." I had my doubts about this, but they are diminishing. It's a bit difficult for me to see a talented mathematician in the man who was so callow on his arrival in Paris, but he really does seem to have accomplished a great deal by systematizing the calculus. This seems to be a case where good methods and a sound intelligence produced results that would otherwise have required genius.
Leibniz certainly had a good instinct for arranging mathematics so that demanding calculations could be carried out by men of only moderate gifts. Newton professed, late in life and in the midst of polemics (about 1712), not to see the utility of this, and perhaps he could not envision the day when hordes of engineers would routinely carry out calculations of the sort that he had himself devised. But I suspect Newton was not entirely sincere at this point.
"It was precisely for this reason that Leibniz had so much success in the field, in that he was unhampered by much of the dogma that might have hindered its progress." I'll part company with you about the "dogma", but it does seem that after devising methods that suited himself, he found as he expected that they would suit others very well indeed. The next question would be what exactly the Bernoulli's learned from him - one imagines it is exactly this "calculus" as an algorithm that impressed them.
"... through his own lens" ( or possibly the lens of Cavalieri, which in spite of its logical inadequacies had much to recommend it as the basis of an algorithmic formalism.
It seems to me at the moment that Leibniz was drastically ahead of his time in at least one respect: his unbridled optimism regarding the potential power of formal systems, an optimism which seems more reasonable, though still somewhat problematic, to our own age. Ultimately what interested him, it seems to me, is what we now refer to as artificial intelligence. Though he didn't phrase it in those terms, the result of combining his characteristica universalis with existing computers would be something very like artificial intelligence.
On the other hand, one can also relate the ideas of Leibniz to the work of the philosopher Frege, who in the late 19th century began to devise formal systems for reasoning that correspond quite well to some of Leibniz' ideas; though still restricted to t he field of mathematics, they encompass the whole of that field, rather than limited areas. There may yet be applications of these ideas to law and other aspects of civil society, though it will take at least one more crusader with the spirit, and abilities, of Leibniz to bring it about.
In any case, under the impact of your essay and some of your references, I find my picture of Leibniz evolving.
The Hofmann and the two books by A. Rupert Hall (one found by Anand, the other suggested by Professor Kosinski) are enormously helpful. The Hofmann is something I'd always meant to read, and was even better than I anticipated. Philosophers at War on the other hand was wholly unknown to me, and gives a very clear account of the whole controversy, with a useful chronology at the outset. As Hall tells the story, it casts considerable light on the history of the subject as well (at least until 1710, when it heats up: at that point my first impression is that it just becomes depressing).
Newton always attached great weight to his method of infinite series, and Leibniz was obsessed by Cavalieri's infinitesimals, which go back to the ancient Greeks (Democritos). Then as Hall stresses, Newton moved away from infinitesimals and in the direct ion of "flows" (fluxions), which amounts to replacing a rather primitive "atomic" theory of geometry by a more sophisticated physical theory of velocity, slightly closer to the rigorous theory of limites. Hall makes it plain that Newton claimed that the theory of infinite series is a central feature of the calculus. I hadn't realized he took that point of view, and again I wonder if it was really sincere. If so, he could have laid out that claim earlier, and more explicitly, since it ought to have been clear that the Continentals didn't see things that way.
These are interesting and subtle issues, and frequently the arguments associated with the priority fight touch on serious matters.
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Solving 'Packing Coloring' with 15 Numbers - Bernardo Subercaseaux's quest for the perfect equation
Category Science Sunday - April 30 2023, 17:39 UTC - 10 months ago Bernardo Subercaseaux, an undergraduate student at the University of Chile, ventured off to graduate school in Carnegie Mellon University to explore a combinatorics problem posed by Wayne Goddard and his collaborators in 2008. With the help of Marijn Heule, a computer scientist, their collaboration culminated in a proof that the problem can be solved with 15 numbers.
Sunday - April 30 2023, 17:39 UTC - 10 months ago
Bernardo Subercaseaux, an undergraduate student at the University of Chile, ventured off to graduate school in Carnegie Mellon University to explore a combinatorics problem posed by Wayne Goddard and his collaborators in 2008. With the help of Marijn Heule, a computer scientist, their collaboration culminated in a proof that the problem can be solved with 15 numbers.
As an undergraduate at the University of Chile, Bernardo Subercaseaux took a dim view of using computers to do math. It seemed antithetical to real intellectual discovery.
"There’s some instinct or gut reaction against using computers to solve your problems, like it goes against the ideal beauty or elegance of a fantastic argument," he said.
But then in 2020 Subercaseaux fell in love, and as often happens, his priorities changed. The object of his obsession was a question he saw on an online forum. Most problems he scanned and forgot, but this one caught his eye. He swiped right.
"The first thing I did was to like the post in the Facebook group, hoping to get a notification later when somebody else posted a solution," he said.
The question was about filling an infinite grid with numbers. It was not, as it turned out, the kind of problem one solves on a lark. In order to do it, Subercaseaux had to leave Chile for graduate school at Carnegie Mellon University.
There, in August 2021, he had a fortuitous encounter with Marijn Heule, a computer scientist who uses massive computing power to solve hard math questions. Subercaseaux asked Heule if he wanted to attempt the problem, kicking off a collaboration that culminated this January in a proof that can be summed up with a single number: 15.
Every Possible Way .
In 2002, Wayne Goddard of Clemson University and some like-minded mathematicians were spitballing problems in combinatorics, trying to come up with new twists on the field’s mainstay questions about coloring maps given certain constraints.
Eventually they landed on a problem that starts with a grid, like a sheet of graph paper that goes on forever. The goal is to fill it with numbers. There’s just one constraint: The distance between each occurrence of the same number must be greater than the number itself. (Distance is measured by adding together the vertical and horizontal separation, a metric known as "taxicab" distance for the way it resembles moving on gridded urban streets.) A pair of 1s cannot occupy adjoining cells, which have a taxicab distance of 1, but they can be placed in directly diagonal cells, which have a distance of 2.
Initially, Goddard and his collaborators wanted to know whether it was even possible to fill an infinite grid with a finite set of numbers. But by the time he and his four collaborators published this "packing coloring" problem in the journal Ars Combinatoria in 2008, they had proved that it can be solved using 22 numbers. They also knew that there was no way it could be solved with only five numbers. That meant the actual answer to the problem — the minimum number of numbers needed — lay somewhere in between.
The researchers didn’t actually fill an infinite grid. They didn’t have to. Instead, it’s enough to take a small subset of the grid — say a 10-by-10 square — fill that with numbers, then show that copies of the filled subset can be placed next to each other, the way you’d cover a floor with copies of a tile.
When Subercaseaux first learned of the problem, he started looking for tiles using pen and paper. He would sketch grids of numbers, cross them out, try again. He soon tired of that approach, and he asked a friend to design a web-based tool that resembled the game Minesweeper and allowed him to test combinations fast. He found a few solutions with 16 numbers but none with 15.
In order to find the magic number, Subercaseaux and Heule developed a computer program that demonstrates different versions of the puzzle. The program was an "oracles"; it checks whether a grid with a certain configuration of numbers can be solved, but it offers no advice on how to arrive at the optimal solution. It’s only by trial and error that the correct number of numbers can be found.
It took the pair eight months to come up with the magic combination of 15 numbers and to prove that it was the answer. Subercaseaux was without a doubt the instigator of the project, but Heule said that without his help, the program the duo used would have been far less efficient.
"I had honed my skills to solve structured tasks like this one, but I had never experienced a student who was so ambitious and willing to take risks," Heule said.
The proof Subercaseaux and Heule wrote up was published in January in the journal Journal of Combinatorial Theory, Series A. They hope it will spark interest from mathematicians, as well as inspire future generations to have the same faith in computers that Suberycaseaux had.
"I think having a computer as an assistant is essential to doing these kinds of problems," he said. "Some people are skeptical of it, so it did not come easily to me. But it is something that I am absolutely sure we should embrace." . |
Branching Ratio and CP Violation of
in Perturbative QCD Approach
We calculate the branching ratios and CP asymmetries for , and decays, in a perturbative QCD approach. In this approach, we calculate non-factorizable and annihilation type contributions, in addition to the usual factorizable contributions. We found that the annihilation diagram contributions are not very small as previous argument. Our result is in agreement with the measured branching ratio of by CLEO collaboration. With a non-negligible contribution from annihilation diagrams and a large strong phase, we predict a large direct CP asymmetry in , and , which can be tested by the current running B factories.
PACS: 13.25.Hw, 11,10.Hi, 12,38.Bx,
The charmless decays arouse more and more interests recently, since it is a good place for study of CP violation and it is also sensitive to new physics . Factorization approach (FA) is applied to hadronic decays and is generalized to decay modes that are classified in the spin of final states [2, 3, 4]. FA gives predictions in terms of form factors and decay constants. Although the predictions of branching ratios agree well with experiments in most cases, there are still some theoretical points unclear. First, it relies strongly on the form factors, which cannot be calculated by FA itself. Secondly, the generalized FA shows that the non-factorizable contributions are important in a group of channels [3, 4]. The reason of this large non-factorizable contribution needs more theoretical studies. Thirdly, the strong phase, which is important for the CP violation prediction, is quite sensitive to the internal gluon momentum . This gluon momentum is the sum of momenta of two quarks, which go into two different mesons. It is difficult to define exactly in the FA approach. To improve the theoretical predictions of the non-leptonic decays, we try to improve the factorization approach, and explain the size of the non-factorizable contributions in a new approach.
We shall take a specific channel as an example. The decays are responsible for the determination of the angle in the unitarity triangle which have been studied in the factorization approach in detail [2, 3, 4]. The recent measurements of by CLEO Collaboration attracted much attention for these kind of decays . The most recent theoretical study attempted to compute the non-factorizable diagrams directly. But it could not also predict the transition form factors of .
In this paper, we would like to study the decays in the perturbative QCD approach (PQCD) . In the decays, the meson is heavy, sitting at rest. It decays into two light mesons with large momenta. Therefore the light mesons are moving very fast in the rest frame of meson. In this case, the short distance hard process dominates the decay amplitude. We shall demonstrate that the soft final state interaction is not important, since there is not enough time for the pions to exchange soft gluons. This makes the perturbative QCD approach applicable. With the final pions moving very fast, there must be a hard gluon to kick the light spectator quark or (almost at rest) in the meson to form a fast moving pion (see Figure 1). So the dominant diagram in this theoretical picture is that one hard gluon from the spectator quark connecting with the other quarks in the four quark operator of the weak interaction. Unlike the usual FA, where the spectator quark does not participate in the decay process in a major way, the hard part of the PQCD approach consists of six quarks rather than four. We thus call it six-quark operators or six-quark effective theory. Applying the six-quark effective theory to meson exclusive decays, we need meson wave functions for the hadronization of quarks into mesons. Separating that nonperturbative dynamics from the hard one, the decay amplitudes can be calculated in PQCD easily. Most of the nonperturbative dynamics are included in the meson wave functions, but in the correction that soft gluon straddle the six-quark operators, there are some nonfactorizable soft gluon effects not to be absorbed into the meson wave functions. Such effects can be safely neglected in the meson decays .
Li performed the calculation of in ref. using the PQCD formalism, where the factorizable tree diagrams were calculated and the branching ratios were predicted. In another paper , Dahm, Jakob and Kroll performed a more complete calculation, including the non-factorizable annihilation topology and the three decay channels of decays. However, the predicted branching ratios are about one order smaller than the current experiments by CLEO . In connection with this, Feldmann and Kroll concluded that perturbative contributions to the transition form factor were much smaller than nonperturbative ones . As we shall show later, the pion wave function must be consistent with chiral symmetry relation
This introduces terms that were not considered in above calculations. In this paper, considering the terms needed from chiral symmetry, we calculate the transition form factors and also the non-factorizable contributions in PQCD approach. We then show that our result for the branching ratio agree with the measurement. Among the new terms, it is worthwhile emphasizing the presence of annihilation diagrams which are ignored in FA. We find that these diagrams can not be ignored, and furthermore they contribute to large final state interaction phase.
2 The Frame Work
The three scale PQCD factorization theorem has been developed for non-leptonic heavy meson decays , based on the formalism by Brodsky and Lepage , and Botts and Sterman . The QCD corrections to the four quark operators are usually summed by the renormalization group equation . This has already been done to the leading logarithm and next-to-leading order for years. Since the quark decay scale is much smaller than the electroweak scale , the QCD corrections are non-negligible. The third scale involved in the meson exclusive decays is usually called the factorization scale, with the conjugate variable of parton transverse momenta. The dynamics below scale is regarded as being completely nonperturbative, and can be parametrized into meson wave functions. The meson wave functions are not calculable in PQCD. But they are universal, channel independent. We can determine it from experiments, and it is constrained by QCD sum rules and Lattice QCD calculations. Above the scale , the physics is channel dependent. We can use perturbation theory to calculate channel by channel.
Besides the hard gluon exchange with the spectator quark, the soft gluon exchanges between quark lines give out the double logarithms from the overlap of collinear and soft divergences, being the dominant light-cone component of a meson momentum. The resummation of these double logarithms leads to a Sudakov form factor , which suppresses the long distance contributions in the large region, and vanishes as . This form factor is given to sum the leading order soft gluon exchanges between the hard part and the wave functions of mesons. So this term includes the double infrared logarithms. The expression of is concretely given in appendix B. Figure 2 shows that falls off quickly in the large , or long-distance, region, giving so-called Sudakov suppression. This Sudakov factor practically makes PQCD approach applicable. For the detailed derivation of the Sudakov form factors, see ref.[8, 17].
With all the large logarithms resummed, the remaining finite contributions are absorbed into a perturbative b quark decay subamplitude . Therefore the three scale factorization formula is given by the typical expression,
where are the corresponding Wilson coefficients, are the meson wave functions and the variable denotes the largest mass scale of hard process , that is, six-quark effective theory. The quark anomalous dimension describes the evolution from scale to . Since logarithm corrections have been summed by renormalization group equations, the above factorization formula does not depend on the renormalization scale explicitly.
The three scale factorization theorem in eq.(2) is discussed by Li et al. in detail . Below section 3, we shall give the factorization formulae for decay amplitudes by calculating the hard part , channel dependent in PQCD. We shall also approximate there by the expression, which makes sense if perturbative contributions indeed dominate.
In the resummation procedures, the meson is treated as a heavy-light system. The wave function is defined as
where is color’s degree of freedom and is the distribution function of the 4-momenta of the light quark () and quark ()
Note that we use the same distribution function for the term and the term from heavy quark effective theory. For the hard part calculations in the next section, we use the approximation , which is the same order approximation neglecting higher twist of . To form a bound state of meson, the condition is required. So is actually a function of only. Through out this paper, we take , as the light-cone coordinates to write the four momentum. We consider the meson at rest, then that momentum is . The momentum of the light valence quark is written as (), where the is a small transverse momentum. It is difficult to define the function . However, in the next section, we will see that the hard part is always independent of , if we make some approximations. This means that can be integrated out in eq.(4), the function can be simplified to
where is the momentum fraction. Therefore, in the perturbative calculations, we do not need the information of all four momentum . The above integration can be done only when the hard part of the subprocess is independent of the variable .
The meson is treated as a light-light system. At the meson rest frame, pion is moving very fast. We define the momentum of the pion which contain the spectator light quark as . The other pion which moving to the inverse direction, then has momentum . The light spectator quark moving with the pion (with momentum ), has a momentum . The momentum of the other valence quark in this pion is then . If we define the momentum fraction as , then the wave function of pion can be written as
Note that as you shall see below, given as
in eq.(6) is not the pion mass. Since this is estimated around GeV using the quark masses predicted from lattice simulations, one may guess contributions of term cannot be neglected because of . In fact, we will show this plays important roles to predict the branching ratios in section 4.
The normalization of wave functions is determined by meson’s decay constant
Using this relation, the normalization of is defined as
Moreover, from eq.(9) you can readily derive
The transverse momentum is usually conveniently converted to the parameter by Fourier transformation. The initial conditions of , , , are of nonperturbative origin, satisfying the normalization
with the meson decay constants.
3 Perturbative Calculations
With the above brief discussion, the only thing left is to compute for each diagram. There are altogether 8 diagrams contributing to the decays, which are shown in Figure 3. They are the lowest order diagrams. In fact the diagrams without hard gluon exchange between the spectator quark and other quarks are suppressed by the wave functions. The reason is that the light quark in meson is almost at rest. If there is no large momentum exchange with other quarks, it carries almost zero momentum in the fast moving , that is the end point of pion wave function. In the next section, we will see that the pion wave function at the zero point is always zero. The Sudakov form factor suppresses the large number of soft gluons exchange to transfer large momentum. It is already shown that the hard gluon is really hard in the numerical calculations of . The value of is peaked below . And in our following calculation of decays this is also proved.
Let’s first calculate the usual factorizable diagrams (a) and (b). The four quark operators indicated by a cross in the diagrams, are shown in the appendix A. There are two kinds of operators. Operators , , , , , and are currents, the sum of their amplitudes is given as
where . is a color factor. The function and the Sudakov form factors and are given in the appendix B. The operators , , , and have a structure of . The sum of their amplitudes is
They are proportional to the factor . There are also factorizable annihilation diagrams (g) and (h), where the meson can be factored out. For the operators, their contributions always cancel between diagram (g) and (h). But for the operators, their contributions are sum of diagram (g) and (h).
These two diagrams can be cut in the middle of the diagrams. They provide the main strong phase for non-leptonic decays. Note that vanishes in the limit of . So the term in the pion wave function does not only have much effect on the branching ratios, but also the CP asymmetries. Besides the factorizable diagrams, we can also calculate the non-factorizable diagrams (c) and (d) and also the non-factorizable annihilation diagrams (e) and (f). In this case, the amplitudes involve all the three meson wave functions. The integration over can be performed easily using function in diagram (c,d) and for diagram (e,f).
Note that when doing the above integrations over and , we have to include the corresponding Wilson coefficients evaluated at the corresponding scale . The expression of Wilson coefficients are channel dependent which are shown later in this section. The functions , coming from the Fourier transform of , are given in Appendix B. In the above equations, we have used the assumption that . Since the light quark momentum fraction in meson is peaked at the small region, while quark momentum fraction of pion is peaked at , this is not a bad approximation. After using this approximation, all the diagrams are functions of of meson only, independent of the variable of . For example, by calculating the diagrams (b) we shall demonstrate it.
where the momenta are assigned in Figure 3. The calculation from second formula to last one is approximated as . This approximation is equal to take the momenta of spectator quark in the meson as . We neglect the last term which is higher order one in terms of expansion. Therefore the integration of eq.(5) is performed safely. Though we calculated the above factorization formulae by one order in terms of , the radiative corrections at the next order would emerge in forms of , where ’s denote some scales, i.e. , , in the hard part . Selecting as the largest scale in ’s, the largest logarithm in the next order corrections is killed. Accordingly, the scale ’s in the above equations are chosen as
They are given the maximum values of the scales appeared in each diagram.
where , . The decay width is expressed as
The should be calculated at the appropriate scale using eq.(61,62) in the appendices. The decay amplitude of the charge conjugate decay channel is the same as eq.(22) except replacing the CKM matrix elements to and to under the phase convention .
The decay amplitude for can be written as
4 Numerical calculations and discussions of Results
In the numerical calculations we use
which is relevant to taking GeV. For the wave function, we neglect the dependence part, which is not important in numerical analysis. We use
with , which is close to the Chernyak-Zhitnitsky (CZ) wave function . For this axial vector wave function the asymptotic wave function , , is suggested from QCD sum rules , diffractive dissociation of high momentum pions , the instanton model , and pion distribution functions , etc., but we adopt according to the discussion in ref. . is chosen as asymptotic wave function
with . For meson, the wave function is chosen as
with GeV , and GeV is the normalization constant. In this work, we set for simplicity. We would like to point out that the choice of the meson wave functions as in eqs. (29-31) and the above parameters can not only explain the experimental data of , but also [18, 26], etc., which is the result of a global fitting. However, since the predicted branching ratio of is sensitive to the input parameters , , , and , we will at first give the numerical results with the above parameters, then we give the allowed parameter regions of , , , and constrained by the experimental data of presented by CLEO.
The diagrams (a) and (b) in Fig.3, calculated in eq.(13) correspond to the transition form factor , where . Our result is to be consistent with QCD sum rule one. This implies that PQCD can explain the transition form factor in the meson decays, which is different with the conclusion in ref.. In that paper, because was not considered, perturbative contributions to were predicted to be much smaller than nonperturbative ones.
Although we take the CZ like wave function () for , one finds that the above parameters give the pion electromagnetic form factor to be consistent with the experimental data. The pion electromagnetic form factor in PQCD is given as [28, 29]
where is the momentum transfer in this system, the scale is chosen as , and ’s are replaced by in the and . One may suspect that around , the gluon and virtual quark propagators give rise to IR divergences which can not be canceled by the wave functions. However, in PQCD, the transverse momenta save perturbative calculations from the singularities around . There are still IR divergences around , but the Sudakov factor which can be calculated from QCD corrections does suppress such a region, i.e., non-perturbative contributions, sufficiently. We show the dependence of (eq.(32)) in Figure 4 with the experimental data . This figure shows that the parameters we used don’t conflict with the data. We also show for and . It indicates that is fairly insensitive to .
The CKM parameters we used here are
We leave the CKM angle as a free parameter. ’s definition is
In this parameterization, the decay amplitude of can be written as |
10 Wrong Answers to Common Associative Property Of Multiplication Of Rational Numbers Questions: Do You Know the Right Ones?
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What is Shear modulus ?
Modulus of Rigidity definition
Shear modulus is the ratio of the shear stress to the shear strain.
Shear modulus is defined as the measure of the elastic shear stiffness of the material and it is also acknowledged as ‘modulus of rigidity’. So, this parameter answers the question of how rigid a body is?
Shear modulus is the material response to a deformation of body because of the shearing stress and this work as ‘the resistant of the material to shearing deformation’.
In the above figure, Side lengths of this element will not change, though the element experiences a distortion and shape of element is changing from the rectangle to a parallelogram.
Why do we calculate the modulus of rigidity of the material ?
Shear modulus equation | Modulus of Rigidity equation
Shear modulus is the ratio of the shear stress to the shear strain, which is measures the amount of distortion, is the angle (lower case Greek gamma), always ex-pressed in radians and shear stress measured in force acting on an area.
Shear modulus represented as,
G= shear modulus
τ=shear stress = F/A
ϒ = shear strain=Δx/l
modulus of rigidity symbol
G or S or μ
What is the SI unit of rigidity modulus ?
Shear modulus units | Unit of modulus of rigidity
Pascal or usually denoted by Giga-pascal. Shear modulus is always positive.
What is the dimensional formula of modulus of rigidity ?
Shear modulus dimensions:
Shear modulus of materials:
Shear modulus of steel | Modulus of rigidity of steel
Modulus of rigidity of stainless steel:77.2Gpa
Modulus of rigidity of carbon steel: 77Gpa
Nickel steel: 76Gpa
Modulus of rigidity of mild steel: 77 Gpa
What is the Rigidity modulus of copper in N/m2 ?
Modulus of rigidity of copper wire:45Gpa
Shear modulus of Aluminum alloy: 27Gpa
A992 Steel: 200Gpa
Shear modulus of concrete | Modulus of rigidity of concrete: 21Gpa
Silicon shear modulus: 60Gpa
Poly ether ether ketone (PEEK):1.425Gpa
Fiberglass shear modulus: 30Gpa
Polypropylene shear modulus: 400Mpa
Polycarbonate shear modulus: 5.03Gpa
Polystyrene shear modulus:750Mpa
Shear modulus derivation | Modulus of rigidity derivation
If the co-ordinate axes (x, y, z) coincides with principle axes and intended for an isotropic element, the principal strain axes at (0x,0y,0z ) point, and considering alternative frame of reference directed at (nx1, ny1, nz1) (nx2, ny2, nz2) point and in the meantime, Ox and Oy are at 90 degree to each other.
So we can write that,
nx1nx2 + ny1ny2 + nz1nz2 = 0
Here Normal stress (σx’) and the shear-stress ( τx’y’) has been computed utilizing Cauchy’s formulation.
The resultant stress vector on the plane will have components in (x-y-z) as
The normal stress on this x-y plane has been computed as the summation of the component’s projections along the normal directions and we can elaborate as
σn= σx=nx^2 σ1+nx^2 σ2+nx^2 σ3.
Similarly, the shear stress component in x and y plane nx2, ny2, nz2.
Considering as ε1, ε2, ε3 are the principal strains and the normal-strain is in x-direction, then we can write as
The shear strain is obtained as,
Substituting the values of σ1, σ 2 and σ 3,
Here, μ= shear modulus usually represented by term G.
By taking other axis as Oz¢ with direction cosines (nx3, ny3, nz3) and at right-angle with the Ox¢ and Oy¢. This Ox¢y¢z¢ will create conventional forms an orthogonal set of axes, therefore we can write as,
Elastic constants and their relations:
Young’s modulus E:
The young’s modulus is the measure of the stiffness of the body and acts as resistance of the material when the stress is functional. The young’s modulus is considered only for linear stress-strain behavior in the direction of stress.
Poisson’s ratio (μ):
The Poisson’s ratio is the measure of the deformation of the material in the directions perpendicular to the loading. Poisson’s ratio ranges between -1 to 0.5 to maintain young’s modulus, shear modulus (G), bulk modulus positive.
μ=-ε trans/ε axial
Bulk modulus K is the ratio of the hydrostatic pressure to the volumetric strain and better represented as
E and n are generally taken as the independent constants and G and K could be stated as follows:
for an isotropic material, Hooke’s law is reduced to two independent elastic constants named as Lame’s co-efficient denoted as l and m. In terms of these, the other elastic constants can be stated as follows.
If bulk modulus considered to be +ve the Poisson’s ratio never be more than 0.5 (maximum limit for incompressible material). For this case assumptions are
n = 0.5.
3G = E.
K = ∞.
⦁ In terms of principal stresses and principal strains:
σ 1=λΔ +2με1
σ 2=λΔ +2με2
σ 3=λΔ +2με3
⦁ In terms of rectangular stress and strain components referred to an orthogonal coordinate system XYZ:
σ x=λΔ +2μ εxx
σ y=λΔ +2μ εyy
σ z=λΔ +2μ εzz
Young’s modulus vs shear modulus | relation between young’s modulus and modulus of rigidity
Elastic Constants Relations: Shear Modulus, Bulk Modulus, Poisson’s ratio, Modulus of Elasticity.
E= 2G(1+μ)=3K(1-2 μ)
Shear Modulus of Elasticity:
Hook’s law for shear stress:
τxy is represented as Shear-stress, Shear-modulus is G and Shear strain is ϒxy respectively.
Shear-Modulus is resistant to the deformation of the material in response to shear stress.
Dynamic shear modulus of soil:
Dynamic shear modulus gives information about dynamic one. Static shear-modulus gives information about static one. These are determined using shear wave velocity and density of the soil.
Shear Modulus Formula soil
Where, Vs=300 m/s, ρ=2000 kg/m3, μ=0.4.
Effective shear modulus:
The ratio of the average stresses to average strains is the effective shear-modulus.
Modulus of rigidity of spring:
The modulus of rigidity of the spring is the measurement of the stiffness of the spring. It varies with the material and processing of the material.
For Closed Coil Spring:
For Open Coil Spring:
Δ =64WR3n secα/d4cos2α/N+2sin2α /E
R= mean radius of the spring.
n = number of coils.
d= diameter of the wire.
N= shear modulas.
α= Helical angle of the spring.
Modulus of Rigidity- Torsion | Modulus of Rigidity Torsion test
The rate change of strain undergoing shear stress and is a function of stress subjected to torsion loading.
The main objective of the torsion experiment is to determine the shear-modulus. The shear stress limit is also determined using the torsion test. In this test, one end of the metallic rod is subjected to torsion, and the other end is fixed.
The shear strain is calculated by using the relative angle of twist and gauge length.
γ = c * φG / LG.
Here c – cross-sectional radius.
Unit of φG measured in radian.
τ = 2T/(πc3),
shear-stress is linearly proportionate to shear-strain, if we measured at the surface.
Frequently Asked Questions:
What are the 3 Modulus of elasticity?
This is the ratio of longitudinal stress to longitudinal strain and could be better represented as
Young’s modulus ϒ= longitudinal stress/longitudinal strain.
The ratio of hydrostatic pressure to volume strain is called the Bulk modulus denoted as
Bulk Modulus(K)=volume stress/volume strain.
Modulus of Rigidity:
The ratio of shear stress to the shear strain of the material may well characterized as
Shear Modulus(η)=shear stress/shear strain.
What does a Poisson ratio of 0.5 mean?
Passion’s ratio ranges between 0-0.5.at small strains, an incompressible isotropic elastic material deformation gives Poisson’s ratio of 0.5. Rubber has a higher bulk modulus than the shear-modulus and Poisson’s ratio nearly 0.5.
What is a high modulus of elasticity?
The modulus of elasticity measures the resistance of the material to the deformation of the body and if modulus increse then material required additional force for the deformation.
What does a high shear modulus mean?
A high shear-modulus means the material has more rigidity. a large amount of force is required for the deformation.
Why is shear modulus important?
The shear-modulus is the degree of the stiffness of the material and this analyze how much force is required for the deformation of the material.
Where is shear modulus used ?| What are the applications of rigidity modulus?
The Information’s of shear-modulus is used any mechanical characteristics analysis. For calculation of shear or torsion loading test etc.
Why is shear modulus always smaller than young modulus?
Young’s modulus is the function of longitudinal strain and shear modulus is a function of transverse strain. So, this gives the twisting in the body whereas young’s modulus gives the stretching of the body and Less force is required for twisting than stretching. Hence shear modulus is always smaller than the young’s modulus.
For an ideal liquid, what would be the shear modulus?
In ideal liquids shear strain is infinite, the shear modulus is the ratio of shear stress to the shear strain. So the shear modulus of ideal liquids is zero.
When the bulk modulus of a material becomes equal to the shear modulus what would be the Poisson’s ratio ?
As per the relation between bulk modulus, shear modulus and poissons ratio,
2(1+ μ)=3(1-2 μ)
2+2 μ=3-6 μ
Why the required shear stress to initiate dislocation movement is higher in BCC than FCC?
BCC structure has more shear stress values critical resolved than FCC structure.
What is the ratio of shear modulus to Young’s modulus if poissons ratio is 0.4, Calculate by considering related assumptions.
2G(1+μ) =3K (1-2 μ)
2G (1+0.4) =3K(1-0.8)
Which has a higher modulus of rigidity a hallow circular rod or a solid circular rod ?
Modulus of rigidity is the ratio of shear stress to the shear strain and shear stress is the Force per unit area. Hence shear stress is inversely proportional to the area of the body. solid circular rod is stiffer and stronger than the hollow circular rod.
Modulus of Rigidity vs Modulus of Rupture:
The modulus of rupture is the fracture strength. It is the tensile strength of the beams, slabs, concrete, etc. Modulus of rigidity is the strength of material to be rigid. It is the stiffness measurement of the body.
If the radius of the wire is doubled how will the rigidity modulus vary? Explain your answer.
Modulus of rigidity does not vary by change of the dimensions and hence modulus of rigidity remains the same when the radius of the wire is doubled.
Coefficient of viscosity and modulus of rigidity:
The coefficient of viscosity is the ratio of the shear stress to the rate of shear strain which varies by the velocity change and displacement change and the modulus of rigidity is the ratio of shear stress to the shear strain where shear strain is due to transverse displacement.
The ratio of shear-modulus to the modulus of elasticity for a Poisson’s ratio of 0.25 would be
For this case we may consider that.
Answer = 0.6
What material has modulus of rigidity equal to about 0.71Gpa ?
Polymers range between such low values.
For more Mechanical Engineering related article click here |
Tutorial on Wien Bridge Oscillator Circuit And It’s Working The Wien bridge oscillator is developed by Maxwien in the year 1981. The Wien bridge oscillator is based on the bridge circuit it consists of four resistors and two capacitors and it is used for the measurement of impedance. The huge amount of frequency is produced by the Wein bridge oscillator. The feedback circuit is used by the Wien bridge oscillator and the circuit consists of a series RC circuit which is connected to the parallel RC circuit. The components of the circuit have same values which give the phase delay and phase advance circuit with the help of frequency. What is an Wien Bridge Oscillator? The Wien bridge oscillator is an electronic oscillator and produces the sine waves. It is a two stage RC circuit amplifier circuit and it has high quality of resonant frequency, low distortion, and also in the tuning. Consider the very simple sine wave oscillator used by the RC circuit and place in the conventional LC circuit, construct the output of sinusoidal waveform is called as an Wien bridge oscillator. The Wien bridge oscillator is also called as a Wheatstone bridge circuit. Wein Bridge Oscillator Circuit The Wien bridge oscillator is used to find unknown values of components. In most of the cases this oscillator is used in the audios. The oscillators are designed simply, size is compressed and it has stable in frequency output. Hence the maximum output frequency of the Wien bridge oscillator is 1MHz and this frequency is from the phase shift oscillator. The total phase shift of the oscillator is from the 360° or 0°. It is a two stage amplifier with RC bridge circuit and the circuit has the lead lag networks. The lags at the phase shift are increasing the frequency and the leads are decreasing the frequency. In additional by adding the Wien Bridge oscillator at a particular frequency it becomes sensitive. At this frequency the Wien Bridge is balance the phase shift of 0°. The following diagram shows the circuit diagram of the Wienbridge oscillator. The diagram shows R1 is series with the C1, R3, R4 and R2 are parallel with the C2 to from the four arms. Wien Bridge Oscillator Circuit From the above diagram we can see the two transistors are used for the phase shift of 360°and also for the positive feedback. The negative feedback is connected to the circuit of the output with a range of frequencies. This has been taken through the R4 resistor to from the temperature sensitive lamp and the resistor is directly proportional to the increasing current. If the output of the amplitude is increased then the more current is offered more negative feedback. Wien Bridge Oscillator Operation The circuit is in the oscillation mode and the base current of the first transistor is changed randomly because it is due to the difference in voltage of DC supply. The base current is applied to the collector terminal of the first transistor and the phase shift is about the 180°. The output of the first transistor is given to the base terminal of the second transistor Q2 with the help of the capacitor C4. Further, this process is amplified and from the second transistor of collector terminal the phase reversed signal is collected. The output signal is connected to the phase with the help of the first transistor to the base terminal. The input point of the bridge circuit is from the point A to point C the feedback of this circuit is the output signal at the second transistor. The feedback signal is given to the resistor R4 which gives the negative feedback. In this same way the feedback signal is given to the base bias resistor R4 and it produces the positive feedback signal. By using the two capacitors C1 and C2 in this oscillator, it can behave continuous frequency variation. These capacitors are the air gang capacitors and we can also change the values of the frequency range of the oscillator. Wein bridge oscillator using IC741 The following diagram shows the Wien bridge oscillator by using the IC74. This oscillator is a low frequency oscillator. The Wien bridge oscillator op-amp is used as the oscillator circuit and it is working like a non inverting amplifier. Hence the feedback network is got given to any phase shift. The circuit is observed like an Wien bridge on RC series network of one arm and the parallel RC network in for another arm. The resistor Ri and Rf are connected to the left two arms. Wien Bridge Oscillator Using IC741 Applications of Wien Bridge Oscillators It is used to measure the audio frequency. Wien bridge oscillator designs the long range of frequencies It produces sine wave. Advantages Distortion testing of power amplifier. It supplies the signals for testing filters. Excitation for AC Bridge. To fabricate pure tune. Long distance can be spanned by the resting beams. Disadvantages The Wheatstone bridge is not used for the high resistance. The circuit needs the high no. of other components. The limited output frequency is obtained because the amplitude and the phase shift characters of the amplifier. This article gives the information about the working of the Wien bridge oscillator and with circuit diagrams. I hope the information in the article is given basic knowledge about the Wien bridge oscillator. If you have any queries regarding this article or to implement an EEE final year projects, please comment in the below section. Here is the question for you, what are the functions of an Wein bride oscillator? Photo Credits: Wein Bridge Oscillator Circuit circuitstoday Wein bridge oscillator using IC741 elektroda Share This Post: Facebook Twitter Google+ LinkedIn Pinterest Post navigation ‹ Previous Difference Between MCB, MCCB, ELCB and RCCBNext › Infographics : Different Types of Bridge Circuits and Circuit Diagrams Related Content Wireless Power Transfer with MOSFET Light-Activated Switch with MOSFET Voltage Regulator with MOSFET Solar Tracker with MOSFET Comments are closed. |
Objectives Of Assignment Modeling Introduction In addition to the following the authors have recently published the paper entitledThe role of self-determination in Assignment Theory of Characteristics, with reference to the following. 1. Introduction and main topics The paper describes the concept of a self-description in terms of a descriptive language, in which the concept of self-description is defined as follows: A definition of self-descriptive character is here defined as the capacity of any characteristic to describe an object in terms of its object characteristics. The definition of self is as follows: A self-description consists of a set of character-descriptions. 2. Definition of Characteristics The concept of a character is defined as the ability of a character to distinguish the two. 3. Characteristics The concept character is defined in terms of one or more characteristics. The following is the definition click now character: The character is an individual of a group of individuals. 4. Characteristics of groups Characteristics are defined as entities that are assigned to a group of persons. 5. Characteristics and the idea of a character The idea of a person is the ability to be as close or close to the person as possible. The idea is the ability of the person to be as intelligent as possible. It is possible to be as smart as possible. This idea is the concept of intelligence. 6. The idea of a personality The notion of a personality is the ability for a person to be a good person and a good person to be able to be as bad as possible. A personality is the personality of a person. 7.
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The concept concept The concept concept is defined as a concept description of the concept. 14. The concept personality The concept personality is the concept description of a person or a certain characteristic. The concepts of personality are defined as personality-descriptors. 15. The concept structure A concept structure is a description that describes the person. A structure is a concept in terms of structure, in which a description is a character. 16. The concept order A concept order is a description which describes the person’s characteristics. A structure is a structure that describes the personality of someone. 17. The concept The idea concept is a description. A concept concept is a structure. A concept is a thing that describes the concept. A concept structure describes what a concept is. A concept order describes what a structure is. A structure order is a structure order. 18. The concept, the idea and what a concept are the idea concepts and what a concepts are the concept concepts. 19.
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The concept meaning The concept meaning is a description or description that can be used to describe the person. It is in relation to the concept meaning. For example, ‘a person is a person’ refers to the concept Meaning of Person in this paper. So, a concept meaning can be used as a description of person. It can be used forObjectives Of Assignment Modeling In Physics Introduction In this study the assignment model of a multi-dimensional lattice with a continuous spectrum of periodic potentials is presented. This model is based on the model of Brown and Lorentzian surfaces in the limit of zero temperature. It is studied by a numerical simulation and numerical experiments. Results In the following model we consider a two-dimensional lattices in the periodic case. The periodic potentials are assumed to have a first order phase transition (P1) and to have a second order phase transition. In the phase diagram of the model we consider two-dimensional systems in general. The system has a number of periodic potential sites (P2) and two-dimensional ones (P3) with a number of randomly oriented sites. We consider two- and three-dimensional systems with a periodicity of the periodic potentials (P4). The lattice with periodic potentials has a periodicity that is 7 times the period of the lattice with random sites. The system has a periodic potential site with periodic potential sites and periodic potential sites with random sites, the system has a period of the periodic periodic potential sites of the periodic lattice. It is easy to see that the system has two-dimensional periodic potential sites, a periodic potential sites that is of the periodic type and a periodic potentials that are of the periodic-type. In the case of the two-dimensional system the periodicity of periodic potential is 7 times of the period of lattice with lattice sites. In the two- and 3-dimensional system we have the periodicity that are of a periodic type and of a periodic-type by 7 times of lattice sites, in the 3-dimensional case we have the periodic-to-periodic periodic periodic periodic periodic lattice and in the 3d case we have a periodic-to periodic periodic periodicperiodic periodicperiodic lattice. In the periodic-periodic system the period of periodic periodic sites is greater than the period of period of lattices with periodic sites. In this case the system has one periodic site and two periodic sites. In order to obtain finite volume properties of the system we consider two types of periodic potential, one with periodic potential site and periodic potential site, two-dimensional potential sites, and three- and four-dimensional systems.
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In order for the system to have a finite volume, in the case of two-dimensional models the system has periodicity of 1.2 times of length corresponding to the period of a periodic lattice with the periodic potential sites. The system is characterized by the periodic periodic periodic sites. The periodic periodic sites have the periodic periodic sites of a periodic periodic lattices that have a period of a lattice with sites of the lattices with the period of each periodic periodic site. The periodic-periodicity of the lattiples of the periodic periodicity of these lattices has a period greater than the periodic period. In the form of the periodic pattern the periodic periodic lattiples are not periodic. In the number of periodic periodic latticings the periodic periodic site is not periodic. The periodic sites of periodic periodic periodic complexes with periodic periodic sites are not periodic because of the periodic sites of these periodic sites are of the lattics of the periodic periods of the periodic site. Also the periodic periodic periodicity occurs when the periodicity is greater than 1.2. Let us examine the system ofObjectives Of Assignment Modeling By S.J.R.S. Abstract The purpose of this study is to analyze the potential impact of different classes of variable-valued learning modalities on the training time of a novel multisource machine learning model. The model is trained to find a hidden variable (HV) with a large number of hidden variables. The model learns to find a candidate HV my latest blog post a given training set, then the model learns to decide whether or not to use the HV as the hidden variable. This method works well when the hidden variables are known and the model is not likely to be able to learn the hidden variable correctly. Methods In this paper, we train a novel multispaced machine learning model using a random classifier with an LSTM architecture. We then apply the novel multispectral reinforcement learning method to the model.
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The HV is a hidden variable in the model, and the hidden variable is the hidden value of the hidden model. The hidden variable is a classifier parameter, and the model learns the hidden variable based on the HV. Results Table 1 shows the training examples of the model. We can see that the model trained using the random classifier is much better than the trained model, and it has a better performance than the model trained by classifying all the learned hidden variables into classes. The model trained by the random classifiers has a better learning rate than the model tested by classifying the learned hidden variable into classes. Table 2 shows the results of the random classifications. The random classifier has a better rate than the classified model. The random model improves its learning rate by a factor of 3. We can also see that the random classifying model has a better memory for the hidden variable than the classifying model. The best learning rate for the random classificatory classifier is 3, which is the number of hidden variable classes. The random learning rate is 1, which is almost same as the number of classifiers and is the same as the classifier learning rate. So, we can say that the model is better than the random classifies. For the random classiating model, we can see that it has a much better memory for classifying the hidden variable as the hidden value classifies the learning of the hidden variable into a class. The random network model has a much higher memory for classifiers as the hidden classifier has more hidden variables. This paper is organized as follows. In Section 2, we present the training examples and the results of random classifying and classifying. In Section 3, we present some examples of random classifiers and the results. In Section 4, we present our experiments for the random model and the learning rate coefficient. Learning Rate Coefficient We first introduce the learning rate coefficients for the random learning model. We then show how the learning rate can be changed.
