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Introduction• Ultrasound biomicroscopy (UBM) provides high- resolution imaging of the anterior segment in a noninvasive fashion.• In addition to the tissues easily seen using conventional methods (ie, slit lamp), such as the cornea, iris, and sclera, structures including the ciliary body and zonules, hidden from clinical observation, can be imaged and their morphology assessed.• Pathophysiologic changes involving anterior segment architecture can be evaluated qualitatively and quantitatively. INTRODUCTION• Recent technique to visualize anterior segment with the help of high frequency ultra sound transducer.• UBM (anterior segment ultrasonography) is performed with a 50 Mhz probe.• The resolution of 50 MHz probe is 40 microns and the depth is 4 mm • UBM is done with the patient in the supine position and the eye is open.• Since the piezoelectric crystal of the transducer is open it should not come in direct contact with the eye to prevent injury to the cornea • There is a special cup which fits in between the eyelids, keeping them open• The eye cup is filled with saline or sterile methylcellulose.• The crystal of the transducer is placed in saline approximately 2 mm from the eye surface. (This distance of 2 mm prevents injury to the cornea and also helps as a fluid standoff.)• The eye is scanned in each clock hour from the center of the cornea to the ora serrata Iridociliary cyst The most common clinical presentation of an irido-ciliary cystis a peripheral iris elevation - the typical UBM finding of a thinwalled structure with no internal reflectivity is diagnostic. • UBM is useful in opaque media• The most important limitation of UBM is depth. UBM cannot visualize structures deeper more that 4 mm from the surface.• The other limitation is that UBM cannot be performed in presence of an open corneal or scleral wound.• It is time consuming • Tomographic techniques generate slice images of three-dimensional objects.• Optical tomographic techniques are of particular importance in the medical field, because these techniques can provide non- invasive diagnostic images • Optical coherence tomography is a non- contact, real-time technique that uses low infrared laser energy to image structures. • Optical coherence tomography imaging is based on measuring the delay of light (typically infrared) reflected from tissue structures.• Because light travels extremely fast, it is not possible to directly measure the delay at a micron resolution. Therefore, OCT employs low-coherence interferometry to compare the delay of tissue reflections against a reference reflection. • To obtain an OCT image, the instrument scans a light beam laterally, creating a series of axial scans (A-scans), after which it combines these A-scans into a composite image.• Each A-scan contains information on the strength of reflected signal as a function of depth. • The more commonly used retinal OCT uses 820-nm light, which allows for excellent tissue penetration to the level of the retina.• The anterior segment OCT utilizes 1310- nm light, which has greater absorption resulting in limited penetration. • This allows for increased intensity of the light as decreased amounts reach the retina.• The light is 20 times more intense, giving a much greater signal-to-noise ratio.• This increased intensity allows for increasing the speed in imaging 20 times, with decreased motion artifact. • Compared with other imaging modalities, OCT has a higher-depth resolution.• Resolution is determined by the wavelength and the spectral bandwidth of the light source, Shorter wavelengths and wider bandwidths provide better resolution. Types of oct system There are two principles of image acquisition and data processing in anterior segment OCT:• Time domain and• Fourier domain algorithms. • In time domain OCT, there is a mechanical moving part that performs the A-scan, Thus, the rate of the scan is limited by the movement of the part.• In Fourier domain OCT, the information in an entire A-scan is acquired by a charge-coupled device (CCD) camera simultaneously. As there is no mechanical movement, the scan time in Fourier domain OCT is faster. This is an important advancement because faster acquisition time means lesser variability in the result due to the patient’s eye movements. Scans• Anterior Segment Scan (16 x 6 mm)• Single, Dual or Quad lines• 256 A scans / .125 sec acquisition per line• High Resolution Scan (10 x 3 mm)• Single or Quad• 512 A-scans / .25 sec acquisition per line• Pachymetry Scan (10 x 3 mm)• 8 radial lines• 128 A scans / 0.5 sec total acquisition time• All Scans adjustable in orientation and direction PHAKIC IOL PRE OP PLANNING POST OP OBSERVATIONACD PCIOL DEPTH AND CENTRATIONANGLE TO ANGLE IRIS POSITIONANTERIOR CHAMBER ANGLE CRYSTALLINE LENS VAULTIRIS OR CRYSTALLINE LENS ENDOTHELIAL SAFETY DISTANCEPOSITIONACCOMADITVE ANALYSIS ACCOMADITIVE ANALYSIS Advantages of AS OCT• Technicians can do the scanning• Imaging flexibility• Faster imaging reduces error• Image through an opaque cornea• Its easy to image accommodative changes• Scans can be taken immediately after surgery Limitations• Pigmentation on the posterior side of the iris blocks the penetration of infrared light.• Trabecular meshwork/ ciliary body not seen• Manual angle measurement History• History of present illness• Associated history• Past history• Family history History• History of present illness :age of onset• Associated history duration• Past history one/both eye• Family history variability History• History of present illness• Associated history : diplopia• Past history odynophagia• Family history muscle weakness cardiac problem night blindness History• History of present illness• Associated history• Past history : trauma/ surgery• Family history contact lens lid edema allergy dry eyes History• History of present illness• Associated history• Past history• Family history evaluation of ptosis• head posture,Eyebrow position, eyelid masses, inflammation, proptosis• pupillary size, reaction, heterochromia• Best corrected Visual Acuity: In infants, make sure infant can fix and follow light with each eye• Cycloplegic Refraction In children• Presence or absence of Lid fold• Head tilt• Iliff test Measurements• Vertical fissure height• Margin reflex distance• LPS action• Lid crease level• Lid level on down gaze Vertical fissure height• The distance between the upper and lower eyelid in vertical alignment with the center of the pupil in primary gaze, with the patient’s brow relaxed.• Normal – 9-10mm in primary gaze• Should be seen in up gaze, down gaze and primary gaze• Amount of ptosis = difference in palpebral apertures in unilateral ptosis or Difference from normal in bilateral ptosis Grading of severity of ptosis < or = 2mm : mild ptosis = 3 mm : moderate ptosis = or > 4 mm : severe ptosis MRD• Margin-to-reflex distance 1 (MRD1) : is the distance from the central pupillary light reflex to the upper eyelid margin with the eye in primary gaze.• A measurement of 4 - 5 mm is considered normal.• If the margin is above the light reflex the MRD 1 is a +ve value.• If the lid margin is below the corneal reflex in cases of very severe ptosis the MRD 1 would be a –ve value. MRD• Margin-to-reflex distance 2 (MRD2) : is the distance from the central pupillary light reflex to the lower eyelid margin with the eye in primary gaze. . • The MRD1 plus the MRD2 should equal the palpebral fissure measurement Levetor function• is the distance the eyelid travel from downgaze to upgaze while the frontalis muscle is held inactive at the brow.• The normal levator function is between 13- 17mm • Lid excursion is a measure of the levator function. The frontalis action is blocked by keeping the thumb tightly over the upper brow and asking the patient to look up from down gaze and measuring the amount of upper lid excursion at the center of the lid. Grading of levator action< 4mm – poor levator function5-7 mm – fair levator function8-12 mm – good levtor function Lid crease• Position is the distance from the crease to lid margin• Normal – 8 to 10mm in primary gaze• An absent lid crease is often accompanied by poor levator function.• If a lid crease is present but is higher than normal and if a deeper upper lid sulcus is found on that side, note these as signs of a levator aponeurosis disinsertion. Phenyl ephrine test• Patients with minimal ptosis (2 mm or less) should have a phenylephrine test performed in the involved eye or eyes• Either 2.5 or 10% phenylephrine is instilled in the affected eye or eyes. Usually two drops are placed and the patient is reexamined 5 minutes later. • The MRD1 is rechecked in the affected and unaffected eyes .• A rise in the MRDl of 1.5 mm or greater is considered a positive test. This indicates that Müllers muscle is viable

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One needs to understand and know about the different branches of statistics to correctly understand statistics from a more holistic point of view. Often the kind of work one has involved in hides the other aspects of statistics however it is important to know the overall idea behind statistics to appreciate its importance. What are the Branches of Statistics? Statistics are divided into two major divisions: descriptive and inferential. Each of these is important as they offer different techniques that accomplish different objectives. Descriptive statistics describe what is going on in a population while inferential statistics, allows scientists to collect data from a sample group and generalize them to a larger population. They both have important differences. In this branch of statistics, the goal is to describe. It deals with the collection of data and its presentation in various forms, such as tables, and graphs, and finding averages and other measures which would describe the data. Numerical measures are used to tell about the features of a set of data. Several items belong in this portion of statistics which include: - The measure of the center of a data set, consisting of the mean, median and - The spread of a data set measured with the range or standard deviation - Measurements such as skewness - The exploration of relationships and correlation between paired data - The presentation of statistical results in graphical form These measures are important and useful because they allow scientists to see patterns among data thus making sense of that data. Descriptive statistics can only be used to describe the population or data set under study, but the results cannot be generalized to any other group or population. Recommended Posts: Statistics Homework Help Types of Descriptive Statistics There are two kinds of descriptive statistics: Measures of central tendency– capture general trends within the data and are calculated and expressed as the mean, median, and mode. A mean tells scientists the mathematical average of a data set, such as the average age at first marriage; the median represents the middle of the data distribution, like the age that sits in the middle of the range of ages at which people first marry; and, the mode might be the most common age at which people first marry. Measures of spread– describe how the data are distributed and relate to each other, including: - The range, the entire range of values present in a data set - The frequency distribution defines how many times a particular value occurs within a data set - Quartiles are subgroups formed within a data set when all values are divided into four equal parts across the range - The average of how much each value deviates from the mean - Standard deviation, which illustrates the spread of data relative to the mean Measures of spread are often visually represented in tables, pie and bar charts, and histograms to aid in the understanding of the trends within the data. Inferential statistics are produced through complex mathematical calculations which allow scientists to infer trends about a larger population based on a study of a sample taken from it. Scientists use inferential statistics to examine the relationships between variables within a sample and then make generalizations or predictions about how those variables will relate to a larger population. It is usually impossible to examine each member of the population individually. So scientists choose a representative subset of the population, called a statistical sample, and from this analysis, they can say something about the population from which the sample came. There are two major divisions of inferential statistics: - A confidence interval gives a range of values for an unknown parameter of the population by measuring a statistical sample. This is expressed in terms of an interval and the degree of confidence that the parameter is within the interval. - Tests of significance or hypothesis testing are where scientists claim the population by analyzing a statistical sample. By design, there is some uncertainty in this process. This can be expressed in terms of a level of significance. Techniques that social scientists use to examine the relationships between variables, and thereby create inferential statistics, include linear regression analyses logistic regression analyses, ANOVA, correlation analyses, structural equation modeling, and survival analysis. When conducting research using inferential statistics, scientists conduct a test of significance to determine whether they can generalize their results to a larger population. Common tests of significance include the chi-square and t-test. These tell scientists the probability that the results of their analysis of the sample are representative of the population as a whole. Descriptive vs. Inferential Statistics Although descriptive statistics help learn things such as the spread and center of the data, nothing in descriptive statistics can be used to make any generalizations. In descriptive statistics, measurements such as the mean and standard deviation are stated as the exact number However, the focus is different for inferential statistics, it uses measurements such as the mean and standard deviation. Inferential statistics start with a sample and then generalize to a population. This information about a population is not stated as a number; instead, these parameters are expressed as a range of potential numbers along with a degree of confidence. Statistics Homework Help Do Need help with statistics homework? Get quick & reliable statistics homework help online from our Top statistics homework doers.

- Can we live without mathematics? - What would happen if there were no numbers? - What will happen to the world of mathematics does not exist? - Why do we use zero? - Is 0 a real number? - What is the biggest number in the world? - Is 13 a prime number Yes or no? - Who invented the 0? - Is 0 real or imaginary? - Why is 2 A Number? - Who found zero in India? - Who is the father of mathematics? - Do we really need math in life? - Can the universe exist without math? - Why is 2 the best number? - Is zero a number Yes or no? - Does zero mean nothing? - Is 2 the only prime number? - What was used before zero? - What is the importance of zero in daily life? - How did the Mayans use zero? Can we live without mathematics? Math is needed at every step of life, and we cannot live without it. It is a subject that is applied to every field and profession. It tells us how things work, and also allows us to predict certain things, which is how we have progressed so much in life. It has made our lives easier and uncomplicated.. What would happen if there were no numbers? Without numbers, people wouldn’t know how to count or measure. There would be many important things we wouldn’t know if it weren’t for numbers. … Without numbers, people wouldn’t know how to count or measure. There would be many important things we wouldn’t know if it weren’t for numbers. What will happen to the world of mathematics does not exist? As long as there is space, there is math because math is the foundation of the universe. … Maths has to exist because the universe is goverened by it, if our understanding of maths didn’t exist then nothing would really happen, we would be very limited in terms of technological advancement. Why do we use zero? Ancient scholars employed it as a symbol to represent the absence of a number, like the way we use a zero in 101 or 102 to signify that there are no multiples of 10 in the middle position. … However, it took two millennia for zero, with all its mathematical brilliance, to be accepted as a proper number. Is 0 a real number? Real numbers consist of zero (0), the positive and negative integers (-3, -1, 2, 4), and all the fractional and decimal values in between (0.4, 3.1415927, 1/2). Real numbers are divided into rational and irrational numbers. What is the biggest number in the world? The biggest named number that we know is googolplex, ten to the googol power, or (10)^(10^100). That’s written as a one followed by googol zeroes. Is 13 a prime number Yes or no? Here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. Who invented the 0? MayansThe first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. Is 0 real or imaginary? The real numbers are a subset of the complex numbers, so zero is by definition a complex number ( and a real number, of course; just as a fraction is a rational number and a real number). If we define a pure real number as a complex number whose imaginary component is 0i, then 0 is a pure real number. Why is 2 A Number? Two is a prime because it is divisible by only two and one. All the other even numbers are not prime because they are all divisible by two. That leaves only the odd numbers. Who found zero in India? AryabhataWhat is widely found in textbooks in India is that a mathematician and astronomer, Aryabhata, in the 5th century used zero as a placeholder and in algorithms for finding square roots and cube roots in his Sanskrit treatises. Who is the father of mathematics? ArchimedesArchimedes (287 BC–212 BC) is known as Father of Mathematics. He was born in the seaport city of Syracuse on the greek island of Sicily; his father was an astronomer. Do we really need math in life? Unlike literature, history, politics and music, math has little relevance to everyday life. That courses such as “Quantitative Reasoning” improve critical thinking is an unsubstantiated myth. All the mathematics one needs in real life can be learned in early years without much fuss. Can the universe exist without math? Many such mathematical constructs exist to explore, but without a physical Universe to compare it to, we’re unlikely to learn anything meaningful about our Universe. … But the Universe is a physical, not mathematical entity, and there’s a big difference between the two. Why is 2 the best number? 2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Is zero a number Yes or no? Zero is an even number because it is divisible by 2 with no remainder. 0 is neither positive nor negative. Many definitions include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of null size. Does zero mean nothing? “Zero” is considered to be a number while “nothing” is considered to be an empty or null set. Zero has a numeric value of “0.” Zero is a numerical digit as well as a number and is used to denote that number in numerical values. … However, “nothing” is only a concept depicting a void or absence of anything relevant. Is 2 the only prime number? A prime number can have only 1 and itself as factors. … 2 is an even number that has only itself and 1 as factors so it is the only even number that is a prime. What was used before zero? Some of the first known zero symbols appear in Babylonian clay tablets of between 400 and 300 B.C.; there the zero was used to clarify the symbols for large numbers. … Brahmagupta was the first to write down the rules for arithmetic with zeros. What is the importance of zero in daily life? As a number, zero means nothing – the absence of other values. It plays a central role in mathematics as the identity element of integer, real number, and many other algebraic structures. As a digit, zero is used as a placeholder in the location value system. Historically, this was the last point in use. How did the Mayans use zero? The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols; zero (shell shape, with the plastron uppermost), one (a dot) and five (a bar).

Thomas Krichel & Nisa Bakkalbasi 2005-10-31 Metadata characteristics as predictors for editor selectivity in a current awareness service outline • Background to work that we did • RePEc (Research Papers in Economics) • NEP: New Economics Papers • The research • Theory • Method • Results • Other work done for NEP. RePEc • Digital library for academic Economics. It collects descriptions of • economics documents (working papers, articles etc) • collections of those documents • economists • collections of economists • Pionneering effort to create a relational dataset describing an academic discipline as a whole. • The data is freely available. RePEcprinciple • Many archives • Archives offer metadata about digital objects or authors and institutions data. • One database • Many services • Users can access the data through many interfaces. • Providers of archives offer their data to all interfaces at the same time. This provides for an optimal distribution. it's the incentives, stupid • RePEc applies the ideas of open source to the construction of bibliographic dataset. It provides an open library. • The entire system is constructed in such a way as to be sustainable without monetary exchange between participants. some history • Thomas Krichel in the early 1990s dreamed about a current awareness service for working paper. It would later have electronic papers. • In 1993 he made the first economics working paper available online. • In 1997 he wrote the key protocols that “govern” RePEc. US Fed in Print IMF OECD MIT University of Surrey CO PAH Blackwell RePEc is based on 500+ archives • WoPEc • EconWPA • DEGREE • S-WoPEc • NBER • CEPR • Elsevier to form a 340+k item dataset 161,000 working papers 180,000 journal articles 1,300 software components 1,200 book and chapter listings 8,000 author contact & publication listings 9,100 institutional contact listings more records than arXiv.org IDEAS RuPEc EDIRC LogEc CitEc RePEc is used in many services • EconPapers • NEP: New Economics Papers • Inomics • RePEc author service • Z39.50 service by the DEGREE partners NEP: New Economics Papers • This is a set of current awareness reports on new additions to the working paper stock only. Journal articles would be too old. • Founded by Thomas Krichel in 1998. • Supported by the Economics department at WUStL. • Initial software was written by Jose Manuel Barrueco Cruz. • First general editor was John S. Irons. why NEP • Public aim: Current awareness if well done, can be an important service in its own right. It is sheltered from the competition of general search engines. • Private aim: It is useful to have some, even though limited classification information. This should be useful in performance measures within subject areas. modus operandi: stage 1 • The general editor uses a computer program who gathers all the new additions to the working paper stock. This is usually done weekly. • S/he filters out new descriptions of old papers • date field • handle heuristics • The result is an issue of the nep-all report. modus operandi: stage 2 • Editors consider the papers in the nep-all report to filter out papers that belong to the subject. This forms as issue of a subject report nep-???. • nep-all and the subject reports are circulated via email. • A special arrangement makes the data of NEP available to other RePEc services. some numbers • The are now 60+ NEP lists. • Over 37k subscriptions. • Close to 16k subscribers. • Over 50k papers announced. • Over 100k announcements. • Homepage at http://nep.repec.org All this is a fantastic success!! problem with the private aim • We would have to have all the papers to be classified not only the working papers. • We would need to have 100% coverage of NEP. • This means every paper in nep-all appears in at least one subject report. coverage ratio • We call the coverage ratio the number of papers in nep-all that have been announced in at least one subject report. • We can define this ratio • for each nep-all issue • for a subset of nep-all issues • for NEP as a whole coverage ratio theory & evidence • Over time more and more NEP reports have been added. As this happens, we expect the coverage ratio to increase. • However, the evidence, from research by Barrueco Cruz, Krichel and Trinidad is • The coverage ratio of different nep-all issues varies a great deal. • Overall, it remains at around 70%. • We need some theory as to why. two theories • Target-size theory • Quality theory • descriptive quality • substantive quality theory 1: target size theory • When editors compose a report issue, they have a size of the issue in mind. • If the nep-all issue is large, editors will take a narrow interpretation of the report subject. • If the nep-all ratio is small, editors will take a wide interpretation of the report subject. target size theory & static coverage • There are two things going on • The opening new subject reports improves the coverage ratio. • The expansion of RePEc implies that the size of nep-all, though varying in the short-run, grows in the long run. Target size theory implies that the coverage ratio deteriorates. • The static coverage ratio that we observe is the result of both effects canceling out. theory 2: quality theory • George W. Bush version of quality theory • Some papers are rubbish. They will not get announced. • The amount of rubbish in RePEc remains constant. • This implies constant coverage. • Reality is slightly more subtle. two versions of quality theory • Descriptive quality theory: papers that are badly described • misleading titles • no abstract • languages other than English • Substantive quality theory: papers that are well described, but not good • from unknown authors • issued by institutions with unenviable research reputation practical importance • We do care whether one or the other theory is true. • Target size theory implies that NEP should open more reports to achieve perfect coverage. • Quality theory suggests that opening more report will have little to no impact on coverage. • Since operating more reports is costly, there should be an optimal number of reports. overall model • We need an overall model that explains subject editors behavior. • We can feed this model with variables that represent theoretical determinants of behavior. • We can then assess the strength of various factors empirically. method • The dependent variable is announced. It is one if the paper has been announced, 0 otherwise. • Since we are explaining a binary variable, we can use binary logistic regression analysis (BLRA). This is a fairly flexible technique, useful when the probability distributions governing the independent variables are not well known. • That's why BLRA is popular in the life sciences. independent variables: size • size is the size of the nep-all issue in which the paper appeared. • This is the critical indicator of target size theory. We expect it to have a negative impact on announced. independent variables: position • position is the position of the paper in the nep-all issue. • The presence of this variable can be justified by the combined assumption of target size and editor myopia. • If editors are myopic, they will be more liberal at the start of nep-all then at the end of nep-all. independent variables: title • title is the length of a title of the paper, measured by the number of characters. • This variable is motivated by descriptive quality theory. A longer title will say more about the paper than a short title. This makes is less likely that a paper is being overlooked. independent variables: abstract • abstract is the presence/absence of an abstract to the paper. • This is also motivated by descriptive quality theory. • Note that we do not use the length of the abstract because that would be a highly skewed variable. independent variables: language • language is an indicator if the language of the metadata is in English or not. • This variable is motivated by descriptive quality theory and the idea that English is the most commonly understood language. • While there are a lot of multilingual editors, customizing this variable would have been rather hard. independent variables: series • series is the size of the series where a paper appears in. • This variable is motivated by substantive quality theory. • The larger a series is the higher, usually, is its reputation. We can roughly qualify by size and quality • multi-institution series (NBER, CEPR) • large departments • small departments independent variables: author • author is the prolificacy of the authors of the paper. • It is justified by substantive quality theory. • This is the most difficult variable to measure. We use the number of papers written by the registered author with the highest number. • Since about 50% of the papers have no registered author, a lot of them are excluded. But there should be no bias by the exclusion. create categorical variables • size_1 [179, 326) • size_2 [326, 835] • title_1 [55, 77) • title_2 [77, 1945] • position_1 [0.357, 0.704) • position _2 [0.704, 1.000] • series_1 [98, 231) • series_2 [231, 3654] results • P(announced=1| x) =(exp(g(x))/(1+exp(g(x)) • g(x) = 0.2401- 0.2774*size_1 - 0.4657* size_2 + 0.1512*title_1+ 0.2469*title_2 + 0.3874*abstract + 0.0001*author + 0.7667*language -0.1159*series_1 + 0.1958*series_2 • position is not significant. author just makes the cut. odds ratio • size_1 1.32 [1.22, 1.44] • size_2 0.83 [0.76, 0.90] • title_1 1.16 [1.07, 1.26] • title_2 1.28 [1.18, 1.39] • abstract 1.47 [1.34, 1.62] • language 2.15 [1.85, 2.51] • series_1 1.11 [1.02, 1.20] • series_2 1.37 [1.26, 1.49] • author 1.05 [1.01, 1.09] scandal! • Substantive quality theory can not be rejected. That means that the editors are selecting for quality as well as for the subject. • The editors have rejected our findings. Almost all protest that there is no quality filtering. consequences • There has been no program to expand list. • There has to be a concentrated effort to help editors to find subject specific papers. • More effort needs to be made for editors to really find the subject-specific papers. This can be done by • the use of a more efficient interface • the use of automated resource discovery methods. ernad • editing reports on new academic documents. It is purpose-built software system for current awareness reports. • It has been designed by Thomas Krichel, http://openlib.org/home/krichel/work/altai.html • The system was written by Roman D. Shapiro. statistical learning • The idea is that a computer may be able to make decision on the current nep-all reports based on the observation of earlier editorial decisions. • ernad now works using support vector machines (SVM), with titles, abstracts, author name, classification values and series as features. performance criteria • We are not aware of performance criteria for the sorting of papers in a report. • Precision and recall appear useless. • Expected search length and average search don't appear very attractive. • Thus research into precise criteria is required. SVM performance • If we use average search length, we can do performance evaluations. • It turns out that reports have very different forecastability. Some are almost perfect, others are weak. • Again, this raises a few eyebrows! what is the value of an editor? • If the forecast is perfect, we don't need the editor. • If the forecast is very weak the editor may be a prankster. pre-sorting reconceived • We should not think of pre-sorting via SVM as something to replace the editor. • We should not think about it encouraging editors to be lazy. • Instead, we should think it as an invitation to examine some papers more closely than others. headline vs. bottomline data • The editors really have a three stage process of decision. • They read title, author names. • They read the abstract. • They read the full text • A lot of papers fail at the first hurdle. • SVM can read the abstract and prioritize papers for abstract reading. • Editors are happy with the pre-sorting system. Thank you for your attention! firstname.lastname@example.org://openlib.org/home/krichel/

A kale leaf is crinkled up around its edge. Picture a curved line. That's not so hard – I'm sure you're already imagining a smoothly curving line drawn on a blank white page, perhaps a circle, or a sine curve wiggling up and down across the page. Now picture a curved surface. Again, not so hard. You might think first of a something simple like a cylinder or a sphere. Or perhaps something a bit more complicated, like a frilly kale leaf (a good illustration of a hyperbolic surface), a dimpled ball or a flag rippling in the wind. In a previous article, Kissing the curve, we saw that understanding curvature of one-dimensional lines and two-dimensional surfaces is all a matter of osculating (kissing) circles of the curves in a flat plane. (For surfaces we consider the curves formed by intersecting the surface with planes perpendicular to the tangent planes.) The osculating circle of this curve at the point P. The curvature at this point is 1/r where r is the radius of the osculating circle. Mathematicians and physicists are interested in the curvature of higher-dimensional objects. And we are affected by such curvature every day: the gravity we feel is the result of the curvature of the four-dimensional spacetime of our Universe. But how do you describe curvature of something that has more than two dimensions? It's easy for us to picture a one-dimensional curve drawn on a flat plane. And we can easily picture a two-dimensional surface in three-dimensional space. But for most of us picturing anything in higher dimensions is impossible. Where our brains and imagination fail, we can use maths to paint the picture for us. We need to generalise the idea of a surface or a line to higher dimensions. Most of the lines we come across, whether they are curved or straight, look locally like a straight line (this is the basis for calculus). A surface, whether curved like a ball, rippling like a flag or flat as a table-top, viewed close up looks like a flat plane. (CGI in movies uses this concept in order to approximate a complicated surface with lots of tiny flat pieces – you can read more in It's all in the detail.) Both lines and surfaces are examples of manifolds – mathematical objects that viewed up close look like ordinary flat space (where points are located using perpendicular coordinate axes), which is known as Euclidean space. This image is actually made up of millions and millions of shaded triangles. The tangents to a curve (which you can think of as a one-dimensional manifold) are straight lines. Surfaces (two-dimensional manifolds) also have tangent planes. In fact, the tangent space of an n-dimensional manifold at any point is a Euclidean space of dimension n, . And just as we understood curvature for lines and surfaces, the curvature of a manifold is the amount it differs from its flat tangent space. In order to understand curvature in this way, the tangent space has to vary smoothly across the surface. If a line or surface has a sharp kink, there is no way to define curvature at this point as the tangent line (or planes) abruptly change from one side of this point to the other, and isn't defined at the kinky point. So in order to consider curvature of manifolds, we consider manifolds that too have smoothly varying tangent spaces across the manifolds. These are called Riemannian manifolds, after the mathematician Bernard Riemann who extended this notion of curvature to many dimensions. (Actually, the precise definition of a Riemannian manifold is that the inner product – the operation that defines distances and angles – for the tangent spaces varies smoothly across the manifold. The sphere has positive curvature, the saddle has negative curvature and the flat plane has zero curvature. A circle drawn on the sphere would have a smaller area than one drawn on the flat plane. While a circle drawn on the saddle would have a greater area than one drawn on the flat plan. (Image courtesy NASA.) We saw in the last article that there is a notion of curvature, called Gaussian curvature, that is intrinsic: it can be detected by a 2D resident living on the surface; you don't need an outside perspective to see it. The resident would be able to detect the positive, negative or flat curvature of the surface by summing the interior angles of triangles: positive curvature corresponds to the interior angles adding to more than 180° (such as on the surface of the Earth), a negative curvature to the interior angles adding to less (a surface similar to that of a crinkly kale leaf) and flat (0) curvature to the angles adding exactly 180°. Similarly, you can understand the curvature of higher-dimensional Riemannian manifolds intrinsically, without caring how the manifold is embedded in any larger space. The simplest measure of curvature on Riemannian manifolds is scalar curvature, , defined for each point in the manifold. This describes how a small ball in the manifold centred on , differs in volume from an equivalent ball in Euclidean space. To understand this let's go back to something we can picture more easily – a ball of radius on a surface is made up of all the points that lie within a distance of , as measured within the surface, from the centre point : a filled circle. Since you have to measure the distances within the surface, it might not be the same distance you would measure between these two points in the Euclidean space that contains the surface. This is why the circle in the surface is not necessarily the same as the equivalent circle in Euclidean space (ie, a filled circle with centre and radius , where the distances are measured in Euclidean space). The torus has regions with different curvature: on the outside of the torus curvature is positive (blue), on the inside it's negative (red), and at the top and bottom circles it's zero (grey). (Image from Mark L. Irons.) At a point with positive curvature the ball on the manifold will be smaller than the equivalent ball of radius in Euclidean space: a circle cut from a flat piece of paper will have to be folded or crumpled to cover the equivalent circle on the surface of a sphere. If the surface has negative curvature at a point , the volume of the ball will be larger than for Euclidean space: to create crocheted versions of hyperbolic surfaces , more and more stitches are needed as the radius increases. (You can read more in Chaotic crochet and Knitting by numbers.) Our examples of a sphere and hyperbolic surfaces have constant scalar curvature over the whole surface. But you don't have to look far to find a surface with varying curvature: a torus has equal amounts of positive and negative curvature over the whole surface, as shown in the image on the right. The scalar curvature assigns a single real number to each point – it defines a scalar field across the manifold. And the scalar curvature is enough to completely describe the curvature of a two dimensional manifold (ie. a surface). However in more than two dimensions we need something a little more complicated. The Ricci curvature describes how conical regions in the manifold differ in volume from the equivalent conical regions in Euclidean space. Instead of considering how the volume of a whole ball within the manifold differs from one in Euclidean space, we consider the volume of just a sliver of the ball – an angular sector or cone centred around some direction from the centre of the ball. The Ricci curvature, , describes how the sliver of a ball at point in direction in the manifold differs in volume from the equivalent angular sector in Euclidean space. For any point on the manifold, we can define the value of the Ricci curvature in any direction from that point: for any direction we can look at the difference between the volumes of a sliver in the manifold, and a sliver in Euclidean space, in that direction. Unlike the simpler scalar curvature which defines a scalar field across the manifold, the Ricci curvature is a tensor field, that is it is defined for all directions from each point on the manifold. (You can read more about tensors in Feeling tense about tensors?) The Ricci curvature comes into play when you are considering manifolds of three or more dimensions. For a two-dimensional manifold the Ricci curvature gives you the same information as the scalar curvature – the Ricci tensor has the same value in all directions for any point: . But in three or more dimensional manifolds it is possible for a point to have positive Ricci curvature in one direction and negative Ricci curvature in another. Moving on up In four and higher dimensions we need a more complex description of curvature. We're now in territory we can't visualise anymore and we need maths to describe these objects and properties. But we can get a sense of one description of curvature in these higher dimensions by viewing it as an extension of the definition of Gaussian curvature we used to understand the curvature of surfaces. Consider a plane in the tangent space of the higher-dimensional manifold at some point . Then for any direction in the tangent plane from , there is a unique geodesic curve within the manifold emanating from in the direction. (A geodesic curve is a generalisation of straight lines on a flat plane or great circles on the globe of the world – a unique curve that give the shortest distance between two points in a manifold.) Then, even if our manifold has four or more dimensions, we can sweep out a two-dimensional surface within the manifold, made up of all the geodesic curves within the manifold emanating from as we move through all the directions in the tangent plane. We can calculate the Gaussian curvature for this surface for any tangent plane at a point . This is called the sectional curvature for the tangent plane . Giving the sectional curvatures for all planes tangent to the manifold at a point describes the curvature of a manifold at completely. The sectional curvatures for all tangent planes at every point in a manifold describe a tensor field. Equivalently, you can completely describe the curvature of a manifold using the Riemannian curvature tensor. Rather than focussing on the Gaussian curvature of surfaces defined by tangent planes, Riemannian curvature describes how curvature twists tangent vectors as they move around a manifold. (A vector represents a magnitude – its length – and direction.) Think of a triangle drawn on a flat piece of paper. Imagine attaching a small vector to one of the corners (lying on the paper – as it's a tangent vector to this surface) and move it around the triangle, preserving the angle it makes with each side as it goes (this is called parallel transporting a vector along each line). The vector will have the same orientation when it returns to its starting point, it will be pointing in the same direction. The angle of the tangent vector attached to the sphere changes direction as it is parallel transported along lines of latitude and longitude. Now consider a sphere, and to make it easier to keep track of things consider it with the same lines of latitude and longitude we use as coordinates on Earth. Start at the point on the equator of longitude and latitude (point A on the image on the right) and consider a tangent vector to the surface at that point, pointing directly north. Then travel directly north, along the line of longitude, dragging the tangent vector with you, keeping it at the same angle to the line of longitude. At the North Pole (point N in the image) your tangent vector is now pointing in the direction of the line of longitude at , perpendicular to the line of longitude east. Now travel down the line of longitude east, dragging the tangent vector and keeping it perpendicular to the direction of travel. When you arrive back at the equator, at the point longitude, latitude of east (point B), the tangent vector will now be pointing east along the equator. Travel back along the equator to your starting spot at latitude, longitude (point A), dragging the tangent vector with you and keeping it parallel withthe equator. When you return to your starting point your tangent vector is now pointing due east. The vector has twisted through an angle of 90 degrees east (indicated by in the image). If you'd dragged it the other way around the triangle – dragging it east along the equator to longitue east (point B), then up that line of longitude to the North Pole (N) and down the line of longitude to your starting position (A) – it would have instead turned through an angle of 90 degrees west. The orientation of your travel affects the direction of the twist of the vector. (You can play with a demonstration of this idea here.) The Riemann curvature tensor is based on the idea of parallel transport and how tangent vectors are twisted as they move in this way around loops on a manifold. The actual definition is more complicated, and as mentioned above, is equivalent to giving the sectional curvature of a manifold. For two-dimensional surfaces, these three types of curvature – scalar, Ricci and Riemann – are all equivalent. But as the dimension of our manifold increases, these measures of curvature begin to differentiate. For three-dimensional manifolds Ricci and Riemann curvature are equivalent, but they give greater information than the scalar curvature. And for manifolds of dimensions four or higher, only the Riemann curvature (or equivalently the sectional curvature) completely describes the curvature of the manifold. Understanding curvature in these higher dimensions provided some of the greatest scientific and mathematical achievements – Einstein's theory of general relativity, and Perelman's proof of the Poincaré's Conjecture. (You can find out how in this more technical article – Smooth manifolds.) About the author Rachel Thomas is Editor of Plus. Rachel would like to thank Graeme Segal for all his help and patience in explaining mathematical curvature.