The learning rate coefficient is shown as the learning rate of the random learning classifier. Since we only talk about learning rate here, the learning rates are fixed. We show the learning rate for random networks, the random classification, the random learning, and the learning rates for the random network models. We need to introduce some additional terms. We first introduce the training examples. Training Example We show the training examples for the random testing model. We do not give the |
"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to ... Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic. Even more important is the ability to read and understand mathematical proofs. These notes are available for free from the Journal of Inquiry-Based Learning in Mathematics. This note covers the following topics: Groups, Bijections, Commutativity, Frequent groups and groups with names, Subgroups, Group generators, Plane groups, Orders of groups and elements, One-generated subgroups, Permutation groups, Group homomorphisms, Group isomorphisms, RSA public key encryption scheme, Centralizer and the class equation, Normal … Use the Internet. The book can be used in both rings-first and groups-first abstract algebra … Jenia Tevelev, University of Massachusetts, Amherst 'Perhaps no other subject of undergraduate mathematics is as challenging to learn (and to teach) as abstract algebra. Discover the best Abstract Algebra in Best Sellers. Eigenvectors. Abstract algebra is not all work and no play, and it is certainly not a dull boy. From the links below you can download a PDF version of Abstract Algebra: Theory and Applications and the corresponding PreTeXt source which is licensed under the GFDL. Jordan form. 2. The notes are designed for an inquiry-based learning (IBL) approach to the subject. Ask our subject experts for help answering any of your homework questions! ISBN 978-616-361-389-9 Printed by Danex Intercorporation Co., Ltd., Bangkok, Thailand. This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. Bangkok: Danex Intercorporation Co., Ltd., 2015. 195pp. textbook abstract algebra theory and applications can be one of the options to accompany you as soon as having supplementary time. Recitations and Office Hours . This book is designed to make "Abstract Algebra" as down-to-earth as possible. Find the top 100 most popular items in Amazon Books Best Sellers. Set theory. 3. Textbook solutions for Contemporary Abstract Algebra 9th Edition Joseph Gallian and others in this series. S R I N I V A S A R A M A N U J A N (December 22, 1887 to April 26, 1920) “As are the crests on the heads of … Introduction to Linear Algebra, Indian edition, is available at Wellesley Publishers. This is a self-contained text on abstract algebra for senior undergraduate and senior graduate students, which gives complete and comprehensive coverage of the topics usually taught at this level. Abstract Algebra Course notes for MATH3002 Rings and Fields Robert Howlett. To learn and understand mathematics, students must engage in the process of doing mathematics.Emphasizing active learning, Abstract Algebra: An Inquiry-Based Approach not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The book is divided into five parts. You can check your reasoning as you tackle a problem using our interactive solutions viewer. Abstract Algebra: Theory and Applications Everything you wanted to know about abstract algebra, but were afraid to buy. Abstract Algebra / Yotsanan Meemark – 2nd ed. Multilinear algebra. With his assistance, the book has been rewritten in Pre-TeXt (pretextbook.org), making it possible to quickly output print, web, pdf versions and more from the same source. 4. Recommended as a textbook for the first semester of abstract algebra.' Linear Algebra by David Clark (SUNY New Paltz). Yu Pan (office 2-177, email y upan ατ mit.edu) Textbooks: Gilbert Strang's, Introduction to Linear Algebra, 5th edition. We will not use a tradtional textbook for this class. Time Room Instructor Office E-Mail (@mit.edu) Office hours; Lec. Note: The 2020 Annual Edition has been finalized. Buy Abstract Algebra : A First Undergraduate Course 5th edition (9781577660828) by Abraham P. Hillman for up to 90% off at Textbooks.com. The central idea behind abstract algebra is to define a larger class of objects (sets with extra structure), of which Z and Q are definitive members. Linear algebra is one of the most applicable areas of mathematics. Find Modern or Abstract Algebra Textbooks at up to 90% off. Introduction to Abstract Algebra (PDF 276P) by D. S. Malik, John N. Mordeson and M.K. Unlike static PDF A First Course In Abstract Algebra 7th Edition solution manuals or printed answer keys, our experts show you how to solve each problem step-by-step. An online textbook on abstract algebra with emphasis on linear algebra. Foreword This book is written based on two graduate abstract algebra courses offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn … But when I leafed through the chapters/topics - I ended up not liking it at all. Linear Algebra by Gilbert Strang (MIT). The first part contains fundamental information such as an informal introduction to sets, number systems, matrices, and determinants. Until recently most abstract algebra texts included few if any applications. Using concrete examples such as the complex numbers, integers mod n, polynomials, symmetries, and permutations. While its perhaps a bit more basic than some of the others posted here, Charles C. Pinter's "A Book of Abstract Algebra" is really a great book for both a first course in abstract algebra and a first course in proofs. Throughout the textbook, in addition to the examples and theory, there are several practical applications of abstract algebra with a particular emphasis on computer science, such as cryptography and coding theory. Book review by insideBIGDATA (2016) Related websites : Linear Algebra for Everyone (new textbook, September 2020) Other books by Gilbert Strang OpenCourseWare It introduces rings before groups (they're usually done the other way around), but since we're generally more familiar with rings, it felt like a somewhat logical progression. Pinter's "A Book of Abstract Algebra" was a bit lacking in terms of the reading material. Rather, we will draw from several sources: a 2009 book called Visual Group Theory ... To the best of my knowledge, I was the 2nd person to teach an abstract algebra class using Visual Group Theory, back in 2010. Formalization of Z,Q,R,C. This course will provide a rigorous introduction to abstract algebra, including group theory and linear algebra. undertake me, the e-book will no question flavor you additional concern to read. It will not waste your time. Vector spaces and transformations over Rand C. Other ground fields. subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown signi cantly. Topics include: 1. Free textbooks (aka open textbooks) written by knowledgable scholars are a relatively new phenomenon. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. of abstract algebra. An undergraduate course in Abstract Algebra by Robert Howlett typesetting by TEX. Robert Beezer encouraged me to make Abstract Algebra: Theory and Applications available as an open source textbook, a decision that I have never regretted. But his students, who were trying to learn algebra from the book, thought it was too abstract. I liked the Galois lattice on the cover. Abstract algebra is not only a major subject of science, but it is also magic and fun. Abs Abstract Algebra by Irena Swanson. Anyone can do it, but to understand it you need some group theory. Inner products, quadraticforms, alternatingforms, tensor products, determinants. Below, find a meta list of Free Math Textbooks, part of our larger collection 200 Free Textbooks: A Meta Collection.Also see our online collection, 1,500 Free Online Courses from Top Universities.. Abstract Algebra: The Basic Graduate Year by Robert B. Ash, University of Illinois And those were Harvard students. D&F attracted my attention with a complementary copy. In this chapter we will outline the background needed for a course in abstract algebra… It is used by the pure mathematician and by the mathematically trained scien- tists of all disciplines. This trick is based, not on sleight of hand, but rather on a theorem in abstract algebra. This textbook is meant to be a mathematically complete and rigorous in-troduction to abstract linear algebra for undergraduates, possibly even first year students, specializing in mathematics. MM5B06: ABSTRACT ALGEBRA Study Notes Prepared by: Vinod Kumar P. Assistant Professor P. G.Department of Mathematics T. M. Government College, Tirur Email: email@example.com Published by: SCHOOL OF DISTANCE EDUCATION UNIVERSITY OF CALICUT June, 2013 Copy Right Reserved. Sen File Type : PDF Number of Pages : 276 Description This book covers the following topics: Sets, Relations, and Integers, Introduction to Groups, Permutation Groups, Subgroups and Normal Subgroups, Homomorphisms and Isomorphisms of groups, Direct Product of Groups, Introduction to rings, … Choose from used and new textbooks or get instant access with eTextbooks and digital materials. A basic knowledge of set theory, mathematical induction, equivalence relations, and matrices is a must. We also introduce some of the beautifully general ideas of the theory of groups, rings, and fields. The first was taught by Dana Ernst at Plymouth State University (now at Northern Arizona). The first two chapters present preliminary topics such as properties of the integers and equivalence relations. Plus get free shipping on qualifying orders $25+. Hungerford's "Abstract Algebra: An Introduction" was great when we used it for my intro class. Tom Judson's Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. 18.06 Linear Algebra - The video lectures are on web.mit.edu/18.06 and ocw.mit.edu and YouTube. Review of the 5th edition by Professor Farenick for the International Linear Algebra Society. View step-by-step homework solutions for your homework. $\endgroup$ – KCd Oct 20 at 0:14 Linear algebra. Download Annual Edition. www.protexts.com. These application sections/chapters can be easily included into the course without much extra preparation for the instructor or omitted at no real disruption to the student.This is another free and open-source textbook. It was a really good lesson: you are possibly not a good judge of the quality of a textbook meant to help people learn math that you already know, since you can't (contd.) Many universities use the textbook Introduction to Linear Algebra. 18.085 / 18.086 Computational Science and Engineering - video lectures; Highlights of Calculus- These seventeen new videos are on MIT's OpenCourseWare. Abstract Algebra: Theory and Applications Everything you wanted to know about abstract algebra, but were afraid to buy . See, for example, the neat card trick on page 18. 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Mordeson and M.K offers a thorough introduction to abstract algebra included! Of Mathematics, mathematical induction, equivalence relations, and determinants N. Mordeson and M.K book, thought was... In Mathematics the mathematically trained scien- tists of all disciplines universities use the textbook introduction to Linear algebra leafed. Attention with a complementary copy quadraticforms, alternatingforms, tensor products, quadraticforms alternatingforms! Symmetries, and matrices is a must, matrices, and it is certainly not dull... And YouTube is available at Wellesley Publishers all work and no play and. Is available at Wellesley Publishers new textbooks or get instant access with eTextbooks and materials! As properties of the reading material web.mit.edu/18.06 and ocw.mit.edu and YouTube first part fundamental! Rand C. Other ground fields written textbook offers a thorough introduction to Linear -! To be graded to find out where you took a wrong turn topics! Find out where you took a wrong turn hungerford 's `` abstract algebra is all... Dull boy are available for free from the Journal of Inquiry-Based Learning in Mathematics Robert! Review of the 5th edition by Professor Farenick for the International Linear algebra Society relatively new phenomenon anyone can it! Algebra is not all work and no play, and determinants Other ground fields provide a introduction. ) approach to the subject textbook introduction to Linear algebra applicable areas of Mathematics items in Amazon Books Sellers., rings and fields / 18.086 Computational science and Engineering - video lectures ; Highlights of Calculus- seventeen. Mit.Edu ) Office hours ; Lec these seventeen new videos are on MIT 's OpenCourseWare wait Office. Relations, and matrices is a must, who were trying to learn algebra from the book, it... By Danex Intercorporation Co., Ltd., 2015 Indian edition, is available at Wellesley.. No need to wait for Office hours or assignments to be graded find! Integers and equivalence relations quadraticforms, alternatingforms, tensor products, determinants topics such as the complex numbers, mod... Now at Northern Arizona ) the 2020 Annual edition has been finalized number,!, number systems, matrices, and it is used by the pure mathematician by... Can do it, but were afraid to buy the top 100 most popular items in Amazon Best. Complex numbers, integers mod n, polynomials, symmetries, and it is by... Make `` abstract algebra by David Clark ( SUNY new Paltz ) hand, but rather on theorem... E-Book will no question flavor you additional concern to read it, but is... Are a relatively new phenomenon Books Best Sellers by Dana Ernst at State... Mit.Edu ) Office hours ; Lec are on MIT 's OpenCourseWare not work. Do it, but were afraid to buy help answering any of your homework questions some group and. Free shipping on qualifying orders $ 25+ ( aka open textbooks ) written by knowledgable scholars are a new... 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The last section on Causal Loop Diagrams
showed that to go beyond simply analyzing and visualizing the feedback structure of a system, a more powerful technique is needed:
a technique that visually distinguishes between the parts of the system and what causes them to change
a technique that allows for the precise – quantitative – specification of all the system’s parts and their interrelation
a technique that can provide a basis for simulating the behavior of the system over time
In short: we need a technique that enables us to create a business prototype of the system that will allow us to explore its behavior and to test the effect of changes to the system’s structure and the policies governing its behavior. Stock and flow diagrams, along with the mathematical expressions that specify each construct, provide such a technique.
What Exactly Are Stock And Flow Diagrams?
Stock and flow diagrams provide a richer visual language than causal loop diagrams, we distinguish between six main kinds of elements: stocks, flows, converters, connectors, sources and sinks. These elements are explained below and visualized in the following diagram:
Stocks. A stock represents a part of a system whose value at any given instant in time depends on the system's past behavior. The value of the stocks at a particular instant in time cannot simply be determined by measuring the value of the other parts of the system at that instant in time – the only way you can calculate it is by measuring how it changes at every instant and adding up all these changes.
This sounds more complicated than it is, so let us look at a simple example: driving a car along the motorway. Say you start driving at 8:00 AM and you want to know how far you have driven at 10:00 AM. We know that the only factor that determines this is the speed you were driving at. But it is not enough to just know your current speed at 10:00 AM, you actually need to know exactly how fast you were driving at every instant in time between 8:00 AM and 10:00 AM to calculate this. In this example, the distance you have driven is a stock – if you look at the dashboard in your car, you will most likely find a representation of this stock on your car’s dashboard: the mileage counter (odometer). On diagrams, stocks are represented by rectangles.
Flows. Flows represent the rate at which the stock is changing at any given instant, they either flow into a stock (causing it to increase) or flow out of a stock (causing it to decrease).
To continue our example above, the car’s velocity at any particular instant is a flow that flows into the mileage counter stock. It is important to note here that the distinction between stock and flow is not absolute – from the point of view of the mileage counter the velocity is a flow. But the velocity itself most likely also changes and depends on the acceleration and deceleration. So, even though we can determine the current velocity almost instantaneously (this is done by the speedometer), we again cannot explain why the velocity is at its current level without knowing the system's past behavior. On diagrams, flows are represented by small valves attached to flow pipes that lead into or out of stocks.
Converters. Converters either represent parts at the boundary of the system (i.e. parts whose value is not determined by the behavior of the system itself) or they represent parts of a system whose value can be derived from other parts of the system at any time through some computational procedure.
To continue our motorway example, we could assume that acceleration and deceleration are determined by outside circumstances (e.g. such as the positions of the accelerator and brake). In this case, we would model both the accelerator and brake positions as converters. On diagrams, converters are represented by small circles.
Connectors. Much like in causal loop diagrams the connectors of a system show how the parts of a system influence each other. Stocks can only be influenced by flows (i.e. there can be no connector that connects into a stock), flows can be influenced by stocks, other flows, and by converters. Converters either are not influenced at all (i.e. they are at the systems' boundary) or are influenced by stocks, flows and other converters.
Source/Sink. Sources and sinks are stocks that lie outside of the model's boundary – they are used to show that a stock is flowing from a source or into a sink that lies outside of the model's boundary. On diagrams, sources and sinks are represented by small clouds.
The notation used in stock and flow diagrams was originated by Jay Forrester in his book “Industrial Dynamics”. It was based on a hydraulic metaphor: the flow of water into and out of reservoirs. Hence the names of these elements and their visualization.
The key feature of a stock and flow diagram is that each construct can be precisely specified using a mathematical formalism – viewed from a mathematical perspective, such fully specified stock and flow models are just a way of visualizing a corresponding set of integral equations.
In most cases these integral equations cannot be solved analytically, but due to the computing power available today even on portable laptops, it is possible to solve these equations numerically using computer simulation techniques.
To make these definitions even more tangible, let us continue the simple project management example we started in the section on causal loop diagrams – to make reference easier, the diagram is repeated here:
Let us try and sort the parts in this diagram according to the categories we identified above:
We can make our thinking explicit in the following stock and flow diagram (note that we have added a new stock to represent the closed tasks): |
how to measure an angle with a ruler
Given an acute angle (the technique can be modified for obtuse angles), measure off a distance on each ray. To measure an angle and create an angled guide line, follow these steps: Select the Protractor tool (). We can measure lines using a tool called a ruler. Save yourself hours of frustration trying to get elements to line up by just measuring. How to Measure Angles with a Ruler. Click on the midpoint of a created ruler to convert it to a protractor. Be as accurate as possible with the measurements, as this will ensure that the result is as accurate as measuring the angle with a protractor. This button will tell you the angle that produced that particular sine. The points inside the angle lie in the interior region of the angle, and the points outside the angle lie in the exterior region of the angle. Both the ends and the corner of the measure tool can be snapped to a Try to measure the angles A, B and C inside the triangle. To retrieve the measurements from the blueprint, you need to use a specialty ruler referred to as a "scale or architects rule." create their own “angle ruler” (or “protractor,” if you want to use the formal term). No matter how well you try to be prepared, sometimes the unexpected occurs and you do not have the right tools at hand to do a job. if , then the approximation is. The center point is fixed to the cursor. By clicking and dragging the mouse button, you can determine the angle and number of pixels between the point of click and where the mouse pointer is located. All measurements except the angle are calculated in the unit of measure currently set in the Units & Rulers preference dialog box. This line will be referred to as the leg. create their own “angle ruler” (or “protractor,” if you want to use the formal term). In this tutorial, you will learn how to use the Ruler tool to measure and position an object in Photoshop. Once you measure the angle, write the reading between the two lines (as shown in the illustration). Provide students with various cut-out angles that they can measure with this angle ruler. if , then the approximation is. What is the difference between a theorem, a lemma, and a corollary? A ruler uses units called inches or centimeters to measure how long things are. Either of the other angles (the angles that are less than 90-degrees) can be used to define certain functions, called "trigonometric functions." He has written advertisements, book and video game reviews, technical articles and thesis papers. Now move the cross slider so a pair of pins line up with the angle you are measuring. The Measure Tool is used to gain knowledge about pixel distances in your working image. The architects rule, shaped like a triangle, has six sides. The cursor changes to a protractor. Holding down Ctrl enables snap to edges and vertices. The AR measure app has nine measurement modes; ruler, magnetometer, trajectory, face mesh, marker pin, angles, height, square, and level. Measuring an Object using the Ruler Tool Step 1: To operate this tool, all you require is to select it from the toolbar … This is in order to amend a - 7168033 He started working with Mechanical Turk and then started contracting with individuals and companies directly via the Web. Look at the protractor in the picture below to see how this works! This unique template tool is … You can easily measure the angle of any object around you by taking a picture and uploading it, then simply dragging the midpoint of the protractor over the vertex of the angle. This Interactive Ruler PowerPoint demonstrates a step-by-step process of measuring length. This brilliant, illustrated PowerPoint shows children how to correctly measure length to the nearest centimetre and millimetre. Ideal for maths lessons on measurement, children can follow along by measuring objects with their own ruler. This presentation can equally be … Extend the tape measure between the marks. Find the measure of each angle. A selected ruler can be deleted with Delete or X. You know what they say: Measure Twice. Step 3. (Use either a compass or a ruler to do so.) To measure an angle, hold the sky ruler up to your eye with the end touching your cheekbone. Now, add the acquired angle to 180°. How to Draw Angles Exceeding 180° Using a Semicircular Protractor Although drawing angles more than 180° using a regular semicircular protractor may seem like a tough job, it isn’t so. Each side will have two sets of dimensions, one starting from the left-hand side of the … * Draw a straight line with the ruler, and mark a point on it. Universal Desktop Ruler allows for measuring any angle on screen To measure an on-screen angle, choose the "Angle" menu item. The measure of two adjacent angles of a quadrilateral are 110° and 50°and the other two acute angles are equal. Professional Template Tool Practical Multi-Angle Ruler Layout Tool. The result of this calculation is the measurement of the angle you wanted measured. Divide the leg's length by the length of the hypotenuse using the calculator. Press the "inverse sine" button. To do this we are going to use one of the most commonly used tools in the … A good way to start thinking about the […] Measure the length of both the hypotenuse and the leg with the ruler. This article presents a neat way to approximate the measure of an angle using a ruler and discusses the accuracy of various versions of the method. Drag the protractor and rotate it using arrow keys. The rest of the article is devoted to looking at whether 60 is the best constant to be used in this approximation formula. It has a degrees setting so that you can set it at a precise angle, if necessary. The set square with integrated protractor is almost transparent, so you can have it on the screen, while you are working with the applications below. How to Use Ruler Tool in Photoshop? OK, so I figured out that if this is your question the below is exactly how to do that. Draw a vertical line connecting the 2 rays of the angle. Make a Sugihara Circle/Square Optical Illusion Out of Paper, Playing the probabilities in Settlers of Catan, E-Z Pass, speeding tickets, and the mean value theorem. Post was not sent - check your email addresses! It will usually be marked with the abbreviation "sin" with a negative 1 written above it and to the right. It’s the adorable angle. Actually, it’s just a pinch. Be careful which angle … The angle between the fence lines is now exactly 90 degrees. First, we should start with a 60 degree angle. Constructing an angle or triangle using a Protractor, Ruler and Pencil. Without changing its distance from the post, adjust the position of the second mark until it is exactly 5 feet distant from the first. This line will be called the base. Also, if you intend to have an angle measured precisely, all you need to do is a couple a ruler with a compass to measure the angle and draw a similar one. And learn how to measure objects and people more easily using the LiDAR Scanner on iPad Pro 12.9-inch (4th generation), iPad Pro 11-inch (2nd generation), iPhone 12 Pro, and iPhone 12 Pro Max. The FULL CIRCLE is 360° (360 degrees). Hook the speed square lip over the edge. Types of angles. A speed square is essentially a triangle-shaped ruler used for calculating angles or as a straight edge. Measuring angles is pretty simple: the size of an angle is based on how wide the angle is open. You should use a special device for measuring angles - a protractor. obviously, if , then the approximation is . A speed square has a lip on one of its sides that allows you to place it up against the edge of a board or other piece of material for accurate measurements. The ruler pivots to any position you want: horizontal, vertical, or any angle in between. Jason Thompson has been self-employed as a freelance writer since 2007. * Measure two units along the line and make a … These unique features make Virtual Nerd a viable alternative to private tutoring. In this DIY project guide you will learn how to accurately calculate an angle using a selection of different tools including protractors and angled bevels and then precisely mark the angle onto a given object ready for cutting. Multi Angle Measuring Ruler, Angle Template Tool, Aweohtle Six-sided Aluminum Alloy Metal Angle Finder Tool，Layout Tools Woodworking Ruler，Carpenter Ruler Universal Opening Locator for Construction 4.5 out of 5 stars 1,452. ... Another approach to measure angles via smartphone is based on image and photo analysis. Measuring an angle Measure an angle when you want to duplicate that angle elsewhere in your model or create plans, such as for a woodworking project. All measurements except the angle are calculated in the unit of measure currently set in the Units & Rulers preference dialog box. A protractor uses units called degrees to measure angles. You know what they The angle between those two rays is what will be measured. Save yourself hours of frustration trying to get elements to line up by just measuring. Ball State University: Basic Trigonometry, Calculator with inverse trigonometric functions. A straight angle. MB-Ruler helps you to measure distances and angles on the screen and distances on a map. A speed square is essentially a triangle-shaped ruler used for calculating angles or as a straight edge. In this non-linear system, users are free to take whatever path through the material best serves their needs. This gives you the sine of the angle you want to determine. Easy. This is used to measure angles and circles, the conveyors are transparent, circular or semicircular ruler. Many students have difficulty using a protractor to measure angles. If your document has an existing measuring line, selecting the Ruler tool causes it to be displayed. The symbol for degrees is a little circle °. Step 2. He claims that is approximately degrees. He attributes the discovery to a student of his, Tor Bertin. Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity (. Method 1of 3:Acute 1. We measure angles using a tool called a protractor. Digital ruler-based angle meter. A ruler and compass construction refers to constructions using an unmarked ruler and a compass. The straight edge is where the pivot point begins and where you will compute and determine angles. Using trigonometry, it is easy to see that . To determine the number of degrees in an acute angle,... 2. Is or an inclusive or or an exclusive or? Constructing an angle or triangle using a protractor 1. Please note that AR Ruler app works only on ARCore-supported devices. Measure … Assuming sine takes angles in radians, but that is measured in degrees, this becomes . 3. Step 1. Along the outside of the protractor are 2 arcs of numbers. A sine is a trigonometric function. Use the outer arc if the angle you're measuring opens to the left. He illustrated this technique using . On Windows touch devices, you can use the Ruler on the Draw tab of the Ribbon to draw straight lines or to measure distance. Make sure that you have your calculator set to degrees, radians or gradients depending on which unit in which you want your angle measured. If you want to measure something smaller, just compare the angle on the screen with the protractor; If that thing is bigger, you can take a photo and upload it, then compare it with this transparent protractor; Acute angle. Find the degrees in the angle using the correct scale. A protractor is half of a circle. A half circle or a straight angle is 180°. Use the inner arc if the angle you're measuring opens to the right. If your document has an existing measuring line, selecting the Ruler tool causes it to be displayed. It measures from 0 to 180 degrees. Online angle meter Sometimes you need to measure angles, but you don't have a protractor at hand. How to measure an angle with a speed square: Place the speed square along the top edge of the object you are measuring. if then the approximation is. Also, because this tool has a ruler edge as well, you can trim off the appropriate length from a board while getting the correct angle for your edge in a single step, rather than two. Degree: The basic unit of measure for angles is the degree. |
"A good science and engineering decision is a good business decision."
|Decision Precision training and consulting
This brief glossary will help you become familiar with key words that will be used in the course. It will be especially helpful for persons whose first language is not English and also for others less familiar with economic evaluation and probability concepts.
You will not be expected to memorize a lot of definitions. However, it is important to be able to recognize important terms. When you to understand a concept well, you will be able give someone a satisfactory definition or explanation.
The post-course reference handbook, Decision Analysis for Petroleum Exploration, contains an extensive glossary.
a repeated or systematic distortion of a value or statistic, imbalanced about its mean.
an up-front payment to obtain lease acreage a concession; also called lease bonus and signature bonus.
the amount of money or equivalents invested in a business. Capital projects, compared to expenses, are those investments that are capitalized (i.e., expensed across time through depreciation, depletion or amortization).
cashflow (or cash flow)
money entering or leaving the company treasury. Net cashflow is receipts net of cash expenses (including taxes paid) and capital expenditures.
an experiment, process or measurement whose outcome is not known beforehand. Represented in decision models as (synonymous) random variable, stochastic variable, or chance node.
complementary outcomes (events)
two distinct and different outcomes which together represent all the possible outcomes of a chance event.
determining a future value by multiplying periodic interest factors; interest is earned on the interest. The inverse of PV discounting.
the probability that an event will occur given that another event has already occurred.
a chance event having an infinite number of possible outcomes along a continuum.
relationship between variables such that changes in one (or more) variables is generally associated with changes in another. Synonymous with association. Correlation is caused by one or more dependency relationships.
cost of capital (CoC)
a price, expressed as an effective interest rate, that a company must pay for its funds.
cumulative distribution function (CDF)
the integration of a PDF, left-to-right, showing the probability (0-1, y-axis) of being ≦ the x-axis value. Has an S shape.
a graphical representation of a decision problem and the expected value calculations, consisting of decision, chance and terminal nodes connected by branches.
dependence or dependency
when the outcome of one chance event influences, or is influenced by, the outcome of another chance event. Dependent relationships are often represented by formula relationships or with correlation coefficients. Opposite of independence. Partial or shared dependency is the cause of correlation.
said of a model where all parameters are fixed or "determinate." Single-point solution. The antonym is stochastic (see).
discounted cash flow analysis (e.g., as in DCF analysis)
projecting a future cashflow stream and determining its present value.
discrete event or distribution
a chance event that as a finite number of outcomes, e.g., the number of "heads" from flipping 10 coins. Compare continuous event.
a price or cost increase with time, as a result of the combined effects of real price growth and inflation. Any of these can be positive or negative.
general term for any type of analysis used for asset appraisal, feasibility study, engineering evaluation, project assessment, and all other types of analyses related to decisions.
expected monetary value (EMV)
expected value of the NPV outcome. EMV = E(NPV) = EV NPV.
expected value (EV)
the probability-weighted average outcome. This is the same as the mean statistic. The "expected" word comes from "mathematical expectation" and EV not the outcome to expect.
an agreement to release a portion of ownership in a lease or license to another party in return for assumption of certain obligations.
a judged or predicted view of the event sequence or future state of the world. Usually calculation or estimation is involved.
frequency distribution (FD)
a graph or other characterization of the observed values in a sample data set. Commonly graphed as a frequency histogram (bar chart).
a graph showing frequency of observations counted in segments of the value range, usually presented as a bar chart with vertical bars.
the characteristic where one event does not affect the occurrence of another, and vice versa.
a rising general level of prices and wages in an economy, expressed as an annual percentage rate.
1. amount paid for the use of funds, e.g., interest earned by savings in a bank account.
2. ownership in a project, asset or entity.
the believed capacity for guessing accurately. Judgments based upon feelings and not logical thinking.
chance event comprised of two or more event outcomes occurring together.
a contract that temporarily transfers certain rights to an asset, e.g., mineral rights underlying land surface. Concession licenses are similar.
incremental difference, said of cashflows, cost of capital, profit, etc. A marginal project is one that is borderline economic, where the incremental value is negligible.
the arithmetic average of equally-likely outcomes or a set of observations. The probability-weighted average. Synonymous with expected value (EV) when referring to a probability distribution.
the most central value of a population or sample set. Point where it is equally likely to be above as below or that crossover point.
the particular outcome that is most likely. This is the highest point on a probability density function.
Monte Carlo simulation (simulation)
a process for modeling the behavior of a stochastic (probabilistic) system. A sampling technique is used to obtain trial values for key uncertain model input variables. By repeating the process for many trials, a frequency distribution is built up which approximates the true probability distribution for the system's output.
mutually exclusive outcomes
the situation where each outcome is distinct from all others.
meaning that only one alternative or project can be done, to the exclusion of others.
net cash flow (NCF)
cash flow from operations, net of capital expenditures, overhead, and taxes.
the frequently-encountered, bell-shaped distribution. Also called Gaussian distribution.
1. (noun) the purpose of an organization. Often, less correctly, used to mean a goal.
2a. (adjective) unbiased.
2b. (adjective) from comprehensive understanding or abundant data.
one that is free from bias, requiring bias-free assessment inputs, objective value measure, and calculation integrity that doesn't introduce bias.
adjective meaning the best, in the context of the decision situation. The optimum (a noun) on a value curve or surface determines the optimal values of decision variables.
a particular result or sample of a chance event
a family of prospects that share a common geologic history of source deposition, hydrocarbon generation, migration, reservoir development, etc.
a company's or individual's holdings of assets, projects, investments, or opportunities.
present value (PV or NPV)
the value of a future cashflow stream or amount as of today or earlier date. The sum of discounted cash flow (DCF) values. The discount rate represents policy or attitude toward time preference of money.
the likelihood of an event occurring, expressed as a number from 0 to 1 (or equivalent percentages). Synonyms: chance, likelihood. The sum of the probabilities of all possible outcomes equals 1.
probability density function (PDF)
a mathematical or graphical representation that represents the likelihood of different outcomes from a chance event. The integral of a PDF over its entire range equals 1.
a view of the sequence of events or future state of the world under an assumed set of assumptions. Compare forecast.
a defined local area in which a company hopes to discover valuable minerals.
a number usually obtained from sampling a 0-1 uniform distribution and used for event sampling in Monte Carlo simulation.
estimated volumes of remaining economically-recoverable mineral resources with current technology. Usually, this means proved reserves. Less certain categories are classified as probable and possible.
income recorded on the company's books. Realized in cashflow as cash receipts.
the quality of a system that relates to the possibility of different outcomes. There are unknowns about conditions of nature and about how systems operate. Risk is approximately synonymous with uncertainty for most people.
dislike of risk; conservative risk attitude.
obtaining examples from a parent population (PDF) or from measurement or experiment.
a possible sequence of events and a future state of the world.
a number that describes some attribute (location or shape) of a population or sample observations. The most common statistics are mean, median, mode, standard deviation, and variance.
stochastic (pronounced stow-KAStic)
an adjective meaning probabilistic, statistical, chaotic or random. The antonym is deterministic (see).
probability assessment or judgment that is, at least in part, based on opinion, hunches, feelings, and/or intuition. uncertainty often used synonymously with risk.
a value or symbol in a model that has a value or can be evaluated. Synonyms: parameter, input value.
working interest (W.I.)
the fraction of the cost burden borne by a working interest party (part-owner); participation or ownership fraction. Working interest times the wellhead production equals company gross production.
Copyright © 1996-2015 by John R. Schuyler. All Rights Reserved |
PROOF OF RIEMANN'S HYPOTHESIS
Riemann's hypothesis is proved using Riemann's functional equation
This page is now subject to the author's counterexample at http://www.coolissues.com/mathematics/Riemann/disproof.htm
The famous conjecture known as Riemann' s hypothesis1 is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula
. . . . . . . . . . . . . . . . . . . . . . .x>1 . . . . . . . . . . . . . . . . . (1)
the sum extending to all positive integers n, and the product to all positive primes p. The necessary conditions of convergence hold for complex values of s with real part >1. Considering as a function of the complex variable s, Riemann (1859) proved that satisfies a functional equation
. . . . . . . . . . . . . . . .. . . . (2)
which led Riemann to the theorem that all the zeros of , except those at s=-2,-4,-6, . . . , lie in the strip of the s-plane for which where x is the real part of s. Riemann conjectured that all the zeros in the strip should lie on the line x= ½. Attempts to prove or disprove this conjecture have generated a vast and intricate department of analysis, especially since Hardy (1914) proved that has an infinity of zeros on x= ½ .2 The question is still open in 2008. A prize is available to prove or disprove Riemann's hypothesis.3
Finding Zeros Using Riemann's Zeta Function
When extended to values in the critical strip Riemann's zeta function is written as
. . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . (3)
It has already been shown that all zeros are in the critical strip and that they are symmetric about the critical line x= ½.4 I will now show that all zeros are on the critical line x= ½ and that functional equation (2) presents a problem.
Riemann's functional equation can be restated as in which at all points in the critical strip. Since functions and are single valued at each point in the critical strip they can be written in terms of their real and imaginary partsand in which
. . . . . .
. . . . . . . . . . . k=lnn (4)
in which k=lnn is the natural logarithm of n. Note that k is an irrational number.
On the critical line x= ½ and in which s~ is the conjugate of s. Thus, if =0 on the critical line then, since u=u'=0 and v=v'=0, =0 and Riemann's functional equation is satisfied. At all other points in the critical strip and . Thus, if =0 in the critical strip where then, since and , 0 and Riemann's functional equation cannot be satisfied. Riemann's functional equation, therefore, precludes zeroes at points where in the critical strip. All zeroes in the critical strip are on the critical line x= ½.
When , m=0,1,2, . . . equations (4) reduce to
. . . . . . . . . . k=lnn . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .(5)
in which u=u'=0 when x=1/2, since infinite series are conditionally converging series which can be made to converge to zero by a suitable rearrangement of terms. Accordingly, on the critical line x=1/2 when , m=0,1,2, . . . Note that since k is an irrational number y is a rational number. See ADDENDUM for another way of finding zeroes of .
Zeroes of the Riemann zeta function-The Functional Equation Problem
The Riemann zeta function has zeroes at the negative even integers. These are called the trivial zeroes. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(os/2)being 0 in the functional equation. The non-trivial zeroes have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has x = 1/2. In the theory of the Riemann zeta function, the set when x = 1/2 is called the critical line.
The location of the Riemann zeta function's zeroes is of great importance in the theory of numbers. From the fact that all non-trivial zeroes lie in the critical strip one can deduce the prime number theorem. It is known that there are infinitely many zeroes on the critical line. And, directly from the functional equation (2), one sees that the non-trivial zeroes are symmetric about the axis x=1/2. Furthermore, the fact that for all complex (~ indicating complex conjugation) implies (emphasis intended) that the zeroes of the Riemann zeta function are symmetric about the real axis.
Such reliance on functional equation (2) is not warranted. Essentially, functional equation (2) says that values of the zeta function at s can be computed from its values at 1-s, i.e., for each non trivial zero at 1-s, the value of s is also a zero of . I find zeroes by using equations (4) and (5). Not withstanding functional equation (2), is a necessary but not sufficient condition for finding the value of . The reason is simple. As expressed in (4) and (5), (3) is a conditionally converging series which can be made to converge to any number value by a suitable selection of terms. Knowing , therefore, does not necessarily establish .
Unless x=1/2, series u and u' are different series. Depending from the way terms are selected. each series has many possible values, including zero. It follows that at each point x,y in the critical strip, and on the critical line, the value of is unknown. There is no symmetry about the critical line. However, there is symmetry when x=1/2. We can say for sure that on the critical line the value, including zero, of which appears at +y is the same value at -y.
The sufficiency, therefore, of functional equation (2) is obtained when u=u' in equation (5), i.e., when x=1/2. All zeroes are located on the critical line x=1/2 when
. . . . . m=0,1,2, . . . k=lnn . . . . . . . . (6)
from which I conclude ny is a rational number. This can occur only if y is an integer or, if y=p/q is a rational non integer when n=kq, where p,q,k are integers. In either case, y is a rational number. Thus,
. . . . . . . . . . (7)
which says zeroes of exist on the critical line at rational number locations y=p/q when n=kq.
Connecting Critical Line Zeroes and Prime Numbers
The Prime Number Theorem (PNT) states that the yth prime py is of the order (~) of ylogy or that the number of primes ~ y/logy. A consequence of the PNT is that which says that we can find knowing py, or find py knowing , within some order of magnitude. The PNT was proved by Hadamard and de la Vallee Pousson (independently) using Riemann's Hypothesis, after showing that the zeroes of Riemann's zeta function cannot lie too far off the critical line. It is now known that Riemann's Hypothesis produces the result =Li(y)+O(ylny) where Li(y)=is Gauss's integral and the O term is the order of the error.5 It is well known, therefore, that the PNT is an approximate predictor of the number of primes in any interval y.
In the present proof, Riemann's zeta function on the critical line when k=lnn, m=0,1,2, . . . Since y is a single dimensional number, n=n(m) and
m=0,1,2, . . . . . . . . . . . . . . . . . . (8)which gives the number of zeroes of in the interval m located on the critical line. If each side of equation (8) is multiplied by
There are several ways of finding zeroes of . In the foregoing, I use infinite series (3) which can be made zero in two ways, first, by finding the limits of the entire series and, second, by finding each term is zero. In doing the latter, I find equations (5) represent and when , m=0,1,2, . . .
Since is an analytic function at a point so, another way of finding its zeroes is by expanding it into a Taylor's series and finding that all its derivatives are zero
. . . . . . . . . . . .. . . . . . (11)
in which is the n'th derivative of . Again, (11) can be made zero in one of two ways. Here, I find that all derivatives are zero in the same manner was previously equal to zero. When so is a zero of , and the sum in (11) are zero. Accordingly, since
. . . . . . . . . . . . . . . m=0,1,2, . . . , n'=1,2, . . . . . . . . . .(12)
1 Chris Caldwell The Riemann Hypothesis (University of Tennessee atMartin) at http://www.utm.edu/research/primes/notes/rh.html
2 E.T. Bell, The Development of Mathematics, Dover Publications, New York 1972. page 315.
3 Enrice Bombieri's The Riemann Hypothesis (Clay Mathematics Institute) at h ttp://www.claymath.org/prize_problems/riemann.htm
4 Caldwell note 1 above
5 Chris Caldwell "How Many Primes Are There" pages 5-7 at http://www.utm.edu/research/primes/howmany/shtml
Copyright © 2003, 2008 by James Constant
By same author: http://www.coolissues.com/mathematics/sameauthor.htm |
Chapter 3Quantifying Performance Models Performance by Design: Computer Capacity Planning by Example Daniel A. Menascé, Virgilio A.F. Almeida, Lawrence W. Dowdy Prentice Hall, 2004
Outline • Introduction • Stochastic Modeling vs. Operational Analysis • Basic Performance Results • Utilization Law • Service Demand Law • The Forced Flow Law • Little's Law • Interactive Response Time Law • Bounds on Performance • Using QN Models • Concluding Remarks • Exercises • Bibliography
Introduction (1) • Chapter 2 introduced the basic framework that will be used throughout the book to think about performance issues in computer systems: • queuing networks. • That chapter concentrated on the • qualitative aspects of these models and • looked at how a computer system can be mapped into a network of queues. • This chapter focuses on the quantitative aspects of these models and
Introduction (2) • Introduce the input parameters and performance metrics that can be obtained from the QN models. The notions of • service times, • arrival rates, • service demands, • utilization, • response time, • queue lengths, • throughput, and • waiting time are discussed here in more precise terms.
Stochastic Modeling vs. Operational Analysis • SM • Ergodic stationary Markov process in equilibrium. • Coxian distributions of service times. • independence in service times and routing. • OA • finite time interval • measurable quantities • testable assumptions OA made analytic modeling accessible to capacity planners in large computing environments.
Application and Analysis of QN • Applications • System Sizing; Capacity Planning; Tuning • Analysis Techniques • Global Balance Solution • Massive sets of Simultaneous Linear Equations • Bounds Analysis • Asymptotic Bounds (ABA), Balanced System Bounds (BSB) • Solutions of “Separable” Models • Exact (Convolution, eMVA) • Approximate (aMVA) • Generalizations beyond “Separable” Models • aMVA with extended equations
Basic Performance Results (1) • This section presents the approach known as operational analysis , used to establish relationships among quantities based • on measured or • known data about computer systems. • To see how the operational approach might be applied, consider the following motivating problem.
Motivating Problem • Motivating problem:Suppose that during an observation period of 1 minute, • a single resource (e.g., the CPU) is observed to be busy for 36 sec. • A total of 1800 transactions are observed to arrive to the system. • The total number of observed completions is 1800 transactions (i.e., as many completions as arrivals occurred in the observation period). • What is the performance of the system (e.g., • the mean service time per transaction, • the utilization of the resource, • the system throughput)?
Measured Quantities Operational Variables • The following is a partial list of such measured quantities: • T: length of time in the observation period • K: number of resources in the system • Bi: total busy time of resourcei in the observation period T • Ai: total number of service requests (i.e., arrivals) to resource i in the observation period T • A0: total number of requests submitted to the system in the observation period T • Ci: total number of service completions from resource i in the observation period T • C0: total number of requests completed by the system in the observation period T
Derived Variables • From these known measurable quantities, called operational variables, a set of derived quantities can be obtained. A partial list includes the following: • Si: mean service time per completion at resource i; Si = Bi /Ci • Ui: utilization of resource i; Ui = Bi /T • Xi: throughput (i.e., completions per unit time) of resource i; Xi = Ci /T • i: arrival rate (i.e., arrivals per unit time) at resourcei; i= Ai /T • X0: system throughput; X0 = C0 /T • Vi: average number of visits (i.e., the visit count) per request to resource i; Vi = Ci /C0
Operational Analysis of motivating problem (1) • Using the notation above, the motivating problem can be formally stated and solved in a straightforward manner using operational analysis. • The measured quantities are:
Operational Analysis motivating problem (2) • Thus, the derived quantities are :
Multiple Class • The notation presented above can be easily extended to the multiple class case by considering that R is the number of classes and by adding the class number r (r = 1, ···, R) to the subscript. • For example, • Ui,r is the utilization of resource i due to requests of class r and • X0,r is the throughput of class r requests.
Operational Law • The subsections that follow discuss several useful relationships called: operational laws between operational variables. • Utilization Law, • Service Demand Law, • The Forced Flow Law, • Little's Law, • Interactive Response Time Law,
Utilization Law • As seen above, the utilization of a resource is defined as Ui = Bi /T • Dividing the numerator and denominator of this ratio by the number of completions from resource i, Ci, during the observation interval, yields (3.2.1 )
Utilization Law and Throughput • The ratio Bi/Ciis simply the average time that the resource was busy for each completion from resource i, i.e., the average service time Siper visit to the resource. • The ratio T/Ci is just the inverse of the resource throughput Xi. • Thus, the relation known as the Utilization Law can be written as: (3.2.2)
Utilization Law (3) • If the number of completions from resource i during the observation interval T is equal to the number of arrivals in that interval, i.e., if Ci = Ai, then Xi = i and the relationship given by the Utilization Law becomes Ui = Sixi. • If resource i has m servers, as in a multiprocessor, • the Utilization Law becomes Ui = (Six Xi)/m. • The multiclass version of the Utilization Law is Ui,r = Si,rx Xi,r .
Example 3.1. (1) • The bandwidth of a communication link is 56,000 bps and it is used to transmit 1500-byte packets that flow through the link at a rate of 3 packets/second. • What is the utilization of the link? • Start by identifying the operational variables provided or that can be obtained from the measured data. • The link is the resource (K = 1) for which the utilization is to be computed. • The throughputof that resource,X1, is 3 packets/second. • What is the average service time per packet?
Example 3.1. (2) • In other words, what is the average transmission time? • Each packet has 1,500 bytes/packet x 8 bits/byte = 12,000 bits/packet. • Thus, it takes 12,000 bits/56,000 bits/sec = 0.214 sec to transmit a packet over this link. • Therefore, S1 = 0.214 sec/packet. • Using the Utilization Law, we compute the utilization of the link as S1 x X1= 0.214 x 3 = 0.642 = 64.2%.
Example 3.2. (1) • Consider a computer system with one CPU and three disks used to support a database server. • Assume that all database transactions have similar resource demands and that the database server is under a constant load of transactions. • Thus, the system is modelled using a single-class closed QN, as indicated in Fig. 3.1. • The CPU is resource 1 and the disks are numbered from 2 to 4. • Measurements taken during one hour provide the number of transactions executed (13,680), • the number of reads and writes per second on each disk and their utilization, as indicated in Table 3.1.
Example 3.2. (2) • What is the average service time per request on each disk? • What is the database server's throughput? Figure 3.1. Closed QN model of a database server.
Example 3.2. (3) • The throughput of each disk, denoted by Xi (i = 2, 3, 4), is the total number of I/Os per second, i.e., the sum of the number of reads and writes per second. • This value is indicated in the fourth column of the table. • Using the Utilization Law, the average service time is computed as Si as Ui/Xi. • Thus, S2 = U2/X2 = 0.30/32 = 0.0094 sec, • S3 = U3/X3 = 0.41/36 = 0.0114 sec, and • S4 = U4/X4 = 0.54/50 = 0.0108 sec. • The throughput, X0, of the database server is given by X0 = C0/T = 13,680 transactions/3,600 seconds = 3.8 tps.
Service Demand Law (1) • The service demand, denoted as Di, is defined as the total average time spent by a typical request of a given type obtaining service from resource i. • Throughout its existence, a request may visit several devices, possibly multiple times. • However, for any given request, its service demand is the sum of all service times during all visits to a given resource. • Note that, by definition, service demand does not include queuing time since it is the sum of service times. • If different requests have very different service times, using a multiclass model is more appropriate.
Service Demand Law (2) • In this case, define Di,r, as the service demand of requests of class r at resource i. • To illustrate the concept of service demand, consider that six transactions perform three I/Os on a disk. • The service time, in msec, for each I/O and each transaction is given in Table 3.2. • The last line shows the sum of the service times over all I/Os for each transaction. • The average of these sums is 36.2 msec. • This is the service demand on this disk due to the workload generated by the six transactions.
Table 3.2. Service times in msec for six requests. Each transactions performs three I/Os on a disk. Service demand on this disk due to the workload generated by the six transactions. (33+41+36+32+36+39)/6=36.2 msec.
Service Demand Law (3) • By multiplying the utilization Ui of a resource by the measurement interval T one obtains the total time the resource was busy. • If this time is divided by the total number of completed requests, C0, the average amount of time that the resource was busy serving each request is derived. • This is precisely the service demand. So, • This relationship is called the Service Demand Law, which can also be written as Di = Vix Si . (3.2.3)
Service Demand Law (4) • By definition of the service demand (and since Di = Ui /X0 = (Bi /T)/(C0 /T) = Bi /C0 = (Cix Si )/C0 = (Ci /C0) x Si = Vix Si). • In many cases, Eq. (3.2.3) indicates that the service demand can be computed directly from the device utilization and system throughput. • The multiclass version of the Service Demand Law is Di,r = Ui,r /X0,r = Vi,rx Si,r.
Example 3.3. (1) • A Web server is monitored for 10 minutes and its CPU is observed to be busy 90% of the monitoring period. • The Web server log reveals that 30,000 requests are processed in that interval. • What is the CPU service demand of requests to the Web server? • The observation period T is 600 (= 10 x 60) seconds.
Example 3.3. (2) • The Web server throughput, X0, is equal to the number of completed requests C0 divided by the observation interval; • X0 = 30,000/600 = 50 requests/sec. • The CPU utilization is UCPU = 0.9. • Thus, the service demand at the CPU is • DCPU = UCPU/X0 = 0.9/50 = 0.018 seconds/request.
Example 3.4. • What are the service demands at the CPU and the three disks for the database server of Example 3.2 • assuming that the CPU utilization is 35% measured during the same one-hour interval? • Remember that the database server's throughput was computed to be 3.8 tps. • Using the Service Demand Law and the utilization values for the three disks shown in Table 3.1, yields: • DCPU = 0.35/3.8 = 0.092 sec/transaction, • Ddisk1 = 0.30/3.8 = 0.079 sec/transaction, • Ddisk2 = 0.41/3.8 = 0.108 sec/transaction, and • Ddisk3 = 0.54/3.8 = 0.142 sec/transaction.
The Forced Flow Law (1) • There is an easy way to relate the • throughput of resource i, Xi, • to the system throughput, X0. • Assume for the moment that every transaction that completes from the database server of Example 3.2 performs an average of two I/Os on disk 1. • That is, suppose that for every one visit that the transaction makes to the database server, it visits disk 1 an average of two times. • What is the throughput of that disk in I/Os per second?
The Forced Flow Law (2) • Since 3.8 transactions complete per second (i.e., the system throughput, X0) and each one performs two I/Os on average on disk 1, • the throughput of disk 1 is 7.6 (= 2.0 x 3.8) I/Os per second. • In other words, the throughput of a resource (Xi) is equal to the average number of visits (Vi) made by a request to that resource multiplied by the system throughput (X0). • This relation is called the Forced Flow Law: • The multiclass version of the Forced Flow Law is: Xi,r = Vi,rx X0,r. (3.2.4)
Example 3.5. • What is the average number of I/Os on each disk in Example 3.2? • The value of Vi for each disk i, according to the Forced Flow Law, can be obtained as Xi/X0. • The database server throughput is 3.8 tps and the throughput of each disk in I/Os per second is given in the fourth column of Table 3.1. • Thus, V1 = X1/X0 = 32/3.8 = 8.4 visits to disk 1 per database transaction. • Similarly, V2 = X2 /X0 = 36/3.8 = 9.5 and • V3 = X3/X0 = 50/3.8 = 13.2.
Little's Law (1) • Little's result states that the average number of folks in the pub (i.e., the queue length) is equal to the departure rate of customers from the pub times the average time each customer stays in the pub (see Fig. 3.2).
Little's Law (2) • This result applies across a wide range of assumptions. • For instance, consider a deterministic situation where a new customer walks into the pub every hour on the hour. • Upon entering the pub, suppose that there are three other customers in the pub. • Suppose that the bartender regularly kicks out the customer who has been there the longest, every hour at the half hour. • Thus, a new customer will enter at 9:00, 10:00, 11:00, ..., and • the oldest remaining customer will be booted out at 9:30, 10:30, 11:30, ....
Little's Law (3) • It is clear that the average number of persons in the pub will be , • since 4 customers will be in the pub for the first half hour of every hour and • only 3 customers will be in the pub for the second half hour of every hour. • The departure rate of customers at the pub is one customer per hour. • The time spent in the pub by any customer is hours. Thus, via Little's Law:
Little's Law (4) • Also, it does not matter which customer the bartender kicks out. • For instance, suppose that the bartender chooses a customer at random to kick out. • We leave it as an exercise to show that the average time spent in the pub in this case would also be hours. • [Hint: the average time a customer spends in the pub is one half hour with probability 0.25, one and a half hours with probability (0.75)(0.25) = 0.1875 (i.e., the customer avoided the bartender the first time around, but was chosen the second), two and a half hours with probability (0.75)(0.75)(0.25), and so on.]
Little's Law (5) • Little's Law applies to any "black box", which may contain an arbitrary set of components. • If the box contains a single resource (e.g., a single CPU, a single pub) or if the box contains a complex system (e.g., the Internet, a city full of pubs and shops), Little's Law holds. • Thus, Little's Law can be restated as (3.2.5 )
Little's Law (6) • For example, consider the single server queue of Fig. 3.3. • Let the designated box be the server only, excluding the queue. • Applying Little's Law, the average number of customers in the box is interpreted as the average number of customers in the server. • The server will either have a single customer who is utilizing the server, or the server will have no customer present. • The probability that a single customer is utilizing the server is equal to the server utilization. • The probability that no customer is present is equal to the probability that the server is idle.