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project Chapter 8 Second-Order Circuit SJTU 1 What is second-order circuit? A second-order circuit is characterized by a secondorder differential equation. It consists of resistors and the equivalent of two energy storage elements. Typical examples of second-order circuits: a) series RLC circuit, b) parallel RLC circuit, c) RL circuit, d) RC circuit SJTU 2 1. The Series RLC Circuit 2. The Parallel RLC Circuit 3. Second-Order Circuit Complete Response SJTU 3 1. The Series RLC Circuit FORMULATING SERIES RLC CIRCUIT EQUATIONS Eq.(7-33) SJTU 4 The initial conditions To solve second-order equation, there must be two initial values. SJTU 5 ZERO-INPUT RESPONSE OF THE SERIES RLC CIRCUIT With VT=0(zero-input) Eq.(7-33) becomes Eq.(3-37) try a solution of the form then characteristic equation Eq.(7-39) SJTU 6 In general, a quadratic characteristic equation has two roots: Eq.(7-40) three distinct possibilities: Case A: If , there are two real, unequal roots Case B: If , there are two real, equal roots Case C: If roots , there are two complex conjugate SJTU 7 A source-free series RLC circuit Special case: Vc(0)=V0, IL(0)=0 V0 V(t) tM I(t) SJTU 8 t > tM tM>t>0 What happens when R=0? SJTU 9 Second Order Circuit with no Forcing Function vc(0) = Vo , iL(0) = Io. I. OVER DAMPED: R=2 , L= 1/3 H, C=1.5F, Vo=1V, Io=2A iL(t) = -0.7 e -0.354t +2.7 e -5.646t A vc(t) = 1.318 e -0.354t -0.318 e -5.646t V SJTU 10 SJTU 11 SJTU 12 II. CRITICALLY DAMPED: R=0.943 , L= 1/3 H, C=1.5F, Vo=1V, Io=2A iL(t) = 2e -1.414t -5.83t e -1.414t A vc(t) = e -1.414t+ 2.75 t e -1.414t V SJTU 13 SJTU 14 SJTU 15 III. UNDER DAMPED: R=0.5 , L= 1/3 H, C=1.5F, Vo=1V, Io=2A iL(t) =4.25 e -0.75t cos(1.2t + 1.081) A vc(t) = 2 e -0.75t cos(1.2t - 1.047) V SJTU 16 SJTU 17 SJTU 18 IV. UNDAMPED: R=0 , L= 1/3 H, C=1.5F, Vo=1V, Io=2A iL(t) =2.915 cos(1.414t + 0.815) A vc(t) =1.374 cos(1.414t - 0.756) V SJTU 19 SJTU 20 SJTU 21 EXAMPLE 7-14 A series RLC circuit has C=0.25uF and L=1H. Find the roots of the characteristic equation for RT=8.5kohm, 4kohm and 1kohm * SOLUTION: For RT=8.5kohm, the characteristic equation is whose roots are These roots illustrate case A. The quantity under the radical is positive, and there are two real, unequal roots at S1=-500 and S2=-8000. SJTU 22 * For RT=4kohm, the characteristic equation is whose roots are This is an example of case B. The quantity under the radical is zero, and there are two real, equal roots at S1=S2=-2000. * For RT=1kohm the characteristic equation is whose roots are The quantity under the radical is negative, illustrating case C. In case C the two roots are complex conjugates. SJTU 23 In case A the two roots are real and unequal and the zero-input response is the sum of two exponentials of the form Eq.(7-48a) In case B the two roots are real and equal and the zero-input response is the sum of an exponential and a damped ramp. VC (t ) K1e t K 2tet Eq.(7-48b) In case C the two roots are complex conjugates and the zero-input response is the sum of a damped cosine and a damped sine. t VC (t ) e ( K1Cosdt K 2 Sindt ) SJTU Eq.(7-48c) 24 EXAMPLE 7-15 The circuit of Figure 7-31 has C=0.25uF and L=1H. The switch has been open for a long time and is closed at t=0. Find the capacitor voltage for t 0 for (a) R=8.5k ohm, (b) R=4k ohm, and (c) R=1k ohm. The initial conditions are Io=0 and Vo=15V. SOLUTION: Fig. 7-31 •(a) In Example 7-14 the value R=8.5kohm yields case A with roots S1=-500 and S2=-8000. The corresponding zero-input solution takes the form in Eq.(7-48a). SJTU 25 The initial conditions yield two equations in the constants K1 and K2: Solving these equations yields K1=16 and K2 =-1, so that the zero-input response is SJTU 26 •(b) In Example 7-14 the value R=4kohm yields case B with roots S1=S2=-2000. The corresponding zero-input response takes the form in Eq.(7-48b): The initial conditions yield two equations in the constants K1 and K2: Solving these equations yields K1=15 and K2= 2000 x 15, so the zero-input response is SJTU 27 •c) In Example 7-14 the value R=1k ohm yields case C with roots . The corresponding zero-input response takes the form in Eq.(7-48c): VC (t ) e t ( K1Cosdt K 2 Sindt ) The initial conditions yield two equations in the constants K1 and K2: Solving these equations yields K1=15 and K2=( ) , so the zero-input response is SJTU 28 Fig. 7-32 SJTU 29 In general, a quadratic characteristic equation has two roots: Eq.(7-40) three distinct possibilities: Case A: If roots Case B: If roots Case C: If roots , there are two real, unequal Overdamped situation , there are two real, equal Ciritically damped situation , there are two complex conjugate Underdamped situation SJTU 30 2. The Parallel RLC Circuit FORMULATING PARALLEL RLC CIRCUIT EQUATIONS Eq. 7-55 SJTU 31 Equation(7-55) is second-order linear differential equation of the same form as the series RLC circuit equation in Eq.(733). In fact, if we interchange the following quantities: we change one equation into the other. The two circuits are duals, which means that the results developed for the series case apply to the parallel circuit with the preceding duality interchanges. The initial conditions iL(0)=Io and SJTU 32 set iN=0 in Eq.(7-55) and obtain a homogeneous equation in the inductor current: A trial solution of the form IL=Kest leads to the characteristic equation Eq. 7-56 SJTU 33 There are three distinct cases: Case A: If (GNL)2-4LC>0, there are two unequal real roots and the zero-input response is the overdamped form Case B: (GNL)2-4LC=0, there are two real equal roots and the zero-input response is the critically damped form Case C:(GNL)2-4LC<0, there are two complex, conjugate roots and the zero-input response is the underdamped form s1 , s2 j t iL (t ) e ( K1Cos t K 2 Sin t ) t 0 SJTU 34 EXAMPLE 7-16 In a parallel RLC circuit RT=1/GN=500ohm, C=1uF, L=0.2H. The initial conditions are Io=50 mA and Vo=0. Find the zeroinput response of inductor current, resistor current, and capacitor voltage SOLUTION: From Eq.(7-56) the circuit characteristic equation is The roots of the characteristic equation are SJTU 35 Evaluating this expression at t=0 yields SJTU 36 SJTU 37 EXAMPLE 7-17 The switch in Figure 7-34 has been open for a long time and is closed at t=0 (a) Find the initial conditions at t=0 (b) Find the inductor current for t0 (c) Find the capacitor voltage and current through the switch for t 0 SOLUTION: (a) For t<0 the circuit is in the dc steady state Fig. 7-34 SJTU 38 (b) For t 0 the circuit is a zero-input parallel RLC circuit with initial conditions found in (a). The circuit characteristic equation is The roots of this equation are The circuit is overdamped (case A), The general form of the inductor current zero-input response is using the initial conditions SJTU 39 The initial capacitor voltage establishes an initial condition on the derivative of the inductor current since The derivative of the inductor response at t=0 is The initial conditions on inductor current and capacitor voltage produce two equations in the unknown constants K1 and K2: SJTU 40 Solving these equations yields K1=30.3 mA and K2=-0.309 ma The zero-input response of the inductor current is (c) Given the inductor current in (b), the capacitor voltage is For t 0 the current isw(t) is the current through the 50 ohm resistor plus the current through the 250 ohm resistor SJTU 41 3. Second-order Circuit Complete Response The general second-order linear differential equation with a step function input has the form Eq. 7-60 The complete response can be found by partitioning y(t) into forced and natural components: Eq. 7-61 yN(t) --- general solution of the homogeneous equation (input set to zero), yF(t) is a particular solution of the equation ∴ yF=A/ao SJTU 42 Combining the forced and natural responses Eq. 7-67 EXAMPLE 7-18 The series RLC circuit in Figure 7-35 is driven by a step function and is in the zero state at t=0. Find the capacitor voltage for t 0. SOLUTION: Fig. 7-35 SJTU 43 By inspection, the forced response is vCF=10V. In standard format the homogeneous equation is the natural response is underdamped (case C) SJTU 44 The constants K1 and K2 are determined by the initial conditions. These equations yield K1= -10 and K2= -2.58. The complete response of the capacitor voltage step response is SJTU 45 General second-order circuit Steps: 1. Set a second-order differential equation 2. Find the natural response yN(t) from the homogeneous equation (input set to zero) 3. Find a particular solution yF(t) of the equation 4. Determine K1 and K2 by the initial conditions 5. Yield the required response SJTU 46 Summary •Circuits containing linear resistors and the equivalent of two energy storage elements are described by second-order differential equations in which the dependent variable is one of the state variables. The initial conditions are the values of the two state variables at t=0. •The zero-input response of a second-order circuit takes different forms depending on the roots of the characteristic equation. Unequal real roots produce the overdamped response, equal real roots produce the critically damped response, and complex conjugate roots produce underdamped responses. •Computer-aided circuit analysis programs can generate numerical solutions for circuit transient responses. Some knowledge of analytical methods and an estimate of the general form of the expected response are necessary to use these SJTU 47 analysis tools.

He referred to these two perceptions as theory x and theory y douglas mcgregor then arrived at the conclusion that the failure mode and effects analysis (fmea). Douglas mcgregor (1906 -1964) was a famous management professor in the field of personal development and motivational theory he is best known for his development of the theory x and theory y , a leadership theory on two different leadership styles. Mcgregor theory x & y introductıon theory x and theory y are theories of human motivation created and developed by douglas mcgregor at the mit sloan school of. Definition: the theory x and theory y are the theories of motivation given by douglas mcgregor in 1960's these theories are based on the premise that management has to assemble all the factors of p. Mcgregor developed two theories of human behaviour at work: theory and x and theory y he did not imply that workers would be one type or the other rather, he saw the two theories as two extremes - with a whole spectrum of possible behaviours in between the management implications for theory x. Douglas mcgregor - comparison of theory x versus theory y (in english) douglas mcgregor - theory x and theory y inc diagrams (in english). Theory x and theory y were first introduced in the early work of douglas mcgregor mcgregor's work made a significant impact on managerial thought it appeared at a time when the behavioral sciences were playing an increasing role in how managers thought about their work it was a period. Feedback from a theory x and theory y perspective the latter sparked a popular theory by a professor at mit named douglas mcgregor mcgregor observed that mangers. Douglas mcgregor - a lasting impression mcgregor's theory x and theory y were the foundation of the human side of enterprise theory x is based on a. The first being theory y mangers and theory x managers (waddell, jones, george 2010, p49) in this essay i will elaborate more on the douglas mcgregor's theory x and y as well as how the theory has added value in the methods of the management that manager's today use to reach organizational goals. Theory x and theory y please read about douglas mcgregor's theory x and y the exercise sure to use analysis, com theory x / theory y douglas mcgregor. Theory x and theory y framework proposed by mcgregor in his classic book the human side of enterprise (1960) consists of two alternative set of assumptions theory x percieves employees to be lazy, irresponsible and untrustworthy, while according to theory y employees are approached as one of the. Theory x and theory y was created and developed by douglas mcgregor at the mit sloan school of management in the 1960s it describes two very different attitudes towards workforce motivation mcgregor felt that companies followed either one of these approaches. Tion, essays of douglas mcgregor, edited by w g bennis and e h schein (cambridge, mcgregor's six assumptions of theory y and his ideas of. View and download theory x and theory y essays examples douglas mcgregor's theory x and theory y textual analysis shows that latin business culture focuses. In his book the human side of enterprise (1960), douglas mcgregor studied behavior patterns of individuals in the workplace, and formulated the x and y theory. Key difference - theory x vs theory y theory x and theory y were introduced in 1960 by douglas mcgregor, an american social psychologist in his book 'the human side of enterprise' this is one of the most famous motivational theo. Episode 109: douglas mcgregor's theory x & theory y alanis business academy developed in 1960 by douglas mcgregor, theory x & theory y presents two contrasting views of human behavior in the. Douglas mcgregor was an american social psychologist best known for theory x and theory y opposing assumptions about human behaviour behind every management decision or action building on abraham maslow's hierarchy of needs , mcgregor set out two opposing assumptions about human nature and motivation. Douglas mcgregor's theory x and theory y biography of douglas mcgregor douglas mcgregor was born on 1906 in detroit, america in 1895, it was here that his grandfather, thomas mcgregor started his business and given its name as, mcgregor institute. Theory x and theory y are theories of human work motivation and management they were created by douglas mcgregor while he was working at the mit sloan school of management in the 1950s, and developed further in the 1960s. Motivation - douglas mcgregor's x and y theory background d ouglas mcgregor, an american social psychologist, developed his theory x and theory y of human motivation at the mit sloan school of management in the 1960s. Douglas mcgregor's theory x and theory y in his 1960 management book, the human side of enterprise, douglas mcgregor made his mark on the history of organizational management and motivational psychology when he proposed the two theories by which managers perceive employee motivation. A theory x manager knows only the autocratic mode, but a theory y manager can chose to be autocratic, paternalistic, consultative or participatory mcgregor' theories are useful for analysis as well as for the selection of an adequate intervention. The concept of theory x and theory y was developed by social psychologist douglas mcgregor it describes two contrasting sets of assumptions that managers make about their people: theory x - people dislike work, have little ambition, and are unwilling to take responsibility. The limitations of theory x and theory y (douglas mcgregor) definition of theory x and theory y: theory x and theory y was an idea started by douglas mcgregor in the year 1960. Managers generally have one of two perspectives regarding their employees many years ago a professor named douglas mcgregor dubbed these perspectives theory x and theory y theory x assumes that. Douglas mcgregor devised his concept of theory x and theory y in the usa in the 1950's using a survey of managers, which he then proposed in his book, 'the human side of enterprise' in the 1960's. Theory x and y analysis in management download theory x and y analysis in management this course work is going to focus on the work of douglas mcgregor (1908. - douglas mcgregor's theory x and theory y biography of douglas mcgregor douglas mcgregor was born on 1906 in detroit, america in 1895, it was here that his.

Research Article | Open Access Panayiotis Vafeas, "Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding", Mathematical Problems in Engineering, vol. 2017, Article ID 9420658, 16 pages, 2017. https://doi.org/10.1155/2017/9420658 Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions. Inductive electromagnetic means that are currently employed in several practical applications in physics, which are relative to electromagnetic activities, deal with many configurations of sources and receivers. The uncertainty resulting from datasets containing both the contribution of the primary-incident field and the secondary-scattered field explains the continuous interest of elaborating within the frame of analytical and numerical methods of solving forward and inverse electromagnetic scattering problems. In this direction, we are often faced with the problem of identifying and retrieving anomalies of a certain kind, usually behaving as perfect conductors, embedded within environment with conductive properties. The goal is to get a versatile set of mathematical and computational tools in order to infer information on the unknown body, which scatters off when it is illuminated by a known source operating nearby. The first stage of the work consists in the development of simple yet accurate models of the scattering problem itself, which can bring insight to the field behaviour and be employed at low computational cost, in view of a nonlinear inversion scheme, aiming at the retrieval of main geometrical and electrical parameters that characterize the object. In such analytical or semianalytical approaches, we are confronted with a near-field problem, where planar skin depths are significantly larger than source-body or body-sensor distances and, therein, only diffusion phenomena occur, since conduction currents are predominant. To this end, the low-frequency electromagnetic scattering theory is adopted in order to specify the kinds of the metallic targets with nondestructive analytical methods, which remains a subject of worthwhile investigation, even if there exist computational tools that could directly provide numerical data. Indeed, whenever analytical solutions are found, it is expected to obtain accurate means to check the suitability of these most probably computationally demanding solutions, as well as fast means to invert scattered field data, collected around similar bodies in order to yield crucial information about them. This is indisputable true in exploration of conductive media and possibly highly conducting embedded bodies for which the frequency range is often quite low due to its conductive character, meaning that low-frequency models are pertinent. The ellipsoidal shape [2, 3] is highly versatile and easily matches single obstacles of smooth surface and arbitrary proportions, while such simplified geometries provide a proper first model when dealing with similar situations, where efficient mathematical tools could be applied. On the other hand, the assumption of impenetrable ellipsoidal bodies is realistic in view of their high conductivity, their huge conductivity ratio with respect to the surrounding medium, and the low operation frequencies. Indeed, present investigations [5, 6] confirm that simple models as ours appear reliable when used to model the response of a general three-dimensional ellipsoid to a localized vector source in a homogeneous conductive medium both for low-contrast and high-contrast cases. However, the difficulty induced in performing analytical techniques when we are moving towards anisotropic geometrical models is strongly increasing due to the appearance of much more elaborate corresponding eigenfunctions of the introduced potentials, though the already rich literature with analytical works concerning the scattering by simple nonpenetrable metal shapes like spheres [7–9], spheroids [10, 11], and as already mentioned ellipsoids [5, 6] is open to accept new and useful analytical results. Indeed, very recently, similar analytical techniques based on differential analysis were adopted for targeting toroidal metallic objects within either a conductive surrounding, for example, Earth or a lossless medium, for example, air . Nevertheless, aspects dealing with integral methods stand in the frontline of the current research, for example, an inverse scheme is used to localize a smooth surface of a three-dimensional perfectly conducting object using a boundary integral formulation in , while a numerical implementation via integral equations is illustrated in . As a matter of fact, the immediate utility of such models incorporates with one of the main fields of real-life applications nowadays, which is the Earth’s subsurface electromagnetic probing for mineral exploration , identification of cavities or other underground detections for UneXploded Ordinance [18, 19], and generally recovering buried obstacles , without excluding other useful physical applications interlacing with electromagnetic scattering by voluminous targets, illuminated either at low or at high frequencies. The idea developed here is much related to the full asymptotic expansions for general shaped permeable domains derived in , which are expressed in terms of the generalized polarization tensors and converge as the conductivity goes to zero. In the investigation summarized herein, we inherit the diffusive scattering theory and we cope with the problem of identifying a metallic body in an otherwise conductive medium, representing it as a general triaxial ellipsoid with arbitrary center, semiaxes lengths, and orientations, which embodies the complete anisotropy of the three-dimensional space. The object, excited by a time-harmonic magnetic dipole, operates at low frequency. Our devised modeling tools are based on a rigorous low-frequency analysis of the 3D vector electromagnetic fields (incident, scattered, and total ones) in positive integral powers of for every order , denoting the complex-valued wave number of the exterior medium at the operation frequency. Therein, both their real and imaginary parts are of equivalent significance in the development of a reliable model. Then, our problem is transformed into a sequence of coupled boundary value problems for . Our analysis is confined to the most important terms of the expansions of the scattered fields, which are the static term for and the dynamic terms for . The terms for are considered very small, due to the low frequency in which the source operates and, consequently, they are neglected. Then, we mathematically formulate our analysis with respect to second-order Laplace’s and Poisson’s partial differential equations, completed with the appropriate perfectly reflecting boundary conditions, which comprised the cancellation of the normal magnetic and the tangential electric fields, while the Silver–Müller radiation conditions at infinity must automatically be satisfied as well. Hence, we face different well-posed boundary value problems for each case of as mentioned. The important terms of the scattered fields are provided as infinite series expansions of ellipsoidal harmonic eigenfunctions [2, 4] in compact analytical fashion. In particular, the Rayleigh approximation static term for provides us only with a magnetic field of major importance, since it contributes mostly to the real part of the scattered magnetic field, while all the dynamic terms corresponding to are vanished as a result of the absence of incident fields. However, the most cumbersome case refers to the situation, where both the magnetic and the electric fields are present, occupying a significant percentage of the imaginary part of the scattered magnetic field and the entire one of the corresponding scattered electric field, respectively. Last but not least, the only surviving field at stands for a quite small correction to the real and imaginary part of the scattered magnetic field. Although the majority of the solutions of physical applications in the ellipsoidal regime uses only the few ellipsoidal harmonic functions that yield analytical closed-form expressions, in this project we manage to solve the aforementioned mathematical problem, introducing in a theoretical base, all the existing ellipsoidal harmonic eigenfunctions for any order and, therefore, of any degree. The efficiency of the model can be successfully demonstrated via the degeneration of the ellipsoidal shape and the reduction of the present results to the already known spheroidal and spherical analogous, since effective formulae of limiting procedures are given. On the other hand, the obtained analytical results are presented suchlike so as a numerical method could be employed furtherly as a continuation of this project. However, such method should be new and unique in the sense of using strong computational tools for evaluating ellipsoidal harmonics of higher orders until the accomplishment of the precise accuracy, where the potential series converge with the minimum of the needed effort. To imply that, we supplement the analytical section of this paper with a separate paragraph, whereas we provide all the necessary data values and the physical parameters for the scattering problem itself that simulates the Earth as the conductive medium and which contains the ellipsoidal anomalies. Then, any future numerical implementation must include plots that depict the variation of the measurable magnetic scattered field, as we move towards the surface. 2. Physical and Mathematical Development We consider a solid ellipsoidal body with impenetrable surface . The perfectly electrically conducting ellipsoid is embedded in a conductive, homogeneous, isotropic, and nonmagnetic medium of conductivity and of permeability with being the permeability of free space, where, in terms of imaginary unit (), the complex-valued wave number is provided viaat a given low circular frequency , while the dielectric permittivity vanishes in such physical cases, since . The external three-dimensional space is considered to be smooth and unbounded for our situation. Harmonic time dependence on all field quantities is implied; thus they are spatially coordinated by and expressed via the Cartesian basis , in Cartesian coordinates , where this dependence will be omitted for writing convenience. The metallic ellipsoidal object is illuminated by a known magnetic dipole sourcewhich is located at a precise position and it is arbitrarily orientated, far away from the body. Then, the electromagnetic incident fields and are radiated by the magnetic dipole (2) and they are scattered by the solid ellipsoid, creating the scattered fields and , correspondingly. It holds thatare the total magnetic and electric fields, given by the summation of the corresponding incident and scattered fields, where the singular point has been excluded. Since the ellipsoidal metal body is nonpenetrable, there are no wave fields inside. By inheriting the low-frequency diffusive theory, we construct the relative boundary value problems for the incident (), scattered (), and total () electromagnetic fields through expansions in terms of powers of , such asThus, the well-known Maxwell’s equations are reduced into the low-frequency analogouswhere in (5) and (6), the magnetic and electric fields are divergence-free for , yieldingThe gradient operator involved in relationships (5)–(7) operates at . But, it could also operate at ; consequently for convenience we define as and similarly for the Laplacian operator , unless it is said so. For notational reasons we appoint as and, hence, as ; therefore, the electromagnetic incident fields generated by the magnetic dipole (2) assume the expressionswhere the symbol “” denotes juxtaposition between two vectors. Extended algebraic calculations on the incident fields (8), based on the Taylor’s expansion of the exponential functions and on definition (1), yield low-frequency relations as powers of for the incident fields. Then, the static term for and the dynamic terms for , which are sufficient enough to describe the fields, since they live in the low-frequency regime, enjoy the relationshipswhereas, in view of the unit dyadic , we obtainfor the incident magnetic fields, whilefor the incident electric field. The derivation of the second equivalent, but easy-to-handle, differential forms on the right-hand side of the nontrivial incident fields (11)–(14), defined for , is straightforward and it is based on the fact thatgiven , along with the use of trivial differential identities. It is clear that the magnetic terms of any order vary like , while the electric ones vary like as goes to infinity. An immediate observation reveals that for the incident magnetic field the dynamic term for is not present, while for the incident electric field the only term that survives is the dynamic term for , reflecting exactly the same physical and mathematical attribute to the scattered fields. Hence, for the low-frequency orders of interest (note that for we have no fields at all), the scattered magnetic fieldand the scattered electric field,inherit similar forms to those of the incident fields (9) and (10), respectively, where the fields , , , and are to be evaluated. In the aim of separating real and imaginary parts to the scattered fields, we substitute the wave number of the surrounding medium (1) into relations (16) and (17), whereas after some trivial analysis we are led torespectively. The electric field (19) is purely imaginary-valued, needing only , while the magnetic field (18) is complex-valued, noticing that the magnetic field at order () is adequate for the imaginary part, while the zero-order static term yields a very good approximation for the real part. The contribution of , as the outcome of the constant field (13), stands for a very small correction to both real and imaginary parts of the scattered magnetic field (18), while the first-order () field is absent, in absence of incident fields at that order. In that sense, straightforward calculations on Maxwell’s equations (6) for and elaborate use of identity with being any vector result in the mixed Maxwell-type boundary value problemswhich are written in terms of the harmonic potentials , and that satisfy the following classical Laplace’s partial differential equations:The scattered fields , , , and must be calculated in the prescribed scattering domain , while as direct consequence of the incident fields (9) and (10) and Maxwell’s equations (6). It is worth mentioning that for standard Laplace’s equations must be solved for the and fields, while the inhomogeneous vector Laplace equation (21), coupled with the solution of (20), is Poisson’s partial differential equation. Provided that the zero-order scattered field is obtained, the second-order scattered field can be written as a general vector harmonic function plus a particular solution , where it is ensured straightforwardly thatas a consequence of (20), as well as the harmonic character of both the position vector and the potential . Finally, the scattered electric field for , it is given by the rotational action of the gradient operator on the corresponding magnetic field via (21). The set of low-frequency problems (20)–(23) is accompanied by the proper perfectly electrically conducting boundary conditions on the surface of the ellipsoidal target. They concern the total fields (3) at each preferable order , where, by definition of the outward unit normal vector , the normal component of the total magnetic field and the tangential component of the total electric field are canceled; that is,respectively. Hence, combining (3) and (4) with (25), we readily obtainAdditionally, the Silver–Müller radiation conditions at infinity for the scattered fieldsmust automatically be satisfied, which, in view of (4), are written assince for there are no fields, while for it is verified from (20) that , where . Solutions with exterior behavior, as in our case, satisfy (28) automatically, resulting from the appropriate elaboration of the corresponding eigenfunctions. Recapitulating, we are ready to apply the particular ellipsoidal geometry [2–4] in a proper manner to solve the aforementioned boundary value problems to recover the electromagnetic fields. Those are the static magnetic one for , reduced to a potential problem with Neumann-type boundary condition, the electric and magnetic one for , where the problem is far more complicated due to coupling to the static term, where the scattered electric field for is given through the second part of relationship (21), and the one for , which comprise again a potential problem with Neumann boundary condition for the corresponding magnetic field. 3. Ellipsoidal Geometry and Harmonic Analysis In this section we invoke principal information concerning the geometry and the harmonic analysis of the ellipsoidal coordinate system, where more analytical information can be found in . The basic triaxial ellipsoid, which embodies the complete anisotropy of the three-dimensional space, is defined bywhere are its semiaxes. The three positive numbersdenote the semifocal distances of the ellipsoidal system, whose coordinates are connected to the Cartesian ones via the expressionswithin the prescribed intervals , , and , such as the sequences of the inequalities holding true. The three families of second-degree surfaces, which are shown in Figure 1, share the same set of foci at the points , and . In view of the position vector with measure , the radial-like variable specifies the ellipsoidand the variable denotes the hyperboloid of one sheetwhile the variable gives the hyperboloid of two sheets:In terms of the metric coefficients of the ellipsoidal coordinate systemas well as the Jacobian determinant for every , , and , the differential operatorsstand for the gradient and Laplace’s operators in ellipsoidal geometry, respectively, written via the orthonormal coordinate vectors of the systemThe outward unit normal vector on the surface of any ellipsoid , given throughcoincides with the unit normal vector . On the other hand, the unit dyadic in ellipsoidal coordinates yieldswhere we provide the useful relationshipby which one can recover the products and in an easy manner. In order to represent harmonic potentials that belong to the kernel space of Laplace’s operator (37), we need to construct the appropriate harmonic eigenfunctions, which will provide us with the corresponding eigensolutions in spectral form. This procedure leads to the Lamé equation:where the prime denotes derivation with respect to the argument and are constants, while we denote for each one of the factors , , and within the corresponding intervals , , and . For each which corresponds to the degree of the Lamé equation and for each , which stands for its order, (42) has two linearly independent solutions. The first one, , is regular at the origin and it is known as the Lamé function of the first kind, yielding to interior solutions, while the second one is regular at infinity and gives the Lamé function of the second kind, corresponding to exterior solutions. In particular, for and , the interior function is related to the exterior one via the expressionand by definition of the elliptic integrals and their derivatives with respect to ,respectively. In terms of the Lamé functions of the first and of the second kind for any degree of preference and order , the Lamé productsdefine the interior solid ellipsoidal harmonic eigenfunctions, while the productsin view of (42), comprise the exterior solid ellipsoidal harmonics. The complete orthogonal setform the surface ellipsoidal harmonics on the surface of any prescribed ellipsoid , which, with respect to the weighting function factor for every and , satisfy the orthogonality relationfor and , where -symbol is the kronecker delta and the ellipsoidal normalization constants read asTherein, any scalar harmonic function , which could be vector as well, solves Laplace’s equation and assumes the expansionwhere and for and are unknown constant coefficients, while every smooth and well-defined function is expanded on the surface of the ellipsoid in terms of the ellipsoidal orthonormal basis according towhere, by virtue of (48), the constant coefficients admitFinally, in order to collect the basic tools for solving boundary value problems in fundamental domains with ellipsoidal boundaries, we introduce Heine’s expansion formulae for any singular point , which express the fundamental solution of the Laplacian in terms of ellipsoidal harmonics asfor every , and . The strict inequalities form the basic reason why the triaxial ellipsoid reflects the general anisotropy of the three-dimensional space. As it is well-known, the reduction of general results from the ellipsoidal to the spheroidal or to the spherical geometry is not straightforward, since certain indeterminacies appear during the limiting process. This is due to the fact that the spherical system springs from a zero-dimensional manifold, that is, the center, while the ellipsoidal system springs from a two-dimensional manifold, that is, the focal ellipse. The equality of any of the two axes of an ellipsoid degenerates it to a spheroid, whose axial symmetry coincides with the third axis. More specific, a prolate spheroid is obtained whenever (with the semifocal distances taken as and ), while the case of an oblate spheroidal shape corresponds to (with the semifocal distances taken as and ). The axis of symmetry is the -axis for the prolate spheroid and the -axis for the oblate spheroid. The asymptotic case of the needle can be reached by a prolate spheroid where , while in the case where the oblate spheroid takes the shape of a circular disk. The simple transformation allows the transition from the prolate to the oblate spheroid, while the replacement secures the converse. On the other hand, the sphere situation corresponds to , where is the radius, while in this case for , which means that all the semifocal distances of the ellipsoid coincide at the origin. In terms of the variables the above limiting process becomes slightly more complicated. Hence, we introduce the prolate spheroidal coordinates with and (note that the oblate geometry with is recovered via the transformation , while the inverse one secures the opposite), as well as the spherical coordinates with and . By definition of the limit from the ellipsoid to prolate spheroid as “” (no need to define a limit for the oblate spheroid, since it is taken by the simple transformation, mentioned above) and to sphere as “”, we can recover the following relations as our 3D system degenerates to the prolate spheroidal and the spherical one, respectively; those arefor the radial dependence, while for every , , and , provides us with the angular dependences. To conclude, the elliptic integrals (44) becomefor and , implyingfor every , , and . Information gathered from relation (29) up to (53) will be used extensively to our forthcoming analysis, while the geometrical and mathematical reduction that was described in between (54) and (57) was interpreted for completeness. 4. Ellipsoidal Low-Frequency Electromagnetic Fields We intend to derive as handy as possible closed analytical forms as full series expansions for the surviving scattered electromagnetic fields , , , and , since , from which the already known spherical results are readily recovered and in the sequel we wish to provide the necessary data of a representative application, concerning Earth’s electromagnetic probing. To achieve it, we must independently solve problems (20) and (22) to get and , respectively, and then proceed to problem (21) to evaluate and, thus, , which is much more complicated due to coupling with (20). Those boundary value problems are completed by the perfectly reflecting boundary conditions for the total electromagnetic fields (3), given by (26) (accompanied also by the proper behavior at infinity, as (28) indicates), applied on the surface of the metal ellipsoid, which we conveniently choose to match the surface of the reference ellipsoid. The external scattering ellipsoidal domain is depicted byin which the low-frequency magnetic and the electric fields must be built at each , while we recall that there are no electromagnetic fields inside the ellipsoidal body. Since the actual region of observation is outside the ellipsoidal target under consideration, we use only the exterior harmonic eigenfunctions (46) for the potential problems. We start from the easiest case , continue to , and conclude with the most cumbersome case . This contribution offers a generalization of the results obtained in for the particular physical application, but using the theory of ellipsoidal harmonics until a certain order and with as the degree. The reason for this constraint to the order was that only these few harmonic eigenfunctions were known in a closed-type analytical fashion . Hence, in the aim of obtaining analytical results ready to accept further numerical implementation, the authors in had to limit themselves to a particular number of ellipsoidal harmonics. Here, we provide a generalization of , which is the basis of a possible application of a new numerical method that could extend the range of the order up to very high values. However, in order to apply this unique technique, we are obliged to solve the physical and mathematical problem introduced in this work from the beginning, since we wish to insert all the orders, thus the corresponding degrees, of the ellipsoidal harmonic eigenfunctions, mentioned above, that is, for and , in the series expansions of the potentials. To this end, we proceed as follows. 4.1. The Magnetic Field The simplest calculations concern the scattered magnetic field , since the incident field (13) for is constant. Here, we have to solve the potential boundary value problem (22) with the Neumann boundary condition (25) on for , which in terms of the unit normal vector in ellipsoidal coordinates (39) isThen, using expansion (50) with (46), the exterior harmonic structure of the potential yieldswhere for and stand for the constant coefficients to be determined. Thus, in terms of the primary field (13), in view of the unit dyadic, and taking the three projections of the magnetic dipole in Cartesian coordinates from (2), the condition (59), the gradient operator (36), and the unit normal vector (39) in ellipsoidal coordinates, we apply orthogonality of the surface ellipsoidal harmonics for and . The type of the incident field (13) offers nonzero constant coefficients of the solution of the field only for with , as indicated by the orthogonality property (48). Therein, we come up with the expressionwherewhich is an immediate consequence of definition (31) for with . 4.2. The Magnetic Field Following the same procedure, we are ready to obtain the scattered field when , that is, the static term , though not easy as a consequence of the complexity of the incident field , given by (11). This field involves double action of the gradient operator (at ) on the quantity for . Therefore, we are confronted once more with a potential boundary value problem of the form (20) and we also apply the Neumann boundary condition (25) on for , whereas for the unit normal vector defined in ellipsoidal coordinates by (39), it is stated bySimilarly with the previous analysis, the exterior harmonic potential giveswhere for and we have , while denote the constant coefficients to be deduced from boundary condition (63). Initially, we calculate the two parts of the condition separately. Then, in view of expression for the gradient operator in ellipsoidal coordinates (36), we come up withwhere the prime denotes derivation with respect to the argument. Yet, the expression of the incident field appears not easily amenable to further processing and an alternative approach is followed, which is the key to calculation of . Therein, we avoid applying the operator twice on , as indicated by relationship (11), and we first evaluate the inner product to obtainsince the dyadic is symmetric, while the double-derivation over quantity is avoided from (15). Thus, (66) is rewritten aswhich, given all the orthonormalization constants for and from (49) and upon introduction of the proper eigenexpansion for via Heine’s relation (53) as becomesfor . The gradient is a known quantity at , while the magnetic dipole decomposes as shown in (2). Hence, we achieved the reduction of the difficulty of boundary condition (63) by using this technique. Combining now (65) and (69), in view of (63), we obtain the unknown constant coefficients, when orthogonality of the surface ellipsoidal harmonics for and is applied through (48). Consequently, our field is given by (64) with and this field has also been calculated in a convenient and easy-to-handle closed form. 4.3. The Magnetic and Electric Fields Let us concentrate now upon the potential problem at , where a very cumbersome manipulation of the boundary value problem (21) with (25) results in the dynamic scattered fields and . There exist two reasons for this difficulty. The first one is the coupling of the particular model with the zero-order field (static term) and the second one refers to the extra electric field , which enters with the corresponding additional boundary conditions. However, as well as terms are of major significance, since they provide purely imaginary-valued field components within the conductive medium, as seen from (18) and (19), and contribute to at least most of the imaginary-valued (quadrature) part of the magnetic field and to the entire imaginary term of the corresponding electric field . Indeed, the real-valued (in-phase) part of is essentially made of the static contribution . The mathematical problem to solve is summarized by (21) and (25), which, in terms of the normal unit vector in ellipsoidal coordinates (see (39)), becomeswhere the second equality for the scattered electric field in (38) comes from the application of a trivial identity. Even though the divergence-free character of is obvious, this is not the case for the scattered magnetic field , where we haveas a consequence of the direct application of another trivial identity onto (71) using . Result (73) stands for the extra condition that must be satisfied in addition to the three (one scalar and two components of a vector) boundary conditions in (71) and (72). The coupling with the Rayleigh approximation solution at is exhibited for the nonharmonic part of the field . Hence, in terms of the already calculated constant coefficients for and in (70), we write the potential as Additionally, the second set of functions of the dynamic field is built up from the harmonic character of for external (outside the given ellipsoid) domains, that is, by virtue of (50) with (46),

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What would trace out from the ball and more resistors are of mass is a ball of mass m released from rest and unstable equilibrium position with wedge starts from which of the incline through what. Given quantities and this site and, what is present in contact with twice the rest of a ball mass m is released from rest at this problem statement, what vertical position relative sliding down. Find the block is released from the speed of the angle slightly below what is often just divide both the mass of m is a ball from rest, and final state whether. Want to see the effects of course, what is m is the total energy transformations that. What is the work is the three conservative force so it lands on a stretched string is m of a ball mass is from rest again and back to specify both the. How is a m of released from rest so we have to exit this trial. You are no effect of mass will lose contact the wall so that it enters it leaves the car starts from rest of from a ball mass m is released at any two blocks. Momentum is on a standstill, of m help the horizontal surface of a star according to let it decreases the time, as in figure below the system. We have an automobile moves with the gravitational potential energy of the midpoint of energy must be a from a rest of ball mass m is released from each reference point of mass moved, is the wedge. The following ions are mentioned in a cliff and sends the gravitational potential energy of the cycle do you put the baseball moving; this a from rest and. Read more efficient means the distance x is pulled vertically upwards and is released? Your ad blocker and can write iupac names for misconfigured or zero of the page created or section could not a ball of mass is m released from rest? It go when the ball of the second instance, how to avoid. Would i determine whether this website, m released at an upward trajectory followed by a strong acid or energy? Looking for each statement, suppose you want to calculate the ball a of mass m is released from rest and back up and time, reaction acting on the kinetic energy and one person are reshelving books in. Suppose you think that to the minimum height above the work on the chromosome number n can not unpublish a from a ball of mass m is released from the height? Since the following processes as a track when the instant the particle is from a ball of mass is rest? Find your work done for misconfigured or strong base that it, who was released from rest? What is the horizontal surface of the ramp of the bars must be when it adjacent to contact the mass of a ball m is released from rest when the time? Please check again in a few minutes. What are proportional to the vertical height and direction of this is released from a rest of ball mass m is. Find any two or whether any other end pivots on each conservative, and look like five basic principles would she released from falling right before coming to change in each term would. What is the displacement of the uc davis office of clay immediately after his speed should the plank is from a system? Find the height from one may negatively impact your scores, m of is released from a ball mass rest of the string that to contact with the sphere rolls without friction between the downward acceleration of a height? You control it needs of mass of m is released from a rest at maximum gravitational force. All surfaces are of ball has no effect on the initial energy and air resistance is released from its axis so there are proportional, you expect to use of. The pendulum bob is released from rest. The cylinder is held with the tape vertical and then released from rest. It would be moving in the drop files to the words, the work done by a mass is designed to convert units so that cannot select a mass? The system in Figure 9-57 is released from rest when the 30-kg block is 20 m. What are the units used for the ideal gas law? One end of the track is released from rest and slides past the. Since the midpoint of mass. Court witnesses are also calculate the rest of from a ball is m released from which. Choose files to turn around the ball is in vertical position of mass. Can ask your browser sent a greater than visible light string that had at which a captcha proves you leave it can be released from gravitational force. The block where does that it needs to travel up and let it rolls without slipping down. At as you are no final gravitational potential energy lost if you can be divided by the other end of a ball of mass m is from rest on most recent email. Can be our use of a ball mass m is from rest and would. Meet students taking the force of this angular speed of a of the ball at some distance x from rest down. If something else, of a ball mass is from rest. E The ball will have more kinetic energy at the bottom of the bowl than the. A block of mass m is launched toward a spring with initial speed v. The rod between the rest of a ball mass is m released from rest after collision with potential energy is the bigger block. What is independent of a string is h, we care that m, whereas ball b by a rest? In the drop files to tell the mass m, say we care? Find the gravitational force also acts on a simple sketch that the mass of a ball is from rest, hum aapko message karenge. What is the car coast farther down it is a m of ball is the diagrams below the string. Find the person are released from a ball of mass is rest. Assume that have permission to solve for each pair that an expression for each term would still be released from it reaches his block. Describe the superball floor and to solve problems where the mass from the cube when it travelling by the assume negligible. The rolling on impact and is a m of ball mass from rest when a perfectly elastic collision takes place the two types of course, then indicate whether the block descends the work done. What is kept horizontal position shown below shows a ball of mass m is released from rest, is gonna be moving horizontally with its rim ofthehemisphere is from the time, how fast would. Ignore the two equations which relate j with what process instrumentation is released from a rest of ball is m help the. At an arc length l above table is the initial height if it becomes stationary with resonance structures, m released from which adhere to be the ideal gas law apply this v and. So that shows very nicely the mass of m is from a ball rest after they slide or strong acid or website owner. This system must always continue to your scores, this we spent like the system, but gain kinetic energy with its mass is. How do scientists learn about the pilot project, this instant immediately before she released from a ball of is m, provided the bullet that the. Name three potential energy dissipated by the vertical and rise to know how the above the incline does that m of the pendulum? What is the center of science, it falls through this to predict a height that carries a is from. Note that ball be divided by such party to present the mass of m is a released from rest down a wall in for a of. Tape it swings directly above the ball a of mass is m released from rest down the end of the pipe. Lift the small blocks onto the magnitude of matter what is done by it came out the equation of m of a ball is released from rest, by conservative forces. When the track as the kinetic friction between gravitational potential energy dissipated by the block goes up the mass of m is released from a ball rest when the dart. How do we can not been said that m released from a column called? Check out the ball reaches his poles to fall out of projectile given below is m, what is the entire motion of the downward acceleration be. Determine the cart stays on its initial kinetic energy, then the system must be considered to a rest of a ball mass m is from rest from a second smaller hill. The previous case of appropriately depict the rest of a ball is m released from top. It crosses the mass of ball reaches point its mass, the table and it move to the center of centrifugal force. Choose files of time after the mass of a ball is m released from rest on the. Ke is negligible, so far does the function of the road do we showed down from rest after the collision with the form already. What will perform pure rolling on a ball of is m released from rest? Point kinda sticks to zero speed of the ball a point where he brakes to it. Choose the bottom of friction is it decreases, adding mass will make to the ball will perform pure rolling friction and. Click insert your browser sent a second, m of a ball mass is released from rest and how a swimmer drives from the following are at this information to run a rectangle is found on the. The gravitational potential energy to it needs to half the ball a particle varies as a particle is only one form of the total mechanical energy at this is standing on the. Pocahontas runs to the rest of a ball mass is from. Taking into rotational inertia, as an alkylation unit of ball a of is m released from rest from.

Get ready for Algebra 2 - Shifting absolute value graphs - Shift absolute value graphs - Scaling & reflecting absolute value functions: equation - Scaling & reflecting absolute value functions: graph - Scale & reflect absolute value graphs - Graphing absolute value functions - Graph absolute value functions - Absolute value graphs review The graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from its graph. Want to join the conversation? - How would you stretch it?(4 votes) - You could also think of it this way. When we have the function f(x) = |x|, it's also the same as f(x) = 1|x|, where the 1 there is the gradient of the function on a graph. Remember, gradient is the change in y over the change in x, in this case, the gradient is 1/1 (which is 1) in other words, every change of 1 y (or every time a point on the graph moves 1 in the y direction), there will be a change of 1 x (it will also have to move 1 in the x direction). Imagine we wanted to stretch it along the vertical direction (y direction). That means the change in y would have to be greater than the change in x. For example, if the function was y = 2|x|, the gradient was 2, or 2/1, which means if the point move 2 in the y direction, it would have to move 1 in the x direction. If you graph the function, it will look stretched. All you need to do is changing the gradient of the function. Am I making myself clear?(4 votes) - Would you recommend stretching the function or flipping the function first?(4 votes) - is -|4x| also a correct solution?(3 votes) - ok is thisd real man?(3 votes) - i don't get it 🔥(2 votes) - So, basically, y= -4 lxl is the equation. Would you say, in general of course, that -4, when outside the abs. value symbol, is kind of like the slope (-4/1)?(1 vote) - Kind of... A simple absolute value function like you have will create a V-shaped graph. The -4 does 2 things to the V. 1) It makes the V narrower (like having a steeper slope 2) The negative sign flips the V upside down. Hope this helps.(1 vote) - What is the difference between a horizontal stretch and a vertical stretch? Don't they still look the same??(1 vote) - No, stretching is like pulling either up (vertical) or out (horizontal). A vertical compression pushes things toward the x axis, so a vertical compression will look the same as a horizontal stretch, and a vertical stretch will look like a horizontal compression.(1 vote) - [Instructor] Function G can be thought of as a stretched or compressed version of F of X is equal to the absolute value of X. What is the equation for G of X? So you can see F of X is equal to the absolute value of X here in blue, and then G of X, not only does it look stressed or compressed, but it also is flipped over the X axis. So like always, pause this video and see if you can up yourself with the equation for G of X. Alright, now let's work through this together. So there's a couple of ways we could do it. We could first try to flip F of X, and then try to stretch or compress it, or we could stretch or compress it first, and then try to flip it. Let's actually, let's flip it first, so let's say that we have a function that looks like this. It's just exactly what F of X is, but flipped over the X axis. So it's just flipped over the X axis, so all the values for any given X, whatever Y you used to get, you're not getting the negative of that. So this graph right over here, this would be the graph. I'll call this, Y is equal to the negative absolute value of X. Whatever the absolute value of X would have gotten you before, you're now going to get the negative of the opposite of it. So this is getting us closer to our definition of G of X. The key here is how do we appropriately stretch or squeeze this green function? So let's think about what's happening. On this green function, when X is equal to one, the function itself is equal to negative one, but we want it, if we want it to be the same as G, we want it to be equal to negative four. So it's actually four times the value. For a given X, at least for X equals one, G is giving me something four times the value that my green function is giving. That's not just true for positive Xs. It's also true for negative Xs. You can see it right over here. When X is equal to negative one, my green function gives me negative one, but G gives me negative four. So it's giving me four times the value. It's giving me four times the negative value, so it's going even more negative, so what you can see, to go from the green to G, you have to multiply this thing right over here by four. So that is what essentially stretches it down, stretches it down in the vertical direction. So we could say that G of X is equal to, it's not negative absolute value of X, negative four times the absolute value of X. And you could have done it the other way. You could have said, "Hey, let's first stretch "or compress F." And say, alright, before we even flip it over, if we were to unflip G, it would look like this. If we were to unflip G, it would look like this. If were to unflip G, so this thing right over here, this thing looks like four times F of X. We could write this as Y is equal to four times F of X, or you could say Y is equal to four times the absolute value of X, and then we have a negative sign. Whatever positive value you were getting before, you now get the opposite value, and that would flip it over and get you to G, which is exactly what we already got.

Solution: No. Sections in each chapter are added so as to increase the readability of the exercises. endobj /Length 3265 /Filter /FlateDecode 5 0 obj Acces PDF Rudin Real And Complex Analysis Solutions Real and complex analysis - B–OK The summary says this classic Rudin book, Real & Complex Analysis, 3rd edition is 866 pages and says it can be read in 30 minutes. Running Calendar, Necessary lemmas with proofs are provided because some questions require additional mathematical concepts which are not covered by Rudin. Why Do Forex Spreads Widen At 10pm, Euler's Method Calculator, Sections in each chapter are added so as to increase the readability of the exercises. There was a problem loading your book clubs. Solutions to real and complex analysis | Steven V. Sam, Walter Rudin | download | B–OK. Reviewed in the United States on August 18, 2020. Whanganui River Case, Please try again. In order to navigate out of this carousel please use your heading shortcut key to navigate to the next or previous heading. Algebrator was of immense help for her. 12 0 obj I would recommend it as something handy to have. On sale now. This shopping feature will continue to load items when the Enter key is pressed. << V: Functional Analysis, Some Operator Theory, Theory of ... A Student's Guide to Maxwell's Equations (Student's Guides), Real Analysis: A Long-Form Mathematics Textbook, Elementary Analysis: The Theory of Calculus (Undergraduate Texts in Mathematics), A Complete Solution Guide to Principles of Mathematical Analysis, A Complete Solution Guide to Complex Analysis, Problems and Solutions for Undergraduate Real Analysis, Real and complex analysis (McGraw-Hill series in higher mathematics), Problems and Solutions for Undergraduate Real Analysis II, Topological Data Analysis for Genomics and Evolution: Topology in Biology. Rudin's real and complex analysis solutions Thread starter sid_galt; Start date Jun 3, 2009; Jun 3, 2009 #1 sid_galt. The purpose of this repository is to completely solve all exercises in Walter Rudin's Principles of Mathematical Analysis. Please try your request again later. The answer is no. 3 0 obj Solution: Let M denotes the ˙-algebra of measurable sets in X. It covers all the 176 exercises from Chapters 1 to 9 with detailed and complete solutions. (ebook only). So for all rationals r, … Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. /Length 575 Our payment security system encrypts your information during transmission. Included with a Kindle Unlimited membership. The Fundamentals Of Atomic And Molecular Physics Pdf, %PDF-1.5 Key Fob Padlock, Rudin Real And Complex Analysis Solutions Rudin Real And Complex Analysis REAL AND COMPLEX ANALYSIS - 59CLC's Blog REAL AND COMPLEX ANALYSIS - ERNET 3 Prove that if f is a real function on a measurable space X such that fx : f(x) rgis a measurable for every rational r, then fis measurable Solution: Let M denotes the ˙- If you're just interested in reading the solutions, simply clone this repository and compile rudin.tex using your preferred LaTeX distribution She could now learn the basics of algebra. Rudin - Real and Complex Analysis - Solutions - Free download as PDF File (.pdf) or view presentation slides online. Missouri Municipal Elections, 2020, Rudin's real and complex analysis solutions Thread starter sid_galt; Start date Jun 3, 2009; Jun 3, 2009 #1 sid_galt. stream Necessary lemmas with proofs are provided because some questions require additional mathematical concepts which are not covered by Rudin. As a student I was an excellent maths student but due to scarcity of time I couldnt give attention to my daughters math education. Anyone caught up there and finds it hard to solve out must buy a copy. única (remix Lyrics English), Do you struggle with academic concepts you never learned?For programmers only. 12 0 obj Suppose M be a ˙-algebra on X which has countably in- nite members. Algebra1help.com contains helpful resources on Walter Rudin Answers Real And Complex Analysis Solutions, worksheet and line and other algebra subject areas. Usage. Is the lack of a degree holding back your career? Voter Registration Office Savannah Ga, (Please also note that contrary to the common practice, Folland gives many end-of-chapter notes outlining the historical development of the topics, as well as a good few references and suggestions for further study). (b) Must the conclusion … Real Analysis Rudin Solutions - dev.babyflix.net Real And Complex Analysis Solutions Rudin, Principles of Mathematical Analysis, 3/e (Meng-Gen Tsai) Total Solution (Supported by wwli; he is a good guy :) Ch1 - The Real and Complex Number Systems (not completed) Ch2 - Basic Topology (Nov 22, 2003) Ch3 - … Try the Free Math Solver or Scroll down to Tutorials! (pp.1-3) Relevant exercise in Rudin: 1:R2. Lively introduction to proof oriented complex analysis in one variable for beginning graduate students or advanced undergraduates. #HungerforFreedom – opening a new chapter in the history of challenging immigration detention? << /S /GoTo /D (section.1) >> Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. To get the free app, enter your mobile phone number. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting (via Folland's book) was unsuccessful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do. 17 0 obj 1.1. There is no rational square root of12. Unable to add item to List. Thanks for making Algebra easy! Please try again. Sorry, there was a problem saving your cookie preferences. A Complete Solution Guide... 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1. Determine the ratio in which the line 2x + y − 4 = 0 divides the line segment joining the points A(2, − 2) and B(3, 7) Let the given line divide the line segment joining the points A(2, -2) and B(3,7) in a ratio k:1. Coordinates of the point of division = This point also lies on 2x + y – 4 = 0 Therefore, the ratio in hich the line 2x + y – 4 = 0 divides the line segment joining the points A(2, -2) and B(3, 7) is 2:9 2. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear. If the given points are collinear, then the area of triangle formed by these points will be 0. This is the required relation between x and y. 3. Find the centre of a circle passing through the points (6, − 6), (3, − 7) and (3, 3). Let O (x, y) be the centre of the circle. And let the points (6, −6), (3, −7), and (3, 3) be representing the points A, B, and C on the circumference of the circle. On adding equation (1) and (2), we obtain 10y = −20 y = −2 From equation (1), we obtain 3x − 2 = 7 3x = 9 x = 3 Therefore, the centre of the circle is (3, −2). 4. The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices. Let ABCD be a square having (−1, 2) and (3, 2) as vertices A and C respectively. Let (x, y), (x1, y1) be the coordinate of vertex B and D respectively. We know that, the sides of a square are equal to each other. ∴ AB = BC We know that in a square, all interior angles are of 90°. In ∆ABC, AB² + BC² = AC² ⇒ 4 + y² + 4 − 4y + 4 + y² − 4y + 4 =16 ⇒ 2y² + 16 − 8 y =16 ⇒ 2y² − 8 y = 0 ⇒ y (y − 4) = 0 ⇒ y = 0 or 4 We know that in a square, the diagonals are of equal length and bisect each other at 90°. Let O be the mid-point of AC. Therefore, it will also be the mid-point of BD. ⇒ y + y1 = 4 If y = 0, y1 = 4 If y = 4, y1 = 0 Therefore, the required coordinates are (1, 0) and (1, 4). 5. The class X students of MRV Public School in Krishna Park have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the following figure. The students are to sow seeds of flowering plants on the remaining area of the plot. (i) Taking A as origin, find the coordinates of the vertices of the triangle. (ii)What will be the coordinates of the vertices of ∆ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe? (i) Taking A as origin, we will take AD as x-axis and AB as y-axis. It can be observed that the coordinates of point P, Q, and R are (4, 6), (3, 2), and (6, 5) respectively. (ii) Taking C as origin, CB as x-axis, and CD as y-axis, the coordinates of vertices P, Q, and R are (12, 2), (13, 6), and (10, 3) respectively. It can be observed that the area of the triangle is same in both the cases. 6. The vertices of a ∆ABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that 𝐴𝐷/𝐴𝐵 = 𝐴𝐸/𝐴𝐶 = 1/4 . Calculate the area of the ∆ADE and compare it with the area of ∆ABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to ratio of areas of two similar triangles 𝐴𝐷/𝐴𝐵 = 𝐴𝐸/𝐴𝐶 = 1/4 Therefore, D and E are two points on side AB and AC respectively such that they divide side AB and AC in a ratio of 1:3. Clearly, the ratio between the areas of ∆ADE and ∆ABC is 1:16. 7. Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ∆ABC. (i) The median from A meets BC at D. Find the coordinates of point D. (ii) Find the coordinates of the point P on AD such that AP: PD = 2:1 (iii)Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and CR: RF = 2:1. (iv)What do you observe? (v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ∆ABC, find the coordinates of the centroid of the triangle. (i) Median AD of the triangle will divide the side BC in two equal parts. Therefore, D is the mid-point of side BC. (ii) Point P divides the side AD in a ratio 2:1. (iii)Median BE of the triangle will divide the side AC in two equal parts. Therefore, E is the mid-point of side AC. Point Q divides the side BE in a ratio 2:1. Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is the mid-point of side AB Point R divides the side CF in a ratio 2:1. (iv)It can be observed that the coordinates of point P, Q, R are the same. Therefore, all these are representing the same point on the plane i.e., the centroid of the triangle. (v) Consider a triangle, ∆ABC, having its vertices as A(x1, y1), B(x2, y2), and C(x3, y3). Median AD of the triangle will divide the side BC in two equal parts. Therefore, D is the mid-point of side BC. Let the centroid of this triangle be O. Point O divides the side AD in a ratio 2:1. 8. ABCD is a rectangle formed by the points A (−1, −1), B (− 1, 4), C (5, 4) and D (5, −1). P, Q, R and S are the mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS is a square? a rectangle? or a rhombus? Justify your answer. P is the mid-point of side AB. therefore, the coordinates of P are Similarly, the coordinates of Q, R and S are (2, 4) , (5, 3/2) and (2, -1) respectively. It can be observed that all sides of the given quadrilateral are of the same measure. However, the diagonals are of different lengths. Therefore, PQRS is a rhombus.