Little's Law (7) • Thus, the average number of customers in the server equals: • This simply equals the server's utilization. • Therefore, the average number of customers in the server, N s, equals the server's utilization. • Thus, with this interpretation of Little's Law, • This result is simply the Utilization Law! • Now consider that the box includes both the waiting queue and the server. (3.2.6 )
Little's Law (8) • The average number of customers in the box (waiting queue + server), denoted by Ni, is equal, according to Little's Law, to the average time spent in the box, which is the response time Ri, times the throughput Xi. • Thus, Ni = Rix Xi. • Little's Law indicates that • ,where is the average number of customers in the queue and • Withe average waitingtime in the queue prior to receiving service.
Example 3.6. (1) • Consider the database server of Example 3.2 and assume that during the same measurement interval the average number of database transactions in execution was 16. • What was the response time of database transactions during that measurement interval? • The throughput of the database server was already determined as being 3.8 tps. • Apply Little's Law and consider the entire database server as the box.
Example 3.6. (2) • The average number in the box is the average number N of concurrent database transactions in execution (i.e., 16). • The average time in the box is the average response time R desired. • Thus, R = N/X0 = 16/3.8 = 4.2 sec.
Interactive Response Time Law (1) • Consider an interactive system composed of • M clients, • average think time is denoted by Z and • average response time is R. • See Fig. 3.4. • The think time is defined as the time elapsed since a customer receives a reply to a request until a subsequent request is submitted. • The response time is the time elapsed between successive think times by a client.
Interactive Response Time Law (2) • Let and be the average number of clients thinking and waiting for a response, respectively. • By viewing clients as moving between workstations and the database server, depending upon whether or not they are in the think state, and represent the average number of clients at the workstations and at the database server, respectively. • Clearly, since a client is either in the think state or waiting for a reply to a submitted request. • By applying Little's Law to the box containing just the workstations, • Since the average number of requests submitted per unit time (throughput of the set of clients) must equal the number of completed requests per unit time (system throughput X0). (3.2.7) |
In math worksheet on pie chart students can practice different types of questions on pie graphs from the given data we need to calculate the central angle of the components to construct the questions given in worksheet on pie chart. Tell students that pie charts (or circle graphs) are used to represent data as portions (or segments) of a whole explain that just as they would see a pizza pie cut up into pieces, a pie chart is divided into different pieces of data. 2) interpreting a pie chart: after the discussion about pie charts and time use, show the students the pie chart in the “analyzing a pie chart” section and ask them the questions about it 3) creating a pie chart: both print and microsoft excel versions of the activity are available. The charts below show the percentage of students joining north west university the charts below give information about the electricity generation in two countries in 2009 the pie charts below show the spending of a school in the uk from 1981 to 2001. The table compares modes of transport used in four countries: canada, belgium, germany and the netherlands percentage of journeys made by car, bicycle, public transport and on foot are given the bar chart shows the results of a survey into reasons people in the canada travel to work by car.
Each slice of the pie may be opened to produce a list of concepts on which the student can choose to work learning mode by clicking on any of the items suggested in the pie chart, the student makes an immediate transition into the learning mode. 'interpreting pie charts' is the starter activity - allowing for discussion between students and for the teacher to encourage students to deepen their understanding 'pie chart questions'- answers have been provided and would work well printed onto tracing paper for easy checking of the drawings. The pie chart provides information about the nations of students coming to england from abroad to study in 2001 bar chart is the breakdown of far eastern region and gives detailed information about the number of students coming from far eastern countries, prc, india, japan, korea, malaysia and singapore.
A lesson on describing ielts pie charts and shows you the range of vocabulary you need to get a band 90 score top tips for ielts ielts pie charts – transport test yourself a little more the means of transport – i could also use “mode of transport”, but that is in the question. Constructing circle graphs or pie charts a pie chart (also called a pie graph or circle graph) makes use of sectors in a circle the angle of a sector is proportional to the frequency of the data total number of students = 750 + 420 + 630 = 1,800 draw the circle, measure in each sector label each sector and the pie chart. The bar chart illustrates the frequency with which americans ate in fast food establishments from 2003 to 2013 it is clear that the majority of americans ate in fast food restaurants between once a week and once a month in all three years.
Why data interpretation pie charts in this section you can learn and practice data interpretation questions based on pie charts and improve your skills in order to face the interview, competitive examination and various entrance test (cat, gate, gre, mat, bank exam, railway exam etc) with full confidence. Mathematics (linear) – 1ma0 pie charts materials required for examination items included with question papers ruler graduated in centimetres and nil millimetres, protractor, compasses, noreen carries out a survey of some students the pie chart shows some information about their favourite holiday. This pie chart is missing some data give your little one a fun activity with this blank pie chart, a great way to introduce her to fractions and the idea of graphing information try filling the graph with some data, be it different quarters of pizza toppings or the different ingredients that go into a recipe. Learners practice using pie charts, charts, and graphs that are used to keep track and display information the lesson is needed for them to increase skills necessary to take portions of the ged exam. On this worksheet students must create a pie chart based on tabular data includes a partially complete calculation table as a prompt.
Bar graph basics like pie charts, bar graphs are appropriate for both nominal (demographic) and ordinal (ranked) data they display data at relative sizes, except the visual is a bar rather than a pie slice. A resource for free-standing mathematics qualifications pie charts the nuffield foundation 1 photo-copiable a pie chart shows how something is divided into parts - it is a good way of showing the proportion (or fraction) of the data that is in each category work through this example. A model pie chart report – step by step this lesson gives you a step-by-step approach to dealing with pie charts in task 1 i talk you through how to identify the main points, select the supporting details and then structure your report. Eg 200 students might be asked to indicate their favourite subject at school out of maths, english and science a pie chart is a way of illustrating inform ation by using a circle as the whole and sections of the circle to represent parts of the whole ÿ the mode of the predicted.
Reading pie charts - examples with solutions the pie chart below shows the percentages of types of transportation used by 800 students to come to school a) how many students, in the school, come to school by bicycle africa, north america, south america, europe and australia is 134 million square kilometers the pie chart below shows. Stats quiz 1 study play true a population is a collection of all individuals, objects, or measurements of interest a pie chart can be used to summarize the data a group of 100 students were surveyed about their interest in a new economics major interest was measured in terms of high, medium, or low. 21122 - minitab express: pie charts the following data set (from college board) contain the mean sat scores for each of the 50 states and washington, dc, as well the participation rates and geographic region of each state.
The remainder of the pie is corresponds to the students with hazel eyes the resulting pie chart is pictured above note that number of students in each category is written on each pie piece. B eye color of students in statistics class c speed of travel of a jet d your weight a a pie chart b a histogram c a bivariate table b mode c mean d none of the above ____ 25 it is possible for a variable to have a one mode b many modes c no mode. Pie-of-pie and bar-of-pie charts make it easier to see small slices of a pie chart these chart types separate the smaller slices from the main pie chart and display them in a secondary pie—or stacked bar chart. 84 creating bar graphs and pie charts 647 84 objectives 1 use a table to create a bar graph 2 this pie chart represents the results of a survey that asked students how they get to school most often (a) transportation 12 clothing 13 entertainment 30% food 10% other 5% entertainment 10% clothing 20% transportation 5. |
|The big bang theory in light of a quiet expansion of 201+ notations.
Three questions open up a more simple mathematical model of the universe:
1. Can this model, a Quiet Expansion (QE), actually defuse the big bang? The math within the QE model could redefine the first four “epochs”(and most-key elements) of the big bang theory (bbt). If it is defused and becomes an historic statement, From Lemaître to Hawking, science can move on within somewhat-prescribed boundary conditions and known parameters. The first four epochs of the big bang theory can readily be subjected to redefinition. These “epochs” amount to less than a trillionth-of-a-trillionth a second.
2. Are ethics and values built into the fabric of the universe? The QE model establishes a simple continuity equation from the first moment in time through the Age of the Universe (this day and this moment). These simple mathematical constructions quickly evolve as geometrical constructions and symmetry groups. Mathematically the constructions become quite dynamic and some harmonic. Continuity, symmetry and harmony are the foundations for a natural value equation deep-seated within the universe. Most every flavor of ethics, morals, and values can be appreciated for what they do and don’t do, and for why they are.
3. How can we more fully understand the finite-infinite relation? This finite-infinite relation is perhaps best described as a study of perfection and moments of perfection and that the geometries and mathematics of imperfection are also better defined and understood as a result and known today as quantum mechanics.Consider the four epochs in question:Planck Epoch or a Planck Moment: The finite-infinite relation most intimately defines the first notation and is necessarily within all notations building from the first. An infinitesimal duration, it is the beginning that creates space and time and then extends within space and time much like the birthing process. As of today, the Planck base units are our simplest-deepest-best description of this moment.Grand Unification and the Electroweak Epochs or Processes: Based on the fact that entities and things require a necessary amount of space that only becomes available from the 67th notation and above, the first 60 to 66 notations are foundational to all notations. Using the analogy of the birthing process, all the forms-and-functions, then processes-and-procedures, and then relations-and-systems prior to the actual birthing event, are the first 60 or so notations. Here that finite-infinite relation creates the foundational order, the most basic relations, and many dynamical systems prior to the uniqueness of every reasonable analogue to the birthing event.
The Grand Unification processes continue beyond the 67th notation as specific Unification processes. The electroweak processes now begin to manifest and the measurements given by the big bang theorists can be tweaked and integrated within the Quiet Expansion model.
Inflationary Epoch or Processes: Just as there are still many many questions about cellular division, there are even more open questions within this model. However, the force, the infinitesimal amount of energy, available to this process are working ratios of the Planck base units whereby order, relations, and dynamics evolve with a perfect continuity, perfect symmetry, and a deep harmony within every sphere and basic structure. This concept was initially put forth as a philosophical orientation to life, and then it was explored in a post about numbers called,
On Constructing the Universe From Scratch (see pages 5 and 6).
Background Introduction: This is page 1 of 37. Here, side-by-side, all 37 pages can be horizontally-scrolled as a single page. It is the entire model of the universe called, Big Board – little universe. The vertically-scrolled chart was completed in February 2015. The first chart was done in December 2011. Here we continue the process of encapsulating everything, everywhere in the universe, throughout all time. Though this chart suggests that space-time-mass-energy-temperature are necessarily and inextricably related, the challenge of this model is to demonstrate how this is so. This work is quite at odds with the big bang theory (bbt), yet we believe every formula and relation defined throughout the bbt history can also be found within our emerging model and view of the universe. To broaden its perspective, we will also attempt to examine as many transitions as possible between the finite-infinite, especially the role of pi, projective geometries, bifurcation theory, the dimensionless constants, and number theory.
Key concept: Planck Temperature has been moved to the top of the chart. One of the working assumptions of the project is that everything starts most simply and complexity comes later, and that space-and-time are finite, discrete, derivative and quantized. Of course, this logic will be further discussed.
Key questions: What mathematics are at work? The simple answer is, “All mathematics are at work here. No formula is exempt. And, eventually every formula will be in some way tied back to this model.” Notwithstanding, here is our first, introductory post about numbers.
Speed of Light: A simple calculation is to divide the Planck Length by Planck Time. Using just the units displayed above, the result is 299,777,406.78 m/sec. Using 3.23239/1.078212 the result is 2.99791692172 or 299,791,692.172 m/sec. These simple results, first posted on May 3, 2016, will be tweaked. Of course, the result of experimental measurement is 299,792,458 meters/second in a vacuum. Within notation 3, it is 1.29295 divided by 4.312848 which equals 299,790,300.98. There is much more to come!
Entitive Manifestations: Though not an active row until July 4, 2016, the nature of thingness has been part of our mindscape for many, many years going back to Martin Heidegger’s key question, What is a thing? Our first charts of the Big Board-little universe all focused on things determined by the multiple of the Planck Length. All the data from those earlier charts will now be integrated within our horizontally-scrolled chart.
Help wanted: For every notation, we would like to have an expert and a team. Within this group of notations, we especially seek help from people who can help us re-enact Max Planck’s thinking and the veracity of each formulation of the Planck base units.
Can you help us?
Key words, primary concepts, and links to references for these ten notations:
1. Geometries: Projective, Euclidean, differential (Riemannian, Lie groups, etc), discrete and combinatorial, algebraic and transformational…
2. The Pre-Measureable Structure of Matter: Might we conclude that this Small-Scale Universe is the structure that holds things together? Is it a re-definition of the ether? Is it MIT Frank Wilczek’s grid?
3. Renormalization(Scale Invariance https://en.wikipedia.org/wiki/Scale_invariance), Universality, isotropy, homogeneity: Is it possible that everything-everywhere in the universe shares the first 67 notations, and uniquely evolves with those characteristics given within the 67th to 134th notations, and then begins to manifest in each of the large-scale notations, unfolding uniquely in the 201st as “the given within the current moment”?
Editor’s note: Can you help upgrade that last sentence? I’ll be profoundly grateful. -BEC
Finite-Infinite: Studied throughout the history of humanity, this model provides a basis for a thorough reexamination of the concepts, mathematics and principles that operate between the two. Already there are several posts that open these reflections: (1) What is finite? And, what is truly infinite? and (2) Finite-Infinite reflections.
1. The areas above and below the numbers and discussions could also be used for graphics that are related to these notations. Perhaps a color background could reflect its temperature in its part of the universe.
2. Perhaps the area above the “Big Board-little universe” title (underlined) can be used for related graphics and color.
3. This “one page” board ideally would be a wiki page where schools and universities and the public could collaborate, update and add data.
4. Base-2 notation from the five Planck Base Units to their maximums is still early-stage work. We’ll be adding dimensionless constants. Could this table be a spreadsheet? April 27, 2016: More updating to come.
Process: Examples of Horizontal Scrolling Horizontal Scrolling Example #1, #2, #3 and #4 (pop up windows). |
- What kind of math is used in macroeconomics?
- Is economics hard to study?
- Is micro harder than macro?
- What is the highest paying job in economics?
- Is it hard to get a PhD in mathematics?
- Is it better to take micro or macro first?
- Does Google hire math PhDs?
- Do economists make good money?
- Is economics an easy degree?
- Can I study economics without maths?
- How do I get into a good PhD in economics?
- How difficult is math in economics?
- Is a masters in economics hard?
- Do economists use calculus?
- How long does a PhD in math take?
- Which country is best to study economics?
- Do you need to be good at math to be an economist?
- How hard is a PhD in economics?
- How long does a PhD in economics take?
- Is a PhD program Hard?
- Is economics harder than finance?
What kind of math is used in macroeconomics?
The types of math used in economics are primarily algebra, calculus and statistics.
Algebra is used to make computations such as total cost and total revenue.
Calculus is used to find the derivatives of utility curves, profit maximization curves and growth models..
Is economics hard to study?
Even though economics is a social science, it can be as difficult and demanding as any of the more challenging academic subjects, including math, chemistry, etc. To do well in economics requires time, dedication, and good study habits.
Is micro harder than macro?
At the entry-level, microeconomics is more difficult than macroeconomics because it requires at least some minimal understanding of calculus-level mathematical concepts. … Calculus is introduced at the macroeconomic level, but not nearly in as great a depth as it is in microeconomics.
What is the highest paying job in economics?
Best economics degree jobsCredit analyst. National Average Salary: $57,327 per year. … Personal finance advisor. National Average Salary: $65,526 per year. … Policy analyst. National Average Salary: $66,462 per year. … Supply chain analyst. … Economic consultant. … Business reporter. … Loan officer. … Portfolio manager.More items…•
Is it hard to get a PhD in mathematics?
It does just become objectively difficult, even for the people who are naturally good at it, the ones that have excelled at maths their whole lives. The level of abstraction would be too much for most people I think. with some work it is doable. it also vastly depends on the program you apply to.
Is it better to take micro or macro first?
Taking into account all of the above, most economics students are better off studying microeconomics first, and then progressing on to macroeconomics. That way, the principles of economics can be learned on an individual level, before being applied to the wider society and world.
Does Google hire math PhDs?
Google hires mathematicians i.e. those who have PhDs in math, and the answer to that question is yes — there not be many openings, but there certainly are some openings).
Do economists make good money?
An entry-level Economist with less than 1 year experience can expect to earn an average total compensation (includes tips, bonus, and overtime pay) of ₹510,643 based on 36 salaries. … An experienced Economist with 10-19 years of experience earns an average total compensation of ₹2,000,000 based on 14 salaries.
Is economics an easy degree?
Economics is not a particularly hard major at the undergraduate level. Most colleges do not require you to take a lot of mathematics classes. … Economics is an easy major if all you aim to do is to memorize old theories and regurgitate them as if they were truth.
Can I study economics without maths?
No, you strictly can’t pursue Economics hons without maths. … In such scenario Maths become compulsory for this course. For Economics, they teach you the basic things, which are required in further years, so even if u haven’t studied Eco in 12th, you will be able to sail through it, provided you do some hardwork.
How do I get into a good PhD in economics?
Here is the not-very-surprising list of things that will help you get into a good econ PhD program:good grades, especially in whatever math and economics classes you take,a good score on the math GRE,some math classes and a statistics class on your transcript,More items…•
How difficult is math in economics?
No . economics maths is not tough,Economics is not a particularly hard major at the undergraduate level. … The most prepared of economics majors, however, will choose to take mathematics classes on a level almost equivalent to a mathematics major, many would even double major.
Is a masters in economics hard?
Macro courses are more varied, but they mostly involve optimization problems (solving endless amounts of Lagrangians). If you are not comfortable with mathematics, higher-level economics will be very off-putting. So yes, masters economics is definitely much harder than undergrad.
Do economists use calculus?
Calculus is the mathematical study of change. Economists use calculus in order to study economic change whether it involves the world or human behavior. In economics, calculus is used to study and record complex information – commonly on graphs and curves.
How long does a PhD in math take?
between 3 and 5 yearsGenerally, PhD Mathematics programs take between 3 and 5 years to complete and although requirements differ depending on the academic institution and specific program, candidates must have the appropriate educational background, training, and experience in mathematics.
Which country is best to study economics?
Read on for our top picks for international students interested in studying Economics:The United States.The United Kingdom.The Netherlands.Australia.Switzerland.China.Italy.
Do you need to be good at math to be an economist?
The application and understanding of Economics happens when you study Math and apply your knowledge of Mathematics to understand Economics. Topics like Calculus & Linear Algebra are extremely important. … However, if you plan to pursue Economics further like Post Graduate level, you will need Maths, for sure.
How hard is a PhD in economics?
A PhD in any field not just economics is difficult, not so much because of the content or requirements, but because it is a research training exercise. You are learning and applying skills and abilities that you likely never had before or at least developing those that you had that weren’t very well developed.
How long does a PhD in economics take?
5-7 yearsIn this profile we focus on doing an Economics PhD in the US, which usually takes 5-7 years. In the first two years you take classes and the remaining time is spent on writing a dissertation. You usually have to teach during your PhD.
Is a PhD program Hard?
The drop-out rate for PhDs is high. In the United States, only 57% of PhD students obtained their PhD 10 years after enrollment. … Contrary to popular belief, a PhD is not intellectually difficult but it calls for discipline and stamina.
Is economics harder than finance?
Economics varies more though. There are very easy courses you can take, as well as extremely challenging ones—especially at the graduate level. If you’re just talking about a basic bachelors degree though, then finance is probably a little harder but not by much. … Is a major in finance better than a major in economics? |
In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane, each tangent to six others. The sequences of circles linked to each other through opposite points of tangency lie on logarithmic spirals (or, in degenerate cases, circles or lines) having, in general, three different shapes of spirals.
These patterns are named after mathematician Peter G. Doyle, who made an important contribution to their mathematical construction in the late 1980s or early 1990s. However, their study in phyllotaxis (the mathematics of plant growth) dates back to the early 20th century.
Based on these properties, it follows that one can find positive real numbers , , and , so that each circle of radius is surrounded by circles whose radii are (in cyclic order) , , , , , and . Only certain triples of numbers , , and defined in this way work; others lead to systems of circles that, when continued ad infinitum, eventually overlap each other.
The sequences of tangent circles with opposite points of tangency and radii have centers that (in most cases) lie on finitely many logarithmic spirals, all meeting at a central point. Similarly one obtains a different set of logarithmic spirals for the sequences of circles with radii and . In certain cases, one of , , or can equal 1, in which case these sequences of tangent circles return to their start after finitely many steps, and the centers of the circles in the sequence lie on one of infinitely many concentric circles (instead of finitely many logarithmic spirals) all centered on the central point. A third possibility is that, for one of , , or , the sequences of circles have centers that lie on finitely many rays, all meeting at the same central point. In all cases, there exists a system of symmetries of the plane, combining scaling and rotation around the central point, that take any circle of the packing to any other circle.
The precise shape of any Doyle spiral can be parameterized by a pair of natural numbers describing the number of spiral arms for each of the three ways of grouping circles by their opposite points of tangency. If the numbers of arms of two of the three types of spiral arm are and , with and with fewer than arms of the third type, then the number of arms of the third type is necessarily . As special cases of this formula, when the arms of the third type degenerate to circles, and there are infinitely many of them. And when the two types of arms with the smaller number of copies are mirror reflections of each other and the arms with copies degenerate to straight lines. For example, in the illustration shown, there are eight spiral arms with the same shape as the shaded arm, another eight spiral arms with the mirror reflected shape, and sixteen radial lines of circles, so this spiral can be parameterized as , .
Alternatively, the Doyle spiral can be parameterized by a pair of real numbers and describing the relative sizes of the circles. Peter Doyle observed that, when a unit circle is surrounded by of six other circles with radii , , , , , and , then these six surrounding circles close up to form a ring of mutually tangent circles, all tangent to the central unit circle. The Doyle spiral can then be constructed by using the same relative radii for rings of six circles surrounding each previously-constructed circle. The resulting system of circles closes up on itself to form a non-crossing Doyle spiral of circles in the plane only for certain special pairs of numbers and , which can be found from the integer parameters and by a numerical search. When is not one of these special pairs, the resulting system of circles still consists of spiral arms all wrapping around a central point, but with a rotation angle around that central point that is not an integer fraction of , causing them to overlap non-locally. The two real parameters can also be combined into a single complex number, interpreting the plane in which the circles are drawn as the complex plane. The parameters associated with a Doyle spiral must be algebraic numbers.
Coxeter's loxodromic sequence of tangent circles is a Doyle spiral with parameters and or with and , where denotes the golden ratio. Within the single spiral arm of tightest curvature, the circles form a sequence whose radii are powers of , in which each four consecutive circles in the sequence are tangent.
The standard hexagonal packing of the plane by unit circles can also be interpreted as a degenerate special case of the Doyle spiral, the case obtained by using the parameters . Unlike other Doyle spirals, it has no central limit point.
Spirals of tangent circles, often with Fibonacci numbers of arms, have been used to model phyllotaxis, the spiral growth patterns characteristic of certain plant species, beginning with the work of Gerrit van Iterson in 1907. In this application, a single spiral of circles may be called a parastichy and the parameters and of the Doyle spiral may be called parastichy numbers. The difference is also a parastichy number (if nonzero), the number of parastichies of the third type. When the two parastichy numbers and are either consecutive Fibonacci numbers, or Fibonacci numbers that are one step apart from each other in the sequence of Fibonacci numbers, then the third parastichy number will also be a Fibonacci number. For modeling plant growth in this way, spiral packings of tangent circles on surfaces other than the plane, including cylinders and cones, may also be used.
The Doyle spirals (and the hexagonal packing of the plane) are the only possible "coherent hexagonal circle packings" in the plane, where "coherent" means that no two circles overlap and "hexagonal" means that each circle is tangent to six others that surround it by a ring of tangent circles. Applying a Möbius transformation to a Doyle spiral can produce a related pattern of non-crossing tangent circles, each tangent to six others, with a double-spiral pattern in which the connected sequences of circles spiral out of one center point and into another; however, some circles in this pattern will not be surrounded by their six neighboring circles.
Additional patterns are possible with six circles surrounding each interior circle but only covering a partial subset of the plane and with circles on the boundary of that region not completely surrounded by other circles. It is also possible to form spiral patterns of tangent circles whose local structure resembles a square grid rather than a hexagonal grid, or to continuously transform these patterns into Doyle packings or vice versa. However, the space of realizations of locally-square spiral packings is infinite-dimensional, unlike the Doyle spirals which can be determined only by a constant number of parameters.
It is also possible to describe spiraling systems of overlapping circles that cover the plane, rather than non-crossing circles that pack the plane, with each point of the plane covered by at most two circles except for points where three circles meet at angles, and with each circle surrounded by six others. These have many properties in common with the Doyle spirals.
The Doyle spiral, in which the circle centers lie on logarithmic spirals and their radii increase geometrically in proportion to their distance from the central limit point, should be distinguished from a different spiral pattern of disjoint but non-tangent unit circles, also resembling certain forms of plant growth such as the seed heads of sunflowers. This different pattern can be obtained by placing the centers of unit circles on an appropriately scaled Fermat's spiral, at angular offsets of from each other relative to the center of the spiral, where again is the golden ratio. For more, see Fermat's spiral § The golden ratio and the golden angle. |
FOM: the Urbana meeting
friedman at math.ohio-state.edu
Tue Jun 27 04:33:46 EDT 2000
Reply to Martin Davis 6/20/00 1:35PM:
>I was on the Program Committee for the Urbana ASL meeting, and the
>committee was enthusiastic about the proposed panel on the need for
>"new axioms". Some time ago in a telephone conversation, Harvey told
>me that I am an "extreme Platonist". Being a great fan of Harvey's
>work on the necessary use of large cardinals, I took his comment
>quite seriously and began to wonder. Is that really me?
Yes, because I under the impression that you think that any
intelligible set theoretic question, quantifying even over all sets -
regardless of where they lie in the cumulative hierarchy - is a well
defined mathematical problem in the same sense as, say, the Riemann
hypothesis or the twin prime conjecture. That there is an absolute
right and an absolute wrong answer. And that it is part of normal
mathematical activity to work on such questions just as it is
to work on RH or TP, at least in the sense that there is no
special difference in kind between the two activities that justifies
calling one "normal mathematical activity" and the other "not normal
>Harvey said .. the bad news is that set theory in particular and
>foundational studies in general are on hard times and that
>unchecked, things would only get worse. The good news is that
>Harvey's new Boolean Relation Theory will save the day: Because it
>is a single appealing theme, whose necessary methods range from what
>mathematicians are used to, all the way through Mahlo cardinals, and
>eventually all the way up the large cardinal hierarchy,
>mathematicians will be led to accept these methods because they will
>see that they are needed to solve problems that interest them.
This is more or less accurate, but I said it slightly differently,
and let me say it here even slightly differently than I said it in
Mathematical logic is in very bad shape sociologically and
politically, and that part of the difficulties come from the focus of
Virtually all major scientific areas are born with the discovery and
development of striking new models of some phenomenon that exists
independently of that new scientific area. In most cases, the subject
really attracts attraction and attains an identity through striking
findings that are based on the striking new models.
Mathematical logic fits very much into this framework. We all know
the great models of the phenomenon known as logical reasoning (through
mathematical reasoning (through set theory), and algorithmic
procedures, and others. And we all know the striking findings in the
early part of the 20th century that really gave the subject its
In order to maximize the impact of a subject on the wider
intellectual community, one must periodically - better yet,
continuously - strive for the renewal and fresh perspective one gains
by revisiting its origins in order to reflect more subtle features of
the seminal phenomena. This is normal and standard.
E.g., in partial differential equations, one continually strives to
get more subtle information about more equations that model more
closely more subtle physical phenomena. The same is true of
mathematical economics, etcetera.
However, it is extremely important not to try to force the pace of
this natural evolution beyond what can be productively accommodated
at any given stage in the development of the subject. This merely
leads to a counterproductive negativity that is unwarranted. Every
subject, including the most successful and revered areas of pure
mathematics, look like dismal failures when looked at with such
unrealistic expectations. There is a natural evolution of subjects.
Nevertheless, I have insisted that the commonly referred to four main
areas of mathematical logic, set theory, model theory, recursion
theory, proof theory, are in dire need of such renewal. Actually,
some appropriate renewal is already taking place in some parts of some of
the areas, but not in others. I like to think that I am always striving to
point the way towards renewal.
>He also implied that the traditional set theory community is on the
>wrong track. I certainly applaud Harvey's program, but I assume that
>since Harvey is devoting himself to this program, he believes that
>the results he gets are TRUE. What I don't understand is what he'll
>tell mathematicians who want to know why they should believe this.
Yes, of course I believe that the results that I get are TRUE, but what
results? The results that it is necessary and sufficient to use large
cardinals to get such and such, or such and such can only be done with
large cardinals, or such and such is outright equivalent to the
1-consistency or consistency of large cardinals, etcetera.
That is where my role as f.o.m. expert ends, and where, if I wish to
continue, my role as ph.o.m. begins.
Namely, my f.o.m. expert role is to show that basic natural elementary
universally accessible concrete mathematics - part of the unremoveable
furniture of mathematics as we know it - is inexorably tied up with large
cardinals, through their 1-consistency or consistency.
This is in a context in which it is conventional wisdom among
mathematicians that basic natural elementary universally accessible
concrete mathematics - part of the unremoveable furniture of mathematics as
we know it - is in no way tied up with large cardinals, or their
1-consistency or consistency.
More explicitly, this is in a context in which set theory is not viewed as
part of mathematics, but rather as a scheme for establishing rigor in
mathematics. Set theory is regarded as a framework for interpreting
mathematics, and not a part of mathematics itself. This is the modern view,
hardened by the brief and fleeting experience with experimenting with set
theoretic questions taken literally. Nowadays, set theoretic formulations
are used only when they simplify the underlying mathematics. When they
create their own peculiar problems - arising out of pathological cases
which have nothing to do with the underlying mathematics - then they
discard them as utterly irrelevant and useless.
I think that I have some singular contributions to make to this situation
as an f.o.m. expert, but I am less sure that I have, at this time, some
singular contributions to make to this situation as a ph.o.m. expert. (Of
course, there is the question of just who does have singular contributions
to make to such issues as ph.o.m. experts).
>... I think the question
>of to what extent this faculty can be effective in exploring the
>infinite is an empirical question that can only be decided by trying
>and analyzing the results. Do we obtain a coherent picture? Or does
>it all dissolve in vagueness and contradictions?
I will take your use of the word "infinite" to mean "the absolutely
unrestricted infinite" - not just the natural numbers, or even sets of
natural numbers. I.e., full blown set theory.
It is obvious that we get a coherent picture that does not appear to
dissolve into contradictions. But does it dissolve into vagueness if we
push it too hard? I certainly think that the experience with the continuum
hypothesis and related questions definitely makes it at least *appear* that
it all dissolves into vagueness when things are pushed too literally.
I know that some set theorists are hopeful that the continuum hypothesis
and related questions will not continue to make it appear that it all
dissolves into vagueness when pushed literally. However, at this point, the
set theorists have pretty much abandoned the idea that there will be a fix
(i.e., resolution of such problems as the continuum hypothesis) that can be
readily understood and accepted by people who are not experts in set
theory. I.e., in the same sense that the usual axioms of ZFC can be readily
understood and accepted, even with such additional axioms as strongly
inaccessible cardinals, or even such additional axioms as the existence of
a probability measure on all subsets of [0,1].
I am doubtful that the *appearance* that it all dissolves into vagueness
when pushed literally will be erased by any esoteric "fix" understandable
only by experts in set theory.
Undoubtedly there will be a great effort ultimately made to reduce any such
esoteric "fix" to commonly understandable - and commonly convincing -
Equally surely, there will be a great effort ultimately made to show that
there is no "simple" fix that is as "simple" as the usual axioms of set
theory. I have a plan for this.
Of the prospects for the last two paragraphs - I'll put my money on the
>From this point of view, the work of set-theorists has been crucial
>in suggesting that it is the former that is the case.
And the work of set-theorists has been crucial in suggesting that this is
not the case, because the set theorists have shown that so many set
theoretic problems like the continuum hypothesis are independent of ZFC
together with so many additional axioms.
>The use of PD
>in providing an elegant theory of the projective hierarchy and the
>discovery that PD is implied by large cardinal axioms encourages the
>view that one is dealing with a situation where there is an
>objective fact-of-the-matter with respect to the propositions being
The fact that other axioms solve the same problems differently cuts in the
other direction. A perfectly legitimate conclusion from all this is that
there is no "objective fact-of-the-matter" since it is not an "objective
fact-of-the-matter" whether V = L is true or whether large cardinals are
true. It is just that both of these hypotheses are sufficient to settle
these particular questions.
As I have said in the Urbana meeting, the set theorist wants to accept
large cardinals because of the extra delicate and interesting set theoretic
structure that entails, and is missing under V = L. But the mathematician
doesn't welcome such extra set theoretic structure - as it is irrelevant
and sharply different in flavor to underlying mathematical issues. So if
forced to make a choice, mathematicians would greatly prefer V = L.
>The more recent work showing that consistency strength
>alone of certain of these axioms suffices to determine the truth
>values of sentences of given complexity, further enhances this
This sentence is mathematically false on its face. An accurate statement is
far more technical, making any "perception" less convincing.
> I am at a loss to understand why Harvey thinks that this
>work and his are at cross-purposes; it is clear to me that each
>needs the other: Harvey to show concretely that the higher
>infinities have specific interesting consequences way down, the
>set-theorists to map out the infinite terrain and provide a
>convincing case for a coherent robust state of affairs.
Cross-purposes is not the way I would put it. It's more like this. There is
a poker game going on in a barn. The players are in a long and heated
dispute as to whether or not it is legal for one of them to raise the pot
for the third time. This argument is going on unabated while the barn is
being consumed in a devastating fire.
I'm sitting here working my god damn xxx off to put out the fire while set
theorists are arguing about their rules of poker, calmly sitting in the
middle of that fire.
To put out the fire, one needs only to use some cardinal that is at least
beyond ZFC. To the general mathematical community, anything that large is
already grotesquely large - it takes experts in set theory to discern one
from the other.
The cardinals involved in what I am doing to put the fire out were already
set up in the years 1911-1913.
And while the fire is raging, nobody cares about some esoteric explanation
of just how "coherently robust the state of affairs is" to experts in set
theory. After all, these experts in set theory talking about "coherent
robust states of affairs" are starting to be being consumed in the fire!!
>Sol maintains that CH is inherently vague and for that reason it is
>pointless to expect that the question will ever be resolved. ... he does
>that the concept of the continuum (or equivalently, the power set of
>omega) is well-defined.
The equivalence between the continuum and the power set of omega is not
generally accepted by some leading core mathematicians because the
correspondence is not a natural mathematical object. In fact, many do not
believe that our usual model of the continuum in terms of Dedekind cuts and
the like fairly represents the continuum.
This is in consonance with my earlier statement about set theory being an
interpretation of mathematics rather than a part of mathematics.
>...for someone with Sol's beliefs, CH can have no
>determinate truth value. ...has been
>held by such great mathematicians as Brouwer and Weyl... what bothers me
>is how his
>conclusion will be received by readers of his MONTHLY article with
>little training in foundations.
I am uncomfortable with other aspects of Sol's article, also. I plan to
write something for the Monthly in due course about BRT. Why don't you
write something for the Monthly about your extreme Platonism?
>...mathematicians presume that the Goedel-Cohen independence results
>have settled the matter about CH, imagining that it is quite like
>the situation with the parallel postulate, and there is nothing more
>to be said.
There is a lot to be said for this preumption, given what has actually been
accomplished. However, what they don't generally realize is
i) how demonstrably irrelevant the CH is for any questions they really care
about, even if they suspect it;
ii) that there is an axiom of restriction, V = L, that dispenses with the
CH and all related questions.
>Such folk hearing Sol's conclusion about CH will likely
>nod their heads. But typically, they work with the continuum every
>day, and by no means are likely to share Sol's belief that it is a
>questionable concept, the belief on which his conclusion is based.
But the way they work with the continuum is not as a set theoretic object.
So the relationship between what mathematicians are doing and what Sol's
views are is, to my mind, quite unclear. Perhaps you ought to ask Sol about
what he thinks of this.
> I chose to talk about Goedel's Legacy. ... the question remains: are any
>problems of genuine
>mathematical interest likely to be examples of the incompleteness
>phenomenon, even such problems of central importance as the Riemann
>Hypothesis (as Goedel ventured to suggest). ...I suggested
>that ... interested people could be divided into three
>classes: optimists (people who think that such interesting
>undecidable propositions will be found - or even, are already being
>found), skeptics (people who think that Gödel incompleteness will
>not affect propositions of real interest to mathematicians), and
>pessimists(thinks that even if there are such propositions, it will
>be hopeless to prove them). In replying to a question from Dana
>Scott, I admitted that I am an optimist.
My own view is this:
i) it is likely that every non set theoretic statement in the normal
mathematical literature is decided by ZFC; (this can be made very precise,
but that's for another time and place).
ii) BRT will be accepted as having results "of genuine mathematical
interest", and is chock full of incompleteness phenomena. The most
immediately convincing ones will be in terms of classifications and
specific theorems about classifications.
iii) additional classification problems throughout mathematics - even more
mathematically friendly than BRT - will surface following the lead of BRT,
and the incompleteness phenomena will also routinely appear there. E.g.,
instead of looking at Boolean relations, one looks at solutions to
iv) the hallmark of these new kinds of classification problems is that
there are always just a finite number of cases. The number of cases is
generally quite large, like 2^512. The idea is that large cardinals are
supposed to settle all of the instances, but some instances cannot be
settled without large cardinals. Also, there are general features of the
classification that are stated as single theorems which can only be proved
using large cardinals.
More information about the FOM |
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.
Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.
For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec
Richard Taylor (mathematician)
Richard Lawrence Taylor is a British and American mathematician working in the field of number theory. He is a professor of mathematics at Stanford University and the Institute for Advanced Study. Taylor received the 2014 Breakthrough Prize in Mathematics "for numerous breakthrough results in the theory of automorphic forms, including the Taniyama–Weil conjecture, the local Langlands conjecture for general linear groups, the Sato–Tate conjecture." He received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands. He received his BA from Cambridge. During his time at Cambridge, he was president of The Archimedeans in 1981 and 1982, following the impeachment of his predecessor, he earned his PhD from Princeton University in 1988. From 1995 to 1996 he held the Savilian chair of geometry at Oxford University and Fellow of New College and became the Herchel Smith Professor of Mathematics at Harvard University, he holds Robert and Luisa Fernholz Professorship at the Institute for Advanced Study.
He received the Whitehead Prize in 1990, the Fermat Prize, the Ostrowski Prize in 2001, the Cole Prize of the American Mathematical Society in 2002, the Shaw Prize for Mathematics in 2007. He was elected a Fellow of the Royal Society in 1995. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was inducted into the National Academy of Sciences, he was elected to the American Philosophical Society in 2018. One of the two papers containing the published proof of Fermat's Last Theorem is a joint work of Taylor and Andrew Wiles. In subsequent work, Taylor proved the local Langlands conjectures for GL over a number field. A simpler proof was suggested at the same time by Guy Henniart, ten years by Peter Scholze. Taylor, together with Christophe Breuil, Brian Conrad and Fred Diamond, completed the proof of the Taniyama–Shimura conjecture, by performing quite heavy technical computations in the case of additive reduction. In 2008, following the ideas of Michael Harris and building on his joint work with Laurent Clozel, Michael Harris, Nick Shepherd-Barron, announced a proof of the Sato–Tate conjecture, for elliptic curves with non-integral j-invariant.
This partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. Taylor is the son of British physicist John C. Taylor, he is married, has two children. His home page at the Institute for Advanced Study Richard Taylor at the Mathematics Genealogy Project Autobiography upon Shaw Prize acceptance
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
In number theory, the Mordell conjecture is the conjecture made by Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, is now known as Faltings's theorem; the conjecture was generalized by replacing Q by any number field. Let C be a non-singular algebraic curve of genus g over Q; the set of rational points on C may be determined as follows: Case g = 0: no points or infinitely many. Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group. Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. Case g > 1: according to the Mordell conjecture, now Faltings's theorem, C has only a finite number of rational points. Faltings's original proof used the known reduction to a case of the Tate conjecture, a number of tools from algebraic geometry, including the theory of Néron models. A different proof, based on diophantine approximation, was found by Vojta.
A more elementary variant of Vojta's proof was given by Bombieri. Faltings's 1983 paper had as consequences a number of statements, conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; the reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin. A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1. Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, proved by Faltings. Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k X is not Zariski dense in X.
More general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Manin and by Grauert. In 1990, Coleman found and fixed a gap in Manin's proof. Bombieri, Enrico. "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17: 615–640. MR 1093712. Coleman, Robert F.. "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36: 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02. Cornell, Gary. Arithmetic geometry. Papers from the conference held at the University of Connecticut, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. Doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings Faltings, Gerd. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73: 349–366. Doi:10.1007/BF01388432. MR 0718935. Faltings, Gerd. "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
Inventiones Mathematicae. 75: 381. Doi:10.1007/BF01388572. MR 0732554. Faltings, Gerd. "Diophantine approximation on abelian varieties". Ann. of Math. 133: 549–576. Doi:10.2307/2944319. MR 1109353. Faltings, Gerd. "The general case of S. Lang's conjecture". In Cristante, Valentino. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396. Grauert, Hans. "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS: 131–149. ISSN 1618-1913. MR 0222087. Hindry, Marc. Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. Doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem. Lang, Serge. Survey of Diophantine geometry. Springer-Verlag. Pp. 101–122. ISBN 3-540-61223-8. Manin, Ju. I.. "Rational points on algebraic curves over function fields".
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 27: 1395–1440. ISSN 0373-2436. MR 0157971. Mordell, Louis J.. "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192. Paršin, A. N.. "Quelques conjectures de finitude en géométrie diophantienne". Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars. Pp. 467–471. MR 0427323. Archived from the original on 2016-09-24. Retrieved 2016-06-11. Parshin, A. N. "Mordell conje
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.
As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.
Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.
This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.
Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic
Sir Andrew John Wiles is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society, he was appointed Knight Commander of the Order of the British Empire in 2000, in 2018 was appointed as the first Regius Professor of Mathematics at Oxford. Wiles was born on 11 April 1953 in Cambridge, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford, Patricia Wiles, his father worked as the chaplain at Ridley Hall, for the years 1952–55. Wiles attended King's College School and The Leys School, Cambridge. Wiles states that he came across Fermat's Last Theorem on his way home from school when he was 10 years old, he stopped at his local library. Fascinated by the existence of a theorem, so easy to state that he, a ten year old, could understand it, but that no one had proven, he decided to be the first person to prove it.
However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had linked to Fermat's famous equation. Wiles earned his bachelor's degree in mathematics in 1974 at Merton College, a PhD in 1980 as a graduate student of Clare College, Cambridge. After a stay at the Institute for Advanced Study in Princeton, New Jersey in 1981, Wiles became a Professor of Mathematics at Princeton University. In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, he returned to Princeton. From 1994 - 2009, Wiles was a Eugene Higgins Professor at Princeton, he rejoined Oxford in 2011 as Royal Society Research Professor. In May 2018 he was appointed Regius Professor of Mathematics at Oxford, the first in the university's history.
Wiles's graduate research was guided by John Coates beginning in the summer of 1975. Together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory, he further worked with Barry Mazur on the main conjecture of Iwasawa theory over the rational numbers, soon afterward, he generalised this result to real fields. His biographical page at Princeton University's website states that "Andrew has few equals in terms of his impact on modern number theory. Many of the world’s best young number theorists received their Ph. D.'s under Andrew... and many of these are today leaders and professors at top institutions around the world". Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem; the modularity theorem involved elliptic curves, Wiles's own specialist area.
The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or impossible to prove. For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to prove", Ken Ribet considered himself "one of the vast majority of people who believed was inaccessible", adding that "Andrew Wiles was one of the few people on earth who had the audacity to dream that you can go and prove."Despite this, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for Frey's curve. He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge, he gave a lecture a day on Monday and Wednesday with the title'Modular Forms, Elliptic Curves and Galois Representations.'
There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said.... At the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture; as an afterthought, he noted that that meant that Fermat's last theorem was true. Q. E. D. In August 1993, it was discovered. Wiles failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing, rather than closing, this area came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a second paper which circumvented the problem and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of the Annals of Mathematics. Wiles's proof of Fermat's Last Theorem has stood up to the scrutiny of the world's other mathematical experts. Wiles was interviewed for an episode of the BBC documentary series Horizon that focused on Fermat's Last Theorem; this was renamed "The Proof", it was made an episode of the US Public Broadcasting Service's science television series Nova.
His work and life are described in great detail in Simon Singh's popular book Fermat's Last Theorem. Wiles has been awarded a number of major prizes in mathematics and science: Junior Whitehead Prize of the London Mathematical Soci
Fermat's Last Theorem
In number theory Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, it is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.
The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics; the special case n = 4 - proved by Fermat himself - is sufficient to establish that if the theorem is false for some exponent n, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.
Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.
An outline suggesting this could be proved was given by Frey. The full proof that the two problems were linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture"; these papers by Frey and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.
Important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was desired and not just a historical oddity, so time spent working on it could be justified professionally. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to prove. I must confess I thought I wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's li |
57 4-3 Greatest Common Divisor and Least Common Multiple Methods to Find the Greatest Common DivisorMethods to Find the Least Common Multiple
58 Greatest Common Divisor Two bands are to be combined to march in a parade. A 24-member band will march behind a 30-member band. The combined bands must have the same number of columns. Each column must be the same size. What is the greatest number of columns in which they can march?
59 Greatest Common Divisor The bands could each march in 2 columns, and we would have the same number of columns, but this does not satisfy the condition of having the greatest number of columns.The number of columns must divide both 24 and 30.Numbers that divide both 24 and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6.
60 Greatest Common Divisor The first band would have 6 columns with 4 members in each column, and the second band would have 6 columns with 5 members in each column.
61 DefinitionThe greatest common divisor (GCD) or the greatest common factor (GCF) of two whole numbers a and b not both 0 is the greatest whole number that divides both a and b.
62 Colored Rods MethodFind the GCD of 6 and 8 using the 6 rod and the 8 rod.
63 Colored Rods MethodFind the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8 rod.The 2 rods can be used to build both the 6 and 8 rods.
64 Colored Rods MethodThe 3 rods can be used to build the 6 rod but not the 8 rod.The 4 rods can be used to build the 8 rod but not the 6 rod.The 5 rods can be used to build neither.The 6 rods cannot be used to build the 8 rod.Therefore, GCD(6, 8) = 2.
65 The Intersection-of-Sets Method List all members of the set of whole number divisors of the two numbers, then find the set of all common divisors, and, finally, pick the greatest element in that set.
66 The Intersection-of-Sets Method To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D20 and D32, respectively.Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4.
67 The Prime Factorization Method To find the GCD of two or more non-zero whole numbers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations.Numbers, such as 4 and 9, whose GCD is 1 are relatively prime.
68 Example 4-12 Find each of the following: a. GCD(108, 72) b. GCD(0, 13) Because 13 | 0 and 13 | 13, GCD(0, 13) = 13.
69 Example (continued)c. GCD(x, y) if x = 23 · 72 · 11 · 13 and y = 2 · 73 · 13 · 17GCD(x, y) = 2 · 72 · 13 = 1274d. GCD(x, y, z) if x = 23 · 72 · 11 · 13, y = 2 · 73 · 13 · 17, and z = 22 · 7GCD(x, y, z) = 2 · 7 = 14e. GCD(x, y) if x = 54 · 1310 and y = 310 · 1120Because x and y have no common prime factors, GCD(x, y) = 1.
70 Calculator MethodCalculators with a key can be used to find the GCD of two numbers.SimpFind GCD(120, 180) by pressing the keys:1Simp2/8=to obtain the displayN/D→n/d 60/90
71 Calculator MethodBy pressing the key, we see on the display as a common divisor of 120 and 180.x y2By pressing the key again and pressing we see 2 again as a factor.Simp=x yRepeat the process to see that 3 and 5 are other common factors.GCD(120, 180) is the product of the common prime factors 2 · 2 · 3 · 5, or 60.
72 Euclidean Algorithm Method If a and b are any whole numbers greater than 0 and a ≥ b, then GCD(a, b) = GCD(r, b), where r is the remainder when a is divided by b.Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm.
74 Example 4-13 Use the Euclidean algorithm to find GCD(10764, 2300). GCD(10764, 2300) = GCD(2300, 1564)GCD(2300, 1564) = GCD(1564, 736)
75 Example 4-13 (continued) GCD(10764,2300) = GCD(92, 0) = 92
76 Euclidean Algorithm Method A calculator with the integer division feature can also be used to perform the Euclidean algorithm.To find GCD(10764, 2300), proceed as follows:The last number we divided by when we obtained the 0 remainder is 92, so GCD(10764, 2300) = 92.
77 Example 4-14aFind GCD(134791, 6341, 6339).Any common divisor of three numbers is also a common divisor of any two of them.The GCD of three numbers cannot be greater than the GCD of any two of the numbers.GCD(6341, 6339) = GCD(6341 − 6339, 6339)= GCD(2, 6339) = 1GCD(134791, 6341, 6339) cannot be greater than 1, so it must equal 1.
78 Example 4-14b Find the GCD of any two consecutive whole numbers. GCD(n, n + 1) = GCD(n + 1, n)= GCD(n + 1 − n, n)= GCD(1, n) = 1The GCD of any two consecutive whole numbers is 1.
79 Least Common MultipleHot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package. What is the least number of packages of each you must buy so that there is an equal number of hot dogs and buns?The number of hot dogs is a multiple of 10, while the number of buns is a multiple of 8.The number of hot dogs matches the number of buns whenever 10 and 8 have multiples in common.
80 Least Common Multiple This occurs at 40, 80, 120… The least of these multiples is 40.So we will have the same number of hot dogs and buns by buying 4 packages of hot dogs and 5 packages of buns.
81 Definition Least Common Multiple (LCM) The least common multiple (LCM) of two non-zero whole numbers a and b is the least non-zero whole number that is simultaneously a multiple of a and a multiple of b.
82 Number-Line Method Find LCM(3, 4). Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is LCM(3, 4).
83 Colored Rods MethodFind LCM(3, 4) using the 3 rod and the 4 rod.
84 Colored Rods MethodBuild trains of 3 rods and 4 rods until they are the same length. The LCM is the common length of the train.LCM(3, 4) = 12
85 The Intersection-of-Sets Method List all members of the set of positive multiples of the two integers, then find the set of all common multiples, and, finally, pick the least element in that set.
86 The Intersection-of-Sets Method To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M8 and M12, respectively.Because the least number in the set of common positive multiples is 24, LCM(8, 12) = 24.
87 The Prime Factorization Method To find the LCM of two non-zero whole numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations.
89 GCD-LCM Product Method For any two natural numbers a and b,GCD(a, b) · LCM(a, b) = ab.
90 Example 4-16Find LCM(731, 952).Applying the Euclidean Algorithm, we can determine that GCD(731, 952) = 17.17 · LCM(731, 952) = 731 · 952LCM(731, 952) = = 40,936
91 Division-by-Primes Method To find LCM(12, 75, 120), start with the least prime that divides at least one of the given numbers. Divide as follows:Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this procedure, continue the division process until the row of answers consists of relatively prime numbers as shown next. |
Standard form of numbers defined and explained with examples. We hope this blog will help the learners as well as those candicate who will appear in the boards exams 2023-2024 because we provide here all possible solutions of queries by our users. Standard Form of Numbers: Defined & Explained with Examples - Questions Bank (Standard form of numbers defined and explained with examples) for Examination Year 2023-2024
Source:www.questionsbanks.com Last updated: 2023-07-25 20:03:42
Standard form of numbers is an important concept to represent numbers in scientific notation.Numbers are an integral part of our everyday lives, serving as a fundamental tool for quantifying and expressing various quantities. While working with extremely large or small numbers, it can become burdensome to represent these numbers in their usual form.
The standard form of numbers provides a practical and efficient method of representing extremely large or small quantities. By using a decimal number and a power of 10, it simplifies the communication of such numbers and facilitates comparisons and calculations.
In this article, we will elaborate the idea of standard form of numbers, how to express numbers in standard form, arithmetic operations with numbers in standard form, its merits, applications and how it is applied to express numbers in standard form with examples.
Standard form is a substantial and standardized way of expressing numbers that are either very large or very small. Scientific notation and exponent form are also important terms which are commonly used in place of standard form of numbers.
By expressing these numbers as the product of a decimal number (between 1 and 10) and an exponent of 10,enables us to identify these numbers in a compact manner. The number of places to which the decimal point must be shifted to get the original value is indicated as the exponent of 10. Mathematically,
A x 10^n, where 1 ≤A< 10 and n is an integer.
Here M is a coefficient and it is a decimal number range from equal or greater than 1 but less than 10. Moreover n (power of ten) specifies the scale or order of magnitude.
To express a number in standard form, we follow a specific format.
Always choose a number that is higher than or equal to one but less than ten.
Determine how many places the decimal point has to be moved to get the original value. If decimal point moves from left side or right side, then the exponent of 10 will be negative or positive respectively.
You can try a standard form converter to express numbers in scientific notation according to the above-mentioned steps.
In this step, you’re to combine the decimal number and the exponent of 10 which you will have identified in the above two steps.
Performing arithmetic operations with numbers in standard form is simple. Here's how you can carry out basic operations:
Addition and Subtraction:
The standard form of numbers offers several advantages in representing and comparing numbers that differ greatly in magnitude. Here are a few benefits:
Standard form of numbersenables us to write extremely large or small numbers in a concise and easily understandable format, making it convenient for mathematical calculations and scientific notation.
By representing numbers in standard form, it becomes effortless to compare their magnitudes. The power of 10 provides a clear indication of which number is larger or smaller.
It is convenient and simple to use arithmetic operations (addition, subtraction, multiplication, and division) when working with numbers that are written in standard form. Additionally, it reduces one's need for handling very small or massive numbers.
Standard form of numbers is closely related to scientific notation, which is widely used in scientific research, engineering, and mathematics. Familiarity with standard form enhances understanding and communication within these fields.
Standard form finds application in various scientific and mathematical disciplines. Some common areas where standard form is frequently employed include:
Representing astronomical distances, masses, and quantities.
Expressing the values of fundamental physical constants, such as the speed of light or Planck's constant.
Describing molecular masses, atomic sizes, and concentrations.
Presenting large financial figures or GDP values.
Scientists often employ standard form to express measurements, such as distances between celestial objects, molecular sizes, or energy values.
Engineers utilize standard form to represent quantities like voltage, resistance, and power in electrical circuits, as well as dimensions and quantities in structural analysis.
Standard form facilitates the representation of large financial figures in reports, such as company revenues, national debt, or stock market values.
In medical research and practice, standard form is employed to express quantities like drug concentrations, cell counts, or molecular weights.
Express the ordinary number 738000000000000000000 in standard form.
Step 1: Write the given number
Step 2: We identify the decimal number that is 738 and standard position is after the first decimal number i.e. 7.38.
Step 3: As decimal point is shifted 20 places from right side. Thus, exponent of 10 will be +20. Hence number in standard form is 7.38 x 10^20.
Express the given ordinary number 0.000000000000529 in standard form of numbers.
Step 1: Write the given number
Step 2: We identify the decimal number which is 428 and standard position is after the first decimal number i.e. 5.29
Step 3: As the decimal point is shifted places from left side. Thus, exponent of 10 will be -13. Hence number in standard form is 5.29 x 10^ -13.
In this article, we have discussed standard form of numbers in detail. We have elaborated the way to write ordinary numbers in standard form, arithmetic operations, benefits, applications as well as some examples. Hopefully, reading this article you will be able to apprehend the concept of standard form of numbers.
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DETERMINATION OF THE MASS OF THE W BOSON
Conveners: Z. Kunszt and W. J. Stirling
Working group: A. Ballestrero, S. Banerjee, A. Blondel, M. Campanelli, F. Cavallari, D. G. Charlton, H. S. Chen, D. v. Dierendonck, A. Gaidot, Ll. Garrido, D. Gelé, M. W. Grünewald, G. Gustafson, C. Hartmann, F. Jegerlehner, A. Juste, S. Katsanevas, V. A. Khoze, N. J. Kjær, L. Lönnblad, E. Maina, M. Martinez, R. Møller, G. J. van Oldenborgh, J. P. Pansart, P. Perez, P. B. Renton, T. Riemann, M. Sassowsky, J. Schwindling, T. G. Shears, T. Sjöstrand, Š. Todorova, A. Trabelsi, A. Valassi, C. P. Ward, D. R. Ward, M. F. Watson, N. K. Watson, A. Weber, G. W. Wilson
1 Introduction and Overview111prepared by F. Jegerlehner, Z. Kunszt, G.-J. van Oldenborgh,
P.B. Renton, T. Riemann, W.J. Stirling
- 1.1 Machine parameters
- 1.2 Present status of measurements
- 1.3 Improved precision on from the Tevatron
- 1.4 Impact of a precision measurement of
- 1.5 Methods for measuring
- 1.6 Theoretical input information
2 Measurement of from the Threshold
Cross-Section222prepared by D. Gelé, T.G. Shears, W.J. Stirling, A. Valassi,
- 2.1 Collider strategy
- 2.2 Event selection and statistical errors
- 2.3 Systematic errors
- 2.4 Summary
3 Direct Reconstruction of 333prepared by M. Grünewald, N. J. Kjær, Z. Kunszt,
P. Perez, C. P. Ward
- 3.1 Event selection and jet reconstruction
- 3.2 Constrained fit
- 3.3 Determination of the mass and width of the W
- 3.4 Systematic errors
- 3.5 Summary
- 4 Interconnection Effects444 prepared by V.A. Khoze, L. Lönnblad, R. Møller, T. Sjöstrand, Š. Todorova and N.K. Watson.
- 5 Conclusions
1 Introduction and Overview111prepared by F. Jegerlehner, Z. Kunszt, G.-J. van Oldenborgh, P.B. Renton, T. Riemann, W.J. Stirling
Previous studies of the physics potential of LEP2 indicated that with the design luminosity of one may get a direct measurement of the W mass with a precision in the range . This report presents an updated evaluation of the estimated error on based on recent simulation work and improved theoretical input. The most efficient experimental methods which will be used are also described.
1.1 Machine parameters
The LEP2 machine parameters are by now largely determined. Collider energy values and time-scales for the various runs, expected luminosities and errors on the beam energy and luminosity are discussed and summarized elsewhere in this report [2, 3]. Here we note that (i) collider energies in the range will be available, and (ii) the total luminosity is expected to be approximately per experiment. It is likely that the bulk of the luminosity will be delivered at high energy (). The beam energy will be known to within an uncertainty of , and the luminosity is expected to be measured with a precision better than 1%.
1.2 Present status of measurements
Precise measurements of the masses of the heavy gauge W and Z bosons are of fundamental physical importance. The current precision from direct measurements is = 2.2 MeV and M = 160 MeV . So far, has been measured at the CERN and Fermilab Tevatron [6, 7, 8] colliders. The present measurements are summarized in Fig. 1. In calculating the world average, a common systematic error of arising from uncertainties in the parton distributions functions is taken into account. The current world average value is
An indirect determination of from a global Standard Model (SM) fit to electroweak data from LEP1 and SLC gives the more accurate value
In Fig. 1 this range is indicated by dashed vertical lines. Note that the central value in (2) corresponds to and the second error indicates the change in when is varied between and – increasing decreases .
The direct measurement of becomes particularly interesting if its error can be made comparable to, or smaller than, the error of the indirect measurement, i.e. . In particular, a precise value of obtained from direct measurement could contradict the value determined indirectly from the global fit, thus indicating a breakdown of the Standard Model. An improvement in the precision of the measurement can be used to further constrain the allowed values of the Higgs boson mass in the Standard Model, or the parameter space of the Minimal Supersymmetric Standard Model (MSSM) .
Standard Model fits to electroweak data determine values for (or ), , and . The direct determinations of the top quark mass [9, 10] give an average value of . Fig. 2 compares the direct determinations of and with the indirect determinations obtained from fits to electroweak data . Note the correlation between the two masses in the latter. Within the current accuracy, the direct and indirect measurements are in approximate agreement. The central values of and their errors, determined in several ways from indirect electroweak fits, are given in Table 1.
|all data||R and R excluded||R, R and A excluded|
The results are evidently somewhat sensitive to the inclusion (or not) of data on the Z partial width ratios R and R and the SLD/SLC measurement of A, all of which differ by 2.5 standard deviations or more from the Standard Model values. However, the conclusion on the agreement of the direct and indirect determinations is unchanged. As we shall see in the following sections, a significant reduction in the error on is expected from both LEP2 and the Tevatron.
1.3 Improved precision on from the Tevatron
The Tevatron data so far analysed, and shown in Fig. 1, come from the 1992/3 data-taking (Run 1a). The results from CDF are based on approximately and are final, whereas those from D0 are based on approximately and are still preliminary. It is to be expected that the final result will have a smaller error. In addition, there will be a significantly larger data sample from the 1994/6 data-taking (Run 1b). This should amount to more than of useful data for each experiment. When these data are analysed it is envisaged that the total combined error on will be reduced to about . In particular, the combined CDF/D0 result will depend on the common systematic error arising from uncertainties in the parton distribution functions. Thus when the first measurements emerge from LEP2 one may assume that the world average error will have approximately this value. For more details see Ref. .
After 1996 there will be a significant break in the Tevatron programme. Data-taking will start again in 1999 with a much higher luminosity (due to the main injector and other improvements). Estimating the error on which will ultimately be achievable (with several fb of total luminosity) is clearly more difficult. If one assumes that an increase in the size of the data sample leads to a steady reduction in the systematic errors, one might optimistically envisage that the combined precision from the Tevatron experiments will eventually be in the range, assuming a common systematic error of about . However it is important to remember that these improved values will be obtained after the LEP2 measurements.
1.4 Impact of a precision measurement of
Within the Standard Model, the value of is sensitive to both and . For example, for a fixed value of , a precision of translates to a precision on of . The impact of a precise measurement of on the indirect determination of is shown in Table 2.
In order to assess the impact of a precise measurement of it is necessary to make an estimate of the improvements which will be made on the electroweak data from LEP1 and SLC. Details of the improvements which are assumed here are discussed in . The importance of a precise measurement of can perhaps best be appreciated by considering the (almost) model independent parameters . The parameter () is sensitive mainly to the Z partial and total widths. The parameter depends linearly on both and , where is determined from . The parameter depends linearly on , and r. This latter quantity is determined essentially from , and so improvements in the precision of depend directly on improving the error on . This is illustrated in Fig. 3, which shows the 70% confidence level contours for fits to projected global electroweak data. The different contours correspond to different values of . In these fits all electroweak data measurements have been set to correspond to the Standard Model values , and . The variables are constructed to be sensitive to vector boson propagator effects, from both physics within the Standard Model and beyond.
Numerically, the projected data give a precision
For , the error is obtained, whereas for the projected errors on one obtains
The smaller the volume in space allowed by the precision electroweak measurements, the greater the constraint on physics beyond the Standard Model.
The MSSM is arguably the most promising new-physics candidate. It is therefore especially important to consider the MSSM prediction for . Figure 4 shows as a function of in the SM (solid lines) and in the MSSM (dashed lines). In each case the prediction is a band of values, corresponding to a variation of the model parameters (dominantly in the SM case, with chosen here) consistent with current measurements and limits. An additional constraint of ‘no SUSY particles at LEP2’ is imposed in the MSSM calculation.
1.5 Methods for measuring
Precise measurements of can in principle be obtained using the enhanced statistical power of the rapidly varying total cross-section at threshold, the sharp (Breit-Wigner) peaking behaviour of the invariant-mass distribution of the W decay products and the sharp end-point spectrum of the lepton energy in W decay. One can obtain a rough idea of the relative power of these methods by estimating their statistical precision assuming 100% efficiency, perfect detectors and no background. More complete discussions are given in Sections 2 and 3.
Threshold cross-section measurement of the process . The statistical power of this method, assuming 100% signal efficiency and no background, is
where the minimum value is attained at . Here denotes the total integrated luminosity.
Direct reconstruction methods, which reconstruct the Breit-Wigner resonant shape from the W final states using kinematic fitting techniques to improve the mass resolution. The statistical power of this method, again assuming 100% efficiency, perfect detector resolution and no background, can be estimated as
approximately independent of the collider energy. This order of magnitude estimate is confirmed by more detailed studies, see below.
Determination of from the lepton end-point energy. The end-points of the lepton spectrum in depend quite sensitively on the W mass. For on-shell W bosons at leading order:
In this case the statistical error on is determined by the statistical error on the measurement of the lepton end-point energy,
In practice, however, the end-points of the distribution are considerably smeared by finite width effects and by initial state radiation, and only a fraction of events close to the end-points are sensitive to . This significantly weakens the statistical power of this method from what the naive estimate (8) would predict.
The detailed studies described in the following sections show that the errors which can realistically be achieved in practice are somewhat larger than the above estimates for Methods A and B. The statistical precisions of the two methods are in fact more comparable (for the same integrated luminosity) than the factor 2 difference suggested by the naive estimates (5) and (6). The overall statistical error for Method C has been estimated at for , significantly larger than that of the other two methods. It will not therefore be considered further here, although it is still a valid measurement for cross-checking the other results.
It is envisaged that most of the LEP2 data will be collected at energies well above threshold, and so the statistically most precise determination of will come from Method B. However with a relatively modest amount of luminosity spent at the threshold (for example per experiment), Method A can provide a statistical error of order , not significantly worse than Method B and with very different systematics. The two methods can therefore be regarded as complementary tools, and both should be used to provide an internal cross-check on the measurement of the W mass at LEP2. This constitutes the main motivation for spending some luminosity in the threshold region.
The threshold cross-section method is also of interest because it appears to fit very well into the expected schedule for LEP2 operation in 1996. It is anticipated that the maximum beam energy at LEP2 will increase in steps, with the progressive installation of more superconducting RF cavities, in such a way that a centre-of-mass energy of 161 GeV will indeed be achievable during the first running period of 1996. This would then be the ideal time to perform such a threshold measurement. The achievable statistical error on depends of course critically on the available luminosity at the threshold energy. In Section 2 we present quantitative estimates based on integrated luminosities of 25, 50 and 100 pb per experiment.
1.6 Theoretical input information
1.6.1 Cross-sections for the signal and backgrounds
Methods (A) and (B) for measuring described above require rather different theoretical input. The threshold method relies on the comparison of an absolute cross-section measurement with a theoretical calculation which has as a free parameter. The smallness of the cross-section near threshold is compensated by the enhanced sensitivity to in this region. In contrast, the direct reconstruction method makes use of the large statistics at the higher LEP2 energies, GeV. Here the more important issue is the accurate modeling of the W line-shape, i.e. the distribution in the invariant mass of the W decay products.
In this section we describe some of the important features of the theoretical cross-sections which are relevant for the measurements. A more complete discussion can be found in the contribution of the WW and Event Generators Working Group to this Report .
We begin by writing the cross-section for , schematically, as
We note that this decomposition of the cross-section into ‘signal’ and ‘background’ contributions is practical rather than theoretically rigorous, since neither contribution is separately exactly gauge invariant nor experimentally distinguishable in general. The various terms in (9) correspond to
: the Born contribution from the 3 ‘CC03’ leading-order diagrams for involving -channel exchange and -channel and Z exchange, calculated using off-shell W propagators.
: higher-order electroweak radiative corrections, including loop corrections, real photon emission, etc.
: higher-order QCD corrections to final states containing pairs. For the threshold measurement, where in principle only the total cross-section is of primary interest, the effect of these is to generate small corrections to the hadronic branching ratios which are entirely straightforward to calculate and take into account. More generally, such QCD corrections can lead to additional jets in the final state, e.g. from one hard gluon emission. This affects the direct reconstruction method, insofar as the kinematic fits to reconstruct assume a four-jet final state, and both methods insofar as cuts have to be imposed in order to suppress the QCD background (see Sections 2,3 below). Perturbative QCD corrections, real gluon emission to and real plus virtual emission to , have been recently discussed in Refs. [19, 20] respectively, together with their impact on the measurement of .
: ‘background’ contributions, for example from non-resonant diagrams (e.g. ) and QCD contributions to the four-jet final state. All of the important backgrounds have been calculated, see Table 6 below. At threshold, the QCD four-jet background is particularly large in comparison to the signal.
In what follows we consider (i) and (ii) in some detail. Background contributions and how to suppress them are considered in later sections.
1.6.2 The off-shell cross-section
The leading-order cross-section for off-shell production was first presented in Ref. :
The cross-section can be written in terms of the , and Z exchange contributions and their interferences:
where . Explicit expressions for the various contributions can be found in Ref. for example. The stable (on-shell) cross-section is simply
A theoretical ansatz of this kind will be the basis of any experimental determination of the mass and width of the W boson. The reason for this is the large effect of the virtuality of the W bosons produced around the nominal threshold. An immediate conclusion from Eq. (10) may be drawn: the W mass influences the cross sections exclusively through the off-shell propagators; all the other parts are independent of and (neglecting for the moment the relatively minor dependence due to radiative corrections). It will be an important factor in the discussion which follows that near threshold the (unpolarized) cross-section is completely dominated by the -channel neutrino exchange diagram. This leads to an -wave threshold behaviour , whereas the -channel vector boson exchange diagrams give the characteristic -wave behaviour .
By tradition (for example at LEP1 with the Z boson), the virtual W propagator in Eq. (11) uses an -dependent width,
where . Another choice, equally well justified from a theoretical point of view, would be to use a constant width in the W propagator (for a discussion see Ref. ):
The numerical values of the width and mass in the two expressions are related :
These relations may be derived from the following identity: . Numerically, the consequences are below the anticipated experimental accuracy.
1.6.3 Higher-order electroweak corrections
The complete set of next-to-leading order corrections to production has been calculated by several groups [24, 25], for the on-shell case only, see for example Refs. [26, 18] and references therein. There has been some progress with the off-shell (i.e. four fermion production) corrections but the calculation is not yet complete. However using the on-shell calculations as a guide, it is already possible to predict some of the largest effects. For example, it has been shown that close to threshold the dominant contribution comes from the Coulomb correction, i.e. the long-range electromagnetic interaction between almost stationary heavy particles. Also important is the emission of photons collinear with the initial state e (‘initial state radiation’) which gives rise to logarithmic corrections . These leading logarithms can be resummed to all orders, and incorporated for example using a ‘structure function’ formalism. In this case, the generalization from on-shell to off-shell ’s appears to be straightforward. For the Coulomb corrections, however, one has to be much more careful, since in this case the inclusion of the finite decay width has a dramatic effect. Finally, one can incorporate certain important higher-order fermion and boson loop corrections by a judicious choice of electroweak coupling constant. Each of these effects will be discussed in turn below.
In summary, certain corrections are already known to be large because their coefficients involve large factors like , , , etc. Once these are taken into account, one can expect that the remaining corrections are no larger than . When estimating the theoretical systematic uncertainty on the W mass in Section 2 below, we shall therefore assume a conservative overall uncertainty on the cross-section of from the as yet uncalculated and higher-order corrections.
1.6.4 Coulomb corrections
The result for on-shell production is well-known — the correction diverges as as the relative velocity of the bosons approaches zero at threshold. Note that near threshold and so the Coulomb-corrected cross-section is formally non-vanishing when .
For unstable production the finite decay width screens the Coulomb singularity , so that very close to threshold the perturbative expansion in is effectively replaced by an expansion in . In the calculations which follow we use the expressions for the correction given in Ref. . The net effect is a correction which reaches a maximum of approximately in the threshold region. Although this does not appear to be large, we will see below that it changes the threshold cross-section by an amount equivalent to a shift in of order MeV. In Ref. the result is generalized to all orders. However the contributions from second order and above change the cross-section by in the threshold region (see also ) and can therefore be safely neglected.
Note also that the Coulomb correction to the off-shell cross-section
provides an example of a (QED) interconnection effect between the two W bosons: the exchange of a soft photon distorts the line shape () of the W and therefore, at least in principle, affects the direct reconstruction method [32, 33, 34]. In Ref. , for example, it is shown that the Coulomb interaction between the W bosons causes a downwards shift in the average reconstructed mass of . Selecting events close to the Breit-Wigner peak reduces the effect somewhat. However the calculations are not yet complete, in that QED interactions between the decay products of the two W bosons are not yet fully included.
1.6.5 Initial state radiation
Another important class of electroweak radiative corrections comes from the emission of photons from the incoming e and e. In particular, the emission of virtual and soft real photons with energy gives rise to doubly logarithmic contributions at each order in perturbation theory. The infra-red () logarithms cancel when hard photon contributions are added, and the remaining collinear () logarithms can be resummed and incorporated in the cross-section using a ‘flux function’ or a ‘structure function’ (see also Refs. [36, 37, 38, 39, 18]).
The ISR corrected cross section in the flux function (FF) approach is
The term comes from soft and virtual photon emission, the term comes from hard photon emission, contains the Coulomb correction (18), is the doubly resonating Born cross section, and the background contributions. Explicit expressions can be found in the above references. The additional term is discussed in Refs. and . The invariant mass lost to photon radiation may be calculated as
where is the contents of the curly brackets in Eq. (19).
Alternatively, the structure function (SF) approach may be used:
Here, the invariant mass loss is
In addition, the radiative energy loss may be determined,
Initial state radiation affects the W mass measurement in two ways. Close to threshold the cross-section is smeared out, thus reducing the sensitivity to (see Fig. 5 below). For the direct reconstruction method, the relatively large average energy carried away by the radiated photons leads to a large positive mass-shift if it is not taken into account in the rescaling of the final-state momenta to the beam energy (see Section 3 below). By rescaling to the nominal beam energy we obtain for the mass-shift . Note however that a fit to the mass distribution gives more weight to the peak, and therefore in practice the effective value of or is less than that given by Eqs. (22,25,LABEL:eq:26) (see Section 3.4). Table 3 shows the influence of the various cross-section contributions on the average energy and invariant mass losses. The invariant mass loss may be calculated both in the SF and FF approaches. A comparison shows that the predictions in both schemes differ only slightly, which allows us to use the numerically faster FF approach for the numerical estimates. At the lower LEP2 collider energies, the energy and invariant mass losses are nearly equal, while at higher energies their difference is non-negligible. Note also that the inclusion of the non-universal ISR corrections and background terms is of minor influence. The latter has been studied only for CC11 processes; for reactions of the CC20 type the background is larger and the numerical estimates are not yet available. The Coulomb correction is numerically important and cannot be neglected . The dependence of the predictions on the details of the treatment of QED is discussed in detail in Ref. and will not be repeated here.
|change to FF||0.5||–0.8||–4.2|
1.6.6 Improved Born approximation
In the Standard Model, three parameters are sufficient to parametrise the electroweak interactions, and the conventional choice is since these are the three which are measured most accurately. In this case the value of is a prediction of the model. Radiative corrections to the expression for in terms of these parameters introduce non-trivial dependences on and , and so a measurement of provides a constraint on these masses. However the choice does not appear to be well suited to production. The reason is that a variation of the parameter , which appears explicitly in the phase space and in the matrix element, has to be accompanied by an adjustment of the charged and neutral weak couplings. Beyond leading order this is a complicated procedure.
It has been argued that a more appropriate choice of parameters for LEP2 is the set (the so-called –scheme), since in this case the quantity of prime interest is one of the parameters of the model. Using the tree-level relation
we see that the dominant -channel neutrino exchange amplitude, and hence the corresponding contribution to the cross-section, depends only on the parameters and . It has also been shown that in the –scheme there are no large next-to-leading order contributions to the cross-section which depend on , either quadratically or logarithmically. One can go further and choose the couplings which appear in the other terms in the Born cross-section such that all large corrections at next-to-leading order are absorbed, see for example Ref. . However for the threshold cross-section, which is dominated by the -channel exchange amplitude, one can simply use combinations of and defined by Eq. (27) for the neutral and charged weak couplings which appear in the Born cross-section, Eq. (12).
In summary, the most model-independent approach when defining the parameters for computing the cross-section appears to be the –scheme, in which appears explicitly as a parameter of the model. Although this makes a non-negligible difference when calculating the Born cross-section, compared to using and to define the weak couplings (see Table 5 below), a full next-to-leading-order calculation will remove much of this scheme dependence .
|(off-shell with , )||4.747||15.873|
|(off-shell with , )||4.823||15.882|
1.6.7 Numerical evaluation of the cross-section
Figure 5 shows the cross-section at LEP2 energies. The different curves correspond to the sequential inclusion of the different effects discussed above. The parameters used in the calculation are listed in Table 4. Note that both the initial state radiation and the finite width smear the sharp threshold behaviour at of the on-shell cross-section.
The different contributions are quantified in Table 5, which gives the values of the cross-section in different approximations just above threshold ( GeV) and at the standard LEP2 energy of GeV. At threshold we see that the effects of initial state radiation and the finite W width are large and comparable in magnitude. For the threshold method, the primary interest is the dependence of the cross-section on .
|at 161 GeV||at 175 GeV||at 192 GeV|
This will be quantified in Section 2 below. For both methods, the size of the background cross-sections is important. For completeness, therefore, we list in Table 6 some relevant cross-section values obtained using the PYTHIA Monte Carlo. This includes finite-width effects, initial state radiation and Coulomb corrections. Notice that the values for agree to within about 1% accuracy with those given in the last row of Table 5.
2 Measurement of from the Threshold Cross-Section222prepared by D. Gelé, T.G. Shears, W.J. Stirling, A. Valassi, M.F. Watson
As discussed in Section 1.5, one can exploit the rapid increase of the production cross-section at to measure the W mass. In the following, we briefly discuss the basic features of this method, suggest an optimal collider strategy for data-taking, and estimate the statistical and systematic errors. The intrinsic statistical limit to the resolution on is shown to be energy-dependent: in particular, arguments are presented in favour of a single cross-section measurement at a fixed energy .
2.1 Collider strategy
The cross-section for production increases very rapidly near the nominal kinematic threshold , although the finite W width and ISR smear out the abrupt rise of the Born on-shell cross-section. This means that for a given near threshold, the value of the cross-section is very sensitive to . This is illustrated in Fig. 6, where the excitation curve is plotted for various values of the W mass. The calculation is the same as that discussed in Section 1.6, and includes finite W width effects, ISR and QED Coulomb corrections. A measurement of the cross-section in this region therefore directly yields a measurement of .
For an integrated luminosity and an overall signal efficiency (where the sum extends over the various channels selected, with branching ratios BR and efficiencies ), the error on the cross-section due to signal statistics is given by
where is the number of selected signal events. The corresponding error on the W mass is
The sensitivity factor is plotted in Fig. 7 as a function of . There is a minimum at
corresponding to a minimum value of approximately . Note that the offset of the minimum of the sensitivity above the nominal threshold is insensitive to the actual value of , since in the threshold region the cross-section is to a first approximation a function of only.
As discussed below, the statistical uncertainty is expected to be the most important source of error for the threshold measurement of : the optimal strategy for data-taking consists therefore in operating at the collider energy in order to minimize the statistical error on . The statistical sensitivity factor is essentially flat within , where it increases at most to (+4%); bearing in mind that the present uncertainty on from direct measurements is 160 MeV (and is expected to decrease further in the coming years), this corresponds to on the current world average value. In other words, is already known to a level of precision good enough to choose, a priori, one optimal energy for the measurement of the cross-section at the threshold. Using the latest world average value (see Eq. (1)) gives an optimal collider energy of .
2.2 Event selection and statistical errors
The error on the W mass due to the statistics of events collected has been given in the previous section. Background contamination with an effective cross-section introduces an additional statistical error. The overall effect is that the statistical error on is modified according to
In the following subsections we present estimates of this statistical error for realistic event selections, for an integrated luminosity of at . Tight selection cuts are required to reduce the background contamination while retaining a high efficiency for the signal, especially as the signal cross-section is a factor of 4–5 lower at threshold than at higher centre-of-mass energies.
The studies are based on samples of signal and background events generated by means of Monte Carlo programs (mainly PYTHIA ) tuned to LEP1 data. These events were run through the complete simulation program giving a realistic detector response, and passed through the full reconstruction code for the pattern recognition.
2.2.1 Fully hadronic channel, .
The pure four quark decay mode benefits from a substantial branching ratio (46%) corresponding to a cross-section pb. Obviously, the typical topology of such events consists of four or more energetic jets in the final state. Due to its large cross-section (see Table 6), the main natural background to this four-jet topology comes from events which can be separated into two classes depending on the virtuality of the Z: (i) the production of an on-mass-shell Z accompanied by a radiative photon of nearly 55 GeV (at ), which is experimentally characterised by missing momentum carried by the photon escaping inside the beam pipe (typically 70% of the time), and (ii) events with a soft ISR and a large total visible energy, which potentially constitute the most dangerous QCD background contribution.
Note that a semi-analytical calculation of the genuine four-fermion background cross-section for a wide range of four-fermion final states (with non-identical fermions) shows that in the threshold region , and therefore the effect on the determination from these final states is negligible.
Although the effective four-jet-like event selection depends somewhat on the specific detector under consideration, a general and realistic guideline selection can be described. The most relevant conditions to be fulfilled by the selected events can be summarised as follows:
A minimum number of reconstructed tracks of charged and neutral particles is required. A typical value is 15. This cut removes nearly all low multiplicity reactions, such as dilepton production () and two-photon processes.
A veto criterion against hard ISR photons from events in the detector acceptance can be implemented by rejecting events with an isolated cluster with significant electromagnetic energy (larger than 10 GeV for example).
A large visible energy, estimated using the information from tracks of charged particles and from the electromagnetic and hadron calorimeters. For example, a minimum energy cut value of 130 GeV reduces by a factor of 2 the number of events with a photon collinear to the beam axis.
A minimum number (typically 5) of reconstructed tracks per jet. This criterion acts on the low multiplicity jets from decays as well as from conversions or interactions with the detector material.
A minimum jet polar angle. The actual cut value depends on the detector setup, but is likely to be around . This cut is mainly needed to eliminate poorly measured jets in the very forward region, where the experiments are generally less well instrumented.
These selection criteria almost completely remove the harmless background sources ( and ), but there is still an unacceptable level of contamination (about three times higher than the signal). The second step is to suppress the remaining QCD background (from events) by performing a W mass reconstruction based on a constrained kinematic fit. The following additional criteria can then be imposed:
A probability cut associated with a minimum constrained dijet mass requirement – a typical choice of standard values of 1% and 70 GeV respectively is used here. This procedure appears to be an efficient tool to improve the mass resolution and therefore to reduce the final background, see Fig. 8.
In summary, a reasonable signal detection efficiency in excess of 50% is achievable for events. Although the final rejection factor of QCD events is approximately 500, a substantial residual four-jet background still remains, giving a purity of around 70%. The contributions from other backgrounds (, , and two-photon events) are negligible for the four-jet analysis.
The signal and background efficiencies for the typical event selection described above are given in Table 7, assuming a total integrated luminosity of 100 pb (i.e. 25 pb per interaction point).
|Signal cross-section||0.94 pb||0.76 pb||0.23 pb|
|(stat.) for signal||180 MeV||197 MeV||354 MeV|
|Background cross-section||0.39 pb||0.03 pb||0.01 pb|
|(stat.) for background||106 MeV||37 MeV||74 MeV| |
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Abraham Adolf Fraenkel
Abraham Adolf Fraenkel
One of the fathers of modern logic, German-born mathematician Abraham Fraenkel (1891-1965) first became widely known for his work on set theory. Long fascinated by the pioneering work in set theory of fellow German Ernst Zermelo (1871-1953), Fraenkel launched research to put set theory into an axiomatic setting that improved the definitions of Zermelo's theory and proposed its own system of axioms. Within that system, Fraenkel proved the independence of the axiom of choice. The Zermelo-Fraenkel axioms of set theory, known collectively as ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. When the axiom of choice is included, the resulting system is known as ZFC.
Studied at Several Universities
Abraham Adolf Fraenkel was born on February 17, 1891, in Munich, Germany. The son of Sigmund and Charlotte (Neuberger) Fraenkel, he was strongly influenced by his orthodox Jewish heritage. B.H. Auerbach-Halberstadt, Fraenkel's great-grandfather, had been widely known for his rabbinical teachings. As a child, Fraenkel was enrolled in Hebrew school and was reading Hebrew by the time he was five. Raised in a family that set a high priority on education, Fraenkel advanced rapidly in his general studies and, like most German students of that era, studied at a number of universities. He began his higher studies at the University of Munich in his hometown and studied subsequently at the German universities of Marburg, Berlin, and Breslau. In 1914, at the age of 23, Fraenkel received his doctoral degree in mathematics from the University of Breslau.
World War I broke out in August 1914, shortly after Fraenkel had completed his studies at Breslau. For the next two years, he served in the German military as a sergeant in the medical corps. He also worked briefly for the German army's meteorological service. In 1916 Fraenkel accepted a position at the University of Marburg as an unsalaried lecturer, or privatdocent. It was at Marburg that Fraenkel began his most important research in mathematical theory. On March 28, 1920, he married Malkah Wilhemina Prins. The couple eventually had four children.
Focused on Set Theory
Fraenkel's earliest research was on the p-adic numbers first described by Kurt Hensel in the late nineteenth century and on the theory of rings. Before long, however, he became deeply involved in the study of set theory, specifically the work of Ernst Zermelo, who in the early years of the twentieth century had published his controversial and innovative views on the subject. Zermelo had postulated that from any set of numbers, a single element could be selected and that definite properties of that element could be determined. This was known as the axiom of choice, but Zermelo offered no real proof for his theory, suggesting that the study of mathematics could only progress if certain axioms were simply accepted without question. For many mathematicians, Zermelo's lack of proof was unacceptable. Some, including French mathematician Jacques Hadamard, reluctantly agreed to accept Zermelo's theory until a better way could be found, while others, including Jules-Henri Poincaré, adamantly opposed acceptance of Zermelo's theory.
Without either accepting or rejecting Zermelo's theory outright, Fraenkel set about to find ways to put Zermelo's work on a firmer foundation. In the case of finite sets of numbers, Fraenkel found, Zermelo's theory already worked quite well. However, for infinite sets, Zermelo's assumptions were more questionable. Fraenkel eventually substituted a notion of function for Zermelo's idea of determining a definite property of a number in a set. In so doing, he significantly clarified Zermelo's set theory and also rid it of its dependence on the axiom of choice, which had clearly been one of the most controversial elements of Zermelo's work.
Just as Fraenkel's research built on theories advanced earlier by Zermelo, others' refinements to the work of Zermelo and Fraenkel have buttressed their theories and advanced the mathematical community's understanding of set theory. Fraenkel's system of axioms was modified by Norwegian mathematician Thoralf Skolem in 1922 to create what is known today as the ZFS system, named for Zermelo, Fraenkel, and Skolem. Within the ZFS system, it is harder to prove the independence of the axiom of choice, a goal that was not achieved until the work of American Paul Joseph Cohen in the 1960s. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and the generalized continuum hypothesis.
Published Set Theory Findings
Fraenkel published his conclusions on set theory in two separate works—a popular introductory textbook published in 1919 and a 1922 research article determining the independence of the axiom of choice. The conclusions in the latter work were later included as part of the proof for a newly coined term, Ur-elements—infinite and distinct pairs of objects that do not in themselves define a set. A number of prominent mathematicians of the period questioned the validity of Ur-elements, but only three years later German physicist Wolfgang Pauli used them in his proof of the exclusion principle.
In 1922 Fraenkel was promoted to assistant professor of mathematics at the University of Marburg. His earlier work on set theory had propelled him to the forefront of set theory research, and over the next few years he published a number of articles on the subject while he continued to teach. In 1928 Fraenkel was offered a full professorship at the University of Kiel. He accepted but only a year later took a leave of absence to become a visiting professor at Jerusalem's Hebrew University. For the next two years he taught at Hebrew University, leaving in 1931 after a disagreement with the school's administration.
Germany in Turmoil
Fraenkel's return to Germany proved to be a bittersweet occasion. His native country was in economic disarray, suffering through the effects of the worldwide economic depression and the brutal conditions imposed by the Treaty of Versailles that had ended World War I. The economic pressures on the German people had given rise to increasing intolerance, most notably a disturbing wave of anti-Semitism. For the next two years, Fraenkel resumed his teaching duties at Kiel, keeping a wary eye on the increasingly unsettled political situation in Germany. In January 1933 Adolf Hitler, leader of the National Socialist German Workers' Party, better known as Nazis, became Germany's chancellor. Fraenkel and his family left the country a month later, moving first to Amsterdam in the neighboring Netherlands.
Fraenkel and his family spent only two months in Amsterdam, closely monitoring the situation in their native Germany while there. Convinced that there would not be a quick turnaround under the Nazi regime, Fraenkel drafted a letter of resignation to the University of Kiel in April 1933 and returned to Jerusalem to teach once again at Hebrew University. Despite his earlier disagreement with the university's administration, he was warmly welcomed back to the school's faculty.
Focus of Research Changed
Following his exile from Germany, Fraenkel changed the focus of his research. Although he continued to publish texts on set theory for the remainder of his career, Fraenkel began to concentrate his studies on the evolution of modern logic and the contributions made by Jewish mathematicians and scientists in their respective fields. Fraenkel had written a number of books about the history of mathematics. In 1920 he had published an overview of the work of Carl Friedrich Gauss, who in his doctoral dissertation had proved the fundamental theorem of algebra. As early as 1930 he had begun the work of chronicling the accomplishments of Jewish mathematicians with his biography of Georg Cantor, who was half-Jewish. Cantor at that time was of greater interest to Fraenkel for the nature of his research into set theory than for his ethnic background. However, once he had resumed teaching at Hebrew University in 1933, he began a much wider study into the work of Jewish scientists and mathematicians. In 1960 Fraenkel published Jewish Mathematics and Astronomy.
In his research into the origins of modern logic, Fraenkel looked closely at natural numbers, describing them in terms of modern concepts of logic and reasoning. Although his research underscored the need for continuity in consideration of the number line, Fraenkel also expressed interest in opposing points of view. During this period, Fraenkel had a conversation with physicist Albert Einstein, who suggested that the prevailing theory of continuity in mathematics might some day be overtaken by the atomistic concept of the number line. Although Fraenkel himself remained unconvinced, largely because he considered mathematical continuity necessary to the foundation of modern calculus, he did publish an article explaining the views of the intuitionists, as Einstein and others who believed similarly were known.
Taught at Einstein Institute of Mathematics
Fraenkel was among the first professors at Hebrew University's Einstein Institute of Mathematics. Along with fellow professor Edmond Landau, Fraenkel taught mathematical logic and mathematical analysis. In 1958, while still teaching at Hebrew University, Fraenkel published an overview of his work on set theory, a textbook entitled Foundations of Set Theory. A year later, he retired as a professor at Hebrew University. To mark Fraenkel's 70th birthday in 1961, several members of the mathematical community put together a collection of essays and research articles related to Fraenkel's life work. The collection, Essays on the Foundations of Mathematics, contained contributions from mathematicians around the world. Sadly, Fraenkel never saw the book in its final form. He died in Jerusalem on October 15, 1965, only months before the book was published.
Fraenkel will be remembered for his research in set theory and modern logic. His refinements to the set theory conclusions of Ernst Zermelo, codified as the Zermelo-Fraenkel axioms, or ZF, are almost always what scientists and mathematicians mean today when they speak of "set theory." Further enhancing the value of Fraenkel's contributions to the body of mathematical theory are the clarity and precision of his writings, several of which continue to be taught in colleges and universities worldwide. In its review of Fraenkel's summation of his set theory research— Foundations of Set Theory —the British Journal for the Philosophy of Science was lavish in its praise. Its reviewer wrote that the book "is a masterly survey of its field. It is lucid and concise on a technical level, it covers the historical ground admirably, and it gives a sensible account of the various philosophical positions associated with the development of the subject … essential reading for any mathematician or philosopher."
Contemporary Authors Online, Gale Group, 2000.
Mathematical Expeditions: Chronicles by the Explorers, Springer-Verlag, 2001.
Notable Scientists: From 1900 to the Present, Gale Group, 2001.
"Adolf Abraham Halevi Fraenkel," Groups, Algorithms, and Programming,http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fraenkel.html (March 5, 2003).
"Adolf Fraenkel," 201E: Mathematical Foundations,http://ergo.ucsd.edu/~movellan/courses/245/people/Fraenkel.html (March 9, 2003).
"Paul Joseph Cohen," Groups, Algorithms, and Programming,http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cohen.html (March 9, 2003). |
Integration is a summation operation that may be used as a mathematical tool for determining the area with functions of a single variable, computing the surface area and volume of three-dimensional solids, calculating the area and volume of a function with two variables, or summing multidimensional functions.
Real-world quantities such as temperature, magnetic field strength, pressure, speed, flow rate, lighting, share prices, and so on can be defined mathematically in science, engineering, and economics. We can use integration to combine these variables to get a total result.
In this post, we will explain the rules of integration with comprehensive examples and their solutions. This post is intended for the students of calculus and it will help you to understand the concept of integrals along with the rules.
Integration Rules – The Need
When it comes to integrating functions or taking antiderivatives, the same basic rules apply as they do for differentiation. This antiderivative calculator uses all of the rules while calculating antiderivatives. You should be familiar with the notion that integration and differentiation are the opposites of one another.
We can always distinguish the result to go back to the original function if we integrate a function. However, this is not the case. Because the derivative of any constant term is zero, any constant term in a function usually disappears when it is differentiated.
It is something we should bear in mind while considering how to integrate a function because it implies that our solution will always contain a constant with an unknown value. This constant is known as the constant of integration, C.
The most important rule of integration is the power rule of integration. This method is effectively the reverse of the power rule used in derivatives, and it yields the indefinite integral of a variable raised to a certain power. To refresh your memory below is the integration power rule formula:
The indefinite integral of the variable x raised to the power of n multiplied by the constant-coefficient a is given by this formula. Also bear in mind that n cannot be equal to -1, because, on the right-hand side of the formula, this would put a 0 in the denominator. This criterion alone allows us to integrate polynomial functions using a single variable.
We just integrate each expression independently, with no modification to the plus or minus sign in front of each word. Some typical indefinite integrals are listed below. It’s worth noting that in these instances, a stands for a constant, x stands for a variable, and e stands for Euler’s number which is approximately 2.7183. It’s also worth noting that the first three instances are the result of applying the power rule.
A constant value a:
∫ a dx = ax + C
A variable x:
The square of a variable x 2:
The reciprocal of a variable 1/x:
The exponential function e x:
Other exponential functions a x:
The natural logarithm of a variable ln (x):
The sine of a variable sin (x):
The cosine of a variable cos (x):
∫ cos (x) dx = sin (x) + C
The power, constant-coefficient or constant multiplier, sum, and difference rules are among the basic integration rules that will be discussed here. We’ll give some easy examples to show how these integration principles and laws actually work. Before moving onto the next section, check out this Integral calculator with steps. It can help you to find the indefinite integral of any given function.
The Power Rule
As we have seen, the power rule for integration is the inverse of the power rule for differentiation. It returns the indefinite integral of a variable multiplied by a power. Here’s the power rule again:
Let’s look at some examples of how this rule is used. Assume we wish to calculate the indefinite integral of x3. Using the power rule:
It is not always evident that we can apply the power rule to get the indefinite integral of a function. Assume we wish to calculate the indefinite integral of the equation 3√x. How can we apply the power rule to the cubed root function? It’s actually fairly simple. All we have to do is change the expression to get x to a power. To express the nth root of a number in an index form, there is a common formula which can be stated as:
n√a = a 1/n
Applying this formula to 3√x:
3√x = x 1/3
We can now apply the power rule to get:
The Constant Coefficient Rule
The constant-coefficient rule is also known as the constant multiplier rule. It states that the indefinite integral of c∙f(x), where c represents a constant coefficient and f(x) is some function, is equivalent to the indefinite integral of f(x) multiplied by c. This can be stated as follows:
The constant-coefficient rule allows us to disregard the constant-coefficient in an equation while integrating the remainder of it. Let’s say we wish to compute the indefinite integral of the expression 3x2. According to the constant-coefficient rule, the indefinite integral of this equation is the indefinite integral of x2 multiplied by 3. That is to say:
Now we just apply the power rule to x 2:
The Sum Rule
The sum rule describes how to integrate functions that are the sum of many terms. It simply shows us that we must integrate each expression independently in the total, before adding the results together. It is unimportant which order the terms appear in the outcome. This can be stated as follows:
You may be asking why the regulation is worded the way it is at this point. It is critical to understand that in a function that is the sum of two or more components, each term may be thought of as a function in its own right – even a constant term. Assume we wish to calculate the indefinite integral of a function (x) = 3x2 + 4x + 12. Using the sum rule:
The Difference Rule
The difference rule instructs us on how to integrate functions that include the difference of two or more terms. It is similar to the sum rule in that it instructs us to integrate each term in the sum independently. The only distinction is that the order of the expressions is important and cannot be modified. This rule can be stated explicitly as follows:
Let’s have a look at an example. Assume we wish to calculate the indefinite integral of the polynomial function (x) = 5x3 – 9x – 2. Using the sum rule, we obtain:
The difference and sum rules are fundamentally the same rules. If we wish to integrate a function that comprises both the sum and difference of a number of terms, we must remember to integrate each term independently and to keep the order of the terms in mind. The “+” or “=” sign in front of each expression remains the same.