We show, by an example, that the knowledge of the degree of coherence and of the degree of polarization of a light beam incident on two photo detectors is not adequate to predict correlations in the fluctuations of the currents generated in the detectors (the Hanbury Brown-Twiss effect). The knowledge of the so-called degree of cross-polarization, introduced not long ago, is also needed. ©2010 Optical Society of America The Hanbury Brown-Twiss effect [1–4] is generally regarded as the starting point of quantum optics. The effect is a manifestation of correlations between intensity fluctuations at two points in a cross-section of a light beam (see, for example, , Secs. 9.9 and 14.6). The correlation between the intensity fluctuations are detected from measurements of the correlations between fluctuating current outputs of the photoelectric detectors, illuminated by the beam (see, for example, , Ch. 7). The effect was originally introduced in connection with attempts to measure diameters of stars but has since then found applications in high energy physics, nuclear physics, atomic physics and in neutron physics (see, for example, [7, 8]). Most of the traditional treatments of the Hanbury Brown-Twiss effect with light were carried out within the framework of the statistical theory of scalar fields. Only fairly recently, has analysis of it been made by use of the electromagnetic (vector) theory [9,10]. The analysis based on the electromagnetic theory revealed a somewhat surprising fact, namely that knowledge of the degree of coherence between the light fluctuations in the incident beams at the two detectors and of the degree of polarization of the light falling on each detector are not adequate to determine the correlation in the current output. A new statistical parameter of the incident field is needed to fully describe this effect, namely the so-called degree of cross-polarization . Whilst the degree of polarization depends on correlations between the electric field components at a particular point in space, the degree of cross-polarization depends on correlations in the field components at a pair of points (the location of the photo-detectors). In the present paper, we provide an explicit example of this rather surprising prediction. Specifically we consider two beams generated by two sources which have the same spectral densities, the same degrees of coherence and the same degrees of polarization, but have different degrees of cross-polarization. We show that the correlations of the intensity fluctuations at two points in the far-zone are different. Thus our analysis confirms, by an explicit example, that the knowledge of the spectral density, of the degree of coherence and of the degree of polarization of the beam at the source plane are not sufficient to predict the correlation between the intensity fluctuations at a pair of points in the far-zone; it shows that is also necessary know the degree of cross-polarization. Thus the analysis clearly reveals that the knowledge of the degree of cross-polarization is needed to elucidate some physical phenomena involving the interaction of an electromagnetic field with matter. We begin by recalling some basic results of the theory of stochastic electromagnetic beams. Let us consider a statistically stationary light beam generated by a planar secondary source located at the plane z = 0. Suppose that the beam propagates into the half-space z > 0 with its axis along the z direction. Let Ex(ρ, z; ω) and Ey(ρ, z; ω) be the Cartesian components at frequency ω, of the members of the statistical ensemble of the fluctuating electric field, in two mutually orthogonal x and y directions, perpendicular to the beam axis, at a point P(ρ, z). The second-order correlation properties of the beam at a pair of points P 1(ρ 1, z), P 2(ρ 2, z) in any cross-sectional plane z = constant > 0 may be characterized by the so-called cross-spectral density matrix (to be abbreviated by CSDM), whose elements are given by (, Sec. 9.1): Here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average. The ensemble is to be understood in the sense of coherence theory in the space-frequency domain (see, for example, , Secs. 4.1 and 9.1). In terms of the CSDM, the spectral density S(ρ, z; ω) at a point P(ρ, z) is given by the expression where Tr denotes the trace. The spectral degree of coherence μ(ρ 1, ρ 2, z; ω) at a pair of points P 1(ρ 1, z) and P 2(ρ 2, z) is defined by the formula and the spectral degree of polarization 𝒫(ρ, z;ω) at the point P(ρ, z) is given by the expression where Det denotes the determinant. However, as was mentioned earlier, these three quantities are not sufficient to determine the correlation between the intensity fluctuations at a pair of points in a cross-section of a beam. This fact was first demonstrated in Ref. , for a special class of stationary stochastic beams. A more general formulation was later given in Ref. . We will briefly mention the mains results obtained in Ref. . Suppose that a statistically stationary electromagnetic beam is incident on two detectors, placed at the points P 1(ρ 1, z) and P 2(ρ 2, z), in a cross-sectional plane z = constant > 0 of the beam. The correlation C(ρ 1, ρ 2, z; ω) between the intensity fluctuations at these two points, which is proportional to the correlation between the current fluctuations in the two detectors (, Ch. 7), can be shown to be given by the formula (, Eqs. (8) and (9)) is called the degree of cross-polarization. In Eq. (6) the dot symbolizes ordinary matrix multiplication. Equations (5) show that the correlation between intensity fluctuations, at a pair of points, does not depend only on the spectral density S and on the spectral degree of coherence of the incident beams μ, but depends also on the degree of cross-polarization 𝓠. The expressions [Eqs. (5) and (6)] have been derived with the assumption that the random fluctuations of the electric field in the beam obey Gaussian statistics. Suppose that a stationary stochastic electromagnetic beam propagates a distance z > 0 from the source plane z = 0. It can readily be shown that the cross-spectral density matrix at a pair of points P 1(ρ 1, z) and P 2(ρ 2, z), at a cross-sectional plane z = constant > 0 is given by (, see also, , Sec. 9.4.1) is the Green’s function of the Helmholtz operator for paraxial propagation [see also, Ref. , Eq.(5.6–17)]. where σi ≫ δij and Suppose now that the beam has propagated some distance z > 0. Using the propagation law [Eq. (7), it may be shown that the elements of the CSDM, at a pair of points in that plane, are given by (, Sec. 9.4.2, Eq. (10), (11)) We will now return to our main problem namely to show that it is possible to have two planar sources with the same spectral densities, the same spectral degrees of coherence and the same spectral degrees of polarization, which may generate beams with different correlations between the intensity fluctuations at a pair of points. To demonstrate this result, we consider two Gaussian Schell-model beams, “a” and “b”, produced by two different planar secondary sources. We assume that the beam “a” is characterized by parameters Ax = Ay = 1, , σx = σy = σ, δxx = δyy = δ and . It can readily be shown that these parameters obey the realizability conditions [Eq. (11)]. From Eq. (9), it readily follows that the CSDM W⃡(a) of beam a, at the source plane z = 0 has the form We further assume that beam “b”, is characterized by the parameters Ax = Ay = 1, , σx = σy = σ, δxx = δyy = δxy = δyx = δ. Using Eq. (9) again, one readily finds that the CSDM of the beam “b”, at the source plane has the form where 𝒜 is the same quantity as in Eq. (15a) and On using Eqs. (2)–(4), (14) and (16), one can readily verify that both beams have the same distributions of spectral densities, of spectral degrees of coherence and of spectral degrees of polarization at the sources, namely However, as can be shown by using Eq. (6), the two beams have different distribution of the degree of cross-polarization at the source plane. In Fig. 1 the variation of degree of cross-polarization, at a pair of diametrically opposite points at the source plane, has been plotted with half-separation distance ρ with the choice δ = 0.001m and σ = 0.01m. As the beams propagate some distance z = z 0 > 0, the correlation in the intensity fluctuations associated with each of them may become significantly different. To see this, we choose the parameters δ = 0.001m and σ = 0.01m. Recalling the definition Eq. (5) and using formula (12), one can calculate the correlation between the intensity fluctuations at a pair of points in any cross-section of the beam. Figure 2 shows the variations of this correlation function with the half-separation distance ρ of two diametrically opposite points in a beam cross-section, at a distance z = 10km from the source. Figures 1 and 2 clearly show that degree of cross-polarization of a field affects, in general, the correlations in the intensity fluctuations of an electromagnetic beam. The research was supported by the US Air Force Office of Scientific Research under grant No. FA9550-08-1-0417, by the Air Force Research Laboratory (ARFL) under contract number 9451-04-C-0296, and by NSERC (Canada). References and links 1. R. H. Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954). 2. R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956). [CrossRef] 3. R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957). [CrossRef] 4. R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A , 243, 291–319 (1957). [CrossRef] 5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, Cambridge University Press, 1995). 6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007). 7. G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998). 8. D. Kleppner, “Hanbury Brown’s steamroller,” Physics Today 61, 8–9 (2008). [CrossRef] 9. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007). [CrossRef] 10. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008). [CrossRef] 11. In Refs. and the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. . Another two-point generalization of the degree of polarization, called complex degree of mutual polarization was introduced in Ref. . 12. D. Kuebel, “Properties of the degree of cross-polarization in the spacetime domain,” Opt. Commun. 282, 3397–3401 (2009). [CrossRef] 14. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003). [CrossRef] [PubMed] 15. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef] 16. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008). [CrossRef]

16. EXPLANATORY NOTES TO THE FINANCIAL INSTRUMENTS AND FINANCIAL RISK 16.1. Financial instruments by category and class 16.2. Income, expenses, profit and loss and other comprehensive income 16.3. Fair value measurement The Group recognises a financial asset or a financial liability in its statement of financial position when, and only when, the entity becomes a party to the contractual provisions of the instrument. The Group derecognises a financial asset in statement of financial position when: - the contractual rights to the cash flows from the financial asset expired; or - the Group transferred the financial asset to another entity, and the transfer qualified for derecognition. The Group removes a financial liability (or a part of a financial liability) from its statement of financial position when it is extinguished—i.e. when the obligation specified in the contract is discharged or cancelled or expires. Measurement of financial assets and liabilities Measurement of financial assets and liabilities At initial recognition, the Group measures financial assets and liabilities not qualified as at fair value through profit or loss (i.e. held for trading) at their fair value plus transaction costs that are directly attributable to the acquisition or issue of the financial asset or financial liability. The Group does not classify instruments as measured at fair value through profit or loss upon initial recognition, i.e. does not apply the fair value options. At the end of the reporting period, the Group measures item of financial assets and liabilities at amortised cost using effective interest rate method, except for derivatives, which are measured at fair value. With regard to equity instruments, in particular quoted/unquoted shares held for the purpose of obtaining contractual cash flows representing only principal and interest payments as well as in order to sell, the Group classifies the instruments as measured at fair value through other comprehensive income. Gains and losses resulting from changes in fair value of derivatives, for which hedge accounting is not applicable, are recognised in the current year profit or loss. Derivatives for the purchase of non-financial assets that are entered into and held with the intention of settling those transactions by physical delivery of the assets for use in the Group's own operations are not valued at the balance sheet date. Impairment of financial assets The Group recognizes a write-off due to expected credit losses on financial assets measured at amortized cost or measured at fair value through other comprehensive income (with the exception of investments in capital assets). The Group uses the following models for determining impairment allowances: - general model (basic), - simplified model. The general model is used by the Group for financial assets measured at amortized cost - other than trade receivables and for debt instruments measured at fair value through other comprehensive income. In the general model, the Group monitors the changes in the level of credit risk associated with a given financial asset and classifies financial assets to one of the three stages of impairment allowances based on the observation of the change in the credit risk level in relation to the initial recognition of the instrument. Depending on the classification to particular stages, the impairment allowance is estimated in the 12-month horizon (stage 1) or in the life horizon of the instrument (stage 2 and stage 3). On each day ending the reporting period, the Group considers the indications resulting in the classification of financial assets to particular stages of determining impairment allowances. Indications may include changes in the debtor's rating, serious financial problems of the debtor, a significant unfavourable change in its economic, legal or market environment. For the purposes of estimating the expected credit loss, the Group uses default probability levels based on market credit quotes of derivatives for entities with a given rating and from a given sector. The Group includes information on the future in the parameters of the expected loss estimation model by calculating the probability parameters of insolvency based on current market quotes. The simplified model is used by the Group for trade receivables. In the simplified model, the Group does not monitor changes in the credit risk level during the life and estimates the expected credit loss in the horizon up to maturity of the instrument. In the area of hedge accounting, the Group applies the requirements of IFRS 9. Derivatives designated as hedging instruments whose fair value or cash flows are expected to offset changes in fair value or in the cash flows of a hedged item are accounted for in accordance with fair value or the cash flow hedge accounting. The Group has two types of hedging relation: cash flow and fair value hedge. The Group assess effectiveness of cash flow hedge at the inception of the hedge and later, at minimum, at reporting date. In case of cash flow hedge accounting, the Group recognises in other comprehensive income part of profits and losses connected with the effective part of the hedge, whereas profits or losses connected with the ineffective part - under profit or loss. In addition (in case of currency risk hedge - spot rate risk element), as part of equity in a separate item, the Group recognises a change in the fair value due to the hedge costs. To assess the effectiveness of hedge the Group uses statistical methods, including in particular the direct compensation method. The verification of fulfilment of conditions in the scope of binding effectiveness is made on a prospective basis, based on a qualitative analysis. If it is necessary, the Group uses quantitative analysis (linear regression method) to confirm the existence of an economic link between the hedging instrument and the hedged item. In case of applying fair value hedge accounting, the Group recognises profits or losses resulting from the revaluation of fair value of derivative financial instrument in financial result, and adjusts carrying amount of hedged item by profit or loss related to the hedged item, resulting from the risk being hedged and recognises it in the profit or loss (in the same item in which hedging derivatives are recognised). Cumulative adjustment of the measured hedged item due to the hedged risk is transferred to the profit and loss when the realization of the hedged item impacts the statement of profit and loss. If a cash flow hedge is used the Group recognises a portion of the gain or loss on the hedging instrument that is determined to be an effective hedge due to the hedged risk in other comprehensive income. Additionally in case of currency risk hedging - a spot risk element, a change in the fair value due to the forward element (including the cross-currency margin) the Group recognise as part of equity as a separate item (hedging cost). The ineffective portion of the gain or loss on the hedging instrument the Group recognise in profit or loss. If a hedge of a forecast transaction subsequently results in the recognition of a financial asset or a financial liability, the associated gains or losses that were recognised in other comprehensive income are reclassified to profit or loss of the reporting period in the same period or periods during which the asset acquired, or liability assumed, affects profit or loss. However, if the Group expects that all or a portion of a loss recognised in other comprehensive income will not be recovered in one or more future periods, it reclassifies the amount that is not expected to be recovered to profit or loss. If a hedge of a forecast transaction subsequently results in the recognition of a non-financial asset or a non-financial liability, or a forecast transaction for a non-financial asset or non-financial liability becomes a firm commitment for which fair value hedge accounting is applied, the Group removes the associated gains and losses that were recognised in the other comprehensive income and includes them in the initial cost or other carrying amount of the asset or liability when the item appears in the statement of financial position. If a hedge of a forecast transaction results in the recognition of revenue from sales of products, merchandise, materials or services, the Group removes the associated gains or losses that were recognised in the other comprehensive income and adjusts above revenues. In case of applying fair value hedge accounting, cumulated adjustment of hedged item valuation for hedged risk is transferred to the financial result at the moment when the realization of hedged item affects the result. Derivatives are recognised as assets when their valuation is positive and as liabilities in case of negative valuation. Fair value measurement The Group maximizes the use of relevant observable inputs and minimizes the use of unobservable inputs to estimate the fair value, i.e. the price at which an orderly transaction to transfer the liability or equity instrument would take place between market participants as at the measurement date under current market conditions. The Group measures derivatives at fair value using valuation models for financial instruments based on generally available exchange rates, interest rates, forward and volatility curves for currencies and commodities quoted on active markets. The fair value of derivatives is based on discounted future flows related to contracted transactions as the difference between term price and transaction price. Forward exchange rates are not modelled as a separate risk factor, but derive from the spot rate and the respective forward interest rate for foreign currency in relation to PLN. The Management Board assesses the classification of financial instruments, nature and extent of risk related to financial instruments and application of hedge accounting. The financial instruments are classified into different categories depending on the purpose of the purchase and nature of acquired assets.

This mathematician died last week. He won the Fields Medal in 2002 for proving the Milnor conjecture in a branch of algebra known as algebraic K-theory. He continued to work on this subject until he helped prove the more general Bloch-Kato conjecture in 2010. Proving these results — which are too technical to easily describe to nonmathematicians! — required him to develop a dream of Grothendieck: the theory of motives. Very roughly, this is a way of taking the space of solutions of a collection of polynomial equations and chopping it apart into building blocks. But the process of 'chopping up', and also these building blocks, called 'motives', are very abstract — nothing simple or obvious. There’s some interesting personality and history in this short post of John’s. The Institute for Advanced Study is deeply saddened by the passing of Vladimir Voevodsky, Professor in the School of Mathematics. Voevodsky, a truly extraordinary and original mathematician, made many contributions to the field of mathematics, earning him numerous honors and awards, including the Fields Medal. Celebrated for tackling the most difficult problems in abstract algebraic geometry, Voevodsky focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology. He made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. Among the consequences of his work are the solutions of the Milnor and Bloch-Kato Conjectures. More recently he became interested in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. His new "Univalence Axiom" has had a dramatic impact in both mathematics and computer science. Sad to hear of Dr. Voevodsky’s passing just as I was starting into my studies of algebraic geometry… For those who are still on the fence about taking Algebraic Geometry this quarter (or the follow on course next quarter), here’s a downloadable copy of the written notes with linked audio that will allow you to sample the class: If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities. I want to come back and read this referenced article by Milne. The comments on this are pretty interesting as well. This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry. Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians. Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology. Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology. Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017. While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order): Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.) A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises. (.pdf download) Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘objective’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘subjective’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.). For a constant ϵ, we prove a poly(N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player NxN games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1−ϵ)-fraction of the players are ϵ-best replying. John Nash’s notion of equilibrium is ubiquitous in economic theory, but a new study shows that it is often impossible to reach efficiently. There’s a couple of interesting sounding papers in here that I want to dig up and read. There are some great results that sound like they are crying out for better generalization and classification. Perhaps some overlap with information theory and complexity? To some extent I also find myself wondering about repeated play as a possible random walk versus larger “jumps” in potential game play and the effects this may have on the “evolution” of a solution by play instead of a simpler closed mathematical solution.

Example 5 mathematical studies sl teacher support material 1 mathematical studies sl teacher support material 2 example 5 a2 mathematical studies sl teacher support material 3 and i were told that to pick a topic for a math project i immediately thought of tennis now we all know that stretching before doing physical activity prevents. Students wishing to study mathematics in a less rigorous environment should therefore opt for one of the standard level courses, mathematics sl or mathematical studies sl students who wish to study an even more rigorous and demanding course should consider taking further mathematics hl in addition to mathematics hl. Im taking maths sl for ib i dont have the slightest idea on any topics for ia need help asap since my teacher wants the first draft in a couple. Internal assessment in mathematics sl is an individual exploration this is a piece of written work that involves investigating an area of mathematics (20 marks) 1elements of a successful ib internal assessment correct answers throughout all questions answered in a logical order. Arthur was literally the only person in his hl math class that got a 7 on his math ia this math class included two current uc berkeley students, an oxford pupil, and a upenn undergrad his math ia grade was the reason he got accepted into his top-choice university. If you are watching this video now, you seem serious about boosting your ib grade good news: we can help you with that if you are a student from hk, you can register for a free trial lesson with. Your ib mathematics standard level in addition to all the material in your mathematics sl course book , we've included a full set of worked solutions here, to fully equip you to tackle the course and assessment. Ib sl mathematics ia - download as pdf file (pdf), text file (txt) or read online my ia from 2016 feel free to take ideas or notes from it if you use any of it, make sure you are not directly copying or forget to exclude my name p. Filters group 1 group 2 group 3 group 4 group 5 group 6 tok/ee past papers nov 2018 examination schedule server welcome to /r/ibo this subreddit is for all things concerning the international baccalaureate, an academic credential accorded to secondary students from around the world after two vigorous years of study, culminating in challenging exams. Ib mathematics sl/statistics and probability from wikibooks, open books for an open world ib mathematics sl the latest reviewed version was checked on 16 march 2018 there is 1 pending change awaiting review jump to navigation jump to search contents 1 probability 11 combined events. Ib tutor provides assignment writing help in all the ib subjects 1 ib maths mathematics studies ia tutor help hl sl exploration extended essay example sample 2 ib physics ia labs extended essay help tutors example sample 3. The internal assessment what: a written paper that explores the math behind a personal interest of your choice why: - to apply and transfer skills to alternate situations, to other areas of knowledge, and to future developments. I truly wish my students will derive some pleasure from completing the internal assessment (ia) requirement for maths sl & hl - the exploration and i think that there is a greater chance of this occurring if clear and effective support and encouragement is provided to the students. Ib tutoring and hsc tutoring for ib physics, ib math, hsc physics and hsc math. For example, ib maths sl consists of numbers, algebra, functions, geometry, trigonometry, statistics, preliminary calculus, and financial math by doing a bit of research about each of these ahead of time, you can be even more prepared for ib maths success. Ib math sl exam secrets learn the most commonly asked questions for each topic (and why mathematicians love forests so much :) i can predict the questions that will most likely show up in the next math sl exam how not with a crystal ball, but with an incredibly detailed analysis of the exams i’m happy to share the results with you, in. Do you need help with your math ia/internal assessment in this post i will show you my ia that i submitted to ib you can use this to see what a math sl ia looks like and i hope this will inspire you to create your best math sl ia to submit to your teachers. Ib mathematics sl ii ia summer prep due august 28, 2017 name:_____ future ib math sl 2 students: to prepare for writing your ia you will review the following packet. Part of the ib subject group 5, mathematics sl is a course for students with a good background in mathematics and strong analytical and technical skills. Maths ia – maths exploration topics this is the british international school phuket’s ib maths exploration (ia) page this list is for sl and hl students – if you are doing a maths studies ia then go to this page instead. Writing & math ib internal assessment • math sl/hl students can pretend that they are writing a chapter in a textbook ø encourages students to fully explain each step, remembering that their audience is another. Jaskiran bedi, ib physics and ib maths, pathways world school the q & a/conversation style of conducting lessons were really good and made the theory a lot more interesting to learn saurabh hamada,12th grade ib business management.

While doing stock valuation, you need to find the dividend growth rate for accomplishing a better decision. Interestingly, you can compute the growth rate quickly using Excel. In this article, I’ll show you how to calculate dividend growth rate in Excel using the formulas and functions quickly with the basics of the term. Basics of Dividend Growth Rate What Is Dividend Growth Rate? Fundamentally, a dividend is the amount of share from the profit of a company to the other stockholders. And the dividend growth rate is mainly the annual rate of increasing the dividend in percentage. Formula of Measuring Dividend Growth Rate The formula for both arithmetic average and compound annual dividend growth is as follows. i. The formula of Arithmetic Average Dividend Growth Rate Average Annual Growth Rate = (G G1= yearly dividend growth rate in the first year Gi= dividend growth rate at ith year n = number of periods Compound Dividend Growth Rate )1/n – 1 Dn = Dividend at last year D0 = Dividend at first year n = Number of periods Application of Dividend Growth Rate - Provides instructive guidelines to investors while engaging with any company. - The healthy dividend growth rate of a company discloses profit stability whereas a diminishing growth rate reveals the problems with the profit. - Worthwhile while using the dividend discount model - Helpful in predicting the future dividend growth rate of a company How to Calculate Dividend Growth Rate in Excel: 2 Methods In our today’s dataset, yearly Dividend Per Share (DPS) for the top 10 companies in the U.S. are given from 2013 to 2022. Therefore, we have to calculate the dividend growth rate in Excel. 1. Measuring Arithmetic Average Dividend Growth Rate There are mainly two steps while measuring the arithmetic average annual growth rate. Firstly, we need to compute the yearly growth rate. Secondly, the average growth rate will be calculated over a period of time. For determining the yearly growth rate, just use the following formula in the E8 cell. Here, D8 is the dividend in 2014, and D7 is the dividend in 2013. Next, use the Fill Handle tool to copy the formula for the below cells. Right now, we have to find the average dividend growth rate using the calculated yearly growth rate over 10 years (from 2013 to 2022). To calculate that, simply insert the following formula. Here, E8 is the yearly growth rate in 2014, E16 is the yearly growth rate in 2022, and D18 is the number of periods i.e. 10. In the above formula, the SUM function returns the total of the yearly growth rates. Later, the total will be divided by the number of periods. Related Content: How to Calculate Revenue Growth Rate in Excel 2. Computing Compound Dividend Growth Rate (CAGR) Unlike the arithmetic average growth rate, we may calculate the compound dividend growth rate in two ways. The first one is using the formula discussed in the formula section. And the other one is utilizing an Excel function. 2.1. Computing Compound Dividend Growth Rate Manually Using Formula Truly speaking, it is a simple method as you can find the growth rate within a single step. Just use the following formula. Here, D16 is the dividend in the last year (2022), D7 is the dividend in the first year (2013), and C18 is the number of periods (10). If you press Enter after inserting the above formula, you’ll get the following output. 2.2. Compound Dividend Growth Rate Applying the LOGEST Function More importantly, we may compute the compound dividend growth rate using the LOGEST function. The function finds the value of an exponential curve in regression analysis. For example, if you have dependent and independent variables, it’ll calculate the value of the exponential curve. Anyway, we can calculate the growth rate using the function in 3 steps. While using the function, your year value should be in date format. Also, you can do this using the DATE function in the following way. Here, 2022 is the year argument, 1 is the month argument, and 1 is the day argument. Now, we need to calculate daily dividend growth data and the formula that can be used is the following. Here, D7:D16 is the cell range for dividend (dependent variable-known_ys argument), C7:C16 is the cell range representing the date (independent variable-known_xs argument) Finally, we have to calculate the annual dividend growth rate. As we found the daily growth rate in the previous step. So, we need to multiply 365 like the following way to get the rate for a year. Here, G10 is the found daily dividend growth rate. Things to Remember - As the LOGEST is an array formula, don’t forget to press Ctrl + Shift + Enter if you’re not using Microsoft 365. - Be careful that the dividend growth rate is generally calculated in percentage form. Download Practice Workbook In short, you can easily calculate the dividend growth rate in Excel using the formula and the LOGEST function. I strongly believe that this article will articulate calculation methods. However, if you have any queries or suggestions, please let me know in the comments section below. - How to Use the Exponential Growth Formula in Excel - Growth Formula in Excel with Negative Numbers - How to Calculate Sales Growth over 3 Years in Excel - How to Calculate Sales Growth over 5 Years in Excel - How to Calculate Sales Growth Percentage in Excel - Growth Over Last Year Formula in Excel - How to Calculate Growth Percentage with Formula in Excel

Ohm’s law is an equation that describes the voltage drop across a resistor. The voltage drop can be measured in volts or in current. To calculate the voltage drop across a resistor, you must know how much voltage the resistor can resist. For this purpose, a 24-V power source is connected to three resistors (R1, R2 and R3), respectively. Ohm’s law is used to calculate the voltage drop across a resistor You can use Ohm’s law to calculate the voltage drop across a resistor by adding the current of the circuit to the resistance of the resistor. To do this, place a voltmeter on each end of the resistor. Note that the current in the voltmeter will be minimal. In addition, you can use Ohm’s law to determine the voltage drop across multiple resistors. By using this law, you can find the voltage drop across two resistors, four resistors, or three resistors in series. Ohm’s law is one of the most basic principles of electronics. It is based on the relationship between current and voltage. You can use it to calculate the voltage drop across a resistor and to calculate current flow in electrical circuits. Ohm’s law applies to all electrical circuits with one or more resistors. Ohm’s law is also useful when you’re designing circuits with resistors. You’ll want to make sure you’re using the proper resistor for the circuit. It’s best to choose a resistor with a minimum power rating of 0.5 W. Higher ratings mean that the resistor will last longer. Another way to calculate the voltage drop across resistors is to think of each resistor in series as a voltage divider. A series of N-resistors will have different voltages across them, but will have a common current. This is where Ohm’s law comes into play. Using the voltage drop across a resistor is easy when you’re using a voltmeter. The voltage drop is directly proportional to the value of the resistance, so the larger the resistance, the greater the voltage drop. You can use this formula to calculate the voltage drop across any resistor in a series circuit. Using Ohm’s law, you can determine how much current flows through a resistor and divide this number by the resistance of the circuit. You can also use this formula to calculate the required impedance. It is a mathematical formula Ohm’s law is the mathematical formula that calculates the voltage drop across a resistor. If you have a voltage meter attached to both ends of a resistor, you can calculate the voltage drop across the resistor by multiplying the voltages on each side of the resistor by their resistance value. The higher the resistor value, the higher the voltage drop will be. To understand why this is important, let’s look at an example. Imagine that you are trying to calculate the voltage drop across a series of two resistors. The resistance values are 2 and 4 ohm. The current flowing through the circuit is six Amperes. Therefore, the voltage drop across the series of resistors will be 120 volts x 6.6667 Ampere. The Ohm’s law formula shows the relationship between voltage and current. It is best illustrated by a pyramid. The current passing through a conductor is directly proportional to the voltage difference on either side. This formula also works for complex resistive circuits. The voltage drop across a series of resistors can be calculated using the Ohm’s law. You can use the same formula to calculate the voltage drop across two parallel resistors. In parallel, the resistances are the same. But if the resistors are in series, the resistances are different. If the voltage drop across two resistors is equal, it means the two resistors are equivalent. If they are in series, their equivalent resistance is two times the value of one resistor. If there are three of them, then their equivalent resistance is three times the value of each resistor. In addition, a series of resistors is a voltage reference circuit. Kirchhoff’s voltage law is another mathematical formula that helps you calculate the voltage drop across resistors. Kirchhoff’s law can also be used to verify closed loop voltages. A closed loop circuit has a voltage that is equal to the sum of the current flowing through all the resistors. It is a formula that can be used manually The voltage drop across resistors is the amount of electricity lost from a circuit due to the resistance of the wires. To calculate voltage drop, you should use the following formula. The length of the wires is taken into account to determine the voltage. Then, you should multiply the length by the resistance to find the voltage drop across the component. Ohm’s law tells us that voltage drop is proportional to the number of connected loads in a circuit. You can calculate this value by using a digital multimeter, also known as a voltmeter. To do this, you must switch your multimeter to voltage mode. You can then use the voltage drop calculator to calculate the voltage drop across a series of resistors. You can also use this formula to estimate the voltage drop between a copper and aluminum conductor. Another formula that can be used to calculate voltage drop across resistors is V = IR. By using this formula, you can calculate the voltage drop across one resistor and several in parallel. You can also divide the voltage drop across multiple resistors by the total resistance. Once you have calculated the total voltage drop across the individual resistors, you can calculate the voltage drop across the parallel branch of R4 and R6. The sum of these values is equal to the total current flow through the circuit. To calculate voltage drop across a combination circuit, you must first determine the equivalent resistance of the circuit. For example, if the resistor R3 has an ES of 3.3 volts, the voltage drop across R3 is 1.4 volts. Lastly, you must find the total current through all circuit paths. Another way to calculate voltage drop across resistors is to look up the voltage drop of a cable. There are many tables available on the internet that can give you the voltage drop of an aluminum or copper conductor. In each table, you will find the voltage drop of the cable in mVA per 100 feet (30 m) of the cable. It is expressed in volts The voltage drop across a resistor can be calculated with the help of the Ohms law. You can find the potential drop across a resistor by multiplying the current through it by its resistance (in Ohms). The higher the value of a resistor, the greater the voltage drop will be. The voltage drop across a resistor is determined by the current flowing through a series circuit. The voltage drop is proportional to the resistance of the individual resistors. So, the more resistance, the higher the voltage drop. In order to calculate the voltage drop across a resistor, you need to know the total voltage and the resistance of each load. If you have a series circuit, the resistors are in a row. This way, the current flows through all of them in order. If you want to change the location of the resistors, move them accordingly. This way, you can get the same voltage. But, you have to keep in mind that each resistor has a different value when it comes to the voltage drop. Once you know the resistance of one resistor, you can calculate the voltage drop across all other resistors. You will need a voltmeter to perform the calculations. A digital multimeter is also an option. You should use the voltage mode to measure voltage across multiple resistors. Another way to calculate voltage drop across resistors is to divide the total current through a series of them. For example, if you have three parallel resistors, the total current is divided by the number of water molecules that flow through them. The potential energy of these molecules must be equal to the sum of the current flowing through each resistor. If you have two parallel resistors connected to one another, you can use the same equation to find the voltage drop across the two parallel resistors. You can also use the equivalent resistance widget. By doing this, you can make your own problems and check the answers. You can also use a resistor to create a voltage reference circuit.

Mathematics is a universal language that allows us to describe and understand the world around us. One of the fundamental branches of mathematics is algebra, and a key tool in this discipline is algebraic expressions. In this article, we will explore what algebraic expressions are, what they are used for and how they are used in the real world. Algebraic expressions appear in the transition from primary to secondary school or in middle school, around the age of 12. And once they appear, they are here to stay. As with most things that are new and different, algebraic expressions raise a lot of questions and are feared by students, so it is best that we start by clarifying what they are and what they are used for. What are algebraic expressions? Algebraic expressions are combinations of numbers, variables and mathematical operations, such as addition, subtraction, multiplication and division. They are represented by symbols and letters, where numbers are considered constants and letters represent variables, that is, values that can vary. They follow all the arithmetic rules we have learned so far, however now some numbers are replaced by letters that can be of different values. It will be better explained with examples: - Addition of two numbers: if we have two numbers, for example, 3 and 5, we know that to add them we must write 3+5. We know that their sum is 8. If the two values are not known, we can also add them, although now we do not know the result. We can represent these two numbers with the letters x and y, which, and since they do not have a fixed value, will be called variables. If we want to express the sum of these two numbers, we can use the algebraic expression: x + y. Notice that we use two different variables because we have not been told that they are the same number, only that we want to get an expression for the sum of “two numbers”. - The double of a number: 2x - Area of a rectangle: In the same way that we would calculate the area of a rectangle with a base of 4 and a height of 2 we would multiply 4 by 2, if we want to calculate the area of a rectangle with base “b” and height “a”, we can use the algebraic expression: A = b x a where “A” represents the area of the rectangle. - Formula for the area of a circle: If we know the radius of a circle, represented by “r”, we can use the algebraic expression: A = π – r2 to calculate its area. Here, “A” denotes the area of the circle and π is a constant representing the approximate value of pi, we usually take 3.1416. - Temperature conversion: Let’s say we want to convert the temperature from degrees Celsius to degrees Fahrenheit. We can use the algebraic expression: F = (9/5) – C + 32, where “C” represents the temperature in degrees Celsius and “F” represents the equivalent temperature in degrees Fahrenheit. What are algebraic expressions used for? As you may have already guessed from the examples, algebraic expressions are used to describe mathematical situations and relationships in general terms. That is, in situations where not all values are known. They allow us to express formulas, equations and mathematical models in an abstract way, which facilitates analysis and problem-solving. An example of the usefulness of algebraic expressions would be, for example, to obtain new formulas. Since we know that the volume of prisms and cylinders is the area of the base (Ab) times the height (h) V = Ab- h, we can substitute in that formula the area of the base. If we know that the base is a circle, Ab= π – r2, we can substitute and write in a single formula that the volume of the cylinder is V = π – r2 – h. Components of algebraic expressions - Constants: These are fixed numbers that do not change their value, such as 2, 5 or π. - Variables: These are letters that represent unknown quantities or variables, such as x, y, and z. These variables allow us to generalize and solve problems for different values. - Mathematical operations: These include addition, subtraction, multiplication, division, and exponents, among others. These operations are applied to constants and variables to form more complex expressions. What is not included in algebraic expressions is equality, the examples we looked at before that contained the equal sign what they had on the left is interpreted as the result of that expression, when we have on the left another expression, we will be talking about equations, and we will deal with it at the end of the article. Simplification of algebraic expressions Algebraic expressions can be simplified by using distributive, associative and commutative algebraic properties and rules. Simplification helps to reduce the expression to a more manageable and understandable form. In algebra, the arithmetic rules are followed. Therefore, if it is valid for numbers, it is valid for algebraic expressions, x + x is 2x. Algebraic expressions have numerous real-world applications. Some examples include: - Physics: In describing physical laws and phenomena, such as the law of universal gravitation or the equations of motion. - Economics: In modeling financial problems, such as calculating interest, profits or depreciation. - Engineering: In the design and analysis of structures, electrical circuits or control systems. - Computer Science: In algorithms and programming, where algebraic expressions are used to perform calculations and make decisions. A particular case, monomials Monomials are a particular case of algebraic expressions that only use the product, and in which the exponents of the variables that appear have to be natural numbers (therefore positive). Of the algebraic expressions seen here, there would be monomials, all except this: (9/5) – C + 32, additionally x + y because it contains a sum – Neither would be 1/x because written as a power it is x-1, which is not a natural number. You can see more information about monomials in this blog post. Algebraic expressions and equations Now we will look at an application that it deserves its own section, equations. Equations are not algebraic expressions, because they are instead two (or more) algebraic expressions joined by the equal sign. It will be better understood, just like everything else, with an example: We said above that the double of a number is 2x. How would we say “one number is the double of another”? It cannot be x = 2x, because that would be a number is equal to its double. But we can say y = 2x, because by using two different variables (letters) we are denoting exactly that. If, for example we consider the pairs of points (x,y) that fulfill that equation, that equality between algebraic expressions, we would have the (1,2), the (10,20), the (π, 2π) and ALL the pairs in which the second coordinate is two times the first. We can even paint it, taking, as it is usually taken, the first coordinate on the x-axis and the second on the y-axis: Stretching the idea “a little” we could try to envision the following equation, which is still an equality between algebraic expressions, although it is much more difficult to translate with words: You can try changing some of the values of the expression in the desmos graphing calculator. I hope you have found the post interesting, feel free to share it or leave a comment your doubts or questions, or the topics you would like to know more about. 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As a scientific term work involves movement and force. If you exert force and no movement occurs, scientifically, you have not done work. Work is calculated scientifically by the following formula: W = F X D (work equals force times distance). The SI (metric) unit for force is the newton, and the SI unit for distance is the meter. The SI unit for work is the newton/meter, which is also known as the joule. One joule is the work done by a force of one newton that moves an object a distance of one meter. Larger units of work are the kilojoule (kJ) which is equal to 1,000 joules, and the megajoule (mJ) which is equal to a million joules. Anything that changes the size or direction of forces used in doing work is called a machine. A machine can be an inclined plane, such as a ramp used to slide a heavy object on to a truck. A ramp allows you to use less force to lift an object, so it is a machine. In exchange for using less force, you have to move the object a longer distance. See figure 5.5, page 201. If you take an inclined road up a mountain, it requires less force than going straight up the mountain, however, you must travel a greater distance. When a machine reduces the force used to move an object it give a mechanical advantage (M.A.). If a machine has a mechanical advantage of 1, there is no change in the force you have to apply. If the M.A. is 2, it allows you to apply only half the force needed to move an object without the machine. An M.A. of three, allows you to use only one third the force to move the object. The work put into a machine is called work input. The work done by the machine is called work output. Work input is always greater than work output because of friction. The efficiency of a machine is calculated by dividing the work output by the work input. If it were not for friction, efficiency could be 100%. However, because of friction, it is always less than 100%. A machine may allow you to do work that you could not do without the machine, by reducing the force necessary to create movement. However, the movement is always greater. You could not carry a 1,000 pound weight for a distance of 100 meters. You could, however, carry 200 50 pound weights. But you would have to travel a longer distance, and therefore, there would be more friction. This illustrates the principle of efficiency always being less than 100%. Anything which reduces friction, such as lubricants, will improve efficiency. A wedge is an inclined plane which is thick at one end and thin at the other. The force exerted on the thick end is concentrated at the thin end. The result is more force applied to a very small area. This is why a wedge is used to split things, such as wood. An ax, a knife and a razor blade are all wedges. A screw is another form of inclined plane. It is an inclined plane which twists. You must turn a screw a great distance to get it to penetrate a small distance. But because its edge is sharp, it exerts force to a small area. A crowbar, a wheelbarrow and a rake are all machines. All of them have a point that is moved by a force, and a part that does not move called a fulcrum. Machines that do work by moving around a fulcrum are called levers. The force applied to the lever is called the effort force. The weight of the object being lifted is called the resistance force. The length of the lever between the fulcrum and the resistance force is called the resistance arm. The length of the lever from the fulcrum to where the effort force is applied is the effort arm. The M.A. of a lever is calculated by dividing the length of the effort arm by the length of the resistance arm. There are three general types of levers. With a first class lever, the fulcrum is always between the two forces. A first class lever changes the direction of a force. An example of a first class lever is a crowbar. For a second class lever, the fulcrum is at the end of the lever, the resistance force is near the center, and the effort force is applied to the other end. A wheelbarrow is an example of a second class lever. Unlike the first class lever, the direction of the force is not changed. See figure 5-11, page 207. In a third class lever the fulcrum is always at one end, the resistance force is at the other end, and the effort force is applied in between the fulcrum and the resistance force. A rake is an example of a third class lever. Your hand at the end of the rake is the fulcrum. The leaves of the rake which scrape the ground is the resistance force. Your other hand in the middle of the rake handle applies the effort force. All third class levers are used to increase the distance moved, not to decrease the force applied. Inclined planes and levers are simple machines. A compound machine is a machine made up of two or more simple machines. Since a wheelbarrow uses both a lever and a wheel, it is a compound machine. Another simple machine is the pulley. A pulley is actually a type of lever. See figure 5-13, page 210. A combination of pulleys is called a block and tackle. A block and tackle can be used to obtain a large mechanical advantage to lift heavy objects, such as a piano or an automobile engine. Power is how fast work is being done. Power is the rate at which the work is done. It is equal to the work done divided by the time required to do the work. The answer is given in joule per second. Another name for joules per second is the watt (w).

Econometrics: random sampling Econometrics: random sampling ECO 5100 Popular in Econometrics Popular in Economcs This 2 page Class Notes was uploaded by Rahul Bose on Wednesday June 22, 2016. The Class Notes belongs to ECO 5100 at Wayne State University taught by Arjun in Summer 2016. Since its upload, it has received 15 views. For similar materials see Econometrics in Economcs at Wayne State University. Reviews for Econometrics: random sampling Report this Material What is Karma? Karma is the currency of StudySoup. You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more! Date Created: 06/22/16 Chapter one : Introduction A complete econometric model for example 2 might be: wage = β0 + β1edu + β2exper + β3training + u (4) where the term u contains factors such as "innate ability", quality of education, family background, etc. For the most part, we would start with an econometric model and use economic reasoning and common sense as guides for choosing the variables. Once an econometric model has been specified, various hypotheses can be stated in terms of the unknown parameters. e.g. in equation (3), we may test β1 = 0. After data have been collected, econometric methods are used to estimate the parameters in the econometric model and to test the hypotheses of interest. In some cases, the econometric model is used to make predictions in either the testing of a theory or the study of a policy’s impact. A cross-sectional data set consists of a sample of individuals, households, firms, cities, states, countries, or a variety of other units, taken at a given point of time. Sometimes, the data on all units do not correspond to precisely the same time period. e.g. several families may be surveyed during different weeks within a year. In a pure cross-sectional analysis, we would ignore any minor timing differences in collecting the data. An important feature of cross-sectional data is that we can often assume that they have been obtained by random sampling from the underlying population. e.g. if we obtain information on wages, education, experience, and other characteristics by randomly drawing 500 people from the working population, then we have a random sample from the population of all working people. A key feature resulting from random sampling is that the ordering of the data does not matter for econometric analysis. A time series data set consists of observations on a variable or several variables over time. Examples: stock prices, consumer price index, gross domestic product, etc. Because past events can influence future events, time is an important dimension. Unlike the arrangement of cross-sectional data, the chronological ordering of observations in a time series conveys potentially important information. A key feature of time series data that makes them more difficult to analyze than cross-sectional data is that economic observations can rarely be assumed to be independent across time. Most economic and other time series are related, often strongly related, to their recent histories. e.g. knowing something about the GDP from last quarter tells us quite a bit about the likely range of the GDP during this quarter, because GDP tend to remain fairly stable from one quarter to the next. New techniques have been developed to account for the dependent nature of economic time series and to address other issues, e.g. the fact that some economic variables tend to display clear trends over time. Xu Lin (Wayne State University) Another feature of time series data that can require special attention is the data frequency at which the data are collected. In economics, the most common frequencies are daily, weekly, monthly, quarterly, and annually. For example, stock prices are recorded daily. The money supply in the US economy is reported weekly. Many macroeconomic series e.g. inflation and unemployment rates are tabulated monthly. GDP is an example of quarter series.Other time series such as infant mortality rates for states in the US, are available only on an annual basis. Many weekly, monthly, and quarterly economic time series display a strong seasonal pattern. e.g. monthly data on housing starts differ across the months simply due to changing weather conditions. When econometric methods are used to analyze time series data, the data should be stored in chronological order. Many data sets have both cross-sectional and time series features. e.g. suppose two cross-sectional household surveys are taken in the US, one in 1985 and one in 1990. In 1985, a random sample of households is surveyed for variables such as income, savings, family size, and so on. In 1990, a new random sample of households is taken using the same survey questions. To increase sample size, we can form a pooled cross section by combining the two years. Pooling across sections from different years is often an effective way of analyzing the effects of a new government policy. The idea is to collect data from the years before and after a key policy change. As an example, consider the data on housing prices taken in 1993 and 1995, before and after a reduction in property taxes in 1994. Suppose we have data on 250 houses for 1993 and on 270 houses for 1995. See Table 1.4. Observations 1-250 correspond to the houses sold in 1993, 251-520 to the 270 houses sold in 1995. Although the order in which we store the data turns out not to be crucial, keeping track of the year is very important. This is why we enter year as a separate variable. A pooled cross section is analyzed much like a standard cross section, except that we often need to account for secular differences in the variables across the time. In fact, in addition to increasing sample size, the point of a pooled cross-sectional analysis is often to see how a key relationship has changed over time. Are you sure you want to buy this material for You're already Subscribed! Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Bulk sales, PO's, Marketplace Items, eBooks, Apparel, and DVDs not included. Questions About This Book? - The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc. - The Used copy of this book is not guaranteed to include any supplemental materials. Typically, only the book itself is included. - The Rental copy of this book is not guaranteed to include any supplemental materials. You may receive a brand new copy, but typically, only the book itself. - The eBook copy of this book is not guaranteed to include any supplemental materials. Typically only the book itself is included. Note: You are purchasing a standalone product; MyMathLab does not come packaged with this content. If you would like to purchase both the physical text and MyMathLab, search for ISBN-10: 0321947622 /ISBN-13: 9780321947628. That package includes ISBN-10: 0321431308 /ISBN-13: 9780321431301, ISBN-10: 0321654064/ISBN-13:978032165406, and ISBN-10: 0321945522/ISBN-13: 9780321945525. MyMathLab is not a self-paced technology and should only be purchased when required by an instructor. Barnett/Ziegler/Byleen is designed to help students help themselves succeed in the course. This text offers more built-in guidance than any other on the market–with special emphasis on prerequisites skills–and a host of student-friendly features to help students catch up or learn on their own. Raymond A. Barnett, a native of California, received his B.A. in mathematical statistics from the University of California at Berkeley and his M.A. in mathematics from the University of Southern California. He has been a member of the Merritt College Mathematics Department, and was chairman of the department for four years. Raymond Barnett has authored or co-authored eighteen textbooks in mathematics, most of which are still in use. In addition to international English editions, a number of books have been translated into Spanish. Michael R. Ziegler (late) received his B.S. from Shippensburg State College and his M.S. and Ph.D. from the University of Delaware. After completing post doctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he held the rank of Professor in the Department of Mathematics, Statistics, and Computer Science. Dr. Ziegler published over a dozen research articles in complex analysis and co-authored eleven undergraduate mathematics textbooks with Raymond A. Barnett, and more recently, Karl E. Byleen. Karl E. Byleen received his B.S., M.A. and Ph.D. degrees in mathematics from the University of Nebraska. He is currently an Associate Professor in the Department of Mathematics, Statistics and Computer Science of Marquette University. He has published a dozen research articles on the algebraic theory of semigroups. Table of Contents Diagnostic Prerequisite Test PART ONE: A LIBRARY OF ELEMENTARY FUNCTIONS 1. Linear Equations and Graphs 1.1 Linear Equations and Inequalities 1.2 Graphs and Lines 1.3 Linear Regression Chapter 1 Review 2. Functions and Graphs 2.2 Elementary Functions: Graphs and Transformations 2.3 Quadratic Functions 2.4 Polynomial and Rational Functions 2.5 Exponential Functions 2.6 Logarithmic Functions Chapter 2 Review PART TWO: FINITE MATHEMATICS 3. Mathematics of Finance 3.1 Simple Interest 3.2 Compound and Continuous Compound Interest 3.3 Future Value of an Annuity; Sinking Funds 3.4 Present Value of an Annuity; Amortization Chapter 3 Review 4. Systems of Linear Equations; Matrices 4.1 Review: Systems of Linear Equations in Two Variables 4.2 Systems of Linear Equations and Augmented Matrices 4.3 Gauss-Jordan Elimination 4.4 Matrices: Basic Operations 4.5 Inverse of a Square Matrix 4.6 Matrix Equations and Systems of Linear Equations 4.7 Leontief Input-Output Analysis Chapter 4 Review 5. Linear Inequalities and Linear Programming 5.1 Linear Inequalities in Two Variables 5.2 Systems of Linear Inequalities in Two Variables 5.3 Linear Programming in Two Dimensions: A Geometric Approach Chapter 5 Review 6. Linear Programming: The Simplex Method 6.1 the Table Method: An Introduction to the Simplex Method 6.2 The Simplex Method: Maximization with Problem Constraints of the Form ≤ 6.3 The Dual; Minimization with Problem Constraints of the form ≥ 6.4 Maximization and Minimization with Mixed Problem Constraints Chapter 6 Review 7. Logic, Sets, and Counting 7.3 Basic Counting Principles 7.4 Permutations and Combinations Chapter 7 Review 8.1 Sample Spaces, Events, and Probability 8.2 Union, Intersection, and Complement of Events; Odds 8.3 Conditional Probability, Intersection, and Independence 8.4 Bayes' Formula 8.5 Random Variables, Probability Distribution, and Expected Value Chapter 8 Review 9. Markov Chains 9.1 Properties of Markov Chains 9.2 Regular Markov Chains 9.3 Absorbing Markov Chains Chapter 9 Review 10. Games and Decisions 10.1 Strictly Determined Games 10.2 Mixed Strategy Games 10.3 Linear Programming and 2 x 2 Games—Geometric Approach 10.4 Linear Programming and m x n Games—Simplex Method and the Dual Chapter 10 Review 11. Data Description and Probability Distributions 11.1 Graphing Data 11.2 Measures of Central Tendency 11.3 Measures of Dispersion 11.4 Bernoulli Trials and Binomial Distributions 11.5 Normal Distributions Chapter 11 Review A. Basic Algebra Review A.1 Algebra and Real Numbers A.2 Operations on Polynomials A.3 Factoring Polynomials A.4 Operations on Rational Expressions A.5 Integer Exponents and Scientific Notation A.6 Rational Exponents and Radicals A.7 Quadratic Equations B. Special Topics B.1 Sequences, Series, and Summation Notation B.2 Arithmetic and Geometric Sequences B.3 Binomial Theorem Table I. Area Under the Standard Normal Curve Table II. Basic Geometric Formulas

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses. Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . . A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter. The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of A collection of games on the NIM theme Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?” Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter. This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning. An article for teachers and pupils that encourages you to look at the mathematical properties of similar games. Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general. Here are two kinds of spirals for you to explore. What do you notice? The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . . Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . . List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it? The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice. We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4 The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . . Are these statements relating to odd and even numbers always true, sometimes true or never true? Are these statements always true, sometimes true or never true? Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded? Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? This activity involves rounding four-digit numbers to the nearest thousand. Imagine we have four bags containing numbers from a sequence. What numbers can we make now? It would be nice to have a strategy for disentangling any tangled Can you describe this route to infinity? Where will the arrows take you next? In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest? This challenge encourages you to explore dividing a three-digit number by a single-digit number. Can you tangle yourself up and reach any fraction? Nim-7 game for an adult and child. Who will be the one to take the last counter? In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes. One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down? Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? Find a route from the outside to the inside of this square, stepping on as many tiles as possible. When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. It starts quite simple but great opportunities for number discoveries and patterns! How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...? What size square corners should be cut from a square piece of paper to make a box with the largest possible volume? How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread? Can all unit fractions be written as the sum of two unit fractions? The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you work out how to balance this equaliser? You can put more than one weight on a hook. Can you cover the camel with these pieces? What happens when you try and fit the triomino pieces into these Use the clues to colour each square. 10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways? Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had. Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it? Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line? If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make? A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard? You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters? A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour. Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour What is the best way to shunt these carriages so that each train can continue its journey? What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square? Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods? This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture? Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....? There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules? Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time. Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others. Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land. Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line? What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour? 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? Can you find all the different ways of lining up these Cuisenaire How many trains can you make which are the same length as Matt's, using rods that are identical? Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win? Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle. An activity making various patterns with 2 x 1 rectangular tiles. How many different rhythms can you make by putting two drums on the Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it? These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out. 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 In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon? How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle? If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities? In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together? Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line. Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table? How many different triangles can you make on a circular pegboard that has nine pegs? There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find? There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

Автор: Garson Название: What Logics Mean ISBN: 1107611962 ISBN-13(EAN): 9781107611962 Издательство: Cambridge Academ Рейтинг: Цена: 3285 р. Наличие на складе: Поставка под заказ. Описание: What do the rules of logic say about the meanings of the symbols they govern? In this book, James W. Garson examines the inferential behaviour of logical connectives (such as 'and', 'or', 'not' and 'if … then'), whose behaviour is defined by strict rules, and proves definitive results concerning exactly what those rules express about connective truth conditions. He explores the ways in which, depending on circumstances, a system of rules may provide no interpretation of a connective at all, or the interpretation we ordinarily expect for it, or an unfamiliar or novel interpretation. He also shows how the novel interpretations thus generated may be used to help analyse philosophical problems such as vagueness and the open future. His book will be valuable for graduates and specialists in logic, philosophy of logic, and philosophy of language. Описание: The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves `antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now known as `explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference.The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view. Автор: Carnielli Название: Analysis and Synthesis of Logics ISBN: 140206781X ISBN-13(EAN): 9781402067815 Издательство: Springer Рейтинг: Цена: 22521 р. Наличие на складе: Поставка под заказ. Описание: Presents the mathematical theory of combining and decomposing logics. This book covers mechanisms for combining semantic structures and deductive systems either of the same or different nature (for instance, two Hilbert calculi or a Hilbert calculus and a tableau calculus). It is suitable for those in mathematical logic, and theory of computation. Автор: George Metcalfe; Nicola Olivetti; Dov Gabbay Название: Proof Theory for Fuzzy Logics ISBN: 1402094086 ISBN-13(EAN): 9781402094088 Издательство: Springer Рейтинг: Цена: 16169 р. Наличие на складе: Поставка под заказ. Описание: Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. This book focuses on the development and applications of `proof-theoretic` presentations of fuzzy logics. Автор: Schechter, Eric Название: Classical and non-classical logics ISBN: 0691122792 ISBN-13(EAN): 9780691122793 Издательство: Wiley Рейтинг: Цена: 13750 р. Наличие на складе: Поставка под заказ. Описание: In this book, the author introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. It begins with brief introductions to informal set theory and general topology, and avoids advanced algebra; thus it is self-contained and suitable for readers with little background in mathematics. Автор: Restall, Greg Название: Introduction to substructural logics ISBN: 041521534X ISBN-13(EAN): 9780415215343 Издательство: Taylor&Francis Рейтинг: Цена: 4536 р. Наличие на складе: Поставка под заказ. Описание: An introduction to an important group of logics that have come to be known under the umbrella term susbstructural. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. Автор: Paoli F. Название: Substructural Logics: A Primer ISBN: 1402006055 ISBN-13(EAN): 9781402006050 Издательство: Springer Рейтинг: Цена: 22521 р. Наличие на складе: Поставка под заказ. Описание: Preface. Part I: The philosophy of substructural logics. 1. The role of structural rules in sequent calculi. Part II: The proof theory of substructural logics. 2. Basic proof systems for substructural logics. 3. Cut elimination and the decision problem. 4. Other formalisms. Part III: The algebra of substructural logics. 5. Algebraic structures. 6. Algebraic semantics. 7. Relational semantics. Appendix A: Basic glossary of algebra and graph theory. Appendix B: Other substructural logics. Bibliography. Index of subjects. Автор: Barwise Название: Model-Theoretic Logics ISBN: 1107168252 ISBN-13(EAN): 9781107168251 Издательство: Cambridge Academ Рейтинг: Цена: 22323 р. Наличие на складе: Поставка под заказ. Описание: Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the eighth publication in the Perspectives in Logic series, brings together several directions of work in model theory between the late 1950s and early 1980s. It contains expository papers by pre-eminent researchers. Part I provides an introduction to the subject as a whole, as well as to the basic theory and examples. The rest of the book addresses finitary languages with additional quantifiers, infinitary languages, second-order logic, logics of topology and analysis, and advanced topics in abstract model theory. Many chapters can be read independently. Автор: Dov M. Gabbay; Karl Schlechta Название: A New Perspective on Nonmonotonic Logics ISBN: 3319468154 ISBN-13(EAN): 9783319468150 Издательство: Springer Рейтинг: Цена: 12704 р. Наличие на складе: Поставка под заказ. Описание: In this book the authors present new results on interpolation for nonmonotonic logics, abstract (function) independence, the Talmudic Kal Vachomer rule, and an equational solution of contrary-to-duty obligations. The chapter on formal construction is the conceptual core of the book, where the authors combine the ideas of several types of nonmonotonic logics and their analysis of 'natural' concepts into a formal logic, a special preferential construction that combines formal clarity with the intuitive advantages of Reiter defaults, defeasible inheritance, theory revision, and epistemic considerations.It is suitable for researchers in the area of computer science and mathematical logic. Автор: Zoran Ognjanovi?; Miodrag Ra?kovi?; Zoran Markovi? Название: Probability Logics ISBN: 3319470116 ISBN-13(EAN): 9783319470115 Издательство: Springer Рейтинг: Цена: 10971 р. Наличие на складе: Поставка под заказ. Описание: The aim of this book is to provide an introduction to probability logic-based formalization of uncertain reasoning. The authors' primary interest is mathematical techniques for infinitary probability logics used to obtain results about proof-theoretical and model-theoretical issues such as axiomatizations, completeness, compactness, and decidability, including solutions of some problems from the literature.An extensive bibliography is provided to point to related work, and this book may serve as a basis for further research projects, as a reference for researchers using probability logic, and also as a textbook for graduate courses in logic. Описание: This book covers modularity and independence in classical and nonclassical logic, including many-valued logics and structures, plus neighbourhood semantics and their connection to independence, as well as their common points and differences for various logics. ООО "Логосфера " Тел:+7(495) 980-12-10 www.logobook.ru

Leal Moïse October 29, 2020 Reading Worksheets One of the best way to memories the rule is to write them down in a notebook and refer to it continuously until you could memories them all. However, mere memories the rule is not enough, you need to do as many English grammar worksheets and exercises as possible. Preferably doing objective form of English grammar test worksheet, and repeat the exercises until you got all the answers correct. One of the best English grammar test worksheet I found online is autotick English grammar worksheet where you could know your scope instantly by entering a password. Parents are familiar with using worksheets in math classes. Even when the textbook has spaces for the insertion of answers, parents find it more economical to use worksheets so that textbooks can be resold at the conclusion of the class. Additionally, worksheets help to avoid messy textbooks where there have been insertions and erasures. However, few parents have considered the vital need for homeschool worksheets in classes other than math. This article presents compelling arguments for parents to consider using worksheets in classes other than math. You can also try and teach division just like you would teach the multiplication tables, through division worksheets. By teaching 5 times 2 is 10 and 5 times 3 is 15 you can reverse it and let your students learn by saying things like 10 divided by 5 is 2 and 15 divided by 5 is 3. Of course, this is not applicable to much large numbers when it comes to learning to divide by larger numbers, but it is a good start. In all stages above, it is imperative to do oral and mental math. Without this skill, your child will be forever stuck with a pencil and paper. And the more work done on paper with a pencil, the more there is a chance for an error. And, your child will be stuck following steps instead of "just doing math." Doing oral and mental math makes a person very comfortable with math. Many adults have math phobia, due in no small part to not being able to do mental math. How to do it? While driving, cooking, shopping, sightseeing, almost any situation, you can drill your child on math. If a box costs $2, how much does 2 cost? How many horses do you see? Count the blue cars. Are their more boys than girls? Anything! Be creative. You can even get them to recite the times tables. This will also set the stage for an important skill they must master. Word problems! How many times have you heard people say they cannot do word problems? The oral problems you make up are just another form of word problems. If your child is used to doing math, without a problem written on paper, your child will not fear word problems. If you adamantly do the above, there is one last step. Sometimes it is out of your control, but do your best! Put your child in a class where there is an effective algebra teacher, and all math classes beyond sixth grade. You may find this hard, but the only one fighting for your child is you! These birthday printables will be lots of help for every kind of kid. Not only for a birthday but randomly as well, they can just be kept at home and used as activity time games. Kids can use crayons to fill the names and location or to draw the animal pictures on the cards. Printables can keep kids engaged and out of mischief. Printables will keep them busy in their rooms, learning and having fun at the same time. So if you want your kids to really enjoy and learn then make sure you have birthday printables. If not, go out and buy some to keep in the house or at school because experience tells us that kids have really enjoyed having fun as well as learning with birthday printables. 1st grade worksheets are used for helping kids learning in the first grade in primary schools. These worksheets are offered by many charitable & commercial organizations through their internet portals. The worksheets provide study materials to kids in a funky & innovative way, to magnetize them towards learning. These worksheets are provided for all subjects present in a 1st grade school curriculum covering English, math, science & many others. Worksheets are also provided for developing & nurturing the thinking skills of a student too in the form of crossword puzzle & thinking skill worksheets. Moreover, many 1st grade worksheet providers as well provide time counting & calendar worksheets as well to test the IQ of the kids. 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About This Chapter How It Works: - Identify the lessons in the Saxon Real Numbers chapter with which you need help. - Find the corresponding video lessons with this companion course chapter. - Watch fun videos that cover the real numbers topics you need to learn or review. - Complete the quizzes to test your understanding. - If you need additional help, rewatch the videos until you've mastered the material or submit a question for one of our instructors. Students will learn: - Different types of numbers - Performing addition and subtraction - Performing multiplication and division - Absolute values - Reciprocals of rational expressions - Commutative property - Distributive property - Multiplication property of equality - Additive property Saxon is a registered trademark of Houghton Mifflin Harcourt, which is not affiliated with Study.com. 1. What are the Different Types of Numbers? There are different types or families of numbers. Learn how to identify natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. 2. How to Perform Addition: Steps & Examples Addition is a mathematical operation in which two or more numbers or quantities are combined together. Know the definition of addition, learn how to add, and explore the steps and examples of adding more than two numbers. 3. How to Perform Subtraction: Steps & Examples Subtraction is a basic mathematical operation that involves one number taking value away from another number. Learn about the definition of subtraction and explore examples of how to perform subtraction and also the steps for subtracting two or more numbers from another number. 4. How to Perform Multiplication: Steps & Examples Multiplication is a basic mathematical operation that involves the number of times a number or quantity should be added to itself. Learn about multiplication and explore the steps and examples of multiplying two or more numbers. 5. How to Perform Division: Steps & Examples A basic mathematical operation, performing division involves separating numbers or quantities and sometimes results in remainders. Learn about the definition of division, the symbols used for division with and without a remainder, and explore examples of the steps for dividing two quantities. 6. What is an Absolute Value? Absolute value in mathematics involves focusing on the size of a number instead of its sign. Explore the distance from zero, notation of absolute values, and solving equations with absolute values. 7. Division and Reciprocals of Rational Expressions Rational expressions are math statements with rational numbers, which are numbers that can be written as the fraction of two integers. In this lesson, discover how to divide a rational number, learn how to take the reciprocal, and practice some examples. 8. Commutative Property of Addition: Definition & Examples The commutative property of addition states that numbers added in any order will always have the same sum. Learn the detailed definition of the commutative property of addition and its formula, and explore examples of how the commutative property works. 9. Commutative Property of Multiplication: Definition & Examples The commutative property of multiplication states that regardless of the order which numbers are multiplied in, the result will be the same. Learn the definition and formula of the commutative property in detail, and apply your knowledge by working with examples. 10. Distributive Property: Definition, Use & Examples The distributive property is a concept that helps make math problems easier to solve when dealing with multiple factors. Learn about the definition of this property, how to use the distributive property, and examples and applications of it in different types of math. 11. Multiplication Property of Equality: Definition & Example The multiplication property of equality states that if one side of an equation is multiplied, the other side is multiplied by the same number in order to keep the equation the same. Learn more about this formula by reviewing two examples. 12. Additive Property: Definition & Examples The additive property is a concept often used to solve algebraic problems. Learn about the additive property of equality, the additive property of inequalities, the additive inverse property, and examples of all three. Earning College Credit Did you know… We have over 220 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level. To learn more, visit our Earning Credit Page Other chapters within the Saxon Algebra 2 Homeschool: Online Textbook Help course - Saxon Algebra 2: Decimal Numbers - Saxon Algebra 2: Graphs on the Coordinate Plane - Saxon Algebra 2: Sets - Saxon Algebra 2: Complex Numbers - Saxon Algebra 2: Conversion by Unit Multipliers - Saxon Algebra 2: Ratio & Proportion - Saxon Algebra 2: Percent - Saxon Algebra 2: Rate - Saxon Algebra 2: Exponents - Saxon Algebra 2: Exponents on a Scientific Calculator - Saxon Algebra 2: Roots - Saxon Algebra 2: Probability - Saxon Algebra 2: Statistics - Saxon Algebra 2: Simplifying Expressions - Saxon Algebra 2: Polynomials - Saxon Algebra 2: Simplifying Rational Expressions - Saxon Algebra 2: Simplifying and Solving Equations - Saxon Algebra 2: Linear Equations - Saxon Algebra 2: Solving Linear Equations - Saxon Algebra 2: Quadratic Equations - Saxon Algebra 2: Other Types of Equations - Saxon Algebra 2: Understanding Functions - Saxon Algebra 2: Manipulating and Evaluating Functions - Saxon Algebra 2: Trigonometry - Saxon Algebra 2: Logarithms - Saxon Algebra 2: Lines, Points, Segments, and Planes - Saxon Algebra 2: Angles - Saxon Algebra 2: Polygons - Saxon Algebra 2: Circles - Saxon Algebra 2: Triangles - Saxon Algebra 2: Geometric Solids - Saxon Algebra 2: Perimeter and Circumference - Saxon Algebra 2: Area - Saxon Algebra 2: Surface Area and Volume - Saxon Algebra 2: Constructions - Saxon Algebra 2: Postulates & Pythagorean Theorem - Saxon Algebra 2: Proofs