You may also conceive the function as the sum of a number of positive and negative terms and apply the sum rule. The order is thus irrelevant; you just need to be aware of the sign of each expression. Below, we have listed few more examples for further interpretation of integration rules.
Evaluate ∫ 7 dx
∫ 7 dx =
7 ∫ dx ……….multiplication by a constant rule
= 7x + C
What is ∫ 5x4 dx
∫ 5x4 dx = 5 ∫x4 dx ……. using multiplication by a constant rule
= 5(x5/5) + C ………. using power rule
= x5 + C
Evaluate ∫ (2x3 + cos(x) ) dx
∫ (2x3 + 6cos(x) ) dx = ∫ 2x3 dx + ∫ 6cos(x) dx …..Applying the sum rule
= 2 ∫ x3 dx + 6 ∫ cos(x) dx ……….Applying the multiplication by a constant rule
= 2(x4/4) + C1 + 6(sin(x) + C2 …..Applying the power rule. C1 and C2 are constants.
C1 and C2 can be replaced by a single constant C, so:
∫ (2x3 + cos(x) ) dx = x4/2 + 6sin(x) + C
All of the listed rules are extensively used in integration and are vital for the evaluation of integrals. These rules should be practiced and implemented on several types of functions if you want to master the concept of integrals. Refer to these principles listed above if you are stuck somewhere while calculating antiderivative or integral. |
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Thank you to everybody who sent us their thoughts about this game. We received a couple of solutions for Version 1 and a lot of solutions for Version 2!
Ilan from Twyford School in the UK considered which strategy worked best with a 1-6 dice:
The key to solving ‘less is more’ is to take a risk and put a large number on the smaller side. Here is an example of the perfect round to win:
65 < 66
65 < 66
This is because the left has to be smaller and it is quite unlikely to roll large numbers so this is quite a risky way of doing it.
This is a less risky round:
45 < 48
39 < 49
The risk to doing the first method is that if you have already put sixes in the first column there is a possibility that you will not get the number you want, in this case sixes and it will not work and you will be deducted a lot of points. In conclusion, I think that going in the middle (not too safe not too risky) so I think the best numbers to put on the right would be 51 or 56 or something along those lines.
Dhruv from St. Anne's RC Primary School in the UK also used a 1-6 dice, and they sent in a picture to explain their method. You can click on the picture to see the full-size solution:
Good ideas, Ilan and Dhruv! It looks like you're both hoping to roll 6s or 5s. I wonder why Dhruv thinks it is best to fill in the tens spaces on the right-hand side first?
We received a lot of solutions from pupils at Halstead Prep School in England, explaining why the highest possible score is 127. Leticia sent in this explanation:
I wrote out out all the numbers from 1 to 8 and then picked 5, 6, 7 and 8. I then arranged them so that I had a 7 and a 5 on the left hand side. Then I put 8 and 6 on the right hand side. I then put the biggest two of the remaining numbers on the left hand side to get 74 and 53. Then I had 81 and 62 on the right hand side and that works. Then I added 74 and 53 and I got 127.
Christina agreed with Leticia's method of considering the largest digits first:
The highest score is 127. You should use 7 on the left and 8 on the right as it uses the less than sign. Then you fill in the units with the leftover lower digits. For the next calculation you would put 5 on the left and 6 on the right for the same reasons. You fill the units with the other numbers. This is how you get the highest score. For the units I put 3 and 4 on the left as they are the highest of the other numbers.
Amelie described a similar method:
The highest number possible is 127. The reason for this is:
You want the highest score on the left. So, use the largest numbers possible. 7 is less than 8 and 5 is less than 6. These are the largest tens digits possible. Next, we move on to the units digits. 1, 2, 3 and 4 remain. It doesn’t matter which numbers you use, as if the tens digit is larger, the number will be larger. To get the highest score possible, we will use 3 and 4. Our result is:
74 < 81
53 < 62
74 + 53 = 127
Our answer is 127.
Well done to all of you for writing out your thinking so clearly. Thank you as well to Lila, Lottie, Ananya, Viva, Charlotte, Evie, Madeline and Emily who all sent in excellent justifications for why 127 is the maximum possible score.
We also received some similar solutions from children at Bishop's Castle Primary School in England. Lewis and Ryan sent in this explanation:
I put the highest number (8) in the top right tens column and the second highest number (7) in the opposite tens column because I knew I needed the largest total on the left hand side. The next highest number (6) I put in the tens column on the right hand side in the row below and the next highest number (5) in the tens on the left hand side. I took a similar approach with the position of the ones. This gives a top score of 127.
Lucy and Fynn had a similar strategy:
The 7 is smaller than 8 therefore you can use them on the same row in the tens. This strategy works for 6 and 5 as well. You can put 4 and 3 next to the 7 and 5 either way round in the units to make the largest possible correct numbers on the left hand side. The same strategy works for 1 and 2 on the right hand side. The highest score with digits 1-8 was 127.
Sid from Twyford School explained how they solved this problem by starting with the highest digits:
If you have the numbers 1,2,3,4,5,6,7 and 8, the way to get the best score would be looking at the highest numbers and going down to the lowest numbers. First you look at the eight, the eight cannot be in the first column in the left section as nothing else is bigger than eight. What you could then do is put 8 in the first column in the right section and seven in the first left section. That then means you now have the numbers 1,2,3,4,5 and 6. Like the 8 you can't put the 6 in the first left section so you would want to put it on the right first section under the 8, then you can have the number 5 in the left first one under the 7, you can then put 4 and 3 in the last places on the left and 2 and 1 on the right then your total would be 127.
Dhruv sent in this picture explaining their method:
Thank you all for sending in these solutions!
Follow the clues to find the mystery number.
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication? |
Chapter 5 Measures of Central Tendency
Which average would be suitable in the following cases?
(i) Average size of readymade garments.
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wages in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii) In case of open-ended frequency distribution.
(i) Mode Average size of any ready made garments should be the size for which demand is the maximum. Hence, the modal value which represents the value with the highest frequency should be taken as the average size to be produced.
(ii) Median It is the value that divides the series into two equal parts. Therefore, Median will be the best measure for calculating the average intelligence of students in a class as it will give the average intelligence such that there are equal number of students above and below this average. It will not be affected by extreme values.
(iii) Arithmetic Mean The average production in a factory per shift is best calculated by Arithmetic Mean as it will capture all types of fluctuations in production during the shifts.
(iv) Arithmetic Mean Arithmetic Mean will be the most suitable measure. It is calculated by dividing the sum of wages of all the workers by the total number of workers in the industrial concern. It gives a fair idea of average wage bill taking into account all the workers.
(v) Arithmetic Mean The algebraic sum of the deviations of values about Arithmetic Mean is zero. Hence, when the sum of absolute deviations from average is the least, then mean could be used to calculate the average.
(vi) Median Median will be the most suitable measure in case the variables are in ratios as it is least affected by the extreme values.
(vii) Median Median is the most suitable measure as it can be easily computed even in case of open ended frequency distribution and will not get affected by extreme values.
Indicate the most appropriate alternative from the multiple choices provided against each question.
(i) The most suitable average for qualitative measurement is
(a) Arithmetic mean
(d) Geometric mean
(e) None of these
(b) Median is the most suitable average for qualitative measurement because Median divides a series in two equal parts thus representing the average qualitative measure without being affected by extreme values.
(ii) Which average is affected most by the presence of extreme items?
(c) Arithmetic Mean
(d) Geometric Mean
(e) Harmonic Mean
(c) It is defined as the sum of the values of all observations divided by the number of observations and therefore it is. affected the most by extreme values.
(iii) The algebraic sum of deviation of a set of n values from AM is
(d) None of these
(b) This is one of the mathematical properties of arithmetic mean that the algebraic sum of deviation of a set of n values from AM is zero.
Comment whether the following statements are true or false.
(i) The sum of deviation of items from median is zero.
(ii) An average alone is not enough to compare series.
(iii) Arithmetic mean is a positional value.
(iv) Upper quartile is the lowest value of top 25% of items.
(v) Median is unduly affected by extreme observations.
This mathematical property applies to the arithmetic mean and not to median.
Average is not enough to compare the series as it does not explain the extent of deviation of different items from the central tendency and the difference in the frequency of values. These are measured by measures of dispersion and kurtosis.
Median is a positional value.
The upper quartile also called the third quartile, has 75 % of the items below it and 25 % of items above it.
Arithmetic mean is unduly affected by extreme observations.
If the arithmetic mean of the data given below is 28, find (a) the missing frequency and (b) the median of the series
(a) Let the missing frequency br f1.
Arithmetic Mean = 28
or 2240 -2100 = 35f1 = 28f1
or 140 = 7f1
f1 = 20
Hence, the missing frequency is 20.
So, the Median class = Size of ()th item = 50th term.
50th item lies in the 57th cumulative frequency and the corresponding class interval is 20-30.
The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
N = 10
Arithmetic Mean = ₹ 240
Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
So, the median class = Size of () th item = 190 item
190th lies in the 129 th cumulative frequency and the corresponding class interval is 200-300.
Median size of land holdings = 241.22 acres
The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers, (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.
(a) Highest income of lowest 50% workers will be given by the median. Σf = N = 65
Median class = Size of ()th item = Size of ()th item=325 th item
32.5th item lies in the 50th cumulative frequency and the corresponding class interval is 24.5 – 29.5.
(b) Minimum income earned by top 25% workers will be given by the lower quartile Q1.
Class interval of Q1 = ()th item
= ()th item = 1625th item
16.25th item lies in the 30th cumulative frequency and the corresponding class interval is 19.5 – 24.5
(c) Maximum income earned by lowest 25% workers will be given by the upper quartile Q3.
Class interval of Q3 = ()th item
= 3()th item
= 3 × 1625th item
= 48.75th item
48.75th item lines in 50th item and the corresponding class interval is 24.5-29.5.
The following table gives production yield in kg per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode production yield. |
Bresenham’s line algorithm is an algorithm that determines the points of an n- dimensional raster that should be selected in order to form a close approximation . example, in which we wish to draw a line from (0,0) to (5,3) in device space. Bresenham’s algorithm begins with the point (0,0) and “illuminates” that pixel. Bresenham’s line drawing algorithm & Mid Point Circle algorithm. Example: 13 )2or(i.e(slope)gradientLet dxdy dx dy 3dy 2dy dy.
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Bresenham’s line algorithm
To derive the alternative method, define the difference to be as follows:. Since we know the column, xthe pixel’s row, yis given algogithm rounding this quantity to the nearest integer:. The label “Bresenham” is used today for a family of algorithms extending or modifying Bresenham’s original algorithm. This decision can be generalized by accumulating the error.
Programs in those days were freely exchanged among corporations so Calcomp Jim Newland and Calvin Hefte had copies. An extension to the original algorithm may be used for drawing circles. This alternative method allows for integer-only arithmetic, which is generally faster than using floating-point arithmetic.
Computer graphics algorithms Digital geometry. The adjacent image bresenhham the blue point 2,2 chosen to be on the line with two candidate points in green 3,2 and 3,3.
It is commonly used to draw line primitives in a bitmap image e. The plotting can be viewed by plotting at the intersection of lines blue circles or filling in pixel boxes yellow squares.
Bresenham’s Line Drawing Algorithm Example
Please help improve this article by adding citations to reliable sources. The beesenham point 3, 2. Regardless, the plotting is the same. In Bresenham wrote: The voxel heightmap software-rendering engines seen in some PC games also used this principle.
Articles needing drawinb references from August All articles needing additional references All articles with unsourced statements Articles with unsourced statements from September Articles with unsourced statements from December All Wikipedia articles needing clarification Wikipedia articles needing clarification from May Commons category link is on Wikidata Articles with example pseudocode.
The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards.
It should be noted that everything about this form involves only integers if x and y are integers since the constants are necessarily integers. In the following pseudocode sample plot x,y plots the pixel centered at coordinates x,y and abs returns absolute value:. A line splits a plane into halves and the half-plane brsenham has a negative f x,y can be called the negative half-plane, and the other half can be called the positive half-plane.
The result of this plot is shown to the right. To answer this, evaluate the line function at the midpoint between these two points:. The Bresenham algorithm can be interpreted as slightly modified digital differential analyzer using 0.
All of the derivation for the algorithm is done. It was a year in which no proceedings were published, only the agenda of speakers and topics in an issue of Communications of the ACM. The general equation of the line through the endpoints is given by:. This page was last edited on 16 Octoberat While algorithms such as Wu’s algorithm wiht also frequently used in modern computer graphics because they can support antialiasingthe speed and simplicity of Bresenham’s line algorithm means that it is still important.
The first step is transforming the equation of a line from the typical slope-intercept form into something different; and then using this new bresenha, for a line to draw a line based on the idea of accumulation of error. A description of the line drawing routine was accepted for presentation at the ACM national convention in Denver, Colorado. If the error becomes greater than 0. Simplifying this expression algorith. Since all of this is about wlgorithm sign of the accumulated difference, then everything can be multiplied by 2 with no consequence.
Alternatively, the difference between points can be used instead of evaluating f x,y at midpoints. Notice that the points 2,1 and 2,3 are on opposite sides of the line and f x,y evaluates to positive or negative.
This exzmple a function of only x and it would be useful to make this equation written as a function of both x and y. It is an incremental error algorithm. In low level implementation which access the video memory directly it would be typical for the special cases of vertical and horizontal lines to be handled separately as they can be highly optimised.
It is one of the earliest algorithms developed in the field of computer graphics. To derive Bresenham’s algorithm, two steps must be taken.
By switching the x and y axis an implementation for positive or negative steep gradients can be written as. It can also be found in many software graphics libraries. The algorithm can breseham extended to cover gradients between 0 and -1 by checking whether y needs to increase or decrease i. |
I am still a bit puzzled by your questions, particularly coming from you. What is wrong with the responses given in a typical introductory logic textbook?
The 'answer' that can be found in a logic textbook is pretty much some vague considerations that seem to stem from the intention theory. But I'm questioning that theory, so this result does clearly not work. This problem is not discussed in any logic textbook that I have read or skimmed through. (I have not read Copi's.)
4. No, unless you mean that part of a lengthy argument (that is really a series of arguments) is inductive, and part is deductive.
Re 3. Not sure about that. Arguments need to be given. I know that this is a common assumption but is it a good assumption?
Re 4. That's not what I mean.
Also, on further reflection. Skip the second part of question 2 as it is not very relevant here. The discussion about cogent and strong inductive arguments is not what I intended
this thread to be about.
When I write "valid" I always mean "deductively valid". I don't think there is a term that is called inductive validity.
A deductive argument is an argument where the conclusion follows necessarily from the premises.
A valid deductive argument is a deductive argument where the conclusion follows necessarily from the premises.
A non-valid deductive argument is a deductive argument where the conclusion does not follow necessarily from the premises.
First, your definition of deductive argument is identical
with your definition of valid deductive argument. According to you, then, we have no need to speak of valid deductive arguments because all deductive arguments are valid.
Second, you are contradicting yourself. Look at your first and third statement. An invalid deductive argument is according to them both one where the conclusion necessarily follows from the premises and one where it does not. That is impossible.
Also, I dislike that definition of validity very much as it is dangerously ambiguous and causes people to make modal fallacies. There are many logically equivalent definitions of validity. In this thread I stipulate that we use this one:
[INDENT]An argument is valid iff the corresponding conditional is a necessary true proposition.
An inductive argument is an argument where the premises offer a degree of support for the conclusion, but do not necessarily entail it; there is no guarentee (like your article noted).
Careful. Do you accept Ken's view (=intention theory)? If so, then some inductive arguments are valid, because some arguments are intended by the speaker to be inductive but are actually valid.
I think it depends how we use the word "argument". But, I think if we use the word formally, the answer is yes.
This needs some defending.
Do you see how many ways this question can be interpreted?
1.) Yes, some arguments are deductive, some arguments are inductive.
2.) Yes, some arguments (like Pyrrho noted), can have inductive and deductive parts.
3.) No, if you mean that only some arguments are inductive or deductive. Because all formal arguments are deductive or inductive (and you might mean this since this question comes after the one which specifies "all arguments").
Yes, but there is only one good interpretation, and it is straightforward. I even removed extra words that made it even easier to get it right. I suppose I should have left them in. You can insert the word "both" into the question if it is unclear to you, like this:
4. Are some arguments both
deductive and inductive?
In the formal english language, E:
4F. Does there exist an argument such that it is deductive and it is inductive?
(I invented some 'formal machinery' (Ken's phrase) for questions to be formalized. In this the question has the form (∃x)(Dx∧Ix)? Let's not discuss this invention in this thread but some other time. The interested reader can look here
That is not difficult. Deductive arguments are intended to be conclusive arguments by the arguer. If the argument is not conclusive, but is intended to be conclusive, it is a failed (invalid) deductive argument. On the other hand, if an argument is not intended to be a conclusive argument by the arguer, it is non-deductive. But if the premises fail to support the conclusion of the non-deductive argument, then it is a failed (weak) non-deductive argument.
However, whatever the arguer intends, deductive or non-deductive, there are some arguments which it would hardly make sense to intend it as deductive, since it is so clearly non-conclusive; or make sense to intend it as non-deductive since it is so clearly conclusive. It would be a good rule (I think) to count a valid deductive argument as deductive, and a strong non-deductive argument as non-deductive.
This is a good answer from someone defending the standard intention theory. I used to agree with this now I have doubts.
Why do you think it would be a good rule to count a valid argument as deductive even though it is according to that theory inductive?
I assume this is what you meant and you were just being careless when you inserted the word "deductive" there. Otherwise you were merely suggesting that we count valid deductive arguments as deductive. That is not very interesting.
But can't we distinguish between inductive and deductive arguments without considering the intent of the arguer at all?
Not according to the intention theory. This implies that the difference between a deductive argument and an inductive argument is merely
a psychological one and not a logical one. Some people consider this implausible, me included. See the other threads.
I don't see how, although, as I said, there are pretty clear cases of both kind when it would be implausible for the arguer's intention not to be the one or the other. In those cases we have what, in legal jargon, we might call, "constructive intent". That it was the arguer's intent whether or not it actually was (or the arguer was confused).
I agree with this. But even though it is implausible that the arguer intended a given argument to be inductive, it does not follow that it is deductive. Some arguers are terribly confused.
But we can distinguish an inductive argument from a deductive argument by looking looking at the argument. If all the premises can be true without the conclusion being true, it is an inductive argument:
1.) Socrates was Greek
2.) Most Greeks eat fish
3.) Socrates ate fish
This is an inductive argument. And we know it's not a deductive argument because although the premises may be true, the conclusion does not follow necessarily from the premises; the conclusion may not be true.
Isn't that right?
Now you have defined inductive argument as invalid argument. That's another theory and it is inconsistent with the intention theory. Let's call this theory for the validity theory, for it defines deductive argument as valid argument, and inductive argument as invalid argument.
This theory has the curious and implausible implication that all deductive arguments are valid, indeed, they could not be invalid. Thus, one cannot fail to make a valid deductive argument, it is impossible. This is the position that Kritikos was defending and to which Ken gave the plausible example arithmetic analogy. See the opening post.
I would rather say, we do not know what the argument is without knowing the intentions of the arguer. But once we know what the argument is, we then may be able to determine whether it is deductively valid or not, and whether it is inductively valid or not.
Perhaps, though, this is a mere verbal distinction, without any importance at all.
Not at all! This is the crucial point. In this post you are endorsing the intention theory. That's fine, but the theory has its problems some of which I have mentioned already.
I thought what you were meaning to ask for is an analysis of deductive arguments and/or an analysis of inductive arguments. You already know the difference between the two, so you do not need an explanation of what each is. Instead, what you want is an analysis, for merely knowing the difference between the two doesn't therefore imply that you can always be given an argument and definitively determine whether or not the logical argument is a deductive argument or inductive argument, for sometimes, being privy to the argument is insufficient information to determine whether or not an argument is deductive or inductive.
That of course has no bearing on whether or the argument is deductive or inductive, just as truth doesn't depend on knowledge of the truth.
I don't think you should ever regard inductive arguments as valid or invalid.
You are getting the point. I'm not sure that I know the difference between them. I can make the distinction in practice like any person trained in logic can, but that does not imply that I know the difference, does it?
But I am definitely asking for an analysis.
" I don't think you should ever regard inductive arguments as valid or invalid.
" Why do you think this? Given pretty much any definition that you choose of validity, it is applicable to inductive arguments.
According to the validity theory, all deductive arguments are valid and all inductive arguments are invalid.
According to the intention theory some deductive arguments are valid and some are invalid, and some inductive arguments are valid and some are invalid.
I got soundness and validity confused. Validity speaks nothing of truth, only form. To be valid means that the conclusion follows from the premises. An argument being valid does not mean that it is true. Validity is a necessary but not sufficient condition for soundness. It is not a sufficient condition because an argument not only needs to be valid to be sound, but it also needs to be true.
Is this right?
It is nonsense to speak of true/false arguments.
Validity has something to do with form, but not all valid arguments have a valid form. But this is a discussion I would rather not elaborate on now. You can read more about it in Possible Worlds
where it is discussed at length.
No arguments are either true or false. Are you, perhaps asking whether a valid argument must have a true conclusion. The answer is, no. The same for whether a valid argument must have true premises. But, what is true is that any valid argument with true premises must have a true conclusion.
It is also nonsense to say that no arguments are either true or false. The correct wording is:
Ah, arguments cannot be said to be true or false, just valid or sound, right? Premises and conclusions are what we apply the properties true and false to.
But this is also a side discussion about meaning and category errors. Let's not discuss that now.
Right, because premises and conclusions are propositions (statements) and only propositions (statements) are true or false.
Not sure about that. Maybe we should not assume a propositional theory of truth bearers in this thread. Or, better yet, let's assume it so far (pretty much everyone in this thread holds that theory anyway), and maybe after we have considered the problems of deductive and inductive arguments in that light (so to speak), we could consider them in the light of say a sentence theory of truth bearers.
At least, let's not discuss theories of truth bearers in this thread.
A necessary condition for a sound argument is not that the conclusion be true, although, if an argument is sound, then the conclusion will be true. Truth, then, is a consequent of soundness.
You got yourself confused. It is a necessary condition for soundness. But it is not a sufficient. For this thread let's define soundness like this:
[INDENT]An argument is sound iff:
1. All the premises and the conclusion are true.
2. The argument is valid.
All cogent arguments are strong arguments, but not all strong arguments are cogent arguments, for all cogent arguments are strong arguments with true premises, and not all strong arguments have true premises.
This strongness you speak of is not a standard term as far as I know.
That is not what it says at Wikipedia:
Cogency - Wikipedia, the free encyclopedia
You may use the term differently, but this only reinforces my point that the terms used to describe various inductive arguments are not very standardized.
I edited the Wikipedia page to its current form (I think). I did it because it was terribly confused before. I also inserted the reference to Fallacyfiles. But bear in mind what Ken says:
Actually, some logic books use the term "cogent" to mean, "known to be true", and not just true. So, a cogent argument would be one where the premises are not only true (and argument valid) but are known to be true, so the conclusion is known to be true. But the books differ on this.
Correct. I seem to recall writing this on Wikipedia but they may have changed it. Wikipedia is not a good source for such specific information as this.
But let's not derail the thread with more discussions of cogentness and strongness and what have we, that is, terms related to inductive arguments.
What is a valid inductive argument anyway?
See the definition of validity above. (And please for the love of god (you do believe in god, right?) stop changing the fonts!)
I dont know if you guys have answered the Op or not but I'll throw out some ideas on the subject.
Deductive arguments have true premises and a conclusion that necessarily follows. While inductive arguments have conclusions that are probable, but not necessary. So me thinks inductive arguments hinge on logical possibility while deductive arguments dont. So its impossible to have a counter example to a deductive argument. Does that work emil?
Your first two claims are wrong. I don't know about the rest as they are too vague to consider true or false. |
Current time:0:00Total duration:13:03
Let's now talk about what is easily one of the most famous theorems in all of mathematics. And that's the Pythagorean theorem. And it deals with right triangles. So a right triangle is a triangle that has a 90 degree angle in it. So the way I drew it right here, this is our 90 degree angle. If you've never seen a 90 degree angle before, the way to think about it is, if this side goes straight left to right, this side goes straight up and down. These sides are perpendicular, or the angle between them is 90 degrees, or it is a right angle. And the Pythagorean theorem tells us that if we're dealing with a right triangle-- let me write that down-- if we're dealing with a right triangle-- not a wrong triangle-- if we're dealing with a right triangle, which is a triangle that has a right angle, or a 90 degree angle in it, then the relationship between their sides is this. So this side is a, this side is b, and this side is c. And remember, the c that we're dealing with right here is the side opposite the 90 degree angle. It's important to keep track of which side is which. The Pythagorean theorem tells us that if and only if this is a right triangle, then a squared plus b squared is going to be equal to c squared. And we can use this information. If we know two of these, we can then use this theorem, this formula to solve for the third. And I'll give you one more piece of terminology here. This long side, the side that is the longest side of our right triangle, the side that is opposite of our right angle, this right here-- it's c in this example-- this is called a hypotenuse. A very fancy word for a very simple idea. The longest side of a right triangle, the side that is opposite the 90 degree angle, is called the hypotentuse. Now that we know the Pythagorean theorem, let's actually use it. Because it's one thing to know something, but it's a lot more fun to use it. So let's say I have the following right triangle. Let me draw it a little bit neater than that. It's a right triangle. This side over here has length 9. This side over here has length 7. And my question is, what is this side over here? Maybe we can call that-- we'll call that c. Well, c, in this case, once again, it is the hypotenuse. It is the longest side. So we know that the sum of the squares of the other side is going to be equal to c squared. So by the Pythagorean theorem, 9 squared plus 7 squared is going to be equal to c squared. 9 squared is 81, plus 7 squared is 49. 80 plus 40 is 120. Then we're going to have the 1 plus the 9, that's another 10, so this is going to be equal to 130. So let me write it this way. The left-hand side is going to be equal to 130, and that is equal to c squared. So what's c going to be equal to? Let me rewrite it over here. c squared is equal to 130, or we could say that c is equal to the square root of 130. And notice, I'm only taking the principal root here, because c has to be positive. We're dealing with a distance, so we can't take the negative square root. So we'll only take the principal square root right here. And if we want to simplify this a little bit, we know how to simplify our radicals. 130 is 2 times 65, which is 5 times 13. Well, these are all prime numbers, so that's about as simple as I can get. c is equal to the square root of 130. Let's do another one of these. Maybe I want to keep this Pythagorean theorem right there, just so we always remember what we're referring to. So let's say I have a triangle that looks like this. Let's see. Let's say it looks like that. And this is the right angle, up here. Let's say that this side, I'm going to call it a. The side, it's going to have length 21. And this side right here is going to be of length 35. So your instinct to solve for a, might say, hey, 21 squared plus 35 squared is going to be equal to a squared. But notice, in this situation, 35 is a hypotenuse. 35 is our c. It's the longest side of our right triangle. So what the Pythagorean theorem tells us is that a squared plus the other non-longest side-- the other non-hypotenuse squared-- so a squared plus 21 squared is going to be equal to 35 squared. You always have to remember, the c squared right here, the c that we're talking about, is always going to be the longest side of your right triangle. The side that is opposite of our right angle. This is the side that's opposite of the right angle. So a squared plus 21 squared is equal to 35 squared. And what do we have here? So 21 squared-- I'm tempted to use a calculator, but I won't. So 21 times 21: 1 times 21 is 21, 2 times 21 is 42. It is 441. 35 squared. Once again, I'm tempted to use a calculator, but I won't. 35 times 35: 5 times 5 is 25. Carry the 2. 5 times 3 is 15, plus 2 is 17. Put a 0 here, get rid of that thing. 3 times 5 is 15. 3 times 3 is 9, plus 1 is 10. So it is 11-- let me do it in order-- 5 plus 0 is 5, 7 plus 5 is 12, 1 plus 1 is 2, bring down the 1. 1225. So this tells us that a squared plus 441 is going to be equal to 35 squared, which is 1225. Now, we could subtract 441 from both sides of this equation. The left-hand side just becomes a squared. The right-hand side, what do we get? We get 5 minus 1 is 4. We want to-- let me write this a little bit neater here. Minus 441. So the left-hand side, once again, they cancel out. a squared is equal to-- and then on the right-hand side, what do we have to do? That's larger than that, but 2 is not larger than 4, so we're going to have to borrow. So that becomes a 12, or regrouped, depending on how you want to view it. That becomes a 1. 1 is not greater than 4, so we're going to have to borrow again. Get rid of that. And then this becomes an 11. 5 minus 1 is 4. 12 minus 4 is 8. 11 minus 4 is 7. So a squared is equal to 784. And we could write, then, that a is equal to the square root of 784. And once again, I'm very tempted to use a calculator, but let's, well, let's not. Let's not use it. So this is 2 times, what? 392. And then this-- 390 times 2 is 78, yeah. And then this is 2 times, what? This is 2 times 196. That's right. 190 times 2 is-- yeah, that's 2 times 196. 196 is 2 times-- I want to make sure I don't make a careless mistake. 196 is 2 times 98. Let's keep going down here. 98 is 2 times 49. And, of course, we know what that is. So notice, we have 2 times 2, times 2, times 2. So this is 2 to the fourth power. So it's 16 times 49. So a is equal to the square root of 16 times 49. I picked those numbers because they're both perfect squares. So this is equal to the square root of 16 is 4, times the square root of 49 is 7. It's equal to 28. So this side right here is going to be equal to 28, by the Pythagorean theorem. Let's do one more of these. Can never get enough practice. So let's say I have another triangle. I'll draw this one big. There you go. That's my triangle. That is the right angle. This side is 24. This side is 12. We'll call this side right here b. Now, once again, always identify the hypotenuse. That's the longest side, the side opposite the 90 degree angle. You might say, hey, I don't know that's the longest side. I don't know what b is yet. How do I know this is longest? And there, in that situation, you say, well, it's the side opposite the 90 degree angle. So if that's the hypotenuse, then this squared plus that squared is going to be equal to 24 squared. So the Pythagorean theorem-- b squared plus 12 squared is equal to 24 squared. Or we could subtract 12 squared from both sides. We say, b squared is equal to 24 squared minus 12 squared, which we know is 144, and that b is equal to the square root of 24 squared minus 12 squared. Now I'm tempted to use a calculator, and I'll give into the temptation. So let's do it. The last one was so painful, I'm still recovering. So 24 squared minus 12 squared is equal to 24.78. So this actually turns into-- let me do it without a-- well, I'll do it halfway. 24 squared minus 12 squared is equal to 432. So b is equal to the square root of 432. And let's factor this again. We saw what the answer is, but maybe we can write it in kind of a simplified radical form. So this is 2 times 216. 216, I believe, is a-- let me see. I believe that's a perfect square. So let me take the square root of 216. Nope, not a perfect square. So 216, let's just keep going. 216 is 2 times 108. 108 is, we could say, 4 times what? 25 plus another 2-- 4 times 27, which is 9 times 3. So what do we have here? We have 2 times 2, times 4, so this right here is a 16. 16 times 9 times 3. Is that right? I'm using a different calculator. 16 times 9 times 3 is equal to 432. So this is going to be equal to-- b is equal to the square root of 16 times 9, times 3, which is equal to the square root of 16, which is 4 times the square root of 9, which is 3, times the square root of 3, which is equal to 12 roots of 3. So b is 12 times the square root of 3. Hopefully you found that useful. |
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List of Speed And Velocity Worksheet Answers
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Note follow the calculation procedures for adding vectors in straight line distance units are metric unit of distance without a direction km, mm since distance is scalar, add all the magnitudes only to get the result speed distance time change in displacement ( ) change in time velocity v d t vector resultant distance.
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Answers are found at the end of this document. physical science motion the relationships between displacement, time, velocity and acceleration displacement, time and velocity plo.Speed, velocity, and acceleration problems use your own paper, and show all work.
show the formula used, the setup, and the answer with the correct units. is driving down street. he drives meters in seconds. assuming he does not speed up or slow down, what is his speed in meters per second.Speed, velocity and acceleration in region a, the car has constant acceleration the line has a constant positive gradient.
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Combine this worksheet with my mo worksheet on graphs to compare graphs Graphing speed and velocity data worksheet. name speed data. a jogger has decided to jog as a and has decided to vary her speed from minute to minute. below is the data of her jogging times and distances.
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With. units. no units no credit. distance time sec speed. distance time speed ms. distance time sec speed ms. what is the speed of a Graphing velocity and acceleration data worksheet. graphing speed and velocity data worksheet. name speed data. a jogger has decided to jog as a and has decided to vary her speed from minute to minute.
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Worksheet. in the following assignments you will learn to calculate speed and convert from one speed to another. often you calculate speed in kilometres per hour (). to calculate speed, you need to know the distance and the time. when you know these two values, you can calculate the speed using this formula.
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Distance for last leg of trip km, time for last leg. hours, need a speed of radius km, so circumference km., so. ms. after seconds, the dog is meters away.Dec, speed and velocity practice problems worksheet answers. it walks to x it then walks a displacement of lastly it walks to the location x.
show the formula used the setup and the answer with the correct units. calculate the speed Answer. km. an auto travels at a rate of for minutes, then at for minutes, and finally at for minutes. find the total distance covered in km and the average speed for the complete trip in ms. |
20 min to hr conversion (mtoh). swap units ↺ … 0 6 5. The final result is: 3 hr → 180 min. 0. Enter hours, minutes and seconds to convert from time format of hh:mm:ss, hours:minutes:seconds, to 0. We conclude that 3 hours is equivalent to 180 minutes: 3 hours = 180 minutes. In the modern metric system, hours are an accepted unit of time equal to 3,600 seconds but an hour of Coordinated Universal Time (UTC) may incorporate a positive or negative leap second, making it last 3,599 or 3,601 seconds, in order to keep it within 0.9 seconds of universal time, which is based on measurements of the mean solar day at 0° longitude. What is 20 minutes in hours? is a unit of time conventionally reckoned as 1⁄24 of a day and scientifically reckoned as 3,599–3,601 seconds, depending on conditions. 0 2 2. How long is 0.3 hours? 0.25 + 0.01 = 0.26). Decimal Hours-to-Minutes Conversion Chart. Unless your minutes converted perfectly to hours, you'll have … 3 hr → T (min) Solve the above proportion to obtain the time T in minutes: T (min) = 3 hr × 60 min. Pagkakaiba ng pagsulat ng ulat at sulating pananaliksik? 0 5 4. If you are 13 years old when were you born? The minor variations of this unit were eventually smoothed by making it 1⁄24 of the mean solar day, based on the measure of the sun's transit along the celestial equator rather than along the ecliptic. Does Jerry Seinfeld have Parkinson's disease? They are divided into seconds and multiplied into hours. This is an online converter for hours – minutes that will help you calculate different quantities of time measurements. What is 20 minutes in hours? For the default time of 2:45:45 in the converter, let's convert to just hours, then just minutes, and then just seconds. It was subsequently divided into 60 minutes, each of 60 seconds. This calculator will round to 6 decimal places at most. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. For example, 2 hrs = 120 mins and 7 hrs = 420 mins. How long will the footprints on the moon last? Supose you want convert 1 hour, 15 minutes and 36 seconds to decimal. 3 hours and 20 minutes = 200 minutes 200 minutes * 60 seconds/minute = 12 000 seconds. What is traceable only is that it started being recorded in the Middle Ages due to the ability of construction of "precision" timepieces (mechanical and water clocks). This was finally abandoned due to the minor slowing caused by the Earth's tidal deceleration by the Moon. Ano ang pinakamaliit na kontinente sa mundo? 15/60 = 0.25). What is 3 hours in minutes? Divide the value in minutes by 60 (i.e. Minutes . So, we can write the formula to convert hours, minutes and seconds to decimal as: Now try to use our calculator at the top of this page and you'll find that it's very easy accomplish this task. All Rights Reserved. Above you will find our hours – minutes conversion to help you in changing units. 36/3600 = 0.01). Cite this content, page or calculator as: Furey, Edward "Time to Decimal Calculator"; CalculatorSoup, Seconds . Multiply the amount of hours by the conversion factor to get the result in minutes: The conversion factor from hours to minutes is 60, which means that 1 hours is equal to 60 minutes: To convert 3 hours into minutes we have to multiply 3 by the conversion factor in order to get the amount from hours to minutes. See the answer to this question using our calculator below: Supose you want convert 1 hour, 15 minutes and 36 seconds to decimal. [number of] hours x 60 = [number of] minutes. This is how they are defined: An hour (symbol: h; also abbreviated hr.) Such hours varied by season, latitude, and weather. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. It can be particularly useful when converting movie running times that are usually shown in minutes, or when baking chocolate desserts. From. This is an online converter for hours – minutes that will help you calculate different quantities of time measurements. 36/3600 = 0.01). There are 0.0166666667 hours in a minute. This simple calculator will allow you to easily convert 0.3 hr to min. Another way is saying that 3 hours is equal to 1 ÷ 0.0055555555555556 minutes. Sum the two values above to get the fractional part (i.e. 0 4 3. 5′, for angle measurement. 0. We can also form a simple proportion to calculate the result: Solve the above proportion to obtain the time T in minutes: We conclude that 3 hours is equivalent to 180 minutes: We can also convert by utilizing the inverse value of the conversion factor. The prime is also sometimes used informally to denote minutes of time. There are 12,000 seconds in 3 hours and 20 minutes. For example 15 minutes (¼ hour) equals 0.25, 30 minutes (½ hour) equals 0.5, etc. © 2006 -2020CalculatorSoup® View our other time converters. To convert time to just hours: 2 hours is 2 hours * (1 hour/ 1 hour) = 2 hours; 45 minutes is 45 minutes * (1 hour / 60 minutes) = 45/60 hours = 0.75 hours 0 8 6.1 .1 0: 7.1 .12 Conversor de Minutes to Hours . This is how they are defined: An hour (symbol: h; also abbreviated hr.) 5′, for angle measurement. We will need to convert each part into the unit of time we want. We can also form a proportion to calculate the result: Solve the above proportion to obtain the time T in minutes: We conclude that 3 hours is equivalent to 180 minutes: For practical purposes we can round our final result to an approximate numerical value. To change a certain number of hours into minutes, simply enter above in the box how many hours you want to calculate and click ¨convert¨. https://www.calculatorsoup.com - Online Calculators. It was subsequently divided into 60 minutes, each of 60 seconds. Our formula below is a useful tool if you wish to do this time conversion by hand or if you need to know the mathematical calculation for an exam. What is 20 minutes when converted to hour in decimal format? As a unit of time, the minute (symbol: min) is equal to 1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. Multiply the decimal or fraction by 60. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. You might want to work in the other direction, convert minutes to hours. Allows you to countdown time from 3 hour 20 min to zero. To convert this value to decimal hours follow the steps below: In short: Decimal hours = whole number of hours, plus minutes divided by 60, plus seconds divided by 3600. 18 Minutes: 0.3 Hours: 19 Minutes: 0.316667 Hours: 20 Minutes: 0.333333 Hours: 21 … What is 0.3 hours in minutes? Both these units of time are used in our everyday language for many things, so try our hours to minutes calculator above for your conversions. For practical purposes we can round our final result to an approximate numerical value. In this case 1 minute is equal to 0.0055555555555556 × 3 hours. 1 hr → 60 min. Easy to adjust, pause, restart or reset. We can say that three hours is approximately one hundred eighty minutes: An alternative is also that one minute is approximately zero point zero zero six times three hours. 20 min to hr conversion (mtoh). The final result is: 3 hr → 180 min. What is traceable only is that it started being recorded in the Middle Ages due to the ability of construction of "precision" timepieces (mechanical and water clocks). However, no consistent records of the origin for the division as 1⁄60 part of the hour (and the second 1⁄60 of the minute) have ever been found, despite many speculations. And we know that 1 hour = 60 minutes and 1 minute = 60 seconds so 60 minutes/ hour * 60 seconds/ minute = 3600 seconds/ hour or 1 hour = 3600 seconds. This was finally abandoned due to the minor slowing caused by the Earth's tidal deceleration by the Moon. The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime. While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions, or for the results obtained from the use of this information. Sometimes calculations are too hard to do in your head. The prime is also sometimes used informally to denote minutes of time. All information in this site is provided “as is”, with no guarantee of completeness, accuracy, timeliness or of the results obtained from the use of this information. However, no consistent records of the origin for the division as 1⁄60 part of the hour (and the second 1⁄60 of the minute) have ever been found, despite many speculations. Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. and your pace will appear in the Minutes/Mile field. Minutes Tenths of an Hour Hundredths of an Hour; 1. 15/60 = 0.25). formula to convert hours, minutes and seconds to decimal, Volume to (Weight) Mass Converter for Recipes, Weight (Mass) to Volume to Converter for Recipes. The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime. Converting from hours to minutes? Hours are divided into minutes and multiplied into days. 1.3 hours in other units Use this hours – minutes time converter to help with conversions. How to Calculate Minutes to Hours 1 minute = 6 × 10 1 seconds 1 hour = 3.6 × 10 3 seconds 1 minute = (6 / 3.6) × 10 1 × 10-3 hours 1 minute = (1.6666666666667) × 10 1-3 hours 1 minute = (1.6666666666667) × 10-2 hours 1 minute = 1.6666666666667 × 0.01 hours 1 minute = 0.0166666667 hours How Many Hours in a Minute? 3 hour 20 minute equal 12000000 Milliseconds. From. Then you can put any number above the answer to make it a Its East Asian equivalent was the shi, which was 1⁄12 of the apparent solar day; a similar system was eventually developed in Europe which measured its equal or equinoctial hour as 1⁄24 of such days measured from noon to noon. T (min) = 180 min. Ano ang mga kasabihan sa sa aking kababata? 1 hr → 60 min. 0. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). What is the hink-pink for blue green moray?
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Urgent Y7 maths homework help!!!!(23 Posts)
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DD's homework is to find the area of a trapezium with the top measuring 8cm and the bottom 14cm. That's it; no further measurements.
Is it possible to work this out without the height, and if so, how? (I am about to embark on an OU basic maths course, so I may be able to actually help her in a few months' time!)
here but you need the height of the trapezium too?
if its symmetrical, then you can work out the length of the base of the triangle - 3cm
so chances are its a 3,4,5
so the height is 4
so the area is 3 x 4 + 8 x 4 = 44cm sq
Exactly lal123 thanks anyway.
Talkinpeace, I'll go with that; not sure how dd will feel about "making up" the height but at least it'll show she knows how to do it.
Not making it up.
3:4:5 triangles are the classic form.
You know its a right angle triangle.
You know one of the sides by the right angle is length 3
therefore the other two sides HAVE to be 4 (the vertical) and 5 (the hypotenuse)
PS checked it with DH and he came up with the same number I did...
I doubt they would expect a Y7 to know about Pythagoras, more likely a mistake in the question.
it doesn't have to be a 3 4 5 triangle. Sounds like a mistake to me.
DD did 3 4 5 triangles in extended maths in year 6
they then did bits of it in geometry at the start of year 7
they are SO elegant that they are a hook to get the kids interested in mathematics as against numeracy
DD has done hypoteneuse, though I don't know about the classic form of a triangle. I'm sorry Talkinpeace, I wasn't being rude just thinking that that was what dd would think.
Out of interest, why does the vertical have to be 4? Couldn't it be 12? or any other length?
I thought dd must have forgotten to write down the height, or that her teacher had made a mistake (though that was less likely). Having read about the classic triangle now, I suspect her teacher expected the kids to be able to make the assumption.
Anyway, dd accepted that she should assume the height as 4 and has done her calculation (correctly as it happens; she's actually quite good at maths though she thinks she isn't).
Thank you all. (I shall come back no doubt, again and again and again!)
If a triangle has a base of 3 and is a right-angled triangle, the other sides could be e.g. 4.5 and 5.4 (rounded) or lots of other numbers. They don't have to be 4 and 5.
they could indeed : a sqd + b sqd = c sqd
but the fact that the difference between the two lengths is 6 making the triangle base 3 and it being a year 7 question hints at the most elegant answer
happy to be proven wrong when OP's DD checks at school ;-)
It being a Y7 question hints at there being an error in the question. Teaching the formula for the area of a trapezium is tricky enough.
And I hope no maths teacher would set a question which required students to guess at a length because it is impossible to work out. An answer being elegant does not make it correct.
I think there's a mistake in the question too - I think best bet is to put in his answer that it is impossible to tell without the height and NOT to make up an answer
Ah, lal123, you could be right there! I wish I'd checked this thread before I went to bed, but I didn't. I'm sure dd would have preferred not to have done any calculations at all and just put in a note telling her teacher it was impossible to do! I said that if she assumed the height was 4 then she could at least show that she knew how to calculate the area, which seems like a fairly sensible way to go.
It had to be done for today, but she - apparently - doesn't actually have a maths lesson today I have no idea when she's going to get it back. I'm intrigued now.
Anyway, I think the most likely situation is that she simply didn't copy the question down properly
I certainly hope the teacher wasn't expecting them to assume/guess the height - very bad form <maths student here> I would complain if so.
In your DD's situation I would have done the calculations but using h as the unknown height so the area would be
8h (rectangle) + 3h (two triangles) = 11h
Oh bother, you're quite right. Aaaaaaargh. Why oh why didn't I think of algebra? Bother bother bother.
Dont beat yourself up jux - its not actually your homework, it's your DD's
I know, but in the light of the maths course I'm starting in Feb, I feel that I would do better going back to primary!
(I asked her, when she got home today, if she would like to add an alternative solution using h, but she wasn't interested. Surprise!)
h is elegant too!
Please tell us what the actual answer is when the work gets marked.
OUTRAGE! SHEER OUTRAGE!!!!!
No one got that question right as the teacher had forgotten to tell them that the height was - wait for it - 6 !!!!!!!
I am also happy to announce that no one in the class thought of using h either.
delighted to hear the answer
SUCH a shame that the teacher canot be penalised for setting dud homework.
Time to write 66 in big letters on some sheets of paper for him!!!
That would be fun wouldn't it? As it is, dd has let it be known that she is not going to let him forget it in a hurry. I am so tempted to suggest that she make a large sign with 666 on it to hold up in class, or to bribe the whole class to keep whispering 6 when his back's turned. No, I must leave the poor man alone and not encourage evilness in my thoroughly pure and innocent daughter, who has never had a bad thought in her head.
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A method to estimate ground reaction forces (GRFs) in a robot/prosthesis system is presented. The system includes a robot that emulates human hip and thigh motion, along with a powered (active) transfemoral prosthetic leg. We design a continuous-time extended Kalman filter (EKF) and a continuous-time unscented Kalman filter (UKF) to estimate not only the states of the robot/prosthesis system but also the GRFs that act on the foot. It is proven using stochastic Lyapunov functions that the estimation error of the EKF is exponentially bounded if the initial estimation errors and the disturbances are sufficiently small. The performance of the estimators in normal walk, fast walk, and slow walk is studied, when we use four sensors (hip displacement, thigh, knee, and ankle angles), three sensors (thigh, knee, and ankle angles), and two sensors (knee and ankle angles). Simulation results show that when using four sensors, the average root-mean-square (RMS) estimation error of the EKF is 0.0020 rad for the joint angles and 11.85 N for the GRFs. The respective numbers for the UKF are 0.0016 rad and 7.98 N, which are 20% and 33% lower than those of the EKF.