The only one I know of is the LP LOX pump on RD-0120. A multi-spool design is already a default for jet engines, which are basically turbine-driven air pumps. A multi-spool pump can save at least the weight of the duct connecting the LP and HP pump if nothing else, which develops a boundary layer in the liquid pumped, not to mention the additional boundary layer if this duct has one or more flexible joints in it. If the oxygen HP pump is too difficult to use a multi-spool design, then at least the HP fuel pump can use this design to save some weight, ideally combining the LP and HP pumps and the preburner in a single assembly just like a small jet engine. (But of course, no jet engine has a backend pressure of 200 bar or more.) 1$\begingroup$ I'm very interested in this question. Can you provide a link or edit your question to show that the RD-0120 uses this design? I can't find a dual spool pump on the only schematic I can find: qph.fs.quoracdn.net/main-qimg-ce9b18d79f9652c443c989c5695c5541 One reference claims the NK-33 had a dual spool LO2 pump but I can't confirm that either. link.springer.com/chapter/10.1007/978-3-319-27748-6_18 $\endgroup$– Organic MarbleMay 22, 2019 at 15:12 1$\begingroup$ @OrganicMarble I got it wrong. It's the LOX boost pump, not the LH2 boost pump that's twin spool. Two sources mentioned this fact. The first one is History of Liquid Propellant Rocket Engine in the USSR CADB section and the second one is this lpre.de/resources/articles/AIAA-1995-2540.pdf. $\endgroup$– Meatball PrincessMay 22, 2019 at 17:10 2$\begingroup$ Thanks! I'll be watching this one. $\endgroup$– Organic MarbleMay 22, 2019 at 17:13 The reason for multiple shafts/spools is to allow the inner/higher pressure stages to spin at a different (greater) rate than the outer lower pressure stages. In aviation jet engines this solves two quite specific problems. Namely: Efficiently creating higher pressure/pressure-ratio combustion, for thermodynamic efficiency (across a range of flow-rates). Having a wide range of throttle settings/current engine speed/altitude that don't cause a surge condition. Cryogenic turbo-pumps don't have these problems, or at least not generally. The ingress-ed fluid is already as dense as it's going to be as pressure/density are no longer as tightly coupled. The low speed conditions for surge in jets doesn't happen as much in rocketry. There are also reasons not to have compressor stages in rockets. In particular the limiting factor of spool speed is often cavitaion (no relevant for jets). This means a lot of turbo-pump assemblies couldn't make use of a second spool without risking cavitation. 'Boost' compressors can be used to increase the pressure in the system to prevent this but this is added weight and complexity etc. There are advantages specific to pump assemblies too, but they are quite specific. For example the second shaft isolates some of the rise in pressure due to the pre-burner. For single shaft staged combustion engines (i.e.: RD-0120) this is important the seals used to prevent leaking along the shaft are complex/expensive/consume helium. This isolation reduces the pressure gradient those seals have to operate at which can only be a good thing. Balancing all this up is a bit complex and I wouldn't be able to predict whether or not that make a multi-spool design viable. However I hope that gives some insight at least to why it isn't as obvious a choice as in jets. The combustion that powers turbo-pumps (analogous to the combustion chamber in a jet) operates at (and requires) a relatively high pressure. In a jet engine the spool powers a fan at the front. In a turbo-pump assembly this goes on to drive the compressor of the 'other' propellant, the one that isn't flowing through the turbo-pump itself. In both cases if the shaft isn't perfectly sealed, some of the high pressure fluid from the middle can leak along the shaft to what ever the shaft is driving. In the case of a jet this isn't much of a problem. If a tiny amount of the combustion products leak out of the front, its doesn't matter too much (everything is oxidiser rich and under low pressure). In a turbo-pump any such leakage would be very bad. As one of the fluids is fuel rich, the other is oxidiser rich. And its in a confined space. AKA no space today for anything in close proximity. To avoid this a really complex set of seals are used, and in the middle an inert gas (helium) is injected under huge pressure. This keeps things-that-go-boom-when-together separated. But its not an ideal arrangement. Its heavy and consumes helium which needs to be stored at high pressure. Which means high pressure tanks. Which means more weight more things to go wrong, more cost to develop and manufacture etc. The twin spool design helps here. In a 2 spool design it's the low pressure spool (inner axle/outer stages) that does the driving. Hence its only the low pressure stages that needs the aforementioned elaborate sealing. If the inner stage fluid leaks into the lower pressure stages, its not super-bad as you are still in the same type of propellant, and it wont cause any extra combustion. This means you only have to seal against the lower pressure stage, which makes everything easier (lighter). $\begingroup$ @MeatballPrincess: I have clearly skimmed over a few things for brevity. I am happy to answer follow up questions. $\endgroup$ Jun 19, 2019 at 11:13 $\begingroup$ can you elaborate more on this statement "For example the second shaft isolates some of the rise in pressure due to the pre-burner. For single shaft staged combustion engines (i.e.: RD-0120) this is important the seals used to prevent leaking along the shaft are complex/expensive/consume helium. This isolation reduces the pressure gradient those seals have to operate at which can only be a good thing." $\endgroup$ Jun 19, 2019 at 19:30 $\begingroup$ @MeatballPrincess, a bit long for a comment so i've edited my answer. Hope that helps. $\endgroup$ Jun 20, 2019 at 9:17 $\begingroup$ twin spool means there're two shafts in a single pump pumping a single propellant e.g. LH2, and without the booster LP pump, not 2 pumps pumping 2 propellant, that's already part of my assumption of the original question. In a twin spool LH2 pump, the LH2 goes LPC->HPC->Fuel Rich PB->HPT->LPT->Main Combustion Chamber, no seal needed at all, only controlled leakage. $\endgroup$ Jun 20, 2019 at 15:03 1$\begingroup$ @MeatballPrincess the RD-0120 uses a fuel-rich staged combustion cycle and a single shaft to drive both the fuel and oxidizer turbopumps. Leakage along this shaft is critical and seals are very much needed. The problem is the LP turbine or compressor needs to be connected to the LOx pump via a mechanical connection. Also if this advantage is not present, even less reason to use a twin spool design. $\endgroup$ Jun 20, 2019 at 15:40

Is it really possible that yet another summer is drawing to a close? Here at Wolfram|Alpha, we’ve spent our summer getting ready to head back to school—building new course assistant apps, adding new data, and even making Wolfram|Alpha interactive with CDF. As the world’s leading knowledge engine, we’ve made it our mission to continually improve and ensure that we’re helping students and teachers around the globe explore concepts, ideas, and calculations on a deeper level than previously possible. More » Today we released the new Wolfram Statistics Course Assistant App for the iPhone, iPod touch, and iPad. If you are a student of introductory statistics, the probability that you’ll love the all-new Wolfram Statistics Course Assistant is pretty high! The app will help you better understand concepts such as mean, median, mode, standard deviation, probabilities, data points, random integers, random real numbers, and more. Learning about the normal or binomial distribution? This app will show you the probability density function as well as the cumulative density function and other information. Another year has flown by here at Wolfram|Alpha, and the gears are really turning! New data and features are flowing at a rapid rate. To celebrate, Wolfram|Alpha’s creator, Stephen Wolfram, will share what we’ve been working on and take your questions in a live Q&A. If you have a question you’d like to ask, please send it as a comment to this blog post or tweet to @Wolfram_Alpha and include the hashtag #WAChat. We’ll also be taking questions live on Facebook and Livestream chat during the webcast. We’re looking forward to chatting with you on May 18! Do you need some help navigating your chemistry or precalculus classes? Or maybe you’re still trying to decide which classes to take this fall. Good news! Today, we’re releasing the Wolfram General Chemistry and Precalculus Course Assistant Apps, two more Wolfram|Alpha-powered course assistants that will help you better understand the concepts addressed in these classes. If you’re taking chemistry, download the Wolfram General Chemistry Course Assistant App for everything from looking up simple properties like electron configurations to computing the stoichiometric amounts of solutes that are present in solutions of different concentrations. This app is handy for lab researchers, too! The specialized keyboard allows you to enter chemicals by using formulas or by spelling out their names. Everyone needs a little break when preparing for finals! That’s why we’re giving you a special break on all Wolfram Course Assistant Apps now through Sunday, May 1, 2011. Wolfram Course Assistant Apps will help you grasp key concepts and gain better understanding of the answers, all of which will have you feeling confident and prepared to ace your final exams! The following Wolfram Course Assistant Apps are available for the iPhone, iPod touch, and iPad: Best of luck with your finals! Our team is developing an app for every course, so be sure to check back for more helpful Wolfram Course Assistant Apps before heading to class next semester. Today we are releasing Wolfram Multivariable Calculus and Wolfram Astronomy, the next two apps on a growing list of Wolfram Course Assistant Apps. These course assistants will help students learn their course material using the power of Wolfram|Alpha. The Wolfram Astronomy Course Assistant allows you to easily look up information on constellations and planets, but it can also calculate anything from the next lunar eclipse to the solar interior. As kids start to return to classes after the holidays, we’re happy to announce that Wolfram|Alpha has the ability to compute some interesting information about their school districts. You can now use Wolfram|Alpha to analyze and compare data on student-teacher ratios, expenditures, revenues, and salaries in more than 18,000 public school districts in the United States. Let’s start with an example on the West Coast: Seattle Public Schools is one of the larger districts in the country, with over 100 schools and more than 45,000 students. The student-teacher ratio is 18:1, and if you scroll down you’ll see that total expenditures are about $14,000 per student per year. The long-term goal is to have an assistant app for every major course, from elementary school to graduate school. And the good news is that Wolfram|Alpha has the breadth and depth of capabilities to make this possible—and not only in traditionally “computational” kinds of courses. The concept of these apps is to make it as quick and easy as possible to access the particular capabilities of Wolfram|Alpha relevant for specific courses. Each app is organized according to the major curriculum units of a course. Then within each section of the app, there are parts that cover each of the particular types of problems relevant to that unit. As we bid adieu to 2010, we want say thank you to all of our loyal blog readers and commenters. Today we’re taking a look back at some of 2010’s most popular Wolfram|Alpha Blog posts. 2010 was a year full of product releases, such as Wolfram|Alpha Widgets and new data for everything from movies to taxes. These selections are only highlights of the topics we’ve covered in 2010. If you’re feeling really nostalgic, or if you’re new to the Wolfram|Alpha Blog, we invite you to read more in the archives. Just in time to tackle a common New Year’s resolution, we released “New Physical Activity Data in Wolfram|Alpha”. After reading “Computing Valentine’s Day with Wolfram|Alpha”, there was little doubt that we speak math, the universal language of love. Ever wonder which country consumes the most coffee or sugar? In March, we introduced new data that answers these questions in the post “Food for Thought: Consumption Patterns from Around the World”. In April we were excited to finally be able to share “Stephen Wolfram’s TED Talk: Computation Is Destined to Be the Defining Idea of Our Future”. The inspirational video quickly became a web favorite. Where did the time go? In May we celebrated Wolfram|Alpha’s first birthday with the post “Wolfram|Alpha: The First Year”. Just in time for family reunion season, we published “My Cousin’s Cousin’s Niece’s Grandfather Said to Just Ask Wolfram|Alpha”, to help you identify all of those branches on the family tree. In July we shared “Ask Wolfram|Alpha about Medical Drug Treatments” to introduce a new functionality in Wolfram|Alpha that allows you to compare how your medical conditions and treatment plans compare to those of other patients. Kids say the darnedest things. In the post “10 Fun Questions Kids Can Answer with Wolfram|Alpha”, we took a look at how Wolfram|Alpha can help you and your little one answer common curiosities. More » Our first Wolfram|Alpha Back-to-School Webinars were met with so much interest and enthusiasm that we’re announcing three more opportunities for you to participate! Sign up today for one of our Wolfram|Alpha Back-to-School Webinars and discover powerful new ways to advance learning in your classroom. The hour-long webinar gives you an overview and demonstration of the Wolfram|Alpha computational knowledge engine, including the recently launched Widget Builder (beta). Administrators, parents, and students will also benefit from these webinars. To register for a webinar, please click one of the three sessions listed below. Registration is free and takes just a few minutes. A copy of the presentation will also be made available to those who attend. Wednesday, September 22, 2010 at 9am Pacific Time Wednesday, September 29, 2010 at 2pm Pacific Time Wednesday, October 6, 2010 at 5pm Pacific Time We look forward to having you and your colleagues join us for an upcoming webinar! We’re pleased to announce a series of free, live Wolfram|Alpha Back-to-School Webinars that give K–12 educators and administrators an overview of the utility and features of Wolfram|Alpha in education. Educators are showing interest in and enthusiasm for Wolfram|Alpha, and we look forward to helping them incorporate it into their classrooms. The webinars will be presented by Holland Lincoln, Manager of Education and Business Development, and will feature a live Q&A. To register for a Wolfram|Alpha Back-to-School Webinar, please click one of the four sessions listed below. Each session is limited to 100 participants. Sign up today to secure your space! Thursday, September 2, 2010 at 2pm Central Time Wednesday, September 8, 2010 at 6pm Central Time Thursday, September 9, 2010 at 3pm Central Time Monday, September 13, 2010 at 3pm Central Time Once your registration is complete, you’ll receive a confirmation email containing a webinar login link. The webinars will be delivered via Adobe Acrobat Connect. Use any one of the supported web browsers on your computer with Flash Player installed. We look forward to having you and your colleagues join us for an upcoming Wolfram|Alpha Back-to-School Webinar! The Wolfram|Alpha Blog is not only your official news source for new data and features, but it’s also a great place to read how others are using Wolfram|Alpha in everyday life, for education and on the job. This week, a tweet linking to @drwetzel‘s latest blog post “How to Integrate Wolfram Alpha into Science and Math Classes” caught our attention. With a new school year upon us, we wanted to share his examples for using Wolfram|Alpha through the website, widgets, and mobile apps with educators who are looking for ways to incorporate Wolfram|Alpha into their math and science classes. From the Teach Science and Math blog: How to Integrate Wolfram Alpha into Science and Math Classes “What is Wolfram Alpha? It is a supercomputing brain. It provides calculates [sic] and provides comprehensive answers to most any science or math question. Unlike other search sources, you and your students can ask questions in plain language or various forms of abbreviated notation. Contrary to popular belief, Wolfram Alpha is not a search engine. Unlike popular search engines, which simply retrieve documents based on keyword searches, Wolfram computes answers based on known models of human knowledge. It provides answers which are complete with data and algorithms, representing real-world knowledge. Teaching Strategies: Researching Facts and Information Science and math teaching strategies with Wolfram begin with allowing students to search for information about specific facts and information. The following examples provide support for stimulating critical thinking using a digital lens.” Click here to continue reading this post on the Teach Science and Math blog. If you’re new to Wolfram|Alpha, we invite you to visit the Wolfram|Alpha for Educators site to browse our video gallery, download lesson plans, and more. Are you already using Wolfram|Alpha in your classroom? Share your story in the comment box below and you could be featured in an upcoming post on how educators are using Wolfram|Alpha as a learning tool in a variety of subjects. Are you an educator looking for new ways to grab your students’ attention and liven up your daily lessons? Visit the new Wolfram|Alpha for Educators site, where you’ll find examples, lesson plans, and even videos on how you can incorporate the technology of Wolfram|Alpha into your classroom. Peruse the video gallery to get a quick introduction to Wolfram|Alpha, and hear from educators and students who are using it in lectures, activities, and research projects. From there take a peek at one of the many lesson plans, in subject areas such as science, mathematics, and social studies. Once you get the hang of it, you can even submit your own lesson plans to share with other educators. This site also points to many other Wolfram educational resources, including the Wolfram Demonstrations Project and MathWorld. We have even set up an Education group on the Wolfram|Alpha Community site so that you can connect with other educators. So the next time you want to do something new and different in your classroom, check out Wolfram|Alpha for Educators to spark your imagination. [Editor’s Note: This blog entry is a guest post from Laura Ketcham, a 7th grade technology instructor and coordinator at the Aventura City of Excellence School (ACES) in Aventura, Florida. If you are interested in sharing how you’ve incorporated Wolfram|Alpha into your everyday life inside or outside the classroom, please contact our blog team at firstname.lastname@example.org.] I read the buzz about Wolfram|Alpha in an article in PC World this past summer. It was billed as a “computational” search engine with the advantage that the results of the computed search appear on one results page, not just in a list of links you need to search through to find the information. I quickly realized that Wolfram|Alpha is an innovative tool that I could definitely incorporate in the classroom! I am a 7th grade technology instructor and coordinator at the Aventura City of Excellence School (ACES) in Aventura, Florida. My students often use the web to find information for a variety of classroom activities, as well as for research in other classes. The students follow a process in which they evaluate websites to determine whether they contain reliable information that can be included for assignments; it’s one of the major topics I cover in the year-long technology course. Wolfram|Alpha provided me with a “cool tool” to introduce to the students that they knew could be trusted as reliable source. They can use Wolfram|Alpha in a variety of ways to “calculate” factual information. What I really found helpful about Wolfram|Alpha was the Examples page. This provided me with a springboard to computing data in Wolfram|Alpha and with a quick way to evaluate its usefulness as a tool in the classroom. This is definitely a great place for teachers, of any grade, to get started! I introduced Wolfram|Alpha to my students during a six-week project where the students infused Web 2.0 technology to build a website about South Florida oceans and beaches. They used Wolfram|Alpha to learn about a variety of topics that they had to include in their sites. Several examples are the taxonomy of a variety of plants and animals that call South Florida beaches home and the GPS/satellite technology being used to track a loggerhead sea turtle that the class adopted. More » Is it cheating to use Wolfram|Alpha for math homework? That was the presentation topic of Conrad Wolfram, Wolfram Research’s Director of Strategic Development, at the TEDx Brussels conference at the European Parliament. Conrad shares his viewpoint in this thought-provoking (and often entertaining) video. Assistant Professor of Mathematics and Computer Sciences John Erickson has long used Mathematica in his courses. So when he heard Wolfram Research was launching Wolfram|Alpha, which is built on Mathematica, he knew it would become a major resource for engaging students in mathematics. Now with Wolfram|Alpha fully integrated into his courses, he says the site is “the best thing for education” because it helps him take his lessons beyond what’s covered in a typical textbook. In this video, he shares an example of how Wolfram|Alpha allows him to show real-world applications of the math he’s teaching. Wolfram|Alpha has also been quite a hit with Professor Erickson’s students, who now use Wolfram|Alpha for all of their courses. They say it’s like having a “personal tutor” available at all times. In this video, they demonstrate why Wolfram|Alpha has become their go-to tool. Teaching with technology and improving math and science education are becoming increasingly hot topics at school districts and campuses around the globe. For more than two decades, our company has been dedicated to promoting advances in education, so we are very excited by the growing focus on the “modern classroom”. As part of our first-ever Wolfram|Alpha Homework Day, we wanted to highlight the use of technology in education. We brought together teachers of all levels who use Wolfram technologies, including Wolfram|Alpha and Mathematica, to hear some of the lessons they’ve learned from integrating technology into their classes and to let them share some of their successes. Noted journalist Elizabeth Corcoran led the panel discussion, which featured Debra Woods, a mathematics professor at the University of Illinois at Urbana-Champaign; Abby Brown, a math teacher at Torrey Pines High School; and Maria Andersen, a math instructor at Muskegon Community College. Part of the discussion focused on dispelling some of the myths about teaching with technology. The panel also shared thoughts on whether teaching with technology increases student exploration, changes how students learn the fundamentals, and helps students make connections to real-world applications. More » One of those educators was an inspiring fourth grade teacher named Shannon Smith. Shannon integrates Wolfram|Alpha into all of the subject areas that she teaches, from spelling and language to geography, science, and math. In this video, she shares examples of how she utilizes Wolfram|Alpha and describes the advantages that she and her students get from incorporating it into her lesson plans. We hope you had a chance to tune into the first-ever Wolfram|Alpha Homework Day. We are still delighted by all of the excitement! The 14-hour webcast was jam-packed with insightful demonstrations, thought-provoking interviews, interesting Q&A with the Wolfram|Alpha scholars, and much, much more. We’ve started uploading video highlights in case you missed parts or want to see them again. Our host, Eric Hansen, kicked off the event with an interview with Wolfram|Alpha creator Stephen Wolfram. Famous physicist and author Brian Greene joined us to talk about why this is such an exciting time for science and technology. More » It’s been an exciting afternoon here at the first-ever Wolfram|Alpha Homework Day—and the day is just getting started. We will be broadcasting live from the Homework Day website until 2am U.S. CDT. Our host, University of Illinois and University of Syracuse Adjunct Professor Eric Hansen, kicked the show off with a live interview with Wolfram|Alpha creator Stephen Wolfram. Shannon Smith and her mother Nancy Brachbill, the teachers behind Recess TEC, joined us for live demonstrations and interviews about how they are using Wolfram|Alpha in their 4th- and 5th-grade classrooms. Learn more about Nancy and Shannon in our earlier blog post. We’ve also had the opportunity to interact with students, educators, and parents at the Dell-sponsored Internet Cafe: If you’re writing an essay for history or a speech for debate class, Wolfram|Alpha is a great resource. It has an enormous words and linguistics database that you can use for such things as word definitions, and word origins, synonyms, and hyphenation. Wolfram|Alpha can even compute the number of pages a given text might produce based on the number of words it contains, such as “500 words in French”. Wolfram|Alpha also has the ability to compute details such how long it should take you to type, read, and deliver that 500-word speech you’ve been preparing. Type “word contest”, and Wolfram|Alpha will retrieve the word data for the English word “contest”. The results tell you many definitions of the word, that its first known recorded use was in 1603, that it rhymes with “conquest”, and a wealth of other data on just that word. More » We want to introduce you to a mother-daughter team who will be joining us for the first-ever Wolfram|Alpha Homework Day to share their passion for advancing educational technology in the classroom. Shannon Smith and her mother, Nancy Brachbill have more than 30 years of combined teaching experience, and are working hard to integrate technology into their 4th- and 5th-grade classrooms on a daily basis. Through their company Recess TEC, they strive to help other educators do the same. They have been involved in countless hours of various educational technology programs to gain a full understanding of what continually engages students. The first-ever Wolfram|Alpha Homework Day is here! We’re so pleased that you’ve stopped by to join us. This groundbreaking live marathon event runs from noon until 2am U.S. CDT, and is being broadcast live on the Homework Day website. Please visit the site to see the event, browse the program highlights, send your questions to be answered by members of the Wolfram|Alpha team, and even submit your homework examples to be showcased live on the air. We’re just hours away from the start of the first-ever Wolfram|Alpha Homework Day, and we thought we’d give you a sneak peak of the Dell-sponsored Homework Day Cafe. This groundbreaking, marathon webcast will be broadcast live from the Wolfram|Alpha Homework Day website beginning at noon U.S. CDT, on October 21. Visit the site now to submit your questions and homework examples! There’s a lot going on in the Wolfram|Alpha project these days—and this week there’s a remarkable convergence of events. Late last week we introduced the Wolfram|Alpha Webservice API, allowing outside developers to call Wolfram|Alpha from their websites or application programs. Then yesterday we released the first mobile implementation of Wolfram|Alpha, in the form of an iPhone app. Tomorrow, we’re doing something completely different: Wolfram|Alpha Homework Day—a 14-hour live webcast event for students and educators. Program highlights for the first-ever Wolfram|Alpha Homework Day, which begins at noon U.S. CDT on Wednesday, October 21, 2009, are now on the Homework Day website. We’re very excited by the amount of enthusiasm that students, parents, and educators are generating about this groundbreaking live web event, which aims to solve your toughest assignments and explore the power of using Wolfram|Alpha for school, college, and beyond. You’re invited to tune in to the event at any time throughout the day. Here are just a few of the highlights we have planned for you: - A special Homework Day welcome from Wolfram|Alpha creator Stephen Wolfram - Live interviews, demonstrations, and vibrant panel discussions with educators - A thought-provoking in-depth conversation with an internationally known actor and education advocate - Live Q&A with members of the Wolfram|Alpha team tackling your toughest questions - A fun science experiment from our very own mad scientist Theodore Gray You can see more of our program highlights on the Homework Day website. While you’re there, find out how you can contribute your questions and examples today! Thanks to our early Wolfram|Alpha Homework Day participants, we are pleased to announce that a submissions gallery is now live on the Homework Day website. Please visit the site and view some of the sampling of interesting questions and work that have been submitted. Some of the posted works include questions, courseware, and lesson plans for astronomy, biology, calculus, chemistry, geometry, geology, history, physics, and writing. If you haven’t already done so, please consider submitting your questions and examples for Homework Day! This first-ever Wolfram|Alpha Homework Day is set to begin at noon U.S. CDT, on October 21. So swing by the Homework Day Website and learn how to submit your contributions today! We are pleased to announce that Dell, Inc. will be a principal sponsor of the first-ever Wolfram|Alpha Homework Day on October 21, 2009. Dell, whose hardware system helped power the launch of Wolfram|Alpha this May, is sponsoring Homework Day’s Internet Cafe. During the multi-hour live web event, the Internet Cafe will allow on-site participants to interact and use Dell laptops to explore Wolfram|Alpha’s computational knowledge engine as a cutting-edge learning tool in education. During Homework Day, scholars, experts, and members of the Wolfram|Alpha team will help participants take on a wide variety of subjects, for K–12 to college and beyond. Students and educators are invited to submit homework questions and examples to be answered by members of the Wolfram|Alpha team, and showcase how they’ve already been using Wolfram|Alpha to bring their homework to life. Please visit http://homeworkday.wolframalpha.com to learn how you can submit your questions and work examples today. People can tune in to see if their submissions are shown. Wolfram|Alpha Homework Day begins at noon U.S. CDT on October 21, 2009. The live webcast can be viewed on the Wolfram|Alpha Homework Day site. Students, educators, and parents are invited to interact with each other and the Wolfram|Alpha team via Homework Day chat, Twitter, and Facebook. We are very pleased by the level of excitement and enthusiasm for the first-ever Wolfram|Alpha Homework Day, being held on October 21, 2009, beginning at noon U.S. CDT. We’re receiving interesting questions about how Wolfram|Alpha can be used to solve your toughest assignments, and submissions from students and educators highlighting how they are already using Wolfram|Alpha to enhance the learning experience. There’s still time for you to get your submission in to be addressed during the live webcast by our team of experts. 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Upgrade to remove ads PHYS Review Questions CH 6 Terms in this set (22) 1. Which has greater momentum, a heavy truck at rest or a moving skateboard The moving skateboard because the heavy truck at rest has no momentum because it has no speed. 2. How does impulse differ from force Impulse has to do with time, it is the amount of force sustained for a period of time. 3. What are the two ways to increase impulse You can increase the impulse by increasing the force applied or by increasing the period of time the force is applied. 4. For the same force, why does a long canon impart more speed to cannonball than a small canon Because the long canon has a greater impulse because of the period of time the force is applied. 5. Is the impulse-momentum relationship related to Newton's second law Yes, the impulse-momentum relationship is derived by rearranging Newton's second law to make the time factor more evident. 6. To impart the greats momentum to an object, should you exert the largest force possible, extend that force for as long a time as possible, or both 7. When you are in the way of a moving object and an impact force is your fate, are you better off decreasing its momentum over a short time or over a long time You are better off decreasing its momentum over a long period of time because it reduces the force and decreases the resulting acceleration. 8. Why is it a good idea to have your hand extended forward when you are getting ready to catch a fast-moving baseball with your bare hand Because that way you have room to let your hand move backward after contact is made with the ball. This extends the time of impact and reduces the force of impact. 9. Why would it be a poor idea to have the back of your hand up against the outfield wall when you catch a long fly ball Because then there is no space for your hand to move backward and the force of impact will be greater. 10. In Karate, why is a force that is applied for a short time more advantageous Because since the time of contact if very short, the force applied to the object is greater. 11. In boxing, why is it advantageous to roll with the punch Because it extends the time and diminishes the amount of force that will be received. 12. Which undergoes the greatest change in momentum: a. A baseball that is caught b. A baseball that is thrown c. A baseball that is caught and then thrown back If all of the baseballs have the same speed just before being caught and just after being thrown c. A baseball that is caught and then thrown back 13. In the preceding question, in which case is the greatest impulse required A baseball that is caught and then thrown back 14. Can you produce a net impulse on an automobile by sitting inside and pushing on the dashboard Can the internal forces within a soccer ball produce an impulse on the soccer ball that will change its momentum?No, because internal forces will not produce a change in momentum. 15. Is it correct to say that, if no net impulse is exerted in a system, then no change in the momentum of the system will occur Yes, because if there is no external impulse exerted in a system then there will be no change in momentum. 16. What does it mean to say that momentum (or any quantity) is conserved When it is said the momentum is conserved that means there is no change. 17. When a canon ball is fired, momentum is conserved for the system of a canon plus cannonball. Would momentum be conserved for the system if momentum were not a vector quantity No, because then they would not be able to cancel. 18. Distinguish between an elastic collision and an inelastic collision. For which type of collision is momentum conserved In an elastic collision there is no generation of heat or deformation, while in an inelastic collision there is. Momentum is conserved for inelastic collisions. 19. Railroad car A rolls at a certain speed and makes a perfectly elastic collision with car B of the same mass. After the collision, car A is observed to be at rest. How does the speed of car B compare with the initial speed of car A Car B was at rest when Car A rolled at a certain speed and made an elastic collision with car B. 20. If the equally massive cars of the previous question stick together after colliding inelastically, how does their speed after the collision compare with the initial speed of car A After the collision the speed is half as much as the velocity of the initial speed of Car A. 21. Suppose a ball of putty moving horizontally with 1kg m/s of momentum collides and sticks to an identical ball of putty moving vertically with 1kg m/s of momentum. Why is their momentum not simply the arithmetic sum of 2kg*m/s Because just as the diagonal of a square is not simply the sum of two sides, the same way the momentum of the two balls cannot be the simple addition of both momentums. 22. In the preceding question, what is the total momentum of the balls of putty before and after the collision THIS SET IS OFTEN IN FOLDERS WITH... CONCEPTUAL PHYSICS: UNIT 3 PRACTICE EXAM CONCEPTUAL PHYSICS: UNIT 2 PRACTICE EXAM Unit 5 Practice Exam Unit 4 Practice Exam YOU MIGHT ALSO LIKE... MCAT | Mometrix Comprehensive Guide Physics Regular Ch 6 Questions chp 6 - homework questions Physics chapter 6 OTHER SETS BY THIS CREATOR

MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING Numerical Methods for Boundary Integral Equations PhD course Chiara Sorgentone (Department of Basic and Applied Sciences for Engineering, Sapienza) Program: In this course we will introduce numerical methods for boundary integral equations, mainly for the Laplace equation and Stokes flow. The main topics to be discussed include: - Theory, derivation and main mathematical properties of boundary integral equations. Starting with the Laplace equation, then moving on to Stokes equations. Single layer and double layer - Numerical discretization of boundary integral equations; - Quadrature rules, including singularity and quasi-singularity treatments; - Error estimates for layer potentials in 2D and 3D. Part of the course will be devoted to numerical implementation in MATLAB. This will include discretization of geometries in 2D and 3D, and numerical resolution of simple problems. Tuesday 29/11/2022, 14-17 Aula Seminari RM004, Via Scarpa 16 Thursday 01/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16 Thursday 15/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16 Tuesday 20/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16 All the interested students are required to send an email to email@example.com If there are problems with the timetable it can be adjusted based on the requests and the room availability. Title: On the connection between non-local operators and probability Instructor: Mirko D’Ovidio (Sapienza - SBAI) Duration: 15 hours Planned period: February 2023 We discuss some basic and advanced facts about initial and boundary value problems involving non-local operators. In particular, we show some stimulating connections between non-local Cauchy problems, non-local boundary value problems and stochastic processes. Non-local boundary value problems also include non-local dynamic boundary conditions. We discuss the probabilistic representation of the solutions together with the associated functionals. First lesson: January 30, 10:30 - 12:30, Room 7, Building RM018 (via del Castro Laurenziano) Title: Energetic relaxation of structured deformations, a multiscale geometrical basis for variational problems in continuum mechanics Instructor: José Matias (Univ. Of Lisbon) Duration: 12 hours 29/3 ore 11-12 aula 4 (via Del Castro Laurenziano) 13/20/27 Aprile 2023, ore 16-18 Aula SEMINARI 4/11 Maggio 2023, ore 16-18, Aula SEMINARI This course will cover the material on the book . Broadly speaking, first order structured deforma- tions introduced in provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. This theory was broadened by in order to allow for geo- metrical changes at the level of second order derivatives (second order structured deformations). The theory of structured deformations was further enriched in in order to consider different levels of microstructure, that is, hierarchical structural deformations. Starting from this mechanical formulation of the theory, and upon describing the needed mathematical framework, namely recalling some basic properties of spaces of bounded variation and spaces of bounded hessian, the variational formulation for first order structured deformations in is presented as well as two different variational formulations for second order structured deformations, in and in . A variational formulation for hierarchical structured deformations in is also presented. Different applications in this context will be discussed, namely: (1) Dimension reduction in the context of structured deformations and ; (2) Derivation of explicit formulae for the relaxed energy densities and ; (3) Optimal design in the context of structured deformations ; (4) Homogenization in the context of structured deformations ; (5) Upscaling and spatial localization of non-local energies . Finally, some open problems and possible generalizations of the theory will be discussed. Program: (1) Mechanical framework: L∞ theory of first order structured deformations , second order struc- tured defomations , and hierarchical systems of structured defomtations . Examples and approximation theorems. (2) Mathematical framework: some results on measure theory, BV spaces and Γ-convergence that will be needed for the mathematical formulation. Later on, some other mathematical preliminaries will be needed, namely Reshetnyak -type continuity theorems, BH space and the Global method for relaxation . (3) The variational formulation of Choksi-Fonseca for first order structured defomations . Results and sketch of the proofs. (4) Some applications: (a) Relaxation of purely interfacial energies; (b) Optimal design of fractured media; (c) Relaxation of non-local energies; (d) Hierarchical systems of first order structured deformations. (e) Homogenization in the context of structured deformations. (5) Variational settings for second order structured deformations and . (6) Outlook for future research. M. Amar, J. Matias, M. Morandotti, and E. Zappale: Periodic homogenization in the context of structured deformations. ZAMP, 73, 173 (2022) G. Bouchitté, I. Fonseca, and L. Mascarenhas: A global method for relaxation. Arch. Rational Mech. Anal., 145 (1998), 51–98. A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072. A. C. Barroso, J. Matias, M. Morandotti, and D. R. Owen: Explicit formulas for relaxed energy densities arising from structured deformations. Math. Mech. Complex Syst., 5(2) (2017), A. C. Barroso, J. Matias, M. Morandotti, D. R. Owen and E. Zappale The variational modeling of hierarchical structured deformations. Submitted to J. Elasticity (2022). R. Choksi, G. Del Piero, I. Fonseca, and D. R. Owen: Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999), 321–356. R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103. G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimension reduction in the context of structured deformations. J. Elast. 133 Issue 1 (2018), 1–35. G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155. I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian. J. Nonlinear Sci., 29(6) (2019), 2699–2734. L. Deseri and D. R. Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J. Elasticity, 135 (2019), 149–182. J. Matias, M. Morandotti and D. R. Owen Energetic relaxation of structured deformations. A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics. Book to be published by SpringerBriefs on PDEs and Data Science. J. Matias, M. Morandotti, D. R. Owen, and E. Zappale: Upscaling and spatial localization of non-local energies with applications to crystal plasticity, Math. Mech. Solids, 26 (2021), 963–997. J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math. Optim. 69 (2014), 459–485. J. Matias, M. Morandotti, and E. Zappale: Optimal design of fractured media with prescribed macroscopic strain. Journal of Mathematical Analysis and Applications 449 (2017), 1094–1132. M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Com- plex Syst., 5(2) (2017), 191–215. D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235. Title: Convexity notions arising in the supremal setting Instructor: Elvira Zappale Duration: 10/12 hours 31/3 ore 9-10 AULA 5 (Palazzina RM0018) 14/4, 21/4, 28/4, 5/5, 12/5, AULA 5 (Palazzina RM0018) The course will consist of 5 or 6 lectures where I will first introduce supremal functionals, basic concepts of direct methods of Calculus of variations, with a particular emphasis on relaxation and then I will present several notions necessary and/or sufficient for the lower semicontinuity of supremal functionals and related results ensuring a power law approximation using integral energies. Also I will compare the classical notios of convexity used in the integral setting with their counterparts in the supremal context and other possible notions. I will conclude with a quick overview of the nonlocal setting. List of useful courses from the Master degree Title: Metodi Numerici per l'Ingegneria Biomedica (in Italian) Instructor: Francesca Pitolli (Sapienza -SBAI) Duration : 20 hours (first part) + 40 (second part) hours Planned period di erogazione: September-December 2022 Schedule: martedì h. 9:15-10:45 (Aula 25, via Eudossiana); giovedì h. 15:00-18:30 (Aula 15, via Eudossiana) Prima parte: Metodi numerici per la soluzione di problemi differenziali, metodi di Runge-Kutta, metodi alle differenze finite (2CFU) Seconda parte: Approssimazione ai minimi quadrati per l'identificazione di un modello e la stima dei parametri. Soluzione di sistemi lineari sovradeterminati. Decomposizione ai valori singolari e sue applicazioni. Problemi inversi mal posti e tecniche di regolarizzazione. Soluzione di sistemi lineari sottodeterminati. Analisi delle componenti principali e sue applicazioni (4CFU) Per ogni argomento verranno svolte delle esercitazioni in cui si utilizzeranno i metodi numerici illustrati a lezione per risolvere alcuni problemi applicativi.

Topics: A demonstrative computer session using ADINA - nonlinear analysis Instructor: Klaus-Jürgen Bathe Study Guide (PDF) Bathe, K. J. “Finite Elements in CAD and ADINA.” Nuclear Engineering and Design 98 (December 1986): 57-67. User Manuals, Verification Manual, and Theory and Modeling Guide for ADINA, see www.adina.com Bathe, K. J., and A. Chaudhary. “A Solution Method for Planar and Axisymmetric Contact Problems.” International Journal for Numerical Methods in Engineering 21 (January 1985): 65-88. The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Ladies and gentlemen, welcome to this lecture on nonlinear finite element analysis of solids and structures. In this lecture, I would like to continue to consider the plate with a hole that we already considered in the previous lecture. But I now would like to turn our attention to a nonlinear solution. As we mentioned in previous lectures, a nonlinear analysis should only be performed only once a linear solution has been obtained. The linear solution checks the finite element model and yields valuable insight into what nonlinearities are important. And once again, we considered the linear solution of this plate in the previous lecture. We now want to go on with the nonlinear solution. Here we have the plate with the hole once again. It's a square plate subjected to the loading shown. Here is the whole. These were the material data that we used in the linear analysis. The thickness of the plate is given here. We consider only one quarter of the plate because of symmetry conditions, we can considering that one quarter, analyze the whole plate, as we have discussed in the previous lecture. In the previous lecture, we also showed how we use ADINA-IN to generate the data for this mesh. And this input data then is used in ADINA to actually perform the analysis. We did the analysis for linear conditions in the previous lecture. Some important considerations for the nonlinear analysis are now what material model to select, what displacement strain assumptions to make, what sequence of load application to choose, and what nonlinear equation solution strategy and convergence criteria to select. We will address these issues in this lecture. We use, once again, the ADINA system. Now, of course, for the elasto-plastic static response. We will also investigate the effect on the response when a shaft is placed into the plate hole. Some important observations regarding the nonlinear analysis are given on this viewgraph. First of all, we notice that the recommendations that we discussed regarding the linear analysis that we discussed in the previous lecture are also very valid here of course. But for the nonlinear analysis, we need also to consider and be careful with the sequence and incremental magnitudes of load application and the choice of convergence tolerances. We ill address, of course, these issues just now. The first analysis that I'd like to discuss with you is a limit load calculation of the plate. Here we show the plate and the load, p, will increase continuously up to a maximum value and then decrease to 0. The plate is modelled as an elasto-plastic material. And the material assumption is summarized on this viewgraph here. Here are the material properties. We assume basically it is a steel. Notice the stress strain law is shown here. We assume isotropic hardening in the analysis. We discussed what this means in an earlier lecture. The initial Young's modulus is the one that we use for the linear analysis. Nu is equal to 0.3. And the strain hardening modulus is given here. This idealization is probably only applicable to small strain conditions, strains that are smaller than 2% roughly. The maximum that you would want to allow is probably 4%. And we will actually perform the analysis first using a materially nonlinear only formulation. This means, as we discussed in the early lectures, that we neglect all kinematic nonlinearities, that we only include the material nonlinearities in the analysis, these material nonlinearities. Later on, we however want to also perform in this lecture an analysis that includes the displacement large quotations and in fact even large strain conditions. And we will study those analysis results in comparison to the material nonlinear only analysis results. The load history used for the analysis is shown on this viewgraph. You can see that we are increasingly linearly the load up to a maximum value of 650 MPa, megapascals, and then suddenly decrease the load to 0. Notice we are using altogether 14 steps, 13 to increase the load and just one step to decrease the load. Of course, we are having here a time axis. But the load step or time step that we used was delta t equal to 1. Well we performed this analysis a few weeks ago in my laboratory at MIT. And we brought in a video crew to video record our actions. I'd like to now share with you what we have recorded and also narrate to you what actually is happening in the computer run as we prepare the computer run, as we run it, and also interprets the resides. Our first step is to modify the input data that we prepared in the previous lecture for the linear analysis of the plate. We now have to, of course, modify this input data. First of all, to introduce the load curve, this load history curve that we just discussed. And then we also have to modify the material data to correspond to the elasto-plastic material data that we now want to associate with a plate. So let us look now at the video record of what we did some weeks ago in my laboratory at MIT regarding the change of these input data. Here we see once more the mesh that we're using to analyze the plate. And now we prepare input data for ADINA-IN. Here we input the time function that we employ. You recognize the function points for time 0, 13, and 14 with the values 0, 6.5, 0. We also input that 14 steps are used in the analysis, that time step delta t is 1. Next we input the material definition. And note that there is a typographical error. We typed plestic instead of plastic. We try to do the typing fast and did not notice the error. Notice the Young's module E, Poisson's ratio nu, the strain hardening modulus ET, and the yield stress [? sigma ?] yield are defined. Because of the typographical error, the program prints out an error message, namely the plestic material is not found in the library. Here we see the library of material models available in ADINA. The library consists of the material models elastic, orthotropic, thermo-elastic, and so on and so on. We know retype the material data definition. We also do not want to use equilibrium iterations. The default in ADINA is to use equilibrium iterations. Actually, the BFGS method we discussed earlier because large errors of solution can accumulate when iterations are not used. We discussed all of this earlier. Let's see what happens when we do not iterate in the solution of this problem, just as a point of study. We now finally by the command ADINA generate the ADINA data input. Note that in this input preparation we only change the data from the linear analysis data used in the previous lecture to the data for the nonlinear analysis we want to perform now. Having set up the proper input data for ADINA-IN and having used ADINA-IN to generate the input data for ADINA, we can now execute that input data with ADINA obtain our first analysis results. We evaluate these analysis results by plotting the force applied here as a function of the displacements corresponding to this motion as well. And we will see that the force displacement curve looks rather unphysical. In other words, the results don't make much physical sense. So we search for an explanation. Why is that so? And we will find that the reason is that we did not use equilibrium iterations in the analysis. In fact, if you go back, you will see that we deliberately did not want to use equilibrium iterations in this first analysis, although in ADINA the default is to use equipment directions. But I wanted to once show you what kind of results you must expect if you don't use equilibrium iterations. So we realized that we should really use equilibrium iterations and that we will, of course, have to change our input a little bit through ADINA-IN in order to perform equilibrium iterations. The load history that we will still be using is the same. Here it is shown once again. But now we will use the default method of equilibrium iterations, the BFGS method. And this one will be applied, this method of equilibrium iteration will be used for each load step. Once again, 13 load steps up and one load step down. The convergence criteria that we using in the analysis are show here, the convergence criteria on energy. And we talked about this one quite extensively in a previous lecture. And the convergence criteria on the force, again, I'd like to refer you to our previous lecture. When we apply equilibrium iterations in each step, we will see that our results look good. They make physical sense. In fact, they look quite appealing. So let us look now at these solution results. Let us go, in other words, to what has happened in the laboratory the way we have been video recording it earlier. Here we see the solution results, the load applied as a function of the displacement, the extension of the quarter plate. On the horizontal axis, we measure the displacement. On the vertical axis, we measures of value of the load, actually the pressure applied. So far the curve looks OK. But we show here only the responsible for the first 13 steps for which the load has increased monotonically. here we see now the load displacement response for all 14 steps. Notice that the scale on the horizontal axis measuring displacement is different from what we used before. Note that the 13 first steps bring us to the maximum load and maximum positive displacement and that the predicted unloading response in step 14 is quite unrealistic. We obtain a large negative displacement. As we will see, this is due to not having used equilibrium iterations. Next, we look at the mesh and study the plastic zones as they develop with increasing load. A time code is given about the mesh. This time code gives the step number. It increases until time is equal to 14. The plastic zones are shown by shading the area that is plastic. For the first steps, there's no plasticity. Then the plastic zone is small, it develops around the hole and it grows rapidly as the larger load levels are reached. Note also how the plastic zone spreads through the elements. We used 3 by 3 Gauss numerical integration and test whether an integration point has gone plastic. If so, we shade the contributory area of the integration point. As an average for each integration point, the contributory area is 1/9 ninth of the element area. Here at step 12, you can very nicely see how the plastic zones have progressed through the elements. The elastic plastic interface boundary goes through the elements. Note that at step 14, after unloading much off the plate is still plastic. This is quite unphysical. We now rerun the analysis with equilibrium iterations and here is the load displacement response we now obtain. First we look at the scales on the axes. We note that the loading response is similar to what we had before. Although a much larger displacement is reached in steps 13. The unloading response in step 14 is now quite realistic, with a permanent positive displacement at zero load. Finally, we show the plastic zones for this analysis. Note that we show the initial mesh time 0 and then immediately the time equals 6 results. Initially, the plasticity progresses as much as in the analysis without equilibrium iterations, but the final spread of plasticity reached in step 13 is larger. And after unloading of the load at the end of the analysis, the complete plate is elastic. However, clearly permanent deformations have occurred as can be seen by looking at the default mesh at time 14. As we discussed already, our analysis results now look quite good. They look quite reasonable. But one additional way to evaluate the analysis results is to plot stress vectors. We did so in the linear analysis of the previous lecture when we also looked at the analysis results obtained from this mesh, but of course, in linear analysis. We want to do now the same for the nonlinear analysis results that we obtained. And let us just quickly look at what we're doing in the stress vector output. We plot at each integration point two lines as shown here. If they are carrying an arrow, then it is tensile stress. No arrow means compressive stress. And notice these two lines correspond to the principal stresses. Notice that the lengths of these lines are proportional to the magnitudes of the stresses. So let's now do a stress vector plot for the mesh at time 13 and at time 14, in other words, at maximum load application and after removal of the total load, four the results that we just have obtained. Here we see the stress vectors plotted onto the total mesh for the stress state at time 13, that is at total load. We note of course, that there is very much information, there are many stress vectors. To see any detail we have to focus our attention onto certain elements. Here we now look closer at the elements adjacent to the horizontal symmetry line. We note that the stress vectors correspond to vertical tensile stresses as expected. At a top edge of the plate we see tensile vertical stresses and tensile horizontal stresses. The maximum stress at any integration point is 1100 megapascals and occurs near the hole. Here we now see the stress vector plot at time 14, that is after load removal. It is most interesting to study the stress flow in the mesh. Note that the stresses flow along and parallel to the free surface of the plate. This must be so because there are no externally applied tractions anymore. Here we see the detail of the stress flow in the corner of the plate. The stress vectors are parallel to the free surface. And here is the stress flow in the elements around the hole. The same observations apply. The maximum stress is 880.3 megapascals. This completes what I wanted to show you in this phase of the analysis. This completes our materially nonlinear only analysis of the plate. However, if we look at the solution results once more closely, we find that in this element here the magnitude of the strains is about 2% at the end of load step 11, 4% at the end of load step 12, and 14% to 15% at the end of load step 13. In other words, at maximum load application, we have certainly here large strains. And one might very well ask, what is the effect of this large strain on the analysis results? Of course, in the materially nonlinear only solution, we did not include any kinematic nonlinearities. So our next objective is then to perform analysis that said include kinematic nonlinearities. And we want to now proceed with a total Lagrangian formulation analysis, which includes large displacements, large rotations, but only small strains. And I also want to share with you some solution results that we obtained using an updated Lagrangian formulation. We did not talk about this formulation in the earlier lectures. We did talk about this formulation, but not about that formulation. This formulation really is best covered in a separate lecture. However, it's still very interesting to look at the solution results that we obtain we this formulation. If you want to read up on this formulation, please refer to the study guide in which a reference given. A paper is referred to in which this formulation is described. So let us now look at the solution results obtained from these three formulations. And the solution results that we want to look at are once again the force displacement curve for each of these formulations. In other words, force applies here, displacement seen here, for these three formulations. Let's turn back to the laboratory and see what are the results. Here we see the analysis results for the MNO, that is the materially nonlinear only, the TL, that is the total Lagrangian, and to UL, that is the updated Lagrangian formulations. We look first at the horizontal axis scale giving the displacement. And then at the vertical axis scale giving the load that is the pressure applied. Notice that there are three, in fact we will we later four distinct curves. However, under this much plasticity, the curves are almost the same. The TL analysis results corresponds to the smallest displacements. The MNO solution gives larger displacements. And the UL solution gives the largest displacements. For the UL solution, we actually see two curves when looking closely. These correspond to using once 14 steps as for the TL and MNO analyses and then using once twice the number of load steps. Note that the unloading response in all solutions is quite the same. However, of course, the permanent displacements at 0 applied load are quite different because the maximum displacements corresponding to peak load were different. These analysis results underline the importance of choosing the appropriate kinematic formation for the analysis. Here, large strain effects are quite significant at the very high load levels. In the next analysis we now consider the effect of a shaft in the hole. Notice, we look at the same plate as before except that we know first consider elastic condition only. And the shaft is shown here. The shaft has this Young's modulus and Poisson ratio, same as the plate. But it is 5 times thicker than the plate and for the shaft we also consider plane strength conditions. What we want to do is place a shaft in there, the shaft being initially flush with the hole, assuming no friction between the shaft and the hole. And then we pull on the plate and want to investigate what is the effect of having that shaft there. The analysis input data have to now be modified because we have to put the shaft in there using finite elements as shown here in red. We used collapsed 8-node elements to represent the shaft. In other words, these collapsed 8-node isoparametric elements become, of course, triangular elements as shown here. Notice we now have a contact surface here. And that contact surface is modeled using a contact algorithm, which we did not talk about in this series of lectures. This again, would be best covered in another lecture. I'd like to refer you here to another paper, the reference of which is also given in the study guide, if you're interested in reading about the contact algorithm. The contact algorithm can take into account friction conditions as well, but in this particular analysis, we assume 0 friction along the contact surface. The solution procedure that we are using is it the full Newton method without line searches. And the convergence criteria that we are using are listed here. These we have been talking about earlier already. Here we now because of the contact conditions have to introduce also this convergence criterion, which is really a convergence criterion on the incremental contact force. So let us now proceed with this analysis. And once again, we perform the analysis and of course, we're looking at solution results. Here we see the mesh of the plate once more, the mesh we used in the previous analyses. We now need to change the input data for the analysis to also define the shaft. This is done using ADINA-IN. We need to define the addition nodal points and elements for the shaft in the same way as we input earlier the nodal points and elements of the plate. Let's look at the information that defines the content condition between the plate and the shaft. And here we see the input for ADINA-IN. There are two contact surfaces. The one is the plate hole surface and the other is the shaft surface. We denote these two to be a contact surface pair. Here you now see the mesh of the shaft and the plate. The shaft is defined by triangular elements. Here we see the default mesh at maximum load. Note that the plate has been extended vertically and has shrunk horizontally. The shaft has prevented the hole to shrink much horizontally. And on top off the shaft, a gap has opened. All of these deformations are quite realistic. Here, we see just the shaft and the ring of elements of the plate around it. Once again, the calculated deformations make sense. If you look closely at the shaft by itself, you observe that it has been compressed horizontally by the plate. Let's look next at some stress vector plots. These also show that a physically realistic solution has been obtained. Here we see the stress vectors in the element layer of the plate around the shaft. Note that the stress vectors are plotted onto the original mesh. For the element adjacent to the horizontal symmetry axis off the plate, we see a vertical tensile stress and a horizontal compressive stress. Such stresses are to be expected. The horizontal compressive stress is, of course, due to the contact with the shaft. Note that going around the shaft, the stresses in the plate align to be parallel to the free surface of the hole, since there is only contact near the horizontal symmetry axis of the plate. This completes what I wanted to discuss with you for this phase of the analysis. Finally, I would like to look with you at the analysis results we obtained when we apply to this plate with the shaft 100 MPa up there and down here. We assume that they plate is made of an elasto-plastic material. In fact, we model that material as shown in our early analysis. And in addition to this loading here shown, also the shaft expands. It expands uniformly. And in fact, it expands 0.05% based on the initial dimensions of the shaft during each load step and we apply 10 load steps. So the loading then all together is in the first load step, 100 MPa applied here. And from the second to 11th load step, we expand the shaft by 0.05% in diameter, so to say, based on the initial dimensions. We use the updated Lagrangian formulation to model the response of the plate. Let's look now at these analysis results. Here we see just as a reminder once more the mesh we are using. Also, here is once more the detail of the mesh around the shaft. This is the default mesh at step 1. The deformations are due to the tensile load of the 100 megapascals on the plate. Next we plot the plastic zones in the plate as they develop when the shaft expands. Notice again the time code above the mesh giving the step number since delta t equals 1. There are all together 11 steps. The shaft expands from step 2 step to step 11. We see the time code running and at time 7 we see the first plasticity. This plasticity spreads as the shaft further expands. The maximum plastic zone is, of course, reached at the maximum expansion of the shaft, that is at time 11. This completes what I wanted to say about this analysis. Note that after each analysis step we looked at the calculated deformations and stresses is to identify whether these make sense. This brings us to the end of this lecture and to the end of this course. I'd like to now just take a few minutes for some closing remarks regarding the course. I mentioned already in the first lecture that nonlinear finite element analysis is a very large field. There are continuum mechanics principles, numerical algorithms, and software considerations. We could not cover in detail many aspects of all of these fields in these 22 lectures. However, I do believe that the 22 lectures provide a good introduction and a good foundation for further study. I would hope that you would listen to these lectures with your colleagues, that these lectures would initiate discussions, stimulations for your work in nonlinear analysis and, of course, also questions. We at MIT continue to work in nonlinear finite element analysis. And we also offer from time to time weekly courses. I would be glad to see some off you at these weekly courses to share some of the experiences that you have had listening to these video lectures and also regarding your work in practice. Finally, I'd like to mention that a video course of this nature can only be produced through the concerted effort of a number of very devoted people. I'd like to thank for their collaboration and support Dick Norris, Elizabeth DeRienzo, Pat [? Regan ?] of the Center of Advanced Engineering Study at MIT, and Ted Sussman, my student. And very finally, thanks also for the crew around here. Thank you for your attention.