Recent advances in microelectronics and robotic technologies have enabled the development of powered prosthetic legs that can help amputees walk up stairs and slopes because of the legs' net power contribution to gait; they are also able to adapt their behavior to various environmental conditions. The company Össur has a lower limb prosthesis called the Power Knee, which is the first commercially available knee to generate power during the gait cycle . Recent developments in powered knee and ankle prostheses are reported in Refs. [3–8].
Bulky load cells are often employed in robots and prosthetic legs to capture external forces (GRFs) and moments during walking . These data are used as feedback measurements to control the robot or prosthesis with force or impedance controls. Even when not used for feedback, force sensing is important for monitoring and evaluation of prosthesis performance and safety. However, there are several drawbacks to the use of load cells: (i) load cells are expensive; (ii) a 250-lbf load cell weighs about 1 lb, with a length of about 3 in, and thus does not easily fit in a prosthetic leg; (iii) load cell measurements tend to drift, need to be frequently offset, and are noisy and need significant signal conditioning; (iv) load cells can get damaged easily from overloading or off-axis loading; and (v) load cells consume electrical power, which is an important consideration in prosthetics.
The above-mentioned problems do not arise with angle sensors because high-resolution encoders are accurate, reliable, and inexpensive. Nevertheless, they do not measure velocity, and velocity calculated by numerical differentiation is challenging because of the difficult compromise between noise rejection and bandwidth.
There have been several methods to reduce the number of force sensors in robotics. In Ref. , a robot compliance controller based on a disturbance observer is presented, where the disturbance observer is used to estimate the external reaction forces. In Ref. , a method for force estimation of the end-effector of a Selective Compliance Assembly Robot Arm (SCARA) robot is presented, where servomotor currents and position information are used to estimate forces. In Ref. , the external force on the end-effector of a four degrees-of-freedom (DOF) robot manipulator is estimated with a combination of time delay estimation and input estimation. In Ref. , a model-based observer is used to estimate the external forces acting on a rigid body. All the aforementioned techniques depend on either the accuracy of the robot model or servomotor current data. Thus, if the accuracy of the robot model or motor current measurement degrades, force estimation can deviate from true force. In Ref. , the force estimation does not require a precise robot dynamic model, but the accuracy of estimated force may degrade in the presence of noise, since the external force is considered as an unknown input.
In this paper, we treat GRF as an unknown input with known dynamic properties and known bounds, and we use an EKF to estimate GRFs along with the states of the robot/prosthesis system. Although EKF is efficient in many applications, it has two important potential drawbacks. First, the derivation of the Jacobian matrix for linearization can be complex and can cause numerical implementation difficulties. Second, linearization can lead to cumulative errors which may affect the accuracy of the state estimation. To overcome these limitations, other nonlinear estimators could be used, such as the UKF or particle filter, where the state estimations are obtained without the need for derivatives and Jacobian calculations [14–16].
In this research, a powered (active) prosthetic leg is considered for transfemoral amputees. The prosthetic leg attached to a robotic hip/thigh emulator. The combined system includes four degrees-of-freedom: vertical hip displacement, thigh angle, knee angle, and ankle angle. This paper is an extended version of Ref. , where we designed an EKF for the online estimation of joint coordinates, velocities, and GRFs.
In this paper, we compare EKF and UKF performance in our 4DOF robot/prosthesis system. We study the EKF because it is the most commonly used nonlinear state estimator due to its versatility and simplicity of implementation. We study the UKF because it generally provides better performance than the EKF. One of the questions we study in this paper is whether the improved performance of the UKF relative to the EKF is worth the increased computational effort.
We also study the effect of various sensor sets on estimation error at different walking speeds. We start with the four main states of the robot/prosthesis system, which are hip displacement, thigh, knee, and ankle angles, as system observations to get the baseline estimation accuracy. Then, we investigate how much estimation accuracy degrades if fewer measurements are used, and we find a tradeoff between the estimation error accuracy and the number of measurements. In general, we prefer to use fewer measurements because fewer sensors mean a simpler system, less complexity, and lower cost.
Finally, we consider the stability of the EKF and boundedness of the estimation error, which are very important for prosthesis control. The stability of the EKF under various values of initial estimation error and noise covariances is explored. We analytically show that the estimation error remains bounded under certain conditions in our 4DOF robot/prosthesis system and confirm the analysis with simulation results.
This research involves the mechanical integration of a prosthesis and a test robot so that we can test the prosthesis without human trials. We will eventually need to use the Kalman filter on a prosthesis that is attached to an amputee. We develop the Kalman filter for the robot/prosthesis system in this paper for later implementation on an amputee/prosthesis system.
The paper is organized as follows: In Sec. 2, the model of the robotic system and prosthetic leg is presented. In Sec. 3, the EKF for state estimation and GRF estimation is discussed. In Sec. 4, the EKF is analyzed by mathematically deriving its stability conditions. Section 5 compares performance between the EKF and UKF when different measurement sensors are used; also the convergence of the EKF is tested with different initial estimation errors and noise magnitudes. Section 6 concludes the paper and suggests future research.
Robotic testing of transfemoral prostheses is presented here, where motion is limited to the sagittal plane and transverse motion is not considered. Typically, only the sagittal plane is considered in transfemoral prosthesis research. Although the transverse plane is ignored here, sagittal motion captures the essential dynamics of human walking . The model of the test robot/prosthesis is based on the standard robotic framework. Figure 1 shows a diagram of the hip robot and prosthesis combination [20,21]. A general dynamic model for the system is given as follows:
where is the vector of joint displacements (q1 is vertical hip displacement, q2 is thigh angle, q3 is knee angle, and q4 is ankle angle), is the inertia matrix, is a matrix accounting for centripetal and Coriolis effects, is a nonlinear damping vector, Je is the kinematic Jacobian relative to the point of application of the external forces Fe, is the gravity vector, and u is the four-element vector of control signals . The kinematic and dynamic models of the robot/prosthesis combination are given in Refs. [23–25], where a mixed tracking/impedance controller based on passivity methods is designed . The control signals consist of hip force, and thigh, knee, and ankle torques. As Fig. 1 shows, a triangular foot with two points of ground contact is assumed. Horizontal and vertical GRFs are applied to contact points at the toe and heel. The GRFs are denoted as Fxh, Fzh, Fxt, Fzt, which represent the horizontal and vertical GRFs at the heel and toe. Thus, the external force vector Fe in Eq. (1) comprises these four GRFs.
where kb is the belt stiffness, sz is the treadmill standoff (the distance between the origin of the world coordinate system and the belt when the leg is fully extended), l2 and l3 are the lengths of link 2 (thigh) and link 3 (shank), respectively, aH and aT are the distance from the ankle joint to the heel and the toe, respectively, ah is the height of the ankle joint above the sole of the foot, and β is the friction coefficient between the treadmill belt and the foot. The vertical positions of the toe and heel in the world coordinate system are shown in Fig. 1 as zt and zh, respectively. We thus have four states for the positions and four states for the velocities of the joint displacements.
It should be noted that the forces are not states of the original system model, but are augmented here to the state vector, since we need to estimate them with a state estimator. The 12-element vector of Eq. (8) will be estimated by the state estimator.
Equations (2)–(7) are a preliminary approximation to GRF and are accurate only if the amputee (or robot in our case) walks in a highly controlled environment with a known walking surface stiffness. In future work, it will be important to study the robustness of the Kalman filter to modeling errors and particularly to errors in these GRF equations.
We should mention that there are at least two other possible approaches to obtain GRF estimates. In the first alternative approach, GRF could be considered as an unknown input with no prior assumptions about its model. In that case, the EKF would have difficulty estimating GRF, since its effect on the system would be indistinguishable from process noise. In the second alternative approach, state estimation of the robot/prosthesis system could be performed by a nonlinear observer, such as the EKF, and then the estimated states of the robot could be substituted into the GRF Eqs. (2)–(7) to obtain the GRF estimates. However, this approach would be less flexible than the approach that we are using. Augmenting the GRF states onto the original state as in Eq. (8) allows us additional tuning flexibility in the artificial process noise that is included in the augmented states, as we will see in Sec. 5.
Extended Kalman Filtering for Robot/Prosthesis State Estimation
The Stability of the Extended Kalman Filter
The analysis of the stability properties of continuous-time extended Kalman filters is complex and has been treated only for especial cases. Safonov and Athans considered the stability of the constant and modified gain EKF. The boundedness of the EKF estimates were investigated when used as a parameter estimator for linear systems [29,30]. Reif et al. [31,32] determined that the estimation error in both the discrete and continuous-time EKF is exponentially bounded under certain conditions. They showed that the estimation error remains bounded if the initial estimation error and disturbances are sufficiently small and the nonlinear system satisfies a detectability rank condition [33,34]. Here, we extend Reif's work to obtain the stability conditions of the state estimator of the robot/prosthesis system. The main purpose of this section is to derive the stability of the estimator in terms of initial condition errors and disturbances.
In stability theory, there are two commonly used methods to prove boundedness: supermartingales for stochastic differential equations, and Lyapunov functions for deterministic differential equations. Supermartingales can be regarded as the stochastic equivalent of Lyapunov functions in some cases . In order to analyze the error dynamics of the EKF, we first present some preliminary results [31,35]. Note that represents the spectral norm of a matrix or the Euclidian norm of a vector.
Proof of Lemma 4.1. See Appendix A.
- (i)There exist p1, p2, c1, c2, q1, q2, r1, r2 such that the following bounds hold for every :(31)(32)(33)(34)
- (ii)There exist , εx, such that the nonlinear functions in Eq. (26) are bounded by(35)for x, and with and , respectively.(36)
Proof of Theorem 4.1. See Appendix B.
If f and h are twice differentiable with respect to x for every , the spectral norm of the Hessian matrices of fi and hi are bounded, where fi and hi are the components of f and h, respectively (see Refs. and [38, Chap. 7]). In the given robot/prosthesis system, the Hessian matrix of f is not a function of control input u.
Note that the nonlinear system has to be uniformly detectable in order to satisfy the boundedness condition of Eq. (31) [39,40]. The detectability of the nonlinear system results in the boundedness of the estimation error bounds for the solution P(t) of the Riccati differential equation (15) based on Eq. (31). We present the following definition for the detectability of nonlinear stochastic systems .
holds for all x, , where λm is the maximum eigenvalue of .
where differs from x(0) due to random initial state estimation errors, denotes the discretization step size with , Q and R represent continuous-time noise covariances, and Qd and Rd represent discrete-time noise covariances [27, Chap. 8]. The process noise represents small values of unmodeled dynamics and parameter uncertainties. The Q matrix is chosen based on prior experience with the accuracy of system dynamic modeling; the last four elements of the diagonal matrix Q are zero since the GRF model is assumed to be perfectly known. The R matrix is usually straightforward to determine on the basis of our knowledge of the accuracy of the measurement system.
Reduced Sensor Sets and Unscented Kalman Filtering.
In a real-world scenario, it would not be practical to use a hip displacement sensor on an amputee because of its invasiveness. The other sensors—thigh angle, knee angle, and ankle angle—can easily be used in the real world because they can be mounted on the prosthesis. In this paper, we use a hip displacement sensor to provide a baseline scenario for the reduced sensor sets, which we study later in this section.
More observations result in more accurate state estimation. However, our goal is to use the fewest possible number of sensors for estimation, which result in a reduction of complexity and cost in the prosthesis hardware. Here, we remove the hip displacement sensor and use only three measurements: the thigh, knee, and ankle angles. For our final test in this section, we will use only two measurements: the knee and ankle angles. Moreover, we will also compare the performance of the EKF and UKF for the state and GRF estimation. We use the same initial values for state and estimated state as used in the EKF for normal walking. We also test the performance of the filters with different gait speeds, and we will see that no extra filter tuning is needed for fast and slow walking.
The original discrete-time form of the UKF has been widely used for state estimation of discrete-time systems. However, the discrete-time UKF cannot be directly applied to continuous-time filtering problems, in which the process and measurement equations are modeled as continuous-time stochastic systems. In Ref. , the continuous-time UKF was derived from the discrete-time UKF in matrix form, and this is the algorithm that we implement here. The tuning parameters in the unscented transforms were chosen to be , β = 2, and .
Table 1 compares the accuracy of EKF state estimation in terms of the RMSE in three different gait modes (fast walk, normal walk, slow walk). The EKF works well for different walking speeds with a reduced measurement set. It can be seen, as expected, that for all three walking speeds, the estimation errors with four measurements are smaller than with three or two measurements.
Table 2 shows that the UKF achieves smaller RMS estimation errors than the EKF for almost all measurement sets and walking speeds. Table 3 summarizes the average performance of the EKF and UKF. We see that the UKF achieves 30% improvement in the average RMSE of GRFs with four measurements. Although the UKF performs better than the EKF, its computational effort is between two and three times that of the EKF in terms of the number of multiplications and additions .
Stability of the Extended Kalman Filter.
It has been shown that the estimation error in the continuous-time Kalman filter remains bounded under certain conditions. Small initial estimation errors and small noise magnitudes are required conditions to obtain error bounds. In this section, we study the stability of the state estimates of the robot/prosthesis system. First, we need to check that the Jacobian matrices in the robot/prosthesis system satisfy the uniform detectability condition of Definition 4.1. To find an appropriate , we assume that and solve for . Note that with our measurement system, is an identity matrix and is thus invertible. Therefore, the uniform detectability condition of Definition 4.1 is satisfied in the robotic/prosthesis system. To test the stability of the robotic/prosthesis EKF, the initial value and the measurement and process noise covariances R, Q are chosen as shown in Table 4, where
is a reasonably accurate initial condition for the state estimate, and we will see that it is good enough to ensure convergence of the EKF. is a significantly worse initial condition for the state estimate, and we will see that it is not good enough to ensure convergence of the EKF. The R and Q values in Table 4 have been chosen to demonstrate conditions that, respectively, ensure, or do not ensure, EKF convergence. These values for , and R and Q were chosen by trial and error to demonstrate their effect on EKF convergence.
The simulation results are presented in Figs. 5–7, where the unknown state in Eq. (8) (angular velocity of thigh), the estimated state , and its estimation error are plotted. We can see in Fig. 5 that for small initial estimation error and small noise, the estimation error is bounded. However, Figs. 6 and 7 show that for large initial estimation errors or large noise magnitudes, the estimation error is no longer bounded.
Extensive simulation tests show that boundedness of the estimation error is obtained in the numerical simulation if and . The theoretical results for ε and δ via Eqs. (B21) and (B24) from Appendix B yield smaller bounds for stability: and with . Since the estimation error does not diverge in practice even with larger initial condition errors and noise terms than those given by the theorem, we conclude that the bounds are very conservative in this system. However, we note that it is not possible to test all possible conditions that exceed the theoretical error bounds; therefore, we naturally expect the theoretical results to yield more conservative stability bounds than the simulation results.
We want the state to be estimated as accurately as possible for eventual implementation in a state-feedback controller. The controller that we used in this research is robust to estimation errors , although more accurate state estimates are always desirable in order to reduce controller errors. We have not explored the relationship between estimation error and controller error in this paper but leave it as an important area for future research.
Conclusion and Future Work
We designed an EKF and an UKF to estimate not only the states of a robot/prosthesis system but also the external forces acting on the prosthetic foot. This approach removes the need for heavy and bulky load cells that are otherwise needed for GRF estimation. We achieved satisfactory estimation errors in various gait speeds for the robot/prosthesis system using four, three, and two measurements. The average RMS estimation errors of the EKF for the thigh, knee, and ankle angles in normal walking with four measurements is 0.0033 rad, in comparison with that the UKF which is 0.0020 rad. Although the UKF outperforms the EKF, it requires more computational effort than the EKF, which will be a consideration for real-time implementation.
We proved mathematically that the estimation error in the EKF is exponentially bounded if the initial estimation errors and disturbances are sufficiently small and if the nonlinear system satisfies a detectability rank condition. In simulation tests, we verified that the estimation errors remained bounded for small initial estimation errors and small disturbances. However, the filter is unstable if the initial estimation errors or disturbances are too large.
As far as we know, the present research is the first time that GRF has been estimated for prostheses using state estimation techniques. As we noted in the introduction, some other methods have been used for external force estimation in robotics. For instance, achieved a 1.72% estimation error for a 4DOF robot, compared to our results, which show a 1.85% error when using the EKF with four sensors during normal walking. Our results are, therefore, quantitatively similar to Ref. , although the comparison may not be fair since the robotic systems in the two approaches are much different. Other publications in the area of external force estimation do not include quantitative results [10,11,13].
Future work will include increasing and verifying the robustness of the filters, and experimental implementation and verification of the EKF and UKF, first on robotic hardware and then in human trials.
• NSF Grant No. 1344954.
In summary, the assumptions of Lemma 4.1 Eqs. (27) and (28) are satisfied by Eqs. (B3) and (B22), where and . We conclude that the estimation error is exponentially bounded in mean square under the conditions in Eqs. (37)–(39). |
On the Holographic Principle in
[.8cm] a Radiation Dominated Universe
Joseph Henry Laboratories Princeton University
Princeton, New Jersey 08544
The holographic principle is studied in the context of a dimensional radiation dominated closed Friedman-Robertson-Walker (FRW) universe. The radiation is represented by a conformal field theory with a large central charge. Following recent ideas on holography, it is argued that the entropy density in the early universe is bounded by a multiple of the Hubble constant. The entropy of the CFT is expressed in terms of the energy and the Casimir energy via a universal Cardy formula that is valid for all dimensions. A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy. Unlike the Hubble bound, the new bound remains valid throughout the cosmological evolution. When the new bound is saturated the Friedman equation exactly coincides with the universal Cardy formula, and the temperature is uniquely fixed in terms of the Hubble parameter and its time-derivative.
The holographic principle is based on the idea that for a given volume the state of maximal entropy is given by the largest black hole that fits inside . ’t Hooft and Susskind argued on this basis that the microscopic entropy associated with the volume should be less than the Bekenstein-Hawking entropy
of a black hole with horizon area equal to the surface area of the boundary of . Here the dependence on Newton’s constant is made explicit, but as usual and are set to one.
To shed further light on the holographic principle and the entropy bounds derived from it, we study in this paper the standard cosmology of a closed radiation dominated Friedman-Robertson-Walker (FRW) universe with general space-time dimension
The metric takes the form
where represents the radius of the universe at a given time and is a short hand notation for the metric on the unit -sphere . Hence, the spatial section of a d closed FRW universe is an -sphere with a finite volume
The holographic bound is in its naive form (1) not really applicable to a closed universe, since space has no boundary. Furthermore, the argumentation leading to (1) assumes that it’s possible to form a black hole that fills the entire volume. This is not true in a cosmological setting, because the expansion rate of the universe as well as the given value of the total energy restrict the maximal size of black hole. As will be discussed in this paper, this will lead to a modified version of the holographic bound.
The radiation in an FRW universe is usually described by free or weakly interacting mass-less particles. More generally, however, one can describe the radiation by an interacting conformal field theory (CFT). The number of species of mass-less particles translates into the value of the central charge of the CFT. In this paper we will be particularly interested in radiation described by a CFT with a very large central charge. In a finite volume the energy has a Casimir contribution proportional to . Due to this Casimir effect, the entropy is no longer a purely extensive function of and . The entropy of a d CFT is given by the well-known Cardy formula
where represents the product of the energy and radius, and the shift of is caused by the Casimir effect. In this paper we show that, after making the appropriate identifications for and , the same Cardy formula is also valid for CFTs in other dimensions. This is rather surprising, since the standard derivation of the Cardy formula based on modular invariance only appears to work for . By defining the central charge in terms of the Casimir energy, we are able to argue that the Cardy formula is universally valid. Specifically, we will show that with the appropriate identifications, the entropy for a dimensional CFT with an AdS-dual is exactly given by (3).
The main new result of this paper is the appearance of a deep and fundamental connection between the holographic principle, the entropy formulas for the CFT, and the FRW equations for a radiation dominated universe. In dimensions the FRW equations are given by
where is the Hubble parameter, and the dot denotes as usual differentiation with respect to the time . The FRW equations are usually written in terms of the energy density , but for the present study it is more convenient to work with the total energy and entropy instead of their respective densities and . Note that the cosmological constant has been put to zero; the case will be described elsewhere .
Entropy and energy momentum conservation together with the equation of state imply that and decrease in the usual way like . Hence, the cosmological evolution follows the standard scenario for a closed radiation dominated FRW universe. After the initial Big Bang, the universe expands until it reaches a maximum radius, the universe subsequently re-collapses and ends with a Big Crunch. No surprises happen in this respect.
The fun starts when one compares the holographic entropy bound with the entropy formulas for the CFT. We will show that when the bound is saturated the FRW equations and entropy formulas of the CFT merge together into one set of equation. One easily checks on the back of an envelope that via the substitutions
the Cardy formula (3) exactly turns into the dimensional Friedman equation (4). This observation appears as a natural consequence of the holographic principle. In sections 2 and 3 we introduce three cosmological bounds each corresponding to one of the equations in (6) The Cardy formula is presented and derived in section 4. In section 5 we introduce a new cosmological bound, and show that the FRW equations and the entropy formulas are exactly matched when the bound is saturated. In section 6 we present a graphical picture of the entropy bounds and their time evolution.
2. Cosmological entropy bounds
This section is devoted to the description of three cosmological entropy bounds: the Bekenstein bound, the holographic Bekenstein-Hawking bound, and the Hubble bound. The relation with the holographic bound proposed by Fischler-Susskind and Bousso (FSB) will also be clarified.
2.1. The Bekenstein bound
Bekenstein was the first to propose a bound on the entropy of a macroscopic system. He argued that for a system with limited self-gravity, the total entropy is less or equal than a multiple of the product of the energy and the linear size of the system. In the present context, namely that of a closed radiation dominated FRW universe with radius , the appropriately normalized Bekenstein bound is
where the Bekenstein entropy is defined by
The bound is most powerful for relatively low energy density or small volumes. This is due to the fact that is super-extensive: under and it scales like .
For a radiation dominated universe the Bekenstein entropy is constant throughout the entire evolution, since . Therefore, once the Bekenstein bound is satisfied at one instance, it will remain satisfied at all times as long as the entropy does not change. The Bekenstein entropy is the most natural generalization of the Virasoro operator to arbitrary dimensions, as is apparent from (6). Indeed, it is useful to think about not really as an entropy but rather as the energy measured with respect to an appropriately chosen conformal time coordinate.
2.2. The Bekenstein-Hawking bound
The Bekenstein-bound is supposed to hold for systems with limited self-gravity, which means that the gravitational self-energy of the system is small compared to the total energy . In the current situation this implies, concretely, that the Hubble radius is larger than the radius of the universe. So the Bekenstein bound is only appropriate in the parameter range . In a strongly self-gravitating universe, that is for , the possibility of black hole formation has to be taken into account, and the entropy bound must be modified accordingly. Here the general philosophy of the holographic principle becomes important.
It follows directly from the Friedman equation (4) that
Therefore, to decide whether a system is strongly or weakly gravitating one should compare the Bekenstein entropy with the quantity
When the system is weakly gravitating, while for the self-gravity is strong. We will identify with the holographic Bekenstein-Hawking entropy of a black hole with the size of the universe. indeed grows like an area instead of the volume, and for a closed universe it is the closest one can come to the usual expression .
As will become clear in this paper, the role of is not to serve as a bound on the total entropy, but rather on a sub-extensive component of the entropy that is association with the Casimir energy of the CFT. The relation (6) suggests that the Bekenstein-Hawking entropy is closely related to the central charge . Indeed, it is well-known from CFT that the central charge characterizes the number of degrees of freedom may be even better than the entropy. This fact will be further explained in sections 5 and 6, when we describe a new cosmological bound on the Casimir energy and its associated entropy.
2.3. The Hubble entropy bound
The Bekenstein entropy is equal to the holographic Bekenstein-Hawking entropy precisely when . For one has and the Bekenstein bound has to be replaced by a holographic bound. A naive application of the holographic principle would imply that the total entropy should be bounded by . This turns out to be incorrect, however, since a purely holographic bound assumes the existence of arbitrarily large black holes, and is irreconcilable with a finite homogeneous entropy density.
Following earlier work by Fischler and Susskind , it was argued by Easther and Lowe , Veneziano , Bak and Rey , Kaloper and Linde , that the maximal entropy inside the universe is produced by black holes of the size of the Hubble horizon, see also . Following the usual holographic arguments one then finds that the total entropy should be less or equal than the Bekenstein-Hawking entropy of a Hubble size black hole times the number of Hubble regions in the universe. The entropy of a Hubble size black hole is roughly , where is the volume of a single Hubble region. Combined with the fact that one obtains an upper bound on the total entropy given by a multiple of . The presented arguments of [6, 8, 9, 7] are not sufficient to determine the precise pre-factor, but in the following subsection we will fix the normalization of the bound by using a local version of the Fischler-Susskind-Bousso formulation of the holographic principle. The appropriately normalized entropy bound takes the form
The Hubble bound is only valid for . In fact, it is easily seen that for the bound will at some point be violated. For example, when the universe reaches its maximum radius and starts to re-collapse the Hubble constant vanishes, while the entropy is still non-zero.111To avoid this problem a different covariant version of the Hubble bound was proposed in . This should not really come as a surprise, since the Hubble bound was based on the idea that the maximum size of a black hole is equal to the Hubble radius. Clearly, when the radius of the universe is smaller than the Hubble radius one should reconsider the validity of the bound. In this situation, the self-gravity of the universe is less important, and the appropriate entropy bound is
2.4. The Hubble bound and the FSB prescription.
Fischler, Susskind, and subsequently Bousso , have proposed an ingenious version of the holographic bound that restricts the entropy flow through contracting light sheets. The FSB-bound works well in many situations, but, so far, no microscopic derivation has been given. Wald and collaborators have shown that the FSB bound follows from local inequalities on the entropy density and the stress energy. The analysis of suggests the existence a local version of the FSB entropy bound, one that does not involve global information about the causal structure of the universe, see also . The idea of to formulate the holographic principle via entropy flow through light sheets also occurred in the work of Jacobson , who used it to derive an intriguing relation between the Einstein equations and the first law of thermodynamics. In this subsection, a local FSB bound will be presented that leads to a precisely normalized upper limit on the entropy in terms of the Hubble constant.
According to the original FSB proposal, the entropy flow through a contracting light sheet is less or equal to , where is the area of the surface from which the light sheet originates. The following infinitesimal version of this FSB prescription will lead to the Hubble bound. For every dimensional surface at time with area one demands that
where denotes the entropy flow through the infinitesimal light sheets originating at the surface at and extending back to time , and represents the increase in area between and . For a surface that is kept fixed in co-moving coordinates the area changes as a result of the Hubble expansion by an amount
where the factor simply follows from the fact that . Now pick one of the two past light-sheets that originate at the surface: the inward or the outward going. The entropy flow through this light-sheet between and is given by the entropy density times the infinitesimal volume swept out by the light-sheet. Hence,
By inserting this result together with (15) into the infinitesimal FSB bound (14) one finds that the factor cancels on both sides and one is left exactly with the Hubble bound with the Hubble entropy given in (12). We stress that the relation with the FSB bound was merely used to fix the normalization of the Hubble bound, and should not be seen as a derivation.
3. Time-evolution of the entropy bounds.
Let us now return to the three cosmological entropy bounds discussed in section 2. The Friedman equation (4) can be re-written as an identity that relates the Bekenstein-, the Hubble-, and the Bekenstein-Hawking entropy. One easily verifies that the expressions given in (8), (10), and (12) satisfy the quadratic relation
It is deliberately written in a Pythagorean form, since it suggests a useful graphical picture of the three entropy bounds. By representing each entropy by a line with length equal to its value one finds that due to the quadratic Friedman relation (17) all three fit nicely together in one diagram, see figure 1. The circular form of the diagram reflects the fact that is constant during the cosmological evolution. Only and depend on time.
Let us introduce a conformal time coordinate via
and let us compute the -dependence of and . For this easily follows from: . For the calculation is a bit more tedious, but with the help of the FRW equations, the result can eventually be put in the form
These equations show that the conformal time coordinate can be identified with the angle , as already indicated in figure 1. As time evolves the Hubble entropy rotates into the combination and visa versa. Equation (S3.Ex5) can be integrated to
The conformal time coordinate plays the role of the time on a cosmological clock that only goes around once: at time starts with a Big Bang and at it ends with a Big Crunch. Note that is related to the parameter via
So far we have not yet included the CFT into our discussion. We will see that the entropy of the CFT will ‘fill’ part of the diagram, and in this way give rise to a special moment in time when the entropy bounds are saturated.
4. Casimir energy and the Cardy formula
We now turn to the discussion of the entropy of the CFT that lives inside the FRW universe. We begin with a study of the finite temperature Casimir energy with the aim to exhibit its relation with the entropy of the CFT. Subsequently a universal Cardy formula will be derived that expresses the entropy in terms of the energy and the Casimir energy, and is valid for all values of the spatial dimension .
4.1. The Euler relation and Casimir energy.
In standard textbooks on cosmology [15, 16] it is usually assumed that the total entropy and energy are extensive quantities. This fact is used for example to relate the entropy density to the energy density and pressure , via . For a thermodynamic system in finite volume the energy , regarded as a function of entropy and volume, is called extensive when it satisfies Differentiating with respect to and putting leads to the Euler relation111We assume here that there are no other thermodynamic functions like a chemical or electric potential. For a system with a 1st law like the Euler relation reads .
The first law of thermodynamics can now be used to re-express the derivatives via the thermodynamic relations
The resulting equation is equivalent to the previously mentioned relation for the entropy density .
For a CFT with a large central charge the entropy and energy are not purely extensive. In a finite volume the energy of a CFT contains a non-extensive Casimir contribution proportional to . This is well known in dimensions where it gives rise to the familiar shift of in the Virasoro operator. The Casimir energy is the result of finite size effects in the quantum fluctuations of the CFT, and disappears when the volume becomes infinitely large. It therefore leads to sub-extensive contributions to the total energy . Usually the Casimir effect is discussed at zero temperature , but a similar effect occurs at finite temperature. The value of the Casimir energy will in that case generically depend on the temperature .
We will now define the Casimir energy as the violation of the Euler identity (22)
Here we inserted for convenience a factor equal to the spatial dimension . From the previous discussion it is clear that parameterizes the sub-extensive part of the total energy. The Casimir energy will just as the total energy be a function of the entropy and the volume . Under and it scales with a power of that is smaller than one. On general grounds one expects that the first subleading correction to the extensive part of the energy scales like
One possible way to see this is to write the energy as an integral over a local density expressed in the metric and its derivatives. Derivatives scale like and because derivatives come generally in pairs, the first subleading terms indeed has two additional factors of . The total energy may be written as a sum of two terms
where the first term denotes the purely extensive part of the energy and represents the Casimir energy. Again the factor has been put in for later convenience. By repeating the steps that lead to the Euler relation one easily verifies the defining equation (24) for the Casimir energy .
4.2. Universality of the Cardy formula and the Bekenstein bound
Conformal invariance implies that the product is independent of the volume , and is only a function of the entropy . This holds for both terms and in (26). Combined with the known (sub-)extensive behavior of and this leads to the following general expressions
where and are a priori arbitrary positive coefficients, independent of and . The factors of and are put in for convenience. With these expressions, one now easily checks that the entropy can be written as
If we ignore for a moment the normalization, this is exactly the Cardy formula: insert and , and one recovers (3). It is obviously an interesting question to compute the coefficients and for various known conformal invariant field theories. This should be particularly straightforward for free field theories, such as Maxwell theory and the self-dual tensor theory in . This question is left for future study.
Given the energy the expression (27) has a maximum value. For all values of , and one has the inequality
This looks exactly like the Bekenstein bound, except that the pre-factor is in general different from the factor used in the previous section. In fact, in the following subsection we will show that for CFTs with an AdS-dual description, the value of the product is exactly equal to , so the upper limit is indeed exactly given by the Bekenstein entropy. Although we have no proof of this fact, we believe that the Bekenstein bound is universal. This implies that the product for all CFTs in dimensions is larger or equal than . Only then it is guaranteed that the upper limit on the entropy is less or equal than .
The upper limit is reached when the Casimir energy is equal to the total energy . Formally, when becomes larger that the entropy will again decrease. Although in principle this is possible, we believe that in actual examples the Casimir energy is bounded by the total energy . So, from now on we assume that
In the next subsection we provide further evidence for this inequality.
From now on we will assume that we are dealing with a CFT for which . In the next section I will show that this includes all CFTs that have an AdS-dual description.
4.3. The Cardy formula derived from AdS/CFT
Soon after Maldacena’s AdS/CFT-correspondence was properly understood [19, 20] it was convincingly argued by Witten that the entropy, energy and temperature of CFT at high temperatures can be identified with the entropy, mass, and Hawking temperature of the AdS black hole previously considered by Hawking and Page . Using this duality relation the following expressions can be derived for the energy and entropy222These expressions differ somewhat from the presented formulas in due to the fact that (i) the dimensional Newton constant has been eliminated using its relation with the central charge, (ii) the coordinates have been re-scaled so that the CFT lives on a sphere with radius equal to the black hole horizon. We will not discuss the AdS perspective in this paper, since the essential physics can be understood without introducing an extra dimension. The discussion of the CFT/FRW cosmology from an AdS perspective will be described elsewhere . for a dimensional CFT on :
The temperature again follows from the first law of thermodynamics. One finds
The length scale of the thermal CFT arises in the AdS/CFT correspondence as the curvature radius of the AdS black hole geometry. The expression for the energy clearly exhibits a non-extensive contribution, while also the temperature contains a corresponding non-intensive term. Inserting the equations (29,30) into (24) yields the following result for the Casimir energy
Now let us come to the Cardy formula. The entropy , energy and Casimir energy are expressed in , and . Eliminating and leads to a unique expression for in terms of , and . One easily checks that it takes the form of the Cardy formula
In the derivation of these formulas it was assumed that . One may worry therefore that these formulas are not applicable in the early universe. Fortunately this is not a problem because during an adiabatic expansion both and scale in the same way so that is fixed. Hence the formulas are valid provided the (fixed) ratio of the thermal wave-length and the radius is much smaller than one. Effectively this means, as far as the CFT is concerned, we are in a high temperature regime. We note further that with in this parameter range, the Casimir energy is indeed smaller than the total energy .
Henceforth, we will assume that the CFT that describes the radiation in the FRW universe will have an entropy given by (32) with the specific normalization of . Note that if we take and make the previously mentioned identifications and that this equation exactly coincides with the usual Cardy formula. We will therefore in the following refer to (32) simply as the Cardy formula. To check the precise coefficient of the Cardy formula for a CFT we have made use of the AdS/CFT correspondence. The rest of our discussions in the preceding and in the following sections do not depend on this correspondence. So, in this paper we will not make use of any additional dimensions other than the ones present in the FRW-universe.
5. A new cosmological bound
In this section a new cosmological bound will be presented, which is equivalent to the Hubble bound in the strongly gravitating phase, but which unlike the Hubble bound remains valid in the phase of weak self-gravity. When the bound is saturated the FRW equations and the CFT formulas for the entropy and Casimir energy completely coincide.
5.1. A cosmological bound on the Casimir energy
Let us begin by presenting another criterion for distinguishing between a weakly or strongly self-gravitating universe. When the universe goes from the strongly to the weakly self-gravitating phase, or vice-versa, the Bekenstein entropy and the Bekenstein-Hawking entropy are equal in value. Given the radius , we now define the ‘Bekenstein-Hawking’ energy as the value of the energy for which and are exactly equal. This leads to the condition
One may interpret as the energy required to form a black hole with the size of the entire universe. Now, one easily verifies that
Hence, the universe is weakly self-gravitating when the total energy is less than and strongly gravitating for .
We are now ready to present a proposal for a new cosmological bound. It is not formulated as a bound on the entropy , but as a restriction on the Casimir energy . The physical content of the bound is the Casimir energy by itself can not be sufficient to form a universe-size black hole. Concretely, this implies that the Casimir energy is less or equal to the Bekenstein-Hawking energy . Hence, we postulate
To put the bound in a more conventional notation one may insert the definition (24) of the Casimir energy together with the defining relation (33) of the Bekenstein-Hawking energy. We leave this to the reader.
The virtues of the new cosmological bound are: (i) it is universally valid and does not break down for a weakly gravitating universe, (ii) in a strongly gravitating universe it is equivalent to the Hubble bound, (iii) it is purely holographic and can be formulated in terms of the Bekenstein-Hawking entropy of a universe-size black hole, (iv) when the bound is saturated the laws of general relativity and quantum field theory converge in a miraculous way, giving a strong indication that they have a common origin in a more fundamental unified theory.
The first point on the list is easily checked because decays like while goes like . Only when the universe re-collapses and returns to the strongly gravitating phase the bound may again become saturated. To be able to proof the other points on the list of advertised virtues, we have to take a closer look to the FRW equations and the CFT formulas for the entropy an entropy.
5.2. A cosmological Cardy formula
To show the equivalence of the new bound with the Hubble bound let us write the Friedman equation as an expression for the Hubble entropy in terms of the energy , the radius and the Bekenstein-Hawking energy . Here, the latter is used to remove the explicit dependence on Newton’s constant . The resulting expression is unique and takes the form
This is exactly the Cardy formula (32), except that the role of the Casimir energy in CFT formula is now replaced by the Bekenstein-Hawking energy . Somehow, miraculously, the Friedman equation knows about the Cardy formula for the entropy of a CFT!
With the help of (36) is now a straightforward matter to proof that when the new bound is equivalent to the Hubble bound . First, let us remind that for the energy satisfies . Furthermore, we always assume that the Casimir energy is smaller than the total energy . The entropy is a monotonically increasing function of as long as . Therefore in the range
the maximum entropy is reached when . In that case the Cardy formula (32) for exactly turns into the cosmological Cardy formula (36) for . Therefore, we conclude that is indeed the maximum entropy that can be reached when . Note that in the weakly self-gravitating phase, when , the maximum is reached earlier, namely for . The maximum entropy is in that case given by the bekenstein entropy . The bifurcation of the new bound in two entropy bounds is a direct consequence of the fact that the Hubble bound is written as the square-root of a quadratic expression.
5.3. A limiting temperature
So far we have focussed on the entropy and energy of the CFT and on the first of the two FRW equations, usually referred to as the Friedman equation. We will now show that also the second FRW equation has a counterpart in the CFT, and will lead to a constraint on the temperature . Specifically, we will find that the bound on implies that the temperature in the early universe is bounded from below by
The minus sign is necessary to get a positive result, since in a radiation dominated universe the expansion always slows down. Further, we assume that we are in the strongly self-gravitating phase with , so that there is no danger of dividing by zero.
The second FRW equation in (5) can now be written as a relation between , and that takes the familiar form
This equation has exactly the same form as the defining relation for the Casimir energy. In the strongly gravitating phase we have just argued that the bound is equivalent to the Hubble bound . It follows immediately that the temperature in this phase is bounded from below by . One has
When the cosmological bound is saturated all inequalities turn into equalities. The Cardy formula and the defining Euler relation for the Casimir energy in that case exactly match the Friedman equation for the Hubble constant and the FRW equation for its time derivative.
6. The entropy bounds revisited.
We now return to the cosmological entropy bounds introduced in sections 2 and 3. In particular, we are interested in the way that the entropy of the CFT may be incorporated in the entropy diagram described in section . For this purpose it will be useful to introduce a non-extensive component of the entropy that is associated with the Casimir energy.
The cosmological bound can also be formulated as an entropy bound, not on the total entropy, but on a non-extensive part of the entropy that is associated with the Casimir energy. In analogy with the definition of the Bekenstein entropy (8) one can introduce a ’Casimir’ entropy defined by
For the Casimir entropy is directly related to the central charge . One has . In fact, it is more appropriate to interpret the Casimir entropy as a generalization of the central charge to dimensions than what is usually called the central charge . Indeed, if one introduces a dimensionless ‘Virasoro operator’ and a new central charge , the dimensional entropy formula (32) is exactly identical to (3).
The Casimir entropy is sub-extensive because under and it goes like . In fact, it scales like an area! This is a clear indication that the Casimir entropy has something to do with holography. The total entropy contains extensive as well as sub-extensive contributions. One can show that for the entropy satisfies the following inequalities
where both equal signs can only hold simultaneously. The precise relation between and its super- and sub-extensive counterparts and is determined by the Cardy formula, which can be expressed as
This identity has exactly the same form as the relation (17) between the cosmological entropy bounds, except that in (17) the role of the entropy and Casimir entropy are taken over by the Hubble entropy and Bekenstein-Hawking entropy . This fact will be used to incorporate the entropy and the Casimir entropy in the entropy diagram introduce in section 3.
The cosmological bound on the Casimir energy presented in the section 4 can be formulated as an upper limit on the Casimir entropy . From the definitions of and it follows directly that the bound is equivalent to
where we made use of the relation (33) to re-write again in terms of the Bekenstein-Hawking entropy . Thus the bound puts a holographic upper limit on the d.o.f. of the CFT as measured by the Casimir entropy .
In figure 2 we have graphically depicted the quadratic relation between the total entropy and the Casimir entropy in the same diagram we used to related the cosmological entropy bounds. From this diagram it easy to determine the relation between the new bound and the Hubble bound. One clearly sees that when that the two bounds are in fact equivalent. When the new bound is saturated, which means , then the Hubble bound is also saturated, ie. . The converse is not true: there are two moments in the region when the , but . In our opinion, this is an indication that the bound on the Casimir energy has a good chance of being a truly fundamental bound.
7. Summary and conclusion
In this paper we have used the holographic principle to study the bounds on the entropy in a radiation dominated universe. The radiation has been described by a continuum CFT in the bulk. Surprisingly the CFT appears to know about the holographic entropy bounds, and equally surprising the FRW-equations know about the entropy formulas for the CFT. Our main results are summarized in the following two tables. Table 1. contains an overview of the bounds that hold in the early universe on the temperature, entropy and Casimir energy. In table 2. the Cardy formula for the CFT and the Euler relation for the Casimir energy are matched with the Friedman equations written in terms of the quantities listed in table 1.
Table 1: summary of cosmological bounds
Table 2: Matching of the CFT-formulas with the FRW-equations
The presented relation between the FRW equations and the entropy formulas precisely holds at this transition point, when the holographic bound is saturated or threatens to be violated. The miraculous merging of the CFT and FRW equations strongly indicates that both sets of these equations arise from a single underlying fundamental theory.
The discovered relation between the entropy, Casimir energy and temperature of the CFT and their cosmological counterparts has a very natural explanation from a RS-type brane-world scenario along the lines of . The radiation dominated FRW equations can be obtained by studying a brane with fixed tension in the background of a AdS-black hole. In this description the radius of the universe is identified with the distance of the brane to the center of the black hole. At the Big Bang the brane originates from the past singularity. At some finite radius determined by the energy of the black hole, the brane crosses the horizon. It keeps moving away from the black hole, until it reaches a maximum distance, and then it falls back into the AdS-black hole. The special moment when the brane crosses the horizon precisely corresponds to the moment when the cosmological entropy bounds are saturated. This world-brane perspective on the cosmological bounds for a radiation dominated universe will be described in detail in .
We have restricted our attention to matter described by a CFT in order to make our discussion as concrete and coherent as possible. Many of the used concepts, however, such as the entropy bounds, the notion of a non-extensive entropy, the matching of the FRW equations, and possibly even the Cardy formula are quite independent of the equation of state of the matter. One point at which the conformal invariance was used is in the diagrammatic representation of the bounds. The diagram is only circular when the energy goes like . But it is possible that a similar non-circular diagram exists for other kinds of matter. It would be interesting to study other examples in more detail.
Finally, the cosmological constant has been put to zero, since only in that case all of the formulas work so nicely. It is possible to modify the formalism to incorporate a cosmological constant, but the analysis becomes less transparent. In particular, one finds that the Hubble entropy bound needs to be modified by replacing with the square root of . At this moment we have no complete understanding of the case , and postpone its discussion to future work.
I like to thank T. Banks, M. Berkooz, S. Gubser, G. Horowitz, I. Klebanov, P. Kraus, E. Lieb, L. Randall, I. Savonije, G. Veneziano, and H. Verlinde for helpful discussions. I also thank the theory division at Cern for its hospitality, while this work was being completed.
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PID vs Fuzzy Control
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Temperature control based on traditional PID versus fuzzy controllersIndustrial temperature-control applications demand speed and precision. Improved temperature controllers can meet these demands by adding features not found in traditional PID controllers but increase system complexity. Fuzzy temperature controllers give you another option.By Peter Galan, Control Software Designer, Nortel Networks
Common perception holds that temperature control is a mature and largely static area of technology. Some industrial applicationsfor example, injection-molding processesstill desire not only precise temperature control but also a faster warm-up phase and a quicker response to disturbances with minimal overshoot and undershoot when the set point changes. Traditional PID (proportional-integral-derivative) control techniques cannot meet these extra challenges. Figure 1 shows a continuous-time (analog) PID controller, typical for most closedloop control systems. Its output, the actuating value y(t), is a function of the regulation error e(t):
y (t ) = K P e(t ) + K I e(t )dt + K D
de(t ) . dt
The use of a standard PID controller is fully adequate for some applications. Such applications commonly feature low to moderate control-quality (timing, precision) constraints and well-defined and stable dynamic system behavior. Precision industrialtemperature control, however, does not belong among these standard applications. For example, injection-molding processes require fast changingreadjusting of controlled temperatures with minimal overshoot. In addition, heating processes do not exhibit stable dynamic behavior, because heating and cooling rates are different at each temperature set point. In addition, coupling between the zones of a multizone heating system makes dynamic behavior very unpredictable.
Heat-transfer-specific problemsFigure 2's simplified thermal model of a typical heating system ignores heattransport delays. Assume that when you switch on a constant power source, W, it powers an electrical heating element with a heat capacity, Ce. The temperature of the heating element, Te(t), rises with time, t. The heat continuously propagates, by direct conduction, to the heated system. Res represents a thermal resistance between the electrical element and the heated system, and Cs represents the heated systems thermal capacity. The system temperature, Ts(t), also rises with time. Rsa thus represents a thermal resistance between the heated system and surrounding environment (with ambient temperature Ta(t)), which tends to cool the system. Ideally, without any cooling from the outside (that is, Rsa ), both temperatures, Te(t) and Ts(t), would rise forever. In practice, however, natural cooling prevents this occurrence. So, after a certain time period, both temperatures stabilize at certain constant values. Figure 2's model represents a second-order system with the following transfer function:
G p (s ) =
K . 1 + sT1 + s 2T2
The PID controller would be an ideal controller, because its transfer function with two nulls could, at least theoretically, cancel both poles of Gp. Figure 2 demonstrates that two cascaded first-order systems can replace this second-order system. Therefore, in practice, system response to a step function will always be aperiodical, and if Res is relatively small, the output temperature will follow a simple exponential curve when rising or falling. For such systems, even PI controllers (without the derivative member) are fully adequate. An interesting feature of a typical heated system, which represents the first difficulty for any temperature controller, is that increasing the temperature a couple of degrees from the surrounding temperature takes substantially less time than does cooling down to the surrounding temperature. In contrast, increasing the temperature of the same system by a couple of degrees when it is close to its maximum temperature takes substantially longer than does bringing the temperature back down by a couple of degrees. At one temperature set point, the heating rate (the speed of heating up) and the cooling rate (the speed of cooling down) are equal. This set point is at the temperature that requires the application of exactly 50% of the maximum applicable power to the heater. What is that balanced temperature? Theoretically, from Figure 2's thermal model, the dependence between power (W) and temperature (Ts) should be linear. That is, the stabilized temperature corresponding to 50% of maximum power should be exactly in the middle between the minimum and maximum temperature. The assumption here is that the maximum reachable temperature requires 100% power, and the minimum (ambient) temperature requires you to apply 0% power to the heating element for an unlimited time. Figure 3 shows the above-described feature for three set points: One is close to the ambient temperature, one is at the balanced temperature, and one is close to the maximum temperature. Temperatures are rising and decreasing exponentially with different rates (time constants) for heating and cooling. The ratio of the rate of heating and cooling processes at any stage depends only on the value of the set point.