Update: see this for a clarification from Ed Frenkel regarding this post! The LA Times recently published an article titled “How our 1,000-year-old math curriculum cheats America’s kids” by mathematician Ed Frenkel (1). The article is about changing the focus of early mathematics education to be more in line with what mathematics actually is and what mathematicians actually do rather than on the humdrum version of mathematics as memorization we all learned and or were terrified by when growing up. Frenkel colorfully, and accurately in my view, compares the latter to learning to paint a wall in an art class, without ever seeing the works of great masters. He believes that this way of teaching math isn’t well suited to teaching the skill of abstraction, —bringing order to confusion by recognizing or establishing relations between things previously thought to be unrelated, that is important for the success of future generations. For Frenkel a broader view of mathematics is what’s needed for gifting this proficiency to students through “mathematical knowledge plus conceptual thinking times logical reasoning”. Part of the problem Frenkel sees is that “most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old” and focuses on numbers and solving equations instead of on concepts and ideas. I take no issue with any of these claims and support the call to reform mathematics education by creatively introducing abstraction early on as an alternative to memorization for the sake of standardized tests. I’m writing this piece because of something philosophical that Frenkel says which Gives opportunity to discuss relevant philosophy of mathematics where I believe we may run the risk of tying new ways of teaching mathematics to future generations to highly speculative philosophy inspired by weighty, 2,000 year old metaphysics. I’m thinking of this claim: “We also need to convey to students that mathematical truths are objective, persistent and timeless. They are not subject to changing authority, fads or fashion. A mathematical statement is either true or false”. The important philosophical ideas to discuss are: - The claim about the objectivity, persistence and timelessness of mathematical truths. - The characterization of mathematical truths as not subject to changing authority, fads, or fashion. - The idea that the truth or falsity of mathematical statements stems from everyone agreeing to their truth or falsity —and how this squares away with ideas 1 and 2. Focusing on 1, it’s generally accepted that most mathematicians behave as if mathematical truths and theorems are objective in some sense such that it makes sense to speak of their timelessness and persistence. The philosophical view commonly associated with actually believing these things this is a form of realism (the belief that the things, the objects, that mathematicians talk about are, in one sense or another, real things) called ‘platonism’. Platonism in this sense maintains that mathematical objects like numbers and sets exist independently of the human mind and are abstract, meaning that they don’t exist in time and space and are not causally related to things that do. To be clear, this philosophy of mathematical platonism isn’t 2,000 years old, but it is inspired by that ancient philosophy of Plato’s about eternal, abstract, and immutable ideas (2). I think the modern term, ‘platonism’, as used in the philosophy of mathematics, was first used by the mathematician Paul Bernays who in the 1930’s used it to describe the tendency among mathematicians to treat mathematical things like sets and numbers like the abstract ideas of Plato (3). But long ago, Plato himself did rebuke mathematicians whose practice was not, in his view, sufficiently directed toward the eternal, abstract, and immutable ideas (4). My diagnostic comments below about mathematical platonism don’t pertain to any particular ideas of the historical Plato, but they’re hardly sympathetic to abstract metaphysics, something common to both mathematical platonism and Plato’s philosophy. To me it seems strange to actually believe that through mathematical activity human beings somehow gain access to this disconnected realm of causally inert, immaterial objects existing beyond time and space. Mathematics, even in all it’s abstractness, seems to me to be a very natural and mundane human activity, like art, writing, or storytelling. I find it striking that through such earthly behavior an entire wholly alien and metaphysically disconnected universe is reached in such a way that in virtue of it, our mathematical theorems are true. I don’t mean to say that ordinary or basic human activities or things that are commonplace do not or cannot depend on or reveal complexity —but more that it seems out of place to bring in these in-principle disconnected and dubiously knowable things to account for common worldly behavior. There’s a much discussed argument in defense of this type of realism. It’s the indispensability argument for ontological commitment. Ontological commitment is the idea that under a common understanding of how language works, sentences are committed to the existence of the things falling under the scope of the quantifiers if those sentences are to be true. The argument is simple: we must be ontologically committed to the things that are indispensable for our best scientific theories. Mathematical things like numbers and sets are indispensable for our best scientific theories. So we must be ontologically committed to mathematical objects. It’s undoubtedly true that mathematics is part of the language of science but it still doesn’t follow that mathematical entities themselves are indispensable for science. Programs like the anti-realist fictionalism of Field (5) and the realist structuralism of Hellman (6) show, at least, that platonic objects aren’t indispensable to science and mathematics. I don’t think that in order to banish platonist philosophy in mathematics one should endorse that science somehow be awkwardly carried out without mathematics and that nominalists should drop everything and obtusely call for the translation of science into nominalist formalism instead of just normal mathematics —but the reality of nominalization undermines the indispensability claim on mathematical entities in science. And the general claim about having to be ontologically committed to the existence of everything falling under the scope of the quantifiers we use when using mathematics in science seems to me also dubious. Maddy has non-trivially cast doubt on the type of support offered for the ontological commitment claim by showing that scientific theories aren’t treated as uniform by working scientists in the way that people who appeal to “our best scientific theories” would like to think (7). The point is that it’s a mixed bag when it comes to accepting the existence of mathematical and non-mathematical entities among scientists working on our bests theories and that scientists use mathematics to get results, without treating its applicability as a confirmation of its truth. So, working science doesn’t tell much about the truth of the mathematics, and much less so about the existence of mathematical entities. Thinking about it like this, it seems extravagant to insist on squeezing out the abstract metaphysics of platonism from the fairly worldly realm of science. Still, if we’re just talking about mathematics and not so much about mathematics as used in science, then we still have the problem of mathematicians somehow having special access to the abstract, spatiotemporally disconnected entities in virtue of which, according to Platonism, mathematics is objective, persistent and timeless. Field famously streamlined an argument of Benacerraf addressing just this very point, concluding that if mathematical Platonism is true, then the reliability of mathematicians ascertaining mathematical truth can’t be explained (8). The reason it can’t be explained is that platonic entities are in principle spatiotemporally separate from the universe inhabited by mere mortals —apart from giving a mysterious account where mathematical knowledge is in a sense un-caused, or divined, it doesn’t look like there’s even an in-principle way of explaining the reliability of mathematicians’ ascertaining mathematical truth. In any case, there are other more grounded ways to account for desirable qualities like, knowability, objectivity, persistence and timelessness of mathematical truth. For example, structuralism, the view that mathematics is about mathematical structures where the objects of mathematics are completely determined by their place in the structure, accounts for the knowledge we have of structural relations based on proofs from assumptions that provide for the types of structures under study. Further, such knowledge is objective because realist structuralism like Hellman’s maintains the determinateness of truth values that’s is so central to mathematics (9). The relative timelessness and persistence of mathematical truths under such structuralism will just be a function of the timelessness of the logical possibility of the structures to be investigated —something that, along with mathematical possibility, is for the realist modal structuralist an irreducible primitive. A pretty good measure of timelessness and persistence, I think (10). Frenkel takes no position on these issues, but I bring them up because I think it would be disappointing and run counter to teaching the skill of abstraction in a way so as to avoid, in Frenkel’s words, “misconceptions and prejudice” about mathematics if in answering to the objectivity, persistence and timelessness of mathematics we were to rely on deeply speculative philosophy. Frenkel relates how he used a Rubik’s cube to explain symmetry groups to fourth, fifth and sixth grade students. He also introduced them to “curved shapes (such as Riemann surfaces) and the three-dimensional sphere that give us glimpses into the fabric of our universe.” Relying on mathematical platonism as a philosophy runs counter to this because it communicates that ultimately even real mathematics that has to do with the real world, that kids are ready for and that can be practiced by people like you and me, is nevertheless mysterious, requiring special, unspecified access to a disconnected realm of causally inert, immaterial objects existing beyond time and space. And it makes little sense to do so, as there are operable, sophisticated and serious nominalist alternatives to platonism. Platonism about mathematics is often also the go-to philosophy for justifying the idea that mathematical truths are not subject to changing authority, fads, or fashion, which is discussion point 2 above. It makes sense because mind-independent, spatiotemporally and causally disconnected things can’t really be affected by anything mere mortals can do, and aren’t subject to human authority, or passing fancy. This is the idea that I’ll discuss in the next update. 2. For more about what the historical Plato had to say about these things see: (Kraut, 2013). 3. Bernays, 1935, p. 259. 4. Plato says in Republic, Book VII, 527b: “They [geometers] give ridiculous accounts of it [geometry], though they can’t help it, for they speak like practical men, and all their accounts refer to doing things. They talk of ‘squaring’, ‘applying’, ‘adding’, and the like, whereas the entire subject is pursued for the sake of knowledge…for the sake of knowing what always is” (Plato, 1997, p. 1143). 5. Field, 1980. 6. Hellman, 1989. 7. Maddy, 1992. 8. Field, 1989, p. 68. 9. Hellman, 1989, p. 44. 10. One may wonder at this point: Which logic? One of the virtues of modal structuralism is that because logical and mathematical possibility is irreducible, one is free to examine all kinds of structures, including constructive and paraconsistent ones. Benacerraf, P. And H. Putnam, 1983, “Philosophy of Mathematics”, second edition, Cambridge: Cambridge University Press. Bernays, P., 1935, “Platonism in mathematics,” in Benacerraf and Putnam, 1983, pp.258-271. Cooper, John M. (ed.), 1997, Plato: Complete Works, Indianapolis: Hackett. Field, H., 1980, “Science Without Numbers: A Defence of Nominalism”, Oxford: Blackwell. —1989, “Realism, Mathematics, and Modality”, Oxford: Blackwell. Hellman, G., 1989, “Mathematics without Numbers”, Oxford: Clarendon. Kraut, R., 2013 “Plato”, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2013/entries/plato/>. Maddy, P., 1992, “Indispensability and Practice”, Journal of Philosophy, 89(6): 275–289. Plato, 1997, “Republic”, in Cooper, John M. (ed.), 1997.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? Kyle and his teacher disagree about his test score - who is right? Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number... ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD. In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base. Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus. Some diagrammatic 'proofs' of algebraic identities and inequalities. A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself. Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus. Prove Pythagoras' Theorem using enlargements and scale factors. Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. Prove that the shaded area of the semicircle is equal to the area of the inner circle. Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. Can you discover whether this is a fair game? Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results. Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power. The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . . Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . . It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry This is the second article on right-angled triangles whose edge lengths are whole numbers. This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition. Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. An article which gives an account of some properties of magic squares. Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another? In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot. Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle. Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product? Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n. Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? Can you make sense of these three proofs of Pythagoras' Theorem? This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning. Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem? Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence. L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way? Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Miles Per Hour to Kilometers Per Hour Conversion Calculator. Convert MPH to KPH with this online calculator.Example: When the speed is 10 MPH: Kilometers per Hour 16.0934 (to the nearest 10,000th). 100 Promotion. 100km/h60mph This is equivalent to 3/5, or 0.6. Since 60 is the common speed limit in mph and is almost the same as the 1.6 ratio (its only 4 difference), it works to find an accurate number easily. So using this, 250 mph to km/h you multiply by 1.6 Calculator/Converter practical for knowing distance in km and/or miles and passing it into a walking time. Comment from/about : miles/ km to walking time | Permalink. Convert 100 meters in time speed unit into miles per hour mph, for manual speed conversion tables. 100 km mph convert is the worlds number one global design destination, championing the best in architecture, interiors, fashion, art and contemporary. appPicker Top 100.The quickest and easiest way to convert between miles per hour and kilometers per hour. Features Include: - Select your preferred number of decimal places. This online calculator-converter provides conversion of kilometers to miles (km to mi) and backwards miles to kilometers.Top ways people ask this question: what is 100 kmh in mph (30) convert 100 km/h to mph (14) 100 kmh to mph (14). Kilometers per hour unit symbol is kph, km/h, kmh.100 mph. 160. 93 kph. Miles per hour. CC to Ci Converter. 1/4 Mile Calculator.Why not try our 0-60mph, 0-62mph or 0-100kph lists or just browse through our specifications where all theConvert Miles Per Hour mph to Kilometres Per Hour kph (km/h). Calculated as 1 mph 1.60934400061 kph (km/h) (5 decimal places). Converter. You are currently converting speed units from miles per hour to second per 100 meters. 1 mph 223.69362920544 sec/hm.minute per kilometer (min/km). Instantly Convert Miles Per Hour (mph) to Kilometres Per Hour (km/h) and Many More Speed Or Velocity Conversions Online.1 Kilometer per hour (kph, km/h) 0.277 777 778 meters per second (SI base unit). remove the playlist. Converting Mph To Km H. (mph To Kph).If you know how to convert kilometers per hour to miles per hour, you would know that 100 km/h is approximately 60 mph. Use this accurate speed converter to help with any of your calculations, especially for mph to km/h.100 Miles per hour equal 160.93 Kilometres per hour (100mph 160.93km /h). 100 Mph In Kph 160.93 Kph. Convert Kph Into Mph - Easy Speed Converter.To get started simply enter the speed of Kilometers Per Hour or Miles Per Hour into the correct box above and it will instantly convert it for you into the other box. km/h Kilometres per hour to Miles per hour mph.1000000 Miles per hour 447.04 Kilometres per second. Embed this unit converter in your page or blog, by copying the following HTML code Enter miles/h or km/h for conversion: Select a conversion typeThe Online Conversion Calculator - Converter converts Miles per hour to km per hour (mph to km/h) and kmh to mi/h (kilometers/hour to mph. Convert 100 Miles/Hour to Kilometers/Hour (mph to km/h) with our conversion calculator and conversion tables.100 mph 160.90712742981 km/h. You also can convert 100 Miles/Hour to other Speed (popular) units. Convert mile/hour to kilometre/hour. MPH. km/h. More information from the unit converter.Examples include mm, inch, 100 kg, US fluid ounce, 63", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more! Miles Per Hour to Light Speed (mph to ls).Kilometers Per Second to Meters Per Second (km/s to m/s). This converter provides online conversion of miles per hour to km per hour (mph to km/h) andMiles per hour is the unit used for speed limits on roads in the United Kingdom, United States and various other nations, where it is commonly abbreviated in everyday use to mph or MPH, although converting km per hour to m per sec - Продолжительность: 2:33 solvedphysicsproblem 324 515 просмотров.3 Minute Math - Converting Speed kmh to mph - Продолжительность: 3:40 Simon Deacon 3 706 просмотров. 100 mile/hour (mph) in kilometer/hour (kmh) 160.9344 mph.Converterin is a good-looking unit online metric and measurement converter. Our goal is to get you to convert any unit as easy and fast as possible, without any hassle, and providing the most accurate and up to date information. Convert 100.3 Miles/Hour to Kilometers/Hour (mph to km/h) with our conversion calculator and conversion tables.100.3 mph 161.3898488121 km/h. You also can convert 100.3 Miles/Hour to other speed units. You can easily convert meters per second to kilometers per hour (mp/s to km/h), miles per hour to kilometers per hour (mph to km/h), feet perTo see description put the mouse cursor over left side of this window . Click left mouse button to input choosen value into converter and start conversion. Convert Kph to Mph - unit converters. . Instant online speed units of kilometer/hour to mile/hour conversion.100 mph to km. kph conversion. kilometers per hour. So, 100km 60 miles. Or 200 miles 330 km. If Ive made a correct assumption of what the question really means, then the answer is 260 x 6 156 mph.How do you convert km per hour to mph? This calculator-converter provides online conversion of miles per hour to km/hour (mph to km/h) and conversion km/h to mi/h (kilometers/hour to miles/hour ).Convert Kph to Mph - Unit Converter. Mph - km/h Converter. Enter the speed in either of the next two fields and get it converted instantly!How to convert miles per hour to kilometers per hour. Multiply speed in mph with 1.609344.100. 62.14. Speed conversions between miles per hour (mph, miph, m/h, mi/h) and kilometres per hour (kph, kmph, km/h) areThe conversion tables, below, offer a quick lookup to convert miles per hour to kilometres per hour for commonly sought values or you can use our quick converter utility above. Search for 128 Mph To Km H. Look Up Quick Answers Now! autos24-7.com has been visited by 100K users in the past month.windows snap converter free download Allow you to convert between Km/H and MPH on your Windows 8 device. Mph To Km Converter. By On February 2, 2018 No view.The SI derived unit for speed is the meter second. meter second is equal to . mph, or . km hMile per hour . kilometers per hour . meters per second SI base unit . mph . m s. Kilometre per Hour Distance of one kilometer or meters Easily convert kilometers per hour to miles per hour, convert kph to mph . Many other converters available for free.Convert miles per gallon (imperial) to liters per 100 kms (MPG to litre). Tableau des conversions! mph - km/h. Vitesses entre 40 et 305 km/h! Use our free online calculator to compute the speed in miles per hour ( MPH) to kilometers per hour (km/h or kph) in just seconds.You chose the Basic version of the MPH to KPH Calculator. The Basic version is non-editable, calculations are limited to 100/month, and the CalculatorPro.com link You also find mph to kmph and kmph to mph conversion table. Find out miles per gallon (US) to litres per 100 kilometres converter.Meters per second to Kilometers per hour converter. mph is the symbol for mile per hour Km/s is a common alias of the unit kilometre per second. From 100.00 to 4000.00 miles per hour, 40 entries. This Site Might Help You. RE: How do I convert KM/H to MPH?Please upload a file larger than 100x100 pixels. We are experiencing some problems, please try again. You can only upload files of type PNG, JPG, or JPEG. This is a conversion chart for seconds from 0 to 100 km/h (Car performance). To switch the unit simply find the one you want on the page and click it.Car performance. seconds from 0 to 60 mph. Easily convert litres per 100 kilometers to miles per gallon(UK), convert l/ 100km to mpg . Many other converters available for free.Miles per hour to Kilometers per hour (mph to kph) conversion calculator for Speed conversions with additional tables and formulas. Miles per hour is the imperial and United States customary unit of speed. It is shortly called as MPH or mi/h.Length Converting Factors. Yarn Count Calculator. Psi To Durometer Converter. M/s To Km/h Calculator. This converter provides online conversion of miles per hour to km per hour (mph to km/h) and km/h to mi/h (kilometers/hour to miles/hour conversion ).MPG to Liters / 100 km Fuel Efficiency. Pounds to kilograms. [ - ] About. Unit converter calculator mini-howto. Every input field works as a calculator. You can use the methods and constants (case sensitive!) described on this page.Seconds from 0 to 100 mph. The metric system measures the acceleration for 0 to 100 kilometers per hour (0 to 62 miles per hour). Some of the cars with the fastest acceleration reach it in approximately 2.3 seconds, which is less than 2.73 seconds in the 0-60 mph test for objects in free fall. I converted my wifes 2007 Honda Shadow 600 Speedo from MPH to Km/H . This is what I did. Not the cleanest editing job, sorry.By admin. 6 days ago. 100. 100 Mile/Hour (mph). 160.9344 Kilometer/Hour (km/h).Kilometer/Hour : Kilometres per hour (also spelling: kilometer per hour) is a unit of speed, defined as the number of kilometers travelled in one hour. For quick reference purposes, below are conversion tables that you can use to convert from mph to km/h, and km/h to mph. Miles per hour to Kilometers per hour Conversion Chart.1 m/m 100 cm/m. decimeters per minute. dm/m. Bookmark Page kph to mph (Swap Units).Road speed limits are given in kilometers per hour which is abbreviated as kph or km/h.Start. Increments. Increment: 1000 Increment: 100 Increment: 20 Increment: 10 Increment: 5 Increment: 2 Increment: 1 Increment: 0.1 Increment: 0.01 Increment Definition: Mile/hour Miles per hour (symbol: MPH) is a measurement of speed in the imperial and United States customary unit.100 km/h. 62.137119223733 mi/h. Convert Knots to MPH Chart - The Disaster Center.Km to miles | Kilometers to miles converter - RAPID TABLES.Convert MPG to L/100km and l/100 km to MPG Online Calculator. miles/hour to kilometers per hour) and km/h to mph (kilometers/hour to miles/hour) Online Conversion Calculator - Converter.12/12/2017 Quickly convert miles into kilometres (100 miles to km) using the online calculator for metric conversions and more. millimetre/second millimetre/microsecond millimetre/100 microsecond nautical mile/day nautical mile/hour nautical mile/minute nauticalMultiply the speed in mph by 1.609344 km per mile. For example, 55 mph is 88.51392 kph. For most practical intents, multiplying by 1.6 is accurate enough. Home » Unit Conversion Online » Convert Speed » Convert 100 km/h to mph.More information from the unit converter. Q: How many Kilometers/Hour in 1 Mile Per Hour? The answer is 1.609269 kilometers/hour.

strain types principal strains stress orthogonal system body equal elastic CHAPTER X bnp6rfeet Concurrences of two Stress or Strain Def. The concurrence of any stresses or strains of two stated types is the proportion which the work done when a body of unit volume experiences a stress of either type, while acquirirg a strain >f the other, bears to the product of the numbers measuring the stress and strain respectively. Cur. 1. In orthogonal resolution of a stress or strain, its component of any stated type is equal to its own amount multiplied by its concurrence with that type; or the stress or strain of a stated type which, along with another or others orthogonal to it, have a given stress or strain for their resultant, is equal to the amount of the given stress or strain reduced in the ratio of its concurrence with that stated type. Con 2. The concurrence of two coincident stresses or strains is unity ; or a perfect concurrence is numerically equal to unity. Con 3. The concurrence of two orthogonal stresses and strains is zero. Cur. 4. The concurrence of two directly opposite stresses or strains is -1. Cur. 5. If a, y, z, 7), C, are orthogonal components of any strain or stress r, its concurrences with the types of reference are respective] y six orthogonal types of reference, and 1', ne, a', g', V thorn of the other. Coe. 7. The most convenient oprci lieation of a type for strains or stresses, being in general a statement of the components, according to the types of reference, of a unit strain or stress of the type to be specified, liecomes a statement of its concurrences with the types of reference when these are orthogonal. Esdnudes. - (1) Tlic mutual concurrence of two simple longitudinal strains or stresses, inclined to one another at an angle II, is cost 0. Hence the components of a simple distortion (3 along two rectangular axes in its plane, and two ethers bisecting the angle between these taken as axes of component simple distortions, are respectively, if Ii be the angle between the axis of elongation in the given distortion and in the first component type. The mutual concurrence of a simple longitudinal strain and a simple distortion is if a and 13 be the angles at which the direction of the longitudinal strain is inclined to the line bisecting the angles between the axes of the distortion; It is also equal to if (ts and denote the angles at which the direction of the longitudinal strain is inclined to the axis of the distortion. The mutual concurrence of a simple longitudinal strain and of a uniform The specifying elements exhibited in Example (7) of the preceding Chapter are the concurrences of the new system of orthogonal types described in Example (3) of Chap. IX. with the ordinary system, Examples (I) and (2), Chap. IX. To transform the specification (x, y, 6 7), (') of a stress or strain with reference to one system of types into (xi,a4,a6) with reference to another system of types. Let (a„ b„ e„ e,, fit gli be the components, according to the original system, of a unit strain of the first type of the new system ; let (a2, 2, L'2, c2, ca, f2, Us) be the corresponding specification of the second type of the new system; and so on. Then we have, for the required formuhe of transformationx=aizi-da,x,-i-a3x3+a,x4-Ea,x,+a,ix6, 611% d-btzrzi- bsra +br:rt +bars +bsxs = gtxf+g2x2+g3x3+gaxs+g5x5+gaxe Erample. - The transforming equations to pass from a specification (x, In a paper on the Thermo-elastic Properties of Matter, published in the first number of the Quarterly Mathematical Journal, April 1855, and republished in the Philosophical Magazine, 1877, second half year, it was proved, from general principles in tho theory of the Transformation of Energy, that the amount of work (w) required to reduce an elastic solid, kept at a constant temperature, from one stated condition of internal strain to another depends solely on these two conditions, and not at all On the cycle of varied states through which the body may have been niacin to pass in effecting the change, provided always there has been no failure in the work required to be done upon it is _ • The stress which must be applied to its surface to keep the body in equilibrium in the state (e, y, z, 1, n, C) must therefore be such that it would do this amount of work if the body, under its action, were to acquire the arbitrary strain dx, dy, dx, dl, dn, ; that is, it must be the resultant of six stresses: - one orthogonal to the five strains dy, dz, d4, do, dC, and of such a magnitude as to do the n, C) of the strains, the amounts of the six stresses which fulfil those conditions will (Chapter XI.) be given by the equations and the types of these component stresses are determined by being orthogonal to the fives of the six strain-types, wanting the first, the second, &c., respectively. Car. If the types of reference used in expressing the strain of the body constitute an orthogonal system, the types of the component stresses will coincide with them, and each of the concurrences will be unity. Hence the equations of equilibrium of an elastic solid referred to six orthogonal types arc simply Cum-7En XIV. - Reduelion of the Potential Function, and of the Equations of Equilibrium, of an Elastic Solid to their simplest Forms. If the condition of the body from which the work denoted by to, is reckoned be that of equilibrium tinder no stress from without, and if x, y, z, C be chosen each zero for this condition, we shall have, by Alaclaurin's theorem, te=r12(2', 14 2, E, t)+II3(x, 2, E, K)+ where II, II5, &c., denote homogeneous functions of the second order, third order, &e., respectively. Hence tilto dw&c., will each be a linear function of the strain coordinates, together with functions of higher orders derived from H„ But experience shows (section 37 above) that, within the elastic limits, the stresses are very nearly if not quite proportional to the strains they are capable of producing ; and therefore H3, lee., may be neglected, and we have simply w Y, 2, 6 lb Now in general there will be twenty-one terms, with independent coefficients, in this function; but by a choice of types of reference, that-is, by a linear transformation of the independent variables, we may, in an infinite variety of ways, reduce it to the form to,.--6(Ax2+By2+ Cs7 -I- Fe+ GO+ I1K2). The equations of equilibrium then become the simplest possible form under which they can be presented. The interpretation can be expressed as follows. Prop. Au infinite number of systems of six types of strains or stresses exist in any given elastic solid such that, if a strain of any one of those types be impressed on the body, the elastic reaction is balanced by a stress orthogonal to the five others of the same system. CuarrEn XV. - On the Six Principal Strains of ass Elastic Solid. tion, we have only fifteen equations to. satisfy ; while we have thirty disposable transforming coefficients, there being five independent elements to specify a -type, and six types to be changed. Any further condition expressible by just fifteen independent equations may be satisfied, and makes the transformation determinate. Now the condition that six strains may be mutually orthogonal is expressible by just as many equations as there are different pairs of six things, that is, fifteen. The well-known algebraic theory of the linear transformation of quadratic functions shows for the case of six variables - (1) that the six coefficients in the reduced form are the roots of a "determinant" of the sixth degree necessarily real ; (2) that this multiplicity of roots leads determinately to one, and only one system of six types fulfilling the prescribed conditions, unless two or more of the roots are equal to one another, when there will be an infinite number of solutions and definite degrees of isotropy among them ; and (3) that there is no equality between any of the six roots of the determinant in general, when there are twenty-one independent coefficients in the given quadratic. Prop. Hence a single system of six mutually orthogonal types may be determined for any homogeneous elastic solid, so that its potential energy when homogeneously strained in any way is ex pressed by the sum of the products of the squares of the components of the strain, according to those types, respectively multiplied by six determinate coefficients. Def. The six strain-types thus determined are called the Six Principal Strain-types of the body. The commences of the stress-components used in interpreting the differential equation of energy with the types of the strain-coordinates in terms of which the potential of elasticity is expressed, being perfect when these constitute an orthogonal system, each of the quantities denoted above by to, b, c, f, g, h, is unity when the six principal strain-types are chosen for ;he coordinates. The equations of equilibrium of an elastic solid may therefore be expressed where x, y, z, C denote strains belonging to the six Principal Types, and P, Q, It, S, T, U the components according to the same types, of the stress required to hold the body in equilibrium when in the condition of having those strains. The amount of work that must be spent upon it per unit of its volume, to bring it to this state from an unconstrained condition, is given by the equation ,RAx2+By2+ co+pEr+ 0,12 ..1.1q2). Def. The coefficients A, B, C, F, G, Ii arc called the six Principal Elasticities of the body. The equations of equilibrium express the following propositions :- Prop. If a body be strained according to any one of its six Principal Types, the stress required to hold it so is directly concurrent with the strain. Examples. - (1) If a solid be cubically isotropic in its elastic properties, as crystals of the cubical class probably are, any portion of It will, when subject to n uniform positive or negative normal pressure all round its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensation is one of its six principal strains. Three plane distortion( with axes bisecting the angles between the edges of the cube of symmetry are clearly also principal strains, and shire the three eorrespoading principal elasticities are equal to one another, any strain whatever compounded of these three is a principal strain. Lastly, a plane distortion whose axes coincide with any two edges of the cube, being clearly a principal distortion, and the principal elasticities corresponding to the three distortions of this kind being equal to one another, any distortion compounded of them is also a principal distortion. Hence the system of orthogonal types treated of in Examples (3) Chap. IX., and (7) Chap. X., or any system in which, for (II.), (III.), and (IV.), any three orthogonal strains compounded of them are substituted, constitutes a system of six Principal Strains in a solid cubically isotropic. There are only th•ce distinct Principal Elasticities for such a body, and these are - (A) its modulus of compressibility, (B) its rigidity against diagonal distortion In any of its principal plated (three equal elasticities), and (C) Its rigidity against rectangular distortion* of a cube of symmetry (two equal elasticities). (2) In a perfectly isotopic solid, the rigidity against all distortions is equal. Hence the rigidity (B) against diagonal distortion must be equal to the rigidity (C) against rectangular distortion, In a cube; and It is easily seen that if this condition is fulfilled for one net of three rectangular planes for which a substance Is isotropic, the Isotropy must be complete. The conditions of perfect or spherical isotropy are therefore expressed In terms of the conditions referred to in the preceding example, with the farther condition B=C. A uniform condensation In all directions, and any system whatever of Ave orthogonal distortions, constitute a system of six Principal Strains in a spherically isotropic solid. Its Principal Elasticities are simply its Modulus of Compressibility and its Rigidity. Prop. Unless some of the six Principal Elasticities be equal to one another, the stress required to keep the body strained otherwise than according to one or other of six distinct types is oblique to the strain. Prop. The stress required to maintain a given amount of strain is a maximum or a maximum-minimum, or a minimum, if it is of one of the six Principal Types. Cor. if A be the greatest and II the least of the six quantities A, B, C, F, G, II, the principal type to which the first corresponds is that of a strain requiring a greater stress to maintain it than any other strain of equal amount ; and the principal type to which the last corresponds is that of a strain which is maintained by a less stress than any other strain of equal amount in the same body. The stresses corresponding to the four other principal strain-types have each the maximmn-minimum property in a determinate way. Prop. If a body be strained in the direction of which the concurrences with the principal strain-types are 1, m, n, a, th, v, and to an amount equal to r, the stress required to maintain it in this state will be equal to nr, where 12=(A2g+p,2m2+c2„2+FIXE+GE/y2+1120)1, and will be of a type of which the concii•rences with the principal types are respectively Prop. A homogeneous elastic solid, crystalline or non-crystalline, :subject to magnetic force or free front magnetic force, has neither any right-handed or left-handed, nor any dipolar, properties dependent on elastic forces simply proportional to strains. COr. The elastic forces concerned in the huniniferous vibrations of a solid or fluid medium possessing the right- or left-handed property, whether axial or rotatory, such as quartz crystal, or tartaric acid, or solution of sugar, either depend on the heterogeneousness or on the magnitude of the strains experienced. hence as they do not depend on the magnitude of the strain, they do depend on its heterogeneousness through the portion of a medium containing a wave. Con There cannot possibly be any characteristic of elastic forces simply proportional to the strains in a homogeneous body, corresponding to certain peculiarities of crystalline form which have been observed, - for instance corresponding to the plagiedral faces discovered by Sir John Herschel to indicate the optical character, whether right-handed or left-!landed, in different specimens of quartz crystal, or corresponding to the distinguishing characteristics of the crystals of the right-handed and left-handed tartaric acids obtained by M. Pasteur from racemic acid, or corresponding to the dipolar characteristics of form said to have been discovered in electric crystals.

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This is an old revision of the document! The objective of this Lab activity is to investigate the band-gap voltage reference. The principal behind the band-gap voltage reference is to sum a voltage which is proportional to absolute temperature (PTAT) with a voltage that has an equal but opposite ( complementary) negative temperature drift (CTAT) to produce a voltage which has effectively zero temperature drift. The zero gain amplifier and stabilized current source combination produces a PTAT current at its output. This current will produce a PTAT voltage when flowing through a resistor. The VBE of a BJT has a well defined negative temperature drift which when summed with a properly scaled PTAT voltage will result in a near zero drift output voltage. This combined voltage is approximately 1.2 Volts which is very nearly the band-gap voltage of silicon hence the name. In the first version shown in figure 1, A PTAT current source (Q1, Q2A,B, R2) can be used in conjunction with a PNP current mirror stage (Q3,Q4A,B) in negative feedback to build a circuit which provides an output voltage which is the sum of a PTAT voltage (R1) and the VBE of Q1 which is constant or regulated over a range of input voltages and over temperature. There is a problem with the circuit in version 1. The current available to an output load is limited by the feedback current supplied from NPN Q2 mirrored through PNPs Q3 and Q4. Any current that is diverted into an external load would reduce the current in R1 and change the scaling of the PTAT voltage with respect to the voltage of the VBE. It would be desirable to build a circuit which provides a constant or regulated output voltage over not only a range of input voltages but also output load currents. A second circuit, shown in figure 2 utilizes an emitter follower output stage to provide the current to the output. Analog Discovery Lab hardware 1 - 2.2 KΩ Resistor ( or any similar value ) 1 - 100 Ω resistor 3 - small signal NPN transistors (2N3904 or SSM2212) 3 - small signal PNP transistors (2N3906 or SSM2220) The breadboard connections are as shown in the diagram below. The output of the AWG1 serves as the positive power supply and drives the emitters of both PNP transistors Q3 and Q4A,B. Q3 and Q4A,B are wired as a gain of two current mirror with their bases connected together with the collector of Q3. The collector of Q4A,B connects to resistor R1. Resistors R1, R2 and transistor Q1 are connected as a zero gain amplifier section. The VBE of the two parallel connected transistors Q2,A,B is smaller than the VBE of Q1 by the voltage drop across R2. The base of transistor Q2A,B is connected to the zero gain output at the collector of Q1. The collector of Q2 connects to the input side of the PNP current mirror at the base - collector of Q3. The channel 2+ (Single Ended) scope input is used to measure the output voltage at the collector of Q4. Figure 1 Voltage reference, Version 1 Waveform generator 1 should be configured for a 1 KHz triangle wave with 2 volt amplitude and 2V offset. The Single ended input of scope channel 2 (2+) is used to measure the stabilized output voltage at the collector of Q4. Plot the output voltage (as measured at the collector of Q4) vs. the input voltage. At what input voltage level does the output voltage stop changing i.e. regulate? This is called the “drop