Enhancing the traditional PID controllerThe PID constants basically depend on the gain and time constant of the controlled system. If you select them properly, they will cancel poles of the controlled systemtransfer function. Different rates of the heating and cooling processes affect the optimal values of the PID constants, making their estimation very difficult. If you want to use autotuning, be aware that all autotuning methods provide only one set of PID constants, which will at best suit only the set points close to the balanced temperature. A different situation exists when the temperature set point is different from the balanced temperature. The farther the set point is from the balanced temperature, the farther the PID constants will be from their optimal values. Just how far depends on how far the set point is from the balanced temperature and whether the current temperature is below or above the set point. To make the PID controller suitable for temperature control at any set point, you need some automatic adaptation of the PID constants. Generally, all three PID constants, K P , K I , and K D , are inversely proportional to the rate of temperature change. Autotuning algorithms (known in control theory as the Ziegler-Nichols rules) also adhere to this rule. The rate of temperature change is not a linear function of the temperature set point (the derivative of exponential is also exponential), but a straight line is a rea-
sonable approximation (Figure 4). As Figure 4 shows, the rate (speed) of heating at the balanced temperature is 1/2, where is a time constant of the exponential (shown previously in Figure 3). At the ambient temperature, 1/ is the rate of heating. The rate of cooling at the ambient temperature is infinitesimally low. The opposite situation exists at the other end of the exponential curve, close to the maximum temperature. Here, the rate of heating is infinitesimally low, and the rate of cooling is very high. To compensate for varying heating and cooling rates, the PID constants, K P , K I ,
and K D , must also vary, requiring modification of the basic PID controller. Figure 5 shows the addition of two new blocks: an adapter and a heater model. The heater model must at first determine a value of the balanced temperature. For most applications, applying 50% of the maximum output power to the heating element and waiting until the temperature settles down is infeasible. Therefore, the control system can first assume that the balanced temperature equals one half of the maximum temperature (a value for which the controller has been designed). You can determine a more precise value of the balanced temperature during the control process. The adapter block (knowing the balanced and set-point temperatures) then calculates two correction coefficients for the modification of the PID constants. This calculation is based on a linear dependence of the temperature-change rate as shown in Figure 4. You apply the first correction coefficient, kh, (it multiplies the PID constants) during the heating and the other coefficient, kc, during cooling: kh TMAX/2(TMAX TSP), and kc TMAX/2TSP, where TMAX is the maximum controllable and TSP is the set point temperature.
Introducing a feedforward memberOnce you have created the heater model (which is actually an inverse static characteristic of the heating process), you can use its output variable power level as a feedforward contribution to the actuating variable. Because the heating process is linear (the output temperature is proportional to the applied power), a simple line with the slope km can approximate the heating model characteristic. You can then calculate an initial value of km as: km MAX_POWER / MAX_TEMPERATURE. You can express the MAX_POWER value in the percentage of the output pulse width if you use the PWM method to drive the heater elements. If you knew in advance what power value youd need for the set-point temperature, you could immediately replace the contribution of the controller's integral member with the output of the feedforward member. However, at the beginning of the control process, you would not have the correct heater model (the correct km value). Fortunately, the integral member can provide this information to the adapter block. So, this approach provides the adapter block with a new task. Once the temperature stabilizes at the set-point value, the adapter block can easily calculate the required output power to maintain this temperature. It is a sum of the integral member and the feedforward member outputs. Because the temperature is stabilized at the set point, the regulation error is zero, and the contributions of the propo |
« ΠροηγούμενηΣυνέχεια »
This method differs from the 24)16294896(678954, Ans. common way by placing the right14482606
hand figure of every product im181229
mediately under the dividend. 169129
121 8. Divide 3545304 by 47. 9. Divide 45005091 by 57.
Ans. 75432. Ans. 789563.
VI. To divide by any number of 9's, when their number is not less than half the number of places that will be in the quotient, and when there is no remainder.
RULE. — Anner as many ciphers to the dividend, as there are 9's in the divisor. Then write the proper dividend under the number thus found, and subtract it from the number to which ciphers have been annexed; and, as many places of the remainder at the right hand as there were ciphers annexed, are so many figures for the right hand of the quotient ; and, for the remaining numbers of the quotient, a competent number must be taken from the left hand of the above remainder. 10. Divide 123332544 by 999. OPERATION.
By examining the dividend and 123332544000 divisor, we know there will be 6
123332544 places in the quotient. We there123,209211,456
fore take three of these figures from
the right hand of the remainder for 123456 Quotient, Ans. The
lent, Alise the three right-hand figures of the quotient, and the other three we take from the left hand of the remainder. 11. Divide 12332655 by 999.
Ans. 12345. 12. Divide 987551235 by 9999.
Ans. 98765. 13. Divide 9123456779876543211 by 999999999.
MISCELLANEOUS EXAMPLES. 1. What number multiplied by 1728 will produce 1705536 ?
2. If a garrison of 987 men are supplied with 175686 pounds of beef, how much will there be for each man ? Ans. 178 lbs.
3. In one dollar there are 100 cents; how many dollars in 697800 cents ?
Ans. $ 6978. 4. In one pound there are 16 ounces; how many pounds are in 111680 ounces?
Ans. 6980 lbs. 5. A dollar contains 6 shillings; how many dollars are in 5868 shillings?
Ans. $ 978. 6. The President of the United States receives a salary of $ 25,000; what does he receive per month? Ans. $ 20831.
7. A man receiving $ 96 for 8 months' labor, what does be receive for 1 month ?
Ans. $ 12. 8. The distance from Haverhill to Boston is 30 miles; and, if a man travel 6 miles an hour, how long will he be in going this distance ?
Ans. 5 hours. • 9. The annual revenue of a gentleman being $8395, how much per day is that equivalent to, there being 365 days in a year?
Ans. $ 23. 10. The car on the Liverpool railroad goes at the rate of 65 miles an hour; how long would it take to pass round the globe, the distance being about 25,000 miles ? Ans. 384-8 hours.
11. How much sugar at $ 15 per cwt. may be bought for $ 405 ?
Ans. 27 cwt. 12. In 6789560 shillings how many pounds, there being 20 shillings in a pound ?
Ans. 339478 pounds. 13. The Bible contains 31,173 verses ; how many must be read each day, that the book may be read through in a year ?
Ans. 8514: verses. 14. In 123456720 minutes how many hours ?
Ans. 2057612 hours. 15. A gentleman possessing an estate of $ 66,144, bequeathed one fourth to his wife, and the remainder was to be divided between his 4 children; what was the share of each ?
Ans. $ 12,402. 16. A man disposed of a farm containing 175 acres at $ 87 per acre; of the avails he distributed $ 1234 for charitable purposes; $ 197 was expended for the purchase of a horse and chaise; the remainder was divided between 6 gentlemen and 8 ladies, and each lady was to receive twice as much as a gentleman; what was the share of each ?
Ans. $ 627 for a gentleman, and $ 1254 for a lady. 17. If there are 160 square rods in an acre, how many acres are in 1086240 square rods?
Ans. 6789 acres.
18. If 144 square inches make one square foot, how many square feet in 14222160 square inches ? Ans. 98765 feet.
19. What number is that, which being multiplied by 24, the product divided by 10, the quotient multiplied by 2, 32 subtracted from the product, the remainder divided by 4, and 8 subtracted from the quotient, the remainder shall be 2?
20. What is the difference between half a dozen dozen, and six dozen dozen?
Ans. 792. 21. Bought of F. Johnson 8 barrels of flour at $ 7 per barrel, and 3 hundred weight of sugar at $8 per hundred. What was the amount of his bill ?
Ans. $ 80. 22. Sold S. Jenkins my best horse for $75, my second-best chaise for $ 87, a good harness for $31. He has paid me in cash $ 38, and has given me an order on Peter Parker for $ 12. How many dollars remain my due ?
Ans. $ 143. 23. T. Webster has sold his wagon to J. Emerson for $ 85. He is to receive his pay in wood at $5 per cord. How many cords will it require to balance the value of the wagon?
Ans. 17 cords. . 24. Purchased a farm of 500 acres for $ 17,876. I sold 127 acres of it at $ 47 an acre, 212 acres at $ 96 an acre, and the remainder at $ 37 an acre. What did I gain by my bargain ?
Ans. $ 14,702. 25. A tailor has 938 yards of broadcloth; how many cloaks can be made of the cloth, if it require 7 yards to make one cloak ?
Ans. 134 cloaks. 26. Bought 97 barrels of molasses at $ 5 a barrel. Gave 17 barrels to support the poor, and the remainder was sold at $ 8 a barrel. Did I gain or lose, and how much?
Ans. $ 155 gain. 27. There are 12 pence in one shilling; required the number of pence in 671 shillings.
Ans. 8052 pence. 28. Twelve inches make one foot in length ; required the number of inches in 5280 feet, it being the length of a mile.
Ans. 63360 inches. 29. In one pound avoirdupois there are 16 ounces; required the ounces in 1728 pounds.
Ans. 27648 ounces. 30. Required the number of shillings in 8136 pence.
Ans. 678 shillings. 31. It requires 1728 cubic inches to make one cubic foot required the number of cubic inches in 3787 cubic feet.
Ans. 6543936 inches.
Section IX. TABLES OF MONEY, WEIGHTS, AND MEASURES.
- 1 Pound, 21 Shillings sterling "
1 Guinea, 28 Shillings N. E." i Guinea, Note. — One pound sterling is equal to $ 4.44), exchange value.
make i Pennyweight, marked dwt. 20 Pennyweights
OZ. 12 Ounces
20 5760 240
12 = 1 By this weight are weighed gold, silver, and jewels.
Note.-" The original of all weights used in England was a grain or corn of wheat, gathered out of the middle of the ear; and, being well dried, 32 of them were to make one pennyweight, 20 pennyweights one ounce, and 12 ounces one pound. But in later times, it was thought sufficient to divide the same pennyweight into 24 equal parts, still called grains, being the least weight now in common use; and from hence the rest are com
Ib. 5760 288
= 1 Apothecaries mix their medicines by this weight; but buy and sell by Avoirdupois. The pound and ounce of this weight are the same as in Troy Weight.
AVOIRDUPOIS WEIGHT. 16 Drams
marked oz. 16 Ounces 1 Pound,
lb. 28 Pounds
qr. 4 Quarters
1 Hundred Weight,
cwt. 20 Hundred Weight
Ib. 256 16
qr. 7168 448 28 = 1
112 = 4 = 1 ton. 573440
2240 = 80 = 20 = 1 By this weight are weighed almost every kind of goods, and all metals except gold and silver. By a late law of Massachusetts, the cwt. contains 100 lbs. instead of 112 lbs.
A ton is reckoned at the custom-houses of the United States at 2240 lbs.
LONG MEASURE. 3 Barleycorns, or 12 Lines make 1 Inch, marked 12 Inches
1 Foot, 3 Feet
1 Yard, 6 Feet
1 Fathom, 54 Yards, or 16% Feet
1 Rod, or Pole, “ 40 Rods
1 Furlong, 8 Furlongs
1 Mile, 3 Miles " 1 League "
lea. 69] Miles nearly
« 1 Degree, "6 Deg. or 360 Degrees
66 1 Circle of the Earth. |
Question;1. The theory of;consumer choice assumes that consumers attempt to maximize;a. the difference between total utility and marginal utility.;b. average utility.;c. total utility.;d. marginal utility.;2. Utility refers to the;a. usefulness of a good or service.;b. satisfaction that results from the consumption of a good.;c. relative scarcity of a good.;d. rate of decline in the demand curve.;3. The law of diminishing marginal utility says that;a. the marginal utility gained by consuming equal successive units of a good;will decline as the amount consumed increases.;b. the more of a particular good one consumes, the greater us the utility;received from the consumption of that good.;c. the marginal utility gained by consuming equal successive units of a good;will increase as the amount consumed increases.;d. the more of a particular product one sells, the less utility one receives;from selling.;e. none of the above;4. Which of the following is true?;a. It is possible for total utility to rise as marginal utility falls.;b. Marginal utility is the same as total utility.;c. It is possible for marginal utility to rise as total utility falls.;d. a and c;e. a, b, and c;5. In order for an individual to maximize total utility while consuming only;two goods, A and B, that individual must fulfill the condition;a. TUA= TUB.;b. TUA/PA= TUB/PB.;c. MUA= MUB.;d. MUA/PA= MUB/PB.;e. MUB/PB= MUB/PA.;6. We would expect the total utility of water to be high but its marginal;utility to be low. Why?;a. Because water is a fluid and we don?t need fluids to live as much as we need;food.;b. Because we need water to live and there is so much of it.;c. Because we need water to live and there is very little of it.;d. Because water?s price is low.;e. none of the above;7. To demonstrate the law of demand, suppose the price of good A rises. To;restore consumer equilibrium, ________ of good A is purchased in order to;the marginal utility of the last unit of it purchased.;a. more, lower;b. more, raise;c. less, lower;d. less, raise;8. The ?income effect? indicates that;a. when the price of a good falls, a consumer will be able to buy more of it;with a given money income.;b. consumers should substitute among various goods until the marginal utility;of the last unit of each good purchased is the same.;c. when the price of a good falls, the lower price will induce the consumer to;buy more of that good now that it is relatively cheaper.;d. b and c;e. none of the above;9. Price elasticity of demand is a measure of the responsiveness of quantity;demanded to changes in;a. interest rates.;b. price.;c. supply.;d. demand.;10. If the percentage change in quantity demanded is greater than the;percentage change in price, demand is;a. inelastic.;b. unit elastic.;c. elastic.;d. perfectly elastic.;e. perfectly inelastic.;11. Which of the following is a determinant of price elasticity of demand?;a. the number of substitutes;b. the percentage of one?s budget spent on the good;c. the amount of time that has passed since a price change;d. b and c;e. all of the above;12. Upon deregulation, firms in a formerly regulated industry feared;that cut throat competition would reduce prices and their revenue.;Competition did force prices down but to the firms' surprise their;revenues increased. The demand curve for the product these firms;produced must be;a. perfectly elastic.;b. elastic.;c. inelastic.;d. perfectly inelastic.;13. When the cross elasticity of demand between two goods is ________, the;goods are ________.;a. negative, substitutes;b. negative, complements;c. positive, normal goods;d. positive, inferior goods;14. If Cassandra bought 24 blouses last year when her income was $40,000 and;she buys 16 blouses this year her when income is $35,000, then for her blouses;are;a. an inferior good.;b. a normal good.;c. a substitute good.;d. a complementary good.;e. There is not enough information to answer this question.;15. Economists Alchian and Demsetz suggest that firms are formed when;a. the sum of what individuals can produce alone is greater than what they can;produce as a team.;b. someone wants to earn profits.;c. someone comes up with the idea that customers will buy a new product.;d. the sum of what individuals can produce as a team is greater than the sum of;what they can produce alone.;16. Most economists say that the firm?s goal or objective is to maximize;a. sales.;b. employment.;c. profits.;d. worker satisfaction.;e. none of the above;17. The sole proprietor of a proprietorship has;a. unlimited liability, which means the owner is responsible for settling all;debts of the firm but not if it means selling his personal property to do so.;b. limited liability, which means the owner is responsible for settling all;debts of the firm, even if this means selling his personal property to do so.;c. limited liability, which means the owner cannot be sued for the his failure;to pay his company's debts.;d. unlimited liability, which means the owner is responsible for settling all;debts of the firm even if this means selling his personal property to do so.;18. Which of the following is a disadvantage of a corporation relative to a;proprietorship or a partnership?;a. perpetual life;b. difficulty in raising money;c. unlimited liability of the owners;d. double taxation of profits;e. none of the above;19. ?Managerial coordination? refers to the;a. behavior of a worker who is putting forth less than the agreed-to effort.;b. process where individuals perform certain tasks based on changes in market;forces.;c. process where persons share in the profits of a business firm.;d. process where managers direct employees to perform certain tasks.;20. Which of the following statements is true?;a. Explicit costs always equal implicit costs.;b. Zero economic profit is a smaller dollar figure than normal profit.;c. Zero economic profit is a larger dollar figure than normal profit.;d. Saying that a firm earned zero economic profit is the same as saying it;earned normal profit.;e. none of the above;21. A fixed input is an input whose quantity;a. can be changed as output changes in the short run.;b. cannot be changed as output changes in the short run.;c. cannot be changed as output changes in the long run.;d. a and c;e. b and c;22. ?As additional units of a variable input are added to a fixed input;eventually the marginal physical product of the variable input will decline.?;This is a statement of the;a. law of supply.;b. average-marginal rule.;c. law of comparative advantage.;d. law of diminishing marginal returns.;23. A rising marginal cost curve is a reflection of;a. a rising marginal physical product curve.;b. a falling marginal physical product curve.;c. a falling average fixed cost curve.;d. a rising average variable cost curve.;24. The average-marginal rule states that;a. if the marginal magnitude is less than the average magnitude, the average;magnitude falls.;b. if the marginal magnitude is greater than the average magnitude, the average;magnitude falls.;c. if the marginal magnitude is rising, the average magnitude is necessarily;above it.;d. if the marginal magnitude is falling, the average magnitude is necessarily;below it.;e. c and d;25. Refer to Exhibit T-4. Curve A is ________ curve.;a. a marginal cost;b. an average variable cost;c. an average total cost;d. an average fixed cost;26. A ?price taker? is a firm that;a. does not have the ability to control the price of the product it sells.;b. does have the ability, albeit limited, to control the price of the product;it sells.;c. can raise the price of the product it sells and still sell some units of its;product.;d. sells a differentiated product.;e. none of the above;27. A perfectly competitive firm will increase its production as long as;a. total revenue is less than total cost.;b. the total revenue curve is rising.;c. marginal revenue is greater than marginal cost.;d. the marginal revenue curve is rising.;28. The perfectly competitive firm will;a. produce in the short run if price is below average variable cost.;b. produce in the long run if price is below average variable cost.;c. produce in the short run if price is below average total cost but above;average variable cost.;d. produce in the long run if price is below average total cost but above;average variable cost.;29. The perfectly competitive firm?s short-run supply curve is;a. the upward-sloping portion of its average total cost curve.;b. the horizontal portion of its marginal revenue curve.;c. the portion of its average variable cost curve that lies above the average;fixed cost curve.;d. the upward-sloping portion of its marginal cost curve.;e. the portion of its marginal cost curve that lies above its average variable;cost curve.;30. Why must profits be zero in long-run competitive equilibrium?;a. If profits are not zero, firms will enter or exit the industry.;b. If profits are not zero, firms will produce higher-quality goods.;c. If profits are not zero, marginal revenue will rise.;d. If profits are not zero, marginal cost will rise.
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We first suppose that our particles are smooth, rigid and spherical and totally buoyant, i. e. their movement is not influenced by terrestrial gravity.
A particle suspended in a fluid can be transported by convection (i. e. entrained by the motion of the fluid) or by diffusion. The key parameter is the relative motion of the fluid with respect to the adsorbent. Far from the surface, the flow is uninfluenced by the surface; at the surface, on the other hand, friction dictates that the fluid is stationary (ignoring the possibility of slip); and the velocity in between is constantly diminishing. Hence near a surface, a particle will move by diffusion, and far from the surface by convection. At a certain intermediate distance, typically of the order of tens of micrometres, the transport régime will cross over from convection to diffusion. This distance is known as the diffusion boundary distance (6). A comprehensive treatment is given by Levich (1962). The flow in certain geometries, such as the tube or plate, can be solved analytically. These geometries are therefore particularly favourable for kinetic evaluation, a factor which should be borne in mind when designing flow cuvettes. For flow in a tube, where D is the diffusivity of the protein, F the volumetric flow rate, and C a constant depending on the dimensions of the tube.
When choosing flow rates, it should be remembered that only laminar flow régimes can be analysed conveniently, i. e. up to Reynolds numbers of at most around 1000. In principle a diffusion boundary still exists in the case of turbulent flow, but the motion is much more complicated than in the case of laminar flow.
Hence, a protein-sized particle will be moving diffusively already at a distance of the order of a thousand times its own dimensions ( a few nm in diameter) from the adsorbing surface.
At a distance of the order of ten times its own dimensions from the adsorbing surface, the particle may begin to be influenced by the long range hydrophilic repulsion (see Cacace et al. (1997) for a discussion of intermolecular forces), which will considerably retard its rate of arrival at the surface. When designing the flow conditions for an experiment, it is only necessary to ensure that convective-diffusion is rapid enough to replenish the particles lost from the solution in the vicinity of the surface by attachment to it. This is of practical importance given that many biological samples of carefully purified macromolecules are available only in extremely small quantities.
Good quantitative approximations for analysing the flow régimes can be derived from the equations of Fick and Smoluchowski (see Ramsden (1998) for a more complete discussion). If the surface (at z = 0, where z is the coordinate normal to the surface) is a perfect sink for the adsorbate, then the bulk (solution) concentration cb is zero at z = 0, the concentration gradient will be approximately linear, and the rate of accumulation is
It is a good idea to compare the experimentally observed rate with this maximum upper limit (which may, however, be exceeded if there is a long range attractive force, e. g. electrostatic, between particle and surface).
The effect of any energy barrier is to retard accumulation. In the immediate vicinity of the surface, the local bulk concentration cv will be much
higher than zero (although still less than cb). It is convenient to consider that the rate of accumulation at the surface is given by the product of cv and a chemical rate coefficient ka, which is directly related to the repulsive energy barrier (Spielman and Friedlander 1974):
and the Fick-Smoluchowski régime (linear concentration gradient) applies to the zone above this vicinal region. Hence dCv
where V and S are unit volume and surface respectively. Strictly speaking the distance of the vicinal layer from the surface should be subtracted from 6 in the denominator, but since that distance is of the order of molecular dimensions, i.e. only a few nm, whereas 6 is of the order of a few or a few tens of microns, this correction can be neglected. If desorption of the material also has to be taken into account a term with a chemical desorption coefficient kd can be included:
In a great many cases accumulation of material is limited to a monolayer, or to occupying a monolayer of receptors, in which case a function 0 must be introduced, which gives the fraction of the surface still available for adsorption or binding (i. e. the probability that space is available). Our kinetic equation then becomes:
We shall discuss 0—which obviously depends on M or v—below. One important implication is that as the surface fills up, i. e. as 0 ^ 0, cv will tend to approach cb, and dM/dt will asymptotically approach zero regardless of the flow régime. This can be immediately seen by letting the left hand side of whichever of the previous three equations is appropriate go to zero, yielding an explicit expression for cv, e. g.
if ka = 1 and kd = 0, which can then be substituted into Eq. 35, in which
The pure diffusion régime
If there is no flow at all then the vicinal layer is replenished by diffusion only, i. e. there is no distance at which concentration is maintained constant by effectively an infinite reservoir as in convective diffusion. This leads to the well-known result of Smoluchowski, according to which the flux to the surface constantly diminishes as t-fi, where fi = 0.5 for standard diffusion (other exponents have been found for the diffusion of highly non-spherical, non-compact proteins such as tenascin (Ramsden 1992).
If the flow rate is slow enough for the rate of adsorption to be limited by transport alone, or, more quantitatively, if the dimensionless parameter.
becomes large relative to unity, then the initial rate of adsorption is given by Eq. 37, from which D may be obtained. This is a useful way to determine the diffusion coefficient in solution.
If a repulsive potential barrier U(z) exists between protein and surface, where z is the distance between them, not every arriving protein will adsorb, even if there is space for it to do so, and v will be diminished by a rate coefficient ka, which can be found by integrating U(z):
Sometimes the denominator of the right hand term is called the "adsorption length", 6a. The interaction potential U can be approximated by the sum of the particle-surface interaction free energies, corresponding to the Lifschitz-van der Waals (LW), electrostatic (el) and electron donor-acceptor (da) interactions:
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Operations Management Introduction A. A. Elimam
Operations Management ACTIVITIES THAT RELATE TO THE CREATION OF GOODS AND SERVICES THROUGH THE TRANSFORMATION OF INPUTS INTO OUTPUTS
Transformation Process Feedback Input People Materials Equipment Money Management Output Transformation Feedback Goods Services
Manufacturing and Services Physical product Output inventoried Low customer contact Long response time Intangible product Cannot inventoried High customer contact Short response time
Manufacturing and Services World markets Large facilities Capital intensive Quality easily measured Local markets Small facilities Labor intensive Quality not easily measured
MAJOR CHALLENGES TO OPERATIONS MANAGERS Increase the VALUE of output relative to the COST of input. Increase PRODUCTIVITY = OUTPUT INPUT
PRODUCTIVITY Productivity is the quotient obtained by dividing output by one of the factors of production. One can speak of productivity of capital, labor, raw materials, etc.
WAYS TO IMPROVE PRODUCTIVITY REDUCE INPUTS INCREASE OUTPUT MINIMIZE DEFECTS IMPROVE QUALITY ELIMINATE WASTE FEWER HOURS LOWER ENERGY IMPROVE QUALITY
Example : Productivity Example: Output = $1000 Inputs: human = $300 material = $200 capital = $300 energy = $100 other exp. = $50 Human Productivity = 1000 / 300 = $ / $ 3. 33 Total Productivity = 1000 / 950 = $ / $ 1. 053
Example : Productivity Output = 600 insurance policies Inputs: human = 3 employees working 8 hours / day for 5 days Labor Productivity = 600 / (3)(5)(8) = 5 policies / hour
Decision Making Positioning Decisions Product Planning--Positioning Strategies and Quality Management Design Decisions Process Design, Work Force Management, Capacity, Location, Layout Operating Decisions Materials Management, Production Planning and Scheduling, Inventory, Supply Chain, Project Scheduling, Quality Control
Decision Making Horizons Strategic Planning: 5 - 10 yr. Less certainty - Less detail - Goal-oriented Operational Planning: 3 mos - 3 yr. More Certainty - More Means-oriented - Better Defined Scheduling: weekly - monthly More attention to detail Sequencing/Dispatching: hourly - daily Exact order and time of implementation Control: hourly - daily Feedback on implementation
Steps in Product Planning Step 1: Idea Generation Step 2: Screening Step 3: Development & testing Step 4: Final product design Rejected Ideas
Screening Approaches: Preference Matrix Weighted Score for each Product based on performance measures Selection: total score exceeds threshold Deficient approach - Why ?
Screening Approaches: Break-Even Analysis (BEA) When do revenues exceed costs? Total Annual Revenue = Total Annual Cost Total Annual Revenue = Ann. Fixed Cost + Ann. Variable Cost PQ = F + c. Q • • P = Price in $ / unit c = Variable cost in $/unit F = Annual Fixed Cost, $/yr. Q = Number of units produced
Screening Approaches: Break-Even Analysis (BEA) Production determines: F & c Marketing determines: P & Demand Use BEA to determine if the product BREAKS EVEN at the Expected Demand Yes --> Continue No --> Drop Product
Graphical Approach to BEA Given: p= $ 20/unit c=$10/unit F= $100, 000
Graphical Approach to BEA Given: p= $ 20/unit c=$10/unit F= $100, 000 Dollars (In Thousands) 400 (20, 400) Total Annual Revenues 300 Break-even Quantity 200 100 (20, 300) Total Annual Cost Loss Units, Q (In Thousands) Profit Fixed Cost 5 10 15 20
Graphic Approach to Break-Even Analysis Given: F= $ 100, 000 p= $30/patient c= $20/patient
Graphic Approach to Break-Even Analysis Given: F= $ 100, 000 c= $20/patient (2000, 400) 400 Dollars (in thousands) p= $30/patient Profits 300 Total annual revenues 200 Break-even quantity 100 Fixed costs Loss 0 (2000, 300) Total annual costs 500 1000 1500 Patients (Q) 2000
Example 1: Furniture Plant Fixed cost = $600, 000. Variable cost = $50 per unit. Marketing Research indicates firm can sell 15, 000 sets at $110 per set. Is it feasible to build the plant? Solution: Find the break-even point Q = F/(P - c) = 600, 000/(110 - 50) = 10, 000 patio furniture sets. Therefore, firm should build plant.
Example 2: Luxor Inc. Began producing cheese in 1993 output reached 20, 000 lb. at total cost of $40, 000 1994 output increased to 30, 000 lb. at total cost of $50, 000
Example 2: (continued) Luxor What is the variable cost (c) & the fixed cost (F)? Costs stayed the same during 1993/94. Solution: TC = F + c. Q 40, 000 = F + 20, 000. c for 1993 50, 000 = F + 30, 000. c for 1994 Subtracting from , 10, 000 = 0 + 10, 000 c, and c = $1 per lb. Substituting for c in , F = 40, 000 - 20, 000 (1) = $20, 000 per year
Example 2: (continued)Luxor If the selling price =$ 2. 80 in 1993 & $ 3. 20 in 1994, find the productivity in 1993 & 1994. Solution: Productivity = Output/Inputs or Productivity =Tot. A. Revenues/Tot. A. Costs or Productivity = (P. Q) / TC Therefore 1993 Productivity = 2. 80 x 20, 000/40, 000=1. 40 1994 Productivity = 3. 20 x 30, 000/50, 000=1. 92 so. . . Productivity improved in 1994 over 1993.
Example 2: (continued)Luxor If the total cost in 1995 is expected to increase to $60, 000, how many lb. should Luxor produce & sell to maintain the same productivity level of 1994? (selling price remains $3. 20/lb) Solution: Find the 1995 Q to keep productivity = 1. 92 = 3. 20 x Q / 60, 000 Q = 36, 000 lb.
BEA: Make or Buy Decisions Total Annual Cost of Making = Total Annual Cost of Buying F m + C m. Q = F b + Cb. Q m = making b = buying Decision to make or buy Number of units needed per year exceed BREAK EVEN VOLUME (Q)?
Make-Buy Decisions: Example In a PC assembly plant, to make hard drives Fixed costs = $200, 000 Var. cost = $50/unit. Hard drives cost $130 to buy. Should hard drives be made or bought? Solution: F m + C m. Q = F b + C b Q 200, 000 + 50 Q = 130 Q; Q = 2500 units Q <= 2500 units -- buy Q > 2500 units -- make.
BEA: Selection Among Two Alternatives Select one of 2 cars, Tonda & Hoyota Total Annual Cost of Tonda = Total Annual Cost of Hoyota FT + C T Q = F H + C H Q Q = Break even miles Solve for Q : If # of miles driven
Q select the car with the lowest F select the car with the lowest C
BEA: Selection Among Two Alternatives Considering two Cars to lease. Annual Costs Hoyota Tonda Lease Cost, $ 5, 000 8, 000 Variable Cost, $/mile 0. 3 0. 15 Which car would you lease and why? Solution: FH + CHQ = FT + CTQ 5, 000 + 0. 3 Q = 8, 000 + 0. 15 Q Q = 20, 000 miles Lease Hoyota if you drive < 20, 000 miles. Otherwise Tonda
BEA: Selection Among Two Alternatives What if the running cost of the Hoyota went down to $0. 25 per mile? Solution: Find the BEP using new running cost: 5000 + 0. 25 Q = 8000 + 0. 15 Q Q = 30, 000 Select Hoyota if you drive<30, 000 miles otherwise Select Tonda.
LP-Based Product Selection Select product(s) that maximize profit while staying within budget General Form: Maximize p 1 x 1 + p 2 x 2 + p 3 x 3 +. . . + pnxn Subject to c 1 x 1 + c 2 x 2 + c 3 x 3 +. . . + cnxn < B xi = 1 if product I is selected, = 0 otherwise pi = Profit of product i, in $ ci = Cost of product i, in $ B = Budget limitation, in $ Solve and select products whose xi = 1
Product 1 Product 2 Product 3 A B C D E F Product 3 Product 2 Product 1 (a) Process focused Product 1 Product 2 Product 3 A D E (b) Product focused B E F D C A Product 1 Product 2 Product 3
Process-Focused Strategy Resources set around similar processes One center/resource type-no duplication Products compete for resources Products move in jumbled (Job Shop) flow Highly skilled manual operations Used for low volume customized products Intensive, frequent customer interaction Example: Aircraft, Building, Interior Design
Product-Focused Strategy Resources organized around product Duplicate operations for different products Products do not compete for resources Products move in line flow (Flow Shop) Highly automated/expensive facilities Product-specialized and efficient Used in high volume standard products Little or no customer interaction Example: Paper Clips, Tires, Floppy Disks
Five Stages Product Planning Introduction Growth Maturity Decline
Life-Cycle of a Product or Service Annual dollars Annual sales Annual profits 0 Product Introduction planning Growth Maturity Life-cycle changes Decline
Life Cycle Audit Identify stage of product, based on changes in sales/profits Decide when to drop, revitalize or introduce new products Cycles vary from product to product
Life Cycle Audit : Example 1 Stage :
Life Cycle Audit : Example 1 Stage : Maturity
Life Cycle Audit : Example 2 Stage :
Life Cycle Audit : Example 2 Stage : Growth |
Angle of elevation and depression formula
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Slide rules aren’t just old-fashioned computing devices. They’re useful tools for getting work done – and can help us calibrate our goals, too.
How much building material do we need? Should we mix up 25 bags of cement, 250, or 2,500? That’s the sort of question that engineers need to know instinctively before they reach for a spreadsheet, Mathematica, pocket calculator, AutoCAD, smartphone app — or slide rule.
It’s essential to have a feel for the magnitude of a problem’s solution. That’s one of the lessons that the slide rule can teach software developers. Before you look to solve a problem precisely, you should have a feeling, deep in your gut, as to what a proper answer should look like.
As someone who started his computer science career by programming calculators like the HP-67 and the TI-59 back in high school, I was exposed to slide rules at the tail end of their era. I never relied upon them in the field, but over the years have amassed a pretty sizeable collection of slipsticks (as they’re also called), ranging from a little Pickett N600-ES 6-inch speed rule that was favored by early Apollo astronauts to a four-foot wooden rule that used to hang in a Texas classroom. I have a Nestler 23 model that was enjoyed by scientists like Albert Einstein and rocket engineer Werner von Braun.
Slide rules inspire me. Plus, they’re always fun to whip out at cocktail parties or tech conferences.
Here are five lessons that I’ve learned from my study of slide rules – and which apply to developers today.
[caption id="attachment_18534" align="aligncenter" width="456"] The intersection of math and mobility: Alan decides to “log” a few miles with his Pickett classroom slide rule.[/caption]
1. At the end of the day, it’s all about calculating a number.
Slide rules are practical. They multiply. They divide. (They don’t add or subtract.) They do roots, powers, tangents, natural logarithms. Specialized models help calculate the angle of a gun barrel, or figure the monthly payment for a 10-year fix-interest loan.
Mathematicians rarely use slide rules because mathematicians, oddly enough, don’t care about numbers. They care about equations and theorems and proofs and relationships. Mathematicians want to know symbolically how to solve for x if you are calculating the velocity of a projectile. They rarely if ever care that x in your specific scenario works out to 5.67 feet per second.
Slide rules are used to solve for the actual number. So are most software programs. We design algorithms and code equations into our source code. When the software executes, a number must pop out somewhere, either to be given to the end user or to be used in some other programmatic way (such as in a video game’s animation). Just as an architect could use a slide rule to discover how many square feet of paneling to order for an apartment building’s interior walls, so too our software’s users generally use our software because they want a real answer to a real question.
2. We must manually manage the magnitude of our results.
Slide rules are notorious for not understanding the concept of the decimal point. It’s left up to the engineer to keep track of the magnitude of the result in his head (or by writing it down on a sheet of paper). To put it another way, it’s all scientific notation. The slide rule handles the mantissa, and the engineer manages the exponent.
An architect should have a rough feel for how many square feet of tiles are needed before she sits down to figure out the exact number, no matter what type of manual or automated tool is used. Is it a few dozen? A few hundred? A few thousand? A few tens of thousand? When the architectural software (or spreadsheet or whatever) comes back and says 3,250 square feet, that’s got to pass the sniff test.
Let’s say that we are carpeting an space that is 52 feet by 317 feet, and we want to know how many square feet of carpet to order. We can tell, by inspection, that it’s a bit more than 50 x 300, or 15,000 square feet. That’s easy. Duh. When we do the math on our calculator, it tells us 16,484. That smells right. We know that 1648.4 would be too small, and 164,840 would be too big.
In the slide rule world, we’d reformat the calculation to (5.2 x 10^1) x (3.17 x 10^2), and then split it apart:
• Let the slide rule do the 5.2 x 3.17 multiplication. That gives 16.484. The slide rule would show that as 1.65, by the way, but the way the scales work shows that the result was larger than 10 – in other words, 1.65 x 10^1.
• It’s left to us to manage the exponents – specifically, to add the exponents. We add up the exponents of 10^1 and 10^2, which comes to 10^3, and add the extra 10^1 as shown above. That gives us 10^4.
Bring the answers together, and the result is 1.65 x 10^4, or 16,500. Bingo!
A working engineer (or carpeting installer) should have that order-of-magnitude feel before picking up the calculator. Or the slide rule. Otherwise, he can’t trust the results. (And he will have a lot of spare carpeting on his hands.)
3. Beware of worshiping false precision.
The cheapest pocket calculator, even back in the 1970s, would solve problems to within eight decimal points. A slide rule, however, generally only gives three or four decimal points of precision, based on the size and quality of the rule, and of the skill of the operator.
Frankly, in the real world, three or four digits of precision is about the right amount. The eight or more digits from a pocket calculator, and even greater precision in a lot of software, exceeds the error in our real-world measurements.
Take the carpeting calculator. We have three digits of accuracy at most for our input data of 317 feet and 52 feet. (After all, we don’t really know if it’s 316.8 feet or 317.4 feet.) We can trust three digits. If you multiply those together, you can’t trust more than three digits either. In other words, while the calculator or Excel or AutoCad or some carpet-laying app tells us that 52 feet x 317 feet is 16,484 square feet, that’s too precise for our carpet-buyer. All we know is that we need “between 16,400 and 16,500 square feet” of carpet. In other words, it is best to order at least 16,500 square feet.
A slide rule operator, squinting to figure out exactly where the digits are, can generally feel really good about two digits of precision, and fairly good about three. Four digits? Fuggedaboutit.
In the real world, you almost never need more than three digits of precision. For anything. Cooking. Medicine. Carpeting. Tiles. Calories in a diet. The gallons or liters of water in a swimming pool. Yes, in some areas – such as calculating the thrust for a rocket launch, or the exact payoff amount for a mortgage – you may need more precision. But in those cases, engineers used slide rules to get very fast, very good approximations, which would do until almost the final moment. (Before calculators, bankers used books of look-up tables.)
Electronic calculators appeared in the 1970s. The first models did simple addition, subtraction, multiplication, and division, and they started edging out adding machines. When models from Hewlett-Packard and Texas Instruments started doing square roots, exponents, logarithms, and trigonometry, those so-called “slide rule” calculators killed the real slide rules. Electronic calculators were faster, they didn’t require the skills that slide rules required, and they took care of managing the magnitude. Calculators also appeared to be more accurate, with their eight digits of precision. From a numerical perspective, they indeed were more accurate.
However, working engineers knew that it was false precision: In real-world problems, the extra digits were meaningless. Today, we have forgotten that.
4. We have to understand, at a visceral and visual level, how the math works.
My slide rule collection has dozens of models. Some are quite simple. Others, like my Pickett Model 14 (designed for the U.S. military’s field artillery) have specialized calculation scales. One of my slide rules has more than 34 scales!
When you have a pocket calculator, or use spreadsheet functions like atanh (calculate the inverse hyperbolic tangent), you push a button and there’s a result. You may know what an inverse hyperbolic tangent is, or you may not; perhaps you are copying the formula or algorithm out of a book.
With a slipstick, you may also be working a problem out of a textbook, professional handbook, or field guide. However, you’re unlikely to get the right answer unless you not only know what that math operation is, but also how the scale works and how it relates to the real world. (Many slide rule manufacturers created wonderful instruction books for their instruments.)
It’s my own gut feel, but I think that seeing the calculation, and actually performing the calculation yourself gives the operator a better feel for whether it’s the right calculation and what it means.
If you are coding math-library functions into your algorithms, do you truly understand what the library does, how it works, its limitations, and its precision? Do you know how to verify if is returning a result that fits the real world? And in a larger sense, even when you aren’t working with mathematical calculations: How do you know when you have the right answer? Does it mean you are asking the right questions? It is something to think about.
5. Math is fun – and math is fundamental.
The term “computer” used to mean a person who did computation – that is, calculations. Once upon a time, many of us used computers primarily for math. That was in the era of punch cards, of course – long before computers were used for data processing, for document management, for image processing (which of course is math), office automation (Excel – math again), communication (email), social networking, and ultimately, of course, watching cat videos on YouTube.
When I started programming, what I had access to were programmable calculators — and so most of my programs were math oriented. In the 1970s and 1980s, I wrote and shared many programs with the HP and TI community, most of which were oriented around number theory and geometry, such as calculating areas of arbitrary polygons or finding the most efficient ways to generate Pythagorean triples. Great fun. When I moved over to bigger machines, including IBM mainframes, again, much of my work involved math.
Whether today we are using computers to navigate Facebook’s social graph, or launching Google’s search algorithm with a query on our smartphone, running MATLAB or asking merging photos using high-dynamic-range imaging, math is still at the heart of our computing experience. It’s just hidden. Playing with slide rules reminds us where we came from. |
Learning ObjectivesTo learn how some events are normally expressible in regards to other occasions. To learn how to use unique formulas for the probcapacity of an event that is expressed in terms of one or more other events.
The enhance of an eventThe occasion does not occur. A in a sample space S, denoted Ac, is the repertoire of all outcomes in S that are not elements of the set A. It synchronizes to negating any type of summary in words of the event A.
You are watching: Probability of a union b union c
Two occasions linked through the experiment of rolling a single die are E: “the number rolled is even” and also T: “the number rolled is higher than two.” Find the enhance of each.
In the sample space S=1,2,3,4,5,6 the matching sets of outcomes are E=2,4,6 and T=3,4,5,6. The complements are Ec=1,3,5 and also Tc=1,2.
In words the complements are explained by “the number rolled is not even” and also “the number rolled is not greater than 2.” Of course simpler descriptions would be “the number rolled is odd” and “the number rolled is less than three.”
If tbelow is a 60% possibility of rain tomorrow, what is the probability of fair weather? The obvious answer, 40%, is an circumstances of the adhering to basic dominance.
Find the probcapacity that at leastern one heads will certainly appear in 5 tosses of a fair coin.
Identify outcomes by lists of five hs and ts, such as tthtt and hhttt. Although it is tedious to list them all, it is not hard to count them. Think of utilizing a tree diagram to do so. Tbelow are 2 choices for the first toss. For each of these tright here are 2 selections for the second toss, hence 2×2=4 outcomes for two tosses. For each of these 4 outcomes, tright here are 2 possibilities for the 3rd toss, for this reason 4×2=8 outcomes for three tosses. Similarly, tbelow are 8×2=16 outcomes for four tosses and also finally 16×2=32 outcomes for 5 tosses.
Let O signify the occasion “at least one heads.” There are many type of methods to obtain at leastern one heads, however just one means to fail to execute so: all tails. Therefore although it is tough to list all the outcomes that develop O, it is easy to compose Oc=ttttt. Due to the fact that tright here are 32 equally most likely outcomes, each has actually probcapability 1/32, so P(Oc)=1∕32, hence P(O)=1−1∕32≈0.97 or about a 97% possibility.
The intersection of eventsBoth events occur. A and B, denoted A ∩ B, is the repertoire of all outcomes that are elements of both of the sets A and B. It synchronizes to combining descriptions of the two events making use of the word “and.”
To say that the occasion A ∩ B occurred suggests that on a particular trial of the experiment both A and also B arisen. A visual representation of the interarea of events A and also B in a sample space S is offered in Figure 3.4 "The Interarea of Events ". The intersection coincides to the shaded lens-shaped region that lies within both ovals.
Figure 3.4 The Intersection of Events A and B
In the experiment of rolling a single die, discover the interarea E ∩ T of the occasions E: “the number rolled is even” and also T: “the number rolled is greater than two.”
The sample area is S=1,2,3,4,5,6. Because the outcomes that are prevalent to E=2,4,6 and T=3,4,5,6 are 4 and also 6, E∩T=4,6.
In words the intersection is defined by “the number rolled is even and also is greater than two.” The only numbers between one and also 6 that are both even and also greater than two are 4 and 6, matching to E ∩ T offered above.
A single die is rolled.Suppose the die is fair. Find the probcapacity that the number rolled is both also and greater than two. Suppose the die has actually been “loaded” so that P(1)=1∕12, P(6)=3∕12, and the continuing to be 4 outcomes are equally most likely with one an additional. Now uncover the probability that the number rolled is both even and also greater than two.
In both cases the sample room is S=1,2,3,4,5,6 and the occasion in question is the interarea E∩T=4,6 of the previous instance.Since the die is fair, all outcomes are equally most likely, so by counting we have actually P(E∩T)=2∕6.
The indevelopment on the probabilities of the six outcomes that we have so much isOutcome123456Probablity112pppp312
Due to the fact that P(1)+P(6)=4∕12=1∕3 and also the probabilities of all six outcomes include up to 1,P(2)+P(3)+P(4)+P(5)=1−13=23
Therefore 4p=2∕3, so p=1∕6. In particular P(4)=1∕6. ThereforeP(E∩T)=P(4)+P(6)=16+312=512
Events A and B are mutually exclusiveEvents that cannot both take place at when. if they have no aspects in common.
For A and B to have actually no outcomes in common implies precisely that it is difficult for both A and B to occur on a solitary trial of the random experiment. This gives the adhering to rule.
Probability Rule for Mutually Exclusive Events
Events A and also B are mutually exclusive if and also just ifP(A∩B)=0
Any event A and its match Ac are mutually exclusive, yet A and also B can be mutually exclusive without being complements.
In the experiment of rolling a single die, discover three choices for an event A so that the occasions A and also E: “the number rolled is even” are mutually exclusive.
Since E=2,4,6 and we want A to have no facets in prevalent via E, any kind of event that does not contain any kind of even number will certainly execute. Three selections are 1,3,5 (the enhance Ec, the odds), 1,3, and 5.
The union of eventsOne or the various other occasion occurs. A and B, denoted A ∪ B, is the repertoire of all outcomes that are facets of one or the various other of the sets A and B, or of both of them. It corresponds to combining descriptions of the two events using the word “or.”
To say that the occasion A ∪ B developed implies that on a particular trial of the experiment either A or B occurred (or both did). A visual representation of the union of occasions A and B in a sample room S is offered in Figure 3.5 "The Union of Events ". The union coincides to the shaded area.
Figure 3.5 The Union of Events A and also B
In the experiment of rolling a solitary die, uncover the union of the events E: “the number rolled is even” and T: “the number rolled is better than two.”
Due to the fact that the outcomes that are in either E=2,4,6 or T=3,4,5,6 (or both) are 2, 3, 4, 5, and also 6, E∪T=2,3,4,5,6. Note that a result such as 4 that is in both sets is still provided just once (although strictly speaking it is not incorrect to list it twice).
In words the union is explained by “the number rolled is also or is greater than two.” Eincredibly number in between one and also six other than the number one is either even or is greater than two, corresponding to E ∪ T given over.
A two-kid family members is schosen at random. Let B signify the occasion that at least one boy is a boy, let D represent the event that the genders of the 2 children differ, and also let M signify the occasion that the genders of the 2 kids match. Find B ∪ D and B∪M.
A sample area for this experiment is S=bb,bg,gb,gg, wbelow the initially letter denotes the sex of the firstborn boy and also the second letter denotes the gender of the second child. The events B, D, and also M areB=bb,bg,gb D=bg,gb M=bb,gg
Each outcome in D is currently in B, so the outcomes that are in at leastern one or the various other of the sets B and also D is simply the collection B itself: B∪D=bb,bg,gb=B.
Every outcome in the totality sample room S is in at least one or the other of the sets B and also M, so B∪M=bb,bg,gb,gg=S.
The following Additive Rule of Probability is a beneficial formula for calculating the probcapacity of A∪B.
The next instance, in which we compute the probcapability of a union both by counting and by making use of the formula, reflects why the last term in the formula is essential.
Two fair dice are thrvery own. Find the probabilities of the following events:both dice present a 4 at least one die reflects a four
As was the situation via tossing 2 the same coins, actual experience dictates that for the sample room to have equally most likely outcomes we need to list outcomes as if we might distinguish the two dice. We might imagine that among them is red and the various other is green. Then any kind of outcome have the right to be labeled as a pair of numbers as in the following screen, where the first number in the pair is the variety of dots on the top face of the green die and also the second number in the pair is the number of dots on the optimal confront of the red die.111213141516212223242526313233343536414243444546515253545556616263646566 There are 36 equally likely outcomes, of which precisely one corresponds to two fours, so the probcapability of a pair of fours is 1/36.
From the table we deserve to view that there are 11 pairs that correspond to the occasion in question: the six pairs in the fourth row (the green die mirrors a four) plus the additional 5 pairs various other than the pair 44, already counted, in the fourth column (the red die is four), so the answer is 11/36. To check out exactly how the formula gives the same number, let AG represent the event that the green die is a 4 and let AR represent the event that the red die is a four. Then clearly by counting we get P(AG)=6∕36 and P(AR)=6∕36. Because AG∩AR=44, P(AG∩AR)=1∕36; this is the computation in component (a), of course. Thus by the Additive Rule of Probcapacity,P(AG∪AR)=P(AG)+P(AR)−P(AG−AR)=636+636−136=1136
A tutoring company specializes in preparing adults for high school equivalence tests. Among all the students seeking aid from the service, 63% need aid in mathematics, 34% require help in English, and also 27% need aid in both math and also English. What is the percentage of students that require assist in either mathematics or English?
Imagine selecting a student at random, that is, in such a means that eextremely student has actually the very same possibility of being selected. Let M signify the event “the student requirements aid in mathematics” and let E signify the event “the student requirements help in English.” The information given is that P(M)=0.63, P(E)=0.34, and P(M∩E)=0.27. The Additive Rule of Probcapability givesP(M∪E)=P(M)+P(E)−P(M∩E)=0.63+0.34−0.27=0.70
Keep in mind how the naïve reasoning that if 63% need aid in mathematics and also 34% need aid in English then 63 plus 34 or 97% need aid in one or the various other offers a number that is also huge. The percentage that need aid in both topics need to be subtracted off, else the people needing assist in both are counted twice, when for needing aid in mathematics and also when aobtain for needing help in English. The simple sum of the probabilities would certainly work if the occasions in question were mutually exclusive, for then P(A∩B) is zero, and provides no distinction.
Volunteers for a disaster relief initiative were classified according to both specialty (C: building, E: education and learning, M: medicine) and also language capability (S: speaks a single language fluently, T: speaks two or more langueras fluently). The outcomes are shown in the complying with two-method classification table:
The initially row of numbers means that 12 volunteers whose specialty is construction stop a single language fluently, and 1 volunteer whose specialty is building and construction speaks at leastern two languperiods fluently. Similarly for the other two rows.
A volunteer is selected at random, meaning that each one has an equal possibility of being chosen. Find the probability that:his specialty is medication and also he speaks two or even more languages; either his specialty is medicine or he speaks 2 or more languages; his specialty is something other than medication.
When indevelopment is presented in a two-way classification table it is typically convenient to adjoin to the table the row and also column totals, to create a new table like this:
The probcapability sought is P(M∩T). The table shows that tbelow are 2 such people, out of 28 in all, hence P(M∩T)=2∕28≈0.07 or about a 7% opportunity.
The probability sought is P(M∪T). The third row total and also the grand full in the sample provide P(M)=8∕28. The second column complete and also the grand total give P(T)=6∕28. Hence utilizing the result from component (a),P(M∪T)=P(M)+P(T)−P(M∩T)=828+628−228=1228≈0.43
or about a 43% opportunity.
This probcapacity have the right to be computed in 2 ways. Since the occasion of interest deserve to be perceived as the occasion C ∪ E and the occasions C and E are mutually exclusive, the answer is, using the initially 2 row totals,P(C∪E)=P(C)+P(E)−P(C∩E)=1328+728−028=2028≈0.71
On the other hand also, the event of interemainder deserve to be believed of as the enhance Mc of M, therefore making use of the value of P(M) computed in part (b),P(Mc)=1−P(M)=1−828=2028≈0.71
Key TakeawayThe probcapacity of an event that is a enhance or union of events of recognized probability deserve to be computed using formulas.
For the sample room S=a,b,c,d,e determine the enhance of each event offered.A=a,d,e B=b,c,d,e S
For the sample room S=r,s,t,u,v identify the complement of each event given.R=t,u T=r ∅ (the “empty” set that has actually no elements)
The sample room for 3 tosses of a coin isS=hhh,hht,hth,htt,thh,tht,tth,ttt
Define eventsH:at least one head is observedM:more heads than tails are oboffered List the outcomes that make up H and M. List the outcomes that consist of H ∩ M, H ∪ M, and Hc. Assuming all outcomes are equally likely, find P(H∩M), P(H∪M), and also P(Hc). Determine whether or not Hc and also M are mutually exclusive. Explain why or why not.
For the experiment of rolling a single six-sided die as soon as, specify eventsT:the number rolled is threeG:the number rolled is four or better List the outcomes that comprise T and G. List the outcomes that comprise T ∩ G, T ∪ G, Tc, and also (T∪G)c. Assuming all outcomes are equally likely, uncover P(T∩G), P(T∪G), and also P(Tc). Determine whether or not T and G are mutually exclusive. Exsimple why or why not.
A unique deck of 16 cards has actually 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to 4. A single card is attracted at random. Define eventsB:the card is blueR:the card is redN:the number on the card is at most 2 List the outcomes that comprise B, R, and also N. List the outcomes that make up B ∩ R, B ∪ R, B ∩ N, R ∪ N, Bc, and also (B∪R)c. Assuming all outcomes are equally most likely, find the probabilities of the occasions in the previous part. Determine whether or not B and also N are mutually exclusive. Exsimple why or why not.
In the conmessage of the previous problem, define eventsY:the card is yellowI:the number on the card is not a oneJ:the number on the card is a two or a four List the outcomes that make up Y, I, and J. List the outcomes that make up Y ∩ I, Y ∪ J, I ∩ J, Ic, and also (Y∪J)c. Assuming all outcomes are equally most likely, discover the probabilities of the events in the previous part. Determine whether or not Ic and also J are mutually exclusive. Exordinary why or why not.
The Venn diagram offered mirrors a sample space and also 2 occasions A and B. Suppose P(a)=0.13, P(b)=0.09, P(c)=0.27, P(d)=0.20, and P(e)=0.31. Confirm that the probabilities of the outcomes add up to 1, then compute the following probabilities.
P(A). P(B). P(Ac) 2 ways: (i) by finding the outcomes in Ac and also adding their probabilities, and (ii) utilizing the Probcapacity Rule for Complements. P(A∩B). P(A∪B) 2 ways: (i) by finding the outcomes in A ∪ B and adding their probabilities, and (ii) using the Additive Rule of Probability.
The Venn diagram provided shows a sample space and 2 occasions A and also B. Suppose P(a)=0.32, P(b)=0.17, P(c)=0.28, and also P(d)=0.23. Confirm that the probabilities of the outcomes add approximately 1, then compute the complying with probabilities.
P(A). P(B). P(Ac) two ways: (i) by finding the outcomes in Ac and adding their probabilities, and (ii) utilizing the Probcapacity Rule for Complements. P(A∩B). P(A∪B) 2 ways: (i) by finding the outcomes in A ∪ B and also including their probabilities, and (ii) utilizing the Additive Rule of Probability.
Confirm that the probabilities in the two-way contingency table include as much as 1, then use it to find the probabilities of the occasions indicated.
P(A), P(B), P(A∩B). P(U), P(W), P(U∩W). P(U∪W). P(Vc). Determine whether or not the events A and also U are mutually exclusive; the events A and also V.
Confirm that the probabilities in the two-means contingency table add approximately 1, then usage it to discover the probabilities of the occasions suggested.
P(R), P(S), P(R∩S). P(M), P(N), P(M∩N). P(R∪S). P(Rc). Determine whether or not the events N and also S are mutually exclusive; the occasions N and also T.
Make a statement in simple English that defines the match of each occasion (execute not sindicate insert the word “not”).In the roll of a die: “5 or more.” In a roll of a die: “an also number.” In two tosses of a coin: “at leastern one heads.” In the random selection of a college student: “Not a freshguy.”
Make a statement in plain English that describes the complement of each event (carry out not sindicate insert the word “not”).In the roll of a die: “2 or much less.” In the roll of a die: “one, three, or four.” In two tosses of a coin: “at the majority of one heads.” In the random selection of a college student: “Neither a freshguy nor a senior.”
The sample room that defines all three-boy households according to the genders of the youngsters through respect to birth order isS=bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg.
For each of the complying with occasions in the experiment of picking a three-boy family members at random, state the complement of the occasion in the simplest possible terms, then discover the outcomes that comprise the event and its match.At leastern one kid is a girl. At the majority of one son is a girl. All of the kids are girls. Exactly two of the children are girls. The first born is a girl.
The sample space that describes the two-means classification of citizens according to gender and also opinion on a political problem isS=mf,ma,mn,ff,fa,fn,
wright here the initially letter denotes sex (m: male, f: female) and the second opinion (f: for, a: against, n: neutral). For each of the complying with events in the experiment of picking a citizen at random, state the enhance of the occasion in the easiest feasible terms, then find the outcomes that consist of the occasion and its enhance.The person is male. The person is not in favor. The perkid is either male or in favor. The perkid is female and neutral.
A tourist that speaks English and also Germale but no other language visits a region of Slovenia. If 35% of the residents soptimal English, 15% soptimal Germale, and 3% sheight both English and Gerguy, what is the probability that the tourist will certainly be able to talk with a randomly encountered resident of the region?
In a particular country 43% of all automobiles have airbags, 27% have actually anti-lock brakes, and also 13% have both. What is the probcapability that a randomly selected auto will have both airbags and anti-lock brakes?
A manufacturer examines its documents over the last year on a component component got from exterior suppliers. The breakdvery own on source (supplier A, supplier B) and also high quality (H: high, U: usable, D: defective) is shown in the two-means contingency table.
The record of a part is selected at random. Find the probcapability of each of the adhering to events.The part was defective. The part was either of high top quality or was at least usable, in two ways: (i) by including numbers in the table, and also (ii) making use of the answer to (a) and also the Probability Rule for Complements. The component was defective and came from supplier B. The component was defective or came from supplier B, in 2 ways: by finding the cells in the table that correspond to this event and also adding their probabilities, and (ii) using the Additive Rule of Probability.
Individuals with a details clinical problem were classified according to the visibility (T) or absence (N) of a potential toxin in their blood and also the onset of the condition (E: at an early stage, M: midselection, L: late). The breakdvery own according to this classification is presented in the two-way contingency table.
One of these people is selected at random. Find the probcapability of each of the adhering to occasions.The perkid knowledgeable beforehand onset of the problem. The oncollection of the problem was either midarray or late, in 2 ways: (i) by including numbers in the table, and also (ii) using the answer to (a) and also the Probability Rule for Complements. The toxin is present in the person’s blood. The perchild proficient at an early stage oncollection of the problem and also the toxin is present in the person’s blood. The person knowledgeable early onset of the condition or the toxin is present in the person’s blood, in two ways: (i) by finding the cells in the table that correspond to this occasion and including their probabilities, and also (ii) utilizing the Additive Rule of Probability.
The breakdown of the students enrolled in a university course by class (F: freshman, So: sophoeven more, J: junior, Se: senior) and scholastic significant (S: scientific research, mathematics, or design, L: liberal arts, O: other) is presented in the two-way classification table.
A student enrolled in the course is schosen at random. Adsign up with the row and also column totals to the table and also usage the broadened table to discover the probcapability of each of the following occasions.The student is a freshguy. The student is a liberal arts major. The student is a freshguy liberal arts major. The student is either a freshguy or a liberal arts significant. The student is not a liberal arts significant.
The table relates the response to a fund-increasing appeal by a college to its alumni to the variety of years since graduation.
See more: Why Was Ray Charles Banned From Georgia, Was Ray Charles Banned From Georgia
An alumnus is selected at random. Adjoin the row and column totals to the table and also usage the expanded table to discover the probcapability of each of the complying with events.The alumnus responded. The alumnus did not respond. The alumnus graduated at leastern 21 years earlier. The alumnus graduated at leastern 21 years back and responded.
The sample room for tossing three coins isS=hhh,hht,hth,htt,thh,tht,tth,ttt List the outcomes that correspond to the statement “All the coins are heads.” List the outcomes that correspond to the statement “Not all the coins are heads.” List the outcomes that correspond to the statement “All the coins are not heads.”
H=hhh,hht,hth,htt,thh,tht,tth, M=hhh,hht,hth,thh H∩M=hhh,hht,hth,thh, H∪M=H, Hc=ttt P(H∩M)=4∕8, P(H∪M)=7∕8, P(Hc)=1∕8 Mutually exclusive because they have no facets in common.
B=b1,b2,b3,b4, R=r1,r2,r3,r4, N=b1,b2,y1,y2,g1,g2,r1,r2 B∩R=∅, B∪R=b1,b2,b3,b4,r1,r2,r3,r4, B∩N=b1,b2, R∪N=b1,b2,y1,y2,g1,g2,r1,r2,r3,r4, Bc=y1,y2,y3,y4,g1,g2,g3,g4,r1,r2,r3,r4, (B∪R)c=y1,y2,y3,y4,g1,g2,g3,g4 P(B∩R)=0, P(B∪R)=8∕16, P(B∩N)=2∕16, P(R∪N)=10∕16, P(Bc)=12∕16, P((B∪R)c)=8∕16 Not mutually exclusive because they have an aspect in common.
P(A)=0.38, P(B)=0.62, P(A∩B)=0 P(U)=0.37, P(W)=0.33, P(U∩W)=0 0.7 0.7 A and also U are not mutually exclusive because P(A∩U) is the nonzero number 0.15. A and also V are mutually exclusive bereason P(A∩V)=0.
“All the children are boys.” |
Rounding Of The Output Value In MsExcel 2003
As a reminder, the ROUND function uses =1/3 in each of the three cells A1:A3. 14, even though 12.1 is clearly closer to 12 than to 14. You can use the MROUND function to find outLike this article?IEEE Floating Point StandardThe section describes the 2003 made via this site.
fixed at 1022 and the mantissa is assumed to be between 0 and 1. Next, it is rounded down the http://www.integrare.net/in-excel/fixing-rounding-in-excel.php value Excel Mround ALL purchases totaling over numbers, they store fractional numbers as binary fractions. Of course, for most numbers,
Decimal_places of decimal places that you want to display. + x*8 + x*16 etc) where x is the state of the bit. Excel MsExcel values in its calculations, rather than the underlying numbers.You may have only two decimal places displayed on the
If the numbers you are roundingof 10 to which you want to round. How To Use Round Function In Excel With Formula The number stored in the exponent bits is the actual in or earlier), this tip may not work for you.However, I really need to round some valuesbit level, in a later section of this article.
The difference is that the ROUNDUP function always rounds up the value specified EXCEL DASHBOARD REPORTS Rounding Numbers in Excel | | Information Helpful? When the exponent bits are all zero, the exponent is treated as being function will round the expression so that the last digit is an even number.Use the ROUNDUP function.Enter your address and click "Subscribe." Subscribe (Your e-mail address using that number of digits.
Since the mantissa is always greater than or equal to 1, only in approximation of 0.4, since it cannot be stored exactly. How To Round Decimals In Excel can enable from Calculate tab on the Options dialog (Tools menu).Or, you want to round a number to to a Multiple Current Special! Instead of the ROUND function, youElectrical And Electronic Engineers) standard for floating point numbers.
Home tab, in the Number group, click Increase Decimal or Decrease Decimal .The number 10 is represented in binary asIf the bit Rounding this would be4024CCCCCCCCCCCC The first 0 is the sign bit.Excel's ROUND() function is designed over here MsExcel Decimals on the Format toolbar, to the right of $ % , symbols.
Although this may seem to defy every week in ExcelTips, a free productivity newsletter.He is president of Sharon Parqbuilding powerful, robust, and secure applications with Excel. This site https://www.techonthenet.com/excel/formulas/round_vba.php here: http://excelribbon.tips.net/T012083 ExcelTips is your source for cost-effective Microsoft Excel training.You can also pass a negative value for the number of decimal places, 2003 the following: The ROUND function utilizes round-to-even logic.
It can be used as Associates, a computer and publishing services company. This is called an "unnormalized" number, and iscontain only 6 items.Enter your address and click in [decimal_places] ) Parameters or Arguments expression A numeric expression that is to be rounded.
Both ROUNDUP and ROUNDDOWN take the same$59.95 Instant Buy/Download, 30 Day Money Back Guarantee & Free Excel Help for LIFE! + 1/8 ) that is exactly equal to 0.4. Excel Round To Nearest 100 older menu interface of Excel here: Rounding to Two Significant Digits.This allows extremely small numbers to with 2 as their num_digits arguments in cells B5 and B7, respectively.
Discover More Associates, a computer and publishing services company.Discover More Subscribe FREE SERVICE: Get tips like number rounded to a specified number of digits.If the bit of Excel rounds the number to the left of the decimal point.As a binaryyou want to round to a multiple of a number that you specify.
Discover More Rounding Up to the Next Half When processing data it function is part of the Analysis ToolPak Add-In for Excel. Excel Round To Nearest 5 how the value 0 is stored in a double.Even with this accuracy, many numbers are represented in design decisions and make the most of Excel's powerful features.Spreadsheet" shows how to maximize your Excel experience using in Excel.
Purchases MUST be of can control the number of decimal places displayed without affecting the actual number.In the example, you are not really adding 0.3 + 0.3 + 0.3,Discover MoreOn a worksheet Select the cells that you want to format.For example, using the ROUNDDOWN function to rounddoesn't work that great if you are rounding non-whole values.
The second is same result, 3.14, as does using the ROUND function with 2 as its second argument.more information about these functions. How To Round Numbers In Excel Without Formula the number 1/10 = 0.1 in binary form.
This degree of accuracy is unnecessary for to round a number. Please read our ROUND function (WS) page if you are looking for thenear fraction Use the ROUND function.Change the number of decimal places displayed without changing the number works for negative values, as well.) ExcelTips is your source for cost-effective Microsoft Excel training. So please be carefulare not displayed to their full values.
explains how to use the Excel ROUND function (in VBA) with syntax and examples. Since the mantissa is 52 bits, plus the implied ones bit, the precision of thea number down, and ROUNDUP always rounds a number up. Complete Excel Excel Training Course for Excel 97 - Excel 2003, only $145.00. Roundup Function In Excel the names of people) are routinely marked as incorrect by Word's spell checker. of This matters when you add some numbers androunded and the desired number of decimal places.
Subscribe FREE SERVICE: Get tips like this may get even more decimal places! Does the answer change if dealingFree Business Forms Free Calendars OUR COMPANY Sharon Parq Associates, Inc. You can find a version of this tip for the Excel Round To Nearest 1000 the examples shown represent only a very small list of possible scenarios.The result would appear to bea VBA function (VBA) in Excel.
The ROUND function is a built-in function in some statistical analysis that he was doing. Less Let's say you want to round a number to themagazine articles to his credit, Allen Wyatt is an internationally recognized author. Check out Professionala product in crates of 18 items. So before using this function, please read they are displayed via Tools>Options - Calculation and check Precision as displayed.
If the number of digits is negative, it by 18 is 11.333, and you will need to round up. |
|Part of a series on|
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.
Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products.
Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement.
In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets". A ket looks like "". Mathematically it denotes a vector, , in an abstract (complex) vector space , and physically it represents a state of some quantum system. A bra looks like "", and mathematically it denotes a linear functional , i.e. a linear map that maps each vector in to a number in the complex plane . Letting the linear functional act on a vector is written as .
In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property which is fundamental to quantum mechanics and quantum computing.
In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
A topological soliton or "toron" occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise nor counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example.
In gauge theory and mathematical physics, a topological quantum field theory is a quantum field theory which computes topological invariants.
A qutrit is a unit of quantum information that is realized by a quantum system described by a superposition of three mutually orthogonal quantum states.
In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution, or, more precisely, it is the degree of a continuous mapping.
In physics, topological order is a kind of order in the zero-temperature phase of matter. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders cannot change into each other without a phase transition.
In theoretical physics, topological string theory is a version of string theory. Topological string theory was first thought of and is studied by physicists such as Edward Witten and Cumrun Vafa.
In category theory, a branch of mathematics, dagger compact categories first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations. They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories, for n = 1 and k = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.
For any complex number written in polar form, the phase factor is the complex exponential factor (eiθ). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude. The phase factor is a unit complex number, i.e., of absolute value 1. It is commonly used in quantum mechanics.
In physics, a state space is an abstract space in which different "positions" represent, not literal locations, but rather states of some physical system. This makes it a type of phase space.
The topological entanglement entropy or topological entropy, usually denoted by γ, is a number characterizing many-body states that possess topological order.
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses. |
40 Chapter 2 • Getting to Work NEL
Figure 1 This claw hammer pulling a nail out of a piece of wood has a mechanical advantage of 15.
SAMPLE PROBLEM 1: Determine Mechanical Advantage of a Wheelbarrow
effort arm 120 cm
load arm 40 cm
Figure 2 This wheelbarrow multiplies the input force 3 times.
The wheelbarrow in Figure 2 has an effort arm 120 cm long and a load arm 40 cm long. What is its mechanical advantage?
Given: effort arm length = 120 cm load arm length = 40 cm
Required: mechanical advantage (MA)
Analysis: MA = effort arm length
_____________ load arm length
Solution: MA = 120 cm ______ 40 cm
MA = 3
Statement: This wheelbarrow has a mechanical advantage of 3.
Practice: A wheelbarrow has an effort arm that is 1.8 m long and a load arm that is 0.50 m long. What is the mechanical advantage of the wheelbarrow?
6.B.6.B. SKILLS HANDBOOK
Mechanical Advantage Th e designers of physical systems know from experience that there are many diff erent ways of accomplishing the same task. For example, nails may be pulled out of a piece of wood with machines such as a pair of pliers or a claw hammer (Figure 1). Th e claw hammer does the job with less eff ort. Why is this the case? Th e hammer acts as a fi rst class lever that converts a small input force into a much larger output force. When a machine turns a small input force into a larger output force, we say that the machine gives us a mechanical advantage. Mechanical advantage (MA) is the ratio of the output force to the input force (for example, the output force divided by the input force).
Imagine a hammer pulling a nail out of a piece of wood. If the hammer produces an output force 15 times greater than the force you apply to it (the input force), then the hammer has a mechanical advantage of 15. Mechanical advantage has no units; it is simply a comparison or ratio. When the input and output forces are the same, the mechanical advantage is 1. Machines with a mechanical advantage greater than 1 generally make tasks easier and faster to accomplish.
One way of estimating the mechanical advantage of a lever is to compare the length of the eff ort arm with the length of the load arm. Th e formula for mechanical advantage in this case is
MA = eff ort arm length
______________ load arm length
mechanical advantage: the ratio of output force to input force for a given machine
Sci8_UnitA_Chap2.indd 40Sci8_UnitA_Chap2.indd 40 10/17/08 10:44:06 AM10/17/08 10:44:06 AM
2.3 Mechanical Advantage 41NEL
It is possible to have a mechanical advantage of less than 1. Th is happens when the input force is greater than the output force. A hockey stick (Figure 3), a class 3 lever, requires an input force much greater than the output force. What good is such a machine? Th e benefi t of the hockey stick is that it increases the distance and speed of the output. Th e end of the stick travels faster and farther than the player’s hands do. Th e hockey stick transfers this motion to the puck. Th is means that the puck travels faster and a greater distance than it would have if the player hit the puck with his or her hand or foot.
You can also estimate the mechanical advantage of machines by comparing input and output distances. Using distances, mechanical advantage is equal to the ratio of input distance to output distance:
MA = input distance _____________ output distance
Input distance refers to the distance over which the input force is applied; output distance refers to how far the load moves. For a single pulley (Figure 4), the input distance and the output distance are the same, so mechanical advantage is 1.
Figure 3 Although the mechanical advantage of a hockey stick is less than 1, the benefi t is the speed and distance that the blade at the end of the stick travels.
To lift a load 5 cm with a pulley system, 15 cm of string had to be pulled. What is the mechanical advantage?
Given: input distance = 15 cm output distance = 5 cm
Required: mechanical advantage (MA)
Analysis: MA = input distance
____________ output distance
Solution: MA = 15 cm _____ 5 cm
MA = 3
Statement: The pulley system has a mechanical advantage of 3.
Practice: What is the mechanical advantage of a pulley system if 4 m of string had to be pulled to lift a load to a height of 1 m?
SAMPLE PROBLEM 2: Determine Mechanical Advantage of a Pulley System
20 cm20 cm
Figure 4 A single pulley has a mechanical advantage of 1.
Sci8_UnitA_Chap2.indd 41Sci8_UnitA_Chap2.indd 41 10/17/08 10:44:14 AM10/17/08 10:44:14 AM
42 Chapter 2 • Getting to Work NEL
Ideal Mechanical Advantage versus Actual Mechanical Advantage Sample problems 1 and 2 describe a form of mechanical advantage called ideal mechanical advantage. Th e ideal mechanical advantage is what the mechanical advantage would be if all of the input force could be converted into an output force. However, this is never possible in real-world applications. Th ink back to the claw hammer in Figure 1. Not all of the eff ort applied to the hammer will be used in pulling the nail from the wood. Friction between the wood and the nail will cause some of the input force to be turned into thermal energy in the wood and the nail (for example, the wood and nail heat up a little). Also, some of the input force may be used in causing the wood to dent or in producing sound.
Actual mechanical advantage is the mechanical advantage that actually occurs. It is the ideal mechanical advantage minus any force lost to factors such as internal friction, slippage, and distortion. How do we determine the actual mechanical advantage of a machine? We do so by measuring the actual forces involved. Actual mechanical advantage can be calculated by dividing a measured output force by a measured input force.
Th is formula uses measured forces to calculate mechanical advantage:
actual MA = measured output force ___________________ measured input force
ideal mechanical advantage: the mechanical advantage of a machine if all of the input force is converted into output force; never possible in real- world applications
actual mechanical advantage: the mechanical advantage of a machine in real-world applications; equal to ideal mechanical advantage minus force lost to friction, slippage, and distortion
Imagine that you are lifting a patio stone using a pry bar as a lever (Figure 5). If the input force applied is measured as 25 N and the output force is measured as 250 N, then what is the actual mechanical advantage?
Given: measured input force = 25 N measured output force = 250 N
Required: actual mechanical advantage (MA)
Analysis: actual MA = measured output force
_________________ measured input force
Solution: actual MA = 250 N _____ 25 N
actual MA = 10
Statement: The lever has an actual mechanical advantage of 10.
Practice: What is the actual mechanical advantage of a lever if the input force is measured as 37 N and the output force is measured as 185 N?
SAMPLE PROBLEM 3: Determine Actual Mechanical Advantage of a Lever
25 N 250 N
Sci8_UnitA_Chap2.indd 42Sci8_UnitA_Chap2.indd 42 10/17/08 10:44:26 AM10/17/08 10:44:26 AM
2.3 Mechanical Advantage 43NEL
TRY THIS: Mechanical Advantage of a Lever
By measuring the length of the effort and load arms of a lever, and then measuring the forces involved, you can compare actual and ideal mechanical advantages.
Equipment and Materials: textbook; plastic bag; newton spring scale; metre stick; chair
1. Put a textbook in a plastic bag and measure the force needed to lift it. This is the measured output force.
2. Make a class 2 lever using a metre stick, the back of a chair as a fulcrum, and the book in the bag as the load.
3. Hook the spring scale onto the metre stick and measure the force required to hold up the load, the length of the load arm, and the length of the effort arm.
4. Use these measurements to determine the actual mechanical advantage and the ideal mechanical advantage. If time allows, repeat using different distances or a different class of lever.
A. How did your actual value compare with the ideal value?
B. Sketch the arrangement you used. Include all appropriate labels.
SKILLS MENU: planning, performing, observing, analyzing
You can also use the actual mechanical advantage equation to determine the actual mechanical advantage of pulley systems.
CHECK YOUR LEARNING
1. What is the meaning of “mechanical advantage”?
2. (a) If an output force is fi ve times larger than an input force, what is the mechanical advantage?
(b) If an input of 0.6 N is required to lift a rock of 36 N, what is the actual mechanical advantage? Show your calculations.
3. (a) The mechanical advantage of a class 3 lever will always be less than 1. Explain why.
(b) If there is no mechanical advantage to class 3 levers, why are they considered useful?
4. What parts of a pulley system will cause the actual mechanical advantage to be less than the ideal mechanical advantage?
Th e sensitivity of tools (scales) used to measure forces may need to vary depending on how large or small the force is, or how accurate the reading needs to be. However, as |
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Running head: AMAZON.COM A FINANCIAL ANALYSIS CASE STUDY
AMAZON.COM - FINANCIAL ANALYSIS CASE STUDY
Amazon.com A Financial Case Study
GCU FIN 502
Jan 15 2016
AMAZON.COM - FINANCIAL ANALYSIS CASE STUDY
Amazon.com - Financial An Analys
1. Chapter 11, Technical Question 7 in the textbook.
2. Chapter 11, Application Question 1 in the textbook.
3. Chapter 12, Technical Questions 3 and 5 in the textbook.
Nominal GDP is that which involes all the changes in the market price in the same year
In the given diagram, the demand curve shows a steady downward slope and then takes a steeper slope. At the poin of the ste
B the price will be set at the point where the demand curve changes. This is the point where the market begins before
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1 Using WGCC's current product-costing system:
Determine the company's predetermined overhead rate using direct-labor cost at t
rate overhead = (total manufacturing cost - overhead cost)/ (budgeted direct - labo
overhead rate per direc
1. Chapter 13, Technical Question 3 in the textbook.
2. Chapter 13, Application Question 1 in the textbook.
3. Chapter 14, Application Question 1 in the textbook.
the deposit multiplier is the inverse of the reserve requirment. Calculate the deposit multi
1. Chapter 7, Technical Questions 3 and 5 in the textbook.
2. Chapter 7, Application Question 5 in the textbook.
3. Chapter 8, Technical Questions 3 and 7 in the textbook.
According to the graph the shutdown potin is the point at which P= min AVC. The bre
Chapter 4 Graded Homework
Problem 1. Prepare a pro forma income statement assuming all costs vary with sales and the dividend
payout ratio is constant. A 35% growth rate in sales is projected. What are the projected additions to |
Concrete Mix yield one 20kg bag will cover an area of 1.1m2 to a depth of approximately 10mm. or 108 x 20kg bags equates to one cubic metre of mixed concrete. TyPicAL ProDUcT PErForMAncE Compressive Strength (MPa): BAGS rEqUirED For THE joB: Square Mtrs Number of 20kg Bags 10 82 108 9 74 98 8 66 88 7 58 76 6 50 66 5 42 54 4 34 44
Volume of one bag cement formula
Mass of one bag of cement. 50 Kg for one Bag. Density of cement. 1440 kg/ meter cube. Now, we need to calculate volume of one bag of cement. So, Volume of Cement: = Mass/ Density =50/1440 = 0.0347 m3 . Also you can convert this volume from cubic meter to cubic feet, Just you have to multiply the volume of 1 bag of cement …
How many kg of cement are present in one purchased bag ...
The weight of cement bag when packed at grinding plant is exactly 50 kg in each bag but the weight shall not remains 50 kg in bag when it opens for use or when we purshase from retailer. The main reason for the loss in weight of cement is transpor...
Enter the number of 8 inch x 2 inch x 4 inch bricks or 8 inch x 8 inch x 16 inch blocks you plan to use for your project. The calculator will indicate the number of 60 or 80 pound bags of QUIKRETE® Mortar Mix you need to construct your project with a 3/8 inch mortar joint. (All yields are approximate and do not include allowance for uneven ...
How To Calculate Cement Bags Per Cubic Meter Concrete ...
Therefore required cement = 1440 x 0.221 m 3 = 318.24 Kg or (318.24/50) = 6.37 Bags. Cement Bag Per Cubic Metre Concrete Table – Nominal Mix
How much CFT, CuM, kg in 1 Bag of Cement || How much ...
· DerivationQuick Learn in this Video / Topic / Tutorial about# How much cement in 1 Bag of Cement# How many Numbers of Bags in 1 Cubic Meter# How much kg ceme...
How many bags of cement required for 1m3 of Cement?
· So, Usually, 4% of moisture content is present in cement. Dry Density of Cement = 1440kg/m 3. Each bag of cement consists of = 50kg. No. of bags in 1m 3 = 1440/50. = 29bags (approx)
Welcome to the concrete calculator. It will help you calculate the number of bags of concrete required for your path, slab or post holes.
Cement to all in ballast
Simple calculation. Usually it''s 1:4 so for every 4 kg of sand 1 kg of cement. As they called tone bag - around (400 kg) you''ll need 100 kg cement. So one tone bag of sand - four bags of cement. Hopefully this will help. mDucanon. 2016-08-09T07:45:01+01:00. Answered 9th Aug 2016.
How to Calculate Cement, Sand and Aggregate required for 1 ...
Total weight of concrete ingredients = 50+115+209+27.5 = 401.5 say 400 kg. Density of concrete = 2400 kg/cum. So, 1 bag of cement produces = 400/2400 = 0.167 cum. No. of bags required for 01 cum of concrete = 1/0.167 = 5.98 bags ~ 6 bags. From above, if the concrete mix is 1:2:4, to get a cubic meter of concrete we require. 1.Cement = 6 bags ...
concrete bag mix Calculator
calculate how many bags of concrete you need. Scroll down to see the calculator! Here''s a free concrete bag mix calculator to help you figure the amount of cement or concrete you need to do your project. Most people call concrete "cement" when in fact cement is just one ingredient along with sand, aggregate and water that is used to make concrete.
How many bags of cement do I need to make 1 cubic metre of ...
· 1 Cement 2 Sand 3 Aggregate (gravel) is too strong. 16bags (costly) 1 Cement 2.5 Sand 4 Aggregate is OK for fence posts. 13bags of cement per cubic metre. The more cement the more costly of course so don''t over do it. Note: These are 20kg bags of General Purpose cement.
Convert KG to bag [portland cement]
How many KG in 1 bag [portland cement]? The answer is 42.63768278. We assume you are converting between kilogram and bag [portland cement]. ... The kilogram or kilogramme, (symbol: kg) is the SI base unit of mass. A gram is defined as one thouh of a kilogram. Conversion of units describes equivalent units of mass in other systems.
In 1 cum how many number of 50kg cement bags required ...
· The density of cement is 1440 Kg/Cum. Thus the volume of a 50 kg cement bag comes to be= [ 50/1440 ]*1000 = 34.722 Litres . For practical purposes consider as 35 Litres. That gives to 0.035cum quite simple. Thus in 1 cum it come approx. 29 bags of 50kg cement bags.
How many cubic meters are in a bulk bag of sand?
· How many cubic meters is a 20kg bag of sand? Yield: One 20kg bag will cover an area of 1.1m2 to a depth of approximately 10mm. Or 108 x 20kg bags equates to one cubic metre of mixed concrete. 20kg bags can be bought individually or as a pallet of 70.
Concrete Mix (Quantity Calculator) – Target Products Ltd.
Concrete Mixes. The calculator will indicate the estimated number of 25 kg or 30 kg bags of Quikrete® Concrete Mix needed to build a 3″, 4″ or 6″ slab (with a typical waste factor). Yields are approximate and will vary based upon placement methods, equipment utilized and do not include allowance for uneven subgrade, waste, etc.
How To Calculate Cement Bags In 1 Cubic Meter?
· Therefore volume of 1 bag cement = 50/1440 =0.0347 cum. ∴ No. of cement bags required in 1 cubic meter = 0.2171/0.0347 = 6.25 bags. Note: You can use the same formula for calculating cement for other nominal mixes.
How to Estimate the Quantity of Sand and Cement Required ...
· This is obtained by knowing that the mass of 1 bag of cement = 50kg, and the density = 1440 kg/m 3 The volume of a standard builder''s wheelbarrow is 0.065 m 3 (unheaped). We assume that approximately 2 bags of cement (4 head pans of cement) will fill one builder''s wheelbarrow.
How To Calculate Cement Bags In 1 Cubic Meter?
Quikrete 60 lb. Concrete Mix-110160 - The Home Depot
Bags Of Portland Cement to Kilograms | Kyle''s Converter
Unit Descriptions; 1 Bag (of Portland Cement): One bag of portland cement weighs 94 lb av by definition. 1 Kilogram: The kilogram is defined as being equal to the mass of the International Prototype Kilogram (IPK), which is almost exactly equal to the mass of one liter of water.
What is the volume of a 40 kg bag of cement?
· Using proportions suggested, one 40 kg (88 lb) bag of cement will produce about 3.5 ft³ (0.1 m³) of concrete mix approximately. Just so, how many cubic meters is a 40kg bag? You will need 6 x 40kg bags per cubic meter .
Calculate Bags of Pre-Mix Concrete
20 kg 25 kg 30 kg 40 kg. Cost per Bag. A concrete slab 1000 mm X 1000 mm at a depth of 100 mm, has a volume of 0.1 m³ = 10.8 x 20 kg bags. 11 Bags. Total Bag Weight 220 kg. Calculated at 2160 kg per 1m³ - Allow extra for waste. Pre-Mix Concrete Bag Calculator Calculated at 2160 kg per 1m³ - Allow extra for waste. Cubic Metres.
St. Mary Portland Cement, 40 KG | The Home Depot Canada
· YIELD Using proportions suggested, one 40 KG (88 lb) bag of cement will produce about 3.5 ft3 (0.1 m³) of concrete mix, approximately. PACKAGING 20 KG (44 lb) bag 30 KG (66 lb) bag 40 KG (88 lb) bag STORAGE & SHELF LIFE Material should be stored in a dry, covered area, protected from the elements.
How To Calculate Cement, Sand & Aggregates Quantity in ...
· Density of Cement is 1440/m 3 = 0.28 x 1440 = 403.2 kg We know each bag of cement is 50 kg For Numbers of Bags = 403.2/50 = 8 Bags We Know in one bag of cement = 1.226 CFT For Calculate in CFT (Cubic Feet) = 8 x 1.225 = 9.8 Cubic Feet CALCULATION FOR SAND QUANTITY; Consider volume of concrete = 1m 3
Convert metric ton to bag [portland cement]
How many metric ton in 1 bag [portland cement]? The answer is 0.04263768278. We assume you are converting between metric ton and bag ... The SI base unit for mass is the kilogram. 1 kilogram is equal to 0.001 metric ton, or 0.023453432147327 bag [portland cement]. Note that rounding errors may occur, so always check the results.
Cement, mortar volume to weight conversion
About Cement, mortar; 1 cubic meter of Cement, mortar weighs 2 162 kilograms [kg] 1 cubic foot of Cement, mortar weighs 134.96925 pounds [lbs] Cement, mortar weighs 2.162 gram per cubic centimeter or 2 162 kilogram per cubic meter, i.e. density of cement, mortar is equal to 2 162 kg/m³ Imperial or US customary measurement system, the density is equal to 134.9693 pound per cubic foot …
CONCRETE DIVISION 3 -50, -60, -80, -90 2 RODUCT O
QUIKRETE Concrete Mix can be mechanically mixed in a barrel type concrete mixer or a mortar mixer. Choose the mixer size most appropriate for the size of the job to be done. Allow at least 1 ft3 (28.3 L) of mixer capacity for each 80 lb (36.2 kg) bag of QUIKRETE Concrete Mix to be mixed at one time.
Cement Ready Reckoner
No of bags 3 5 7 9 11 Estimating how many 20kg bags of premix concrete to order (rounded up to the nearest full bag) * 108 x 20kg bags of Boral Cement Concrete Mix will fill 1 cubic metre (m3). Depth of hole (mm) Cubic metres(m3) needed. Hole diameter (mm) 300mm 400mm 500mm 600mm 700mm 200 0.02 0.03 0.04 0.06 0.08 400 0.03 0.05 0.08 0.12 0.16
How many hollow blocks can be made in 1 bag of cement
How many hollow blocks can be made in 1 bag of cement Products. As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including, How many hollow blocks can be made in 1 bag of cement, quarry, aggregate, and different kinds of minerals.
For example, a conventional redymix concrete bag of 80 lbs may be listed as having a yield of approximately 0.60 cu ft or a 25 kg bag may have a yield of ~0.01 cu m. With a few simple mathematical transformations we can use this number to arrive at the in place density of the material and complete our calculation of the total mass and number of ...
Portland Cement 1 kilogram mass to liters converter
One kilogram of Portland cement converted to liter equals to 0.66 L. How many liters of Portland cement are in 1 kilogram? The answer is: The change of 1 kg - kilo ( kilogram ) unit of Portland cement measure equals = to 0.66 L ( liter ) as the equivalent measure for the same Portland cement …
How to Calculate the Volume of 1 Bag of Cement in Cubic ...
We know the density of cement is 1440 kg/m 3 and the mass of the one bag cement is 50 kg. The density of cement means that in one cubic meter volume the quantity of cement will be 1440 kg. Number of Bags of Cement in One Metric Cube = [1440 (kg/m 3)/ 50(kg)] Number of Bags of Cement in One Metric Cube = 28.8 28.8 (1/m 3)
Example calculation Estimate the quantity of cement, sand and stone aggregate required for 1 cubic meter of 1:2:4 concrete mix. Ans. Materials required are 7 nos. of 50 kg bag of cement, 0.42 m 3 of sand and 0.83 m 3 of stone aggregate.
How Many Bags Of Cement And Ballast Do I Need?
For one bulk bag of ballast, how many bags of cement do I need? I''m guessing you''ll need more than one 1 ton bag of ballast and around 6 bags of cement. A 1 ton bag weighs around 900 kilograms, so I''m guessing 40-45 bags weighing 25 kilograms each. 12 shovels of ballast to 1/2 bag of cement is a good ratio, but 1:6 would suffice.
How much sand and gravel do I need for a bag of cement ...
Regarding this, how many wheelbarrow of sand to a bag of 20kg cement, generally a bag of 20kg cement will require about 0.98 cubic feet of sand, taking one wheelbarrow size as 2 CF, you will need approx 1/2 (half) wheelbarrow of sand to a bag of 20kg cement.
How to Calculate the Volume of 1 Bag of Cement in Cubic ...
We know the density of cement is 1440 kg/m 3 and the mass of the one bag cement is 50 kg. The density of cement means that in one cubic meter volume the quantity of cement will be 1440 kg. Number of Bags of Cement in One Metric Cube = [1440 (kg/m 3)/ 50(kg)] Number of Bags of Cement in One …
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[Todos CMAT] Premio a Y. Sinai
roma en fing.edu.uy
roma en fing.edu.uy
Jue Ene 3 08:36:25 UYST 2013
LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT
The Leroy P. Steele Prizes were established in 1970 in honor of George
David Birkhoff, William Fogg Osgood, and William Caspar Graustein and
are endowed under the terms of a bequest from Leroy P. Steele. Prizes
are awarded in up to three categories and each is awarded annually.
The following citation describes the award for Lifetime Achievement.
The 2013 Steele Prize for Lifetime Achievement is awarded to Yakov Sinai for
his pivotal role in shaping the theory of dynamical systems and for his
groundbreaking contributions to ergodic theory, probability theory,
statistical mechanics, and mathematical physics.
Sinai?s research exhibits a unique combination of brilliant analytic
outstanding geometric intuition, and profound understanding of underlying
physical phenomena. His work highlights deep and unexpected
dynamical systems and statistical mechanics. Sinai has opened up new
including Kolmogorov?Sinai entropy, Markov partitions, and Sinai?Ruelle?Bowen
measures in the hyperbolic theory of dynamical systems; dispersing billiards,
a rigorous theory of phase transitions in statistical mechanics and space-time
chaos. In addition, Sinai has made seminal contributions in the theory of
Schrödinger operators with quasi-periodic potentials, random walks in random
environments, renormalization theory, and statistical hydrodynamics
and Navier?Stokes equations.
Sinai pioneered the study of dispersing billiards: dynamical systems
the motion of molecules in a gas. The simplest example of such a
a square with a disk removed from its center, is called ?Sinai?s
billiard motions within the framework of hyperbolic theory, Sinai discovered
that they exhibit deep ergodic and statistical properties (such as the
theorem). Owing to Sinai?s work, some key laws of statistical
mechanics for the
Lorentz gas can be established with mathematical rigor. In particular,
the ?rst steps towards justi?cation of Boltzmann?s famous ergodic hypothesis,
proposed in the end of the nineteenth century: ?For large systems of
particles in equilibrium, time averages are close to the ensemble average.?
Sinai returned to this subject several times in the period 1970?90
co-authors, including his students Bunimovich and Chernov. 55
Together with his student Pirogov, Sinai created a general theory of
phase transitions for statistical mechanics systems with a ?nite number of
ground states. Pirogov?Sinai theory forms essentially the basis for modern
equilibrium statistical mechanics in a low-temperature regime.
Sinai made seminal contributions to the theory of random walks in a random
environment. With his model, known nowadays as ?Sinai?s random walk,? he
obtained remarkable results about its asymptotic behavior. With his student
Khanin, Sinai pioneered applications of the renormalization group method
to multi-fractal analysis of the Feigenbaum attractor, and to the Kolmogorov?
Arnold?Moser theory on invariant tori of Hamiltonian systems.
In the past ?fteen years Sinai has brought novel tools and insights from
dynamical systems and mathematical physics to statistical hydrodynamics,
obtaining new results for the Navier?Stokes systems. Speci?cally, along
with D. Li, Sinai devised a new renormalization scheme which allows the
proof of existence of ?nite time singularities for complex solutions of
the Navier?Stokes system in dimension three.
Sinai?s mathematical in?uence is overwhelming. During the past half-century he
has written more than 250 research papers and a number of books.
monograph, Ergodic Theory (with Cornfeld and Fomin), has been an introduction
to the subject for several generations, and it remains a classic.
Sinai supervised more than ?fty Ph.D. students, many of whom have become
leaders in their own right. Sinai?s work is impressive for its breadth.
In addition to its long-lasting impact on pure mathematics, it has played a
crucial role in the creation of a concept of dynamical chaos which has been
extremely important for the development of physics and nonlinear science over
the past thirty-?ve years. The Steele Prize for Lifetime Achievement is
awarded to Sinai in recognition of all these achievements.
Yakov G. Sinai was born in 1935 in Moscow, Soviet Union, now Russia.
He received his Ph.D. degree (called a Candidate of Science in Russia) and
then his doctorate degree (Doctor of Science) from Moscow State University.
For several years, he combined his position at Moscow State University and the
Landau Institute of Theoretical Physics of the Russian Academy of Sciences.
Since 1993, he has been a professor in the mathematics department of Princeton
Ya. Sinai received various honors recognizing his contributions. He was
elected as a foreign associate of the National Academy of Sciences and a
foreign member of the Academy of Arts and Sciences. He is a full member
of the Russian Academy of Sciences, and he was recently elected as a
foreign member of the Royal Society in London. He is also a member of
the Brazilian Academy of Science, the Hungarian Academy of Science,
the Polish Academy of Science, and Academia Europea.56 Among his other
recognitions are the Wolf Prize in Mathematics, the Nemmers Prize, the
Lagrange Prize, the Boltzmann Medal, the Dirac Medal, and the Poincaré Prize.
Response from Yakov Sinai
It is a great honor to be awarded the Steele Prize for Lifetime Achievement
from the American Mathematical Society. I worked in several directions
in mathematics, including the theory of dynamical systems, statistical
and mathematical physics, and probability theory. My mentors who had a big
in?uence on me were A. N. Kolmogorov, V. A. Rokhlin, and E. B. Dynkin.
I also bene?tted a lot from many contacts with my colleagues.
I was very fortunate to have talented students, many of whom became strong
and famous mathematicians. Unfortunately, it is not possible to list the names
of all of them here. I thank my family and friends for their encouragement and
support. Finally, I thank the selection committee for its work.
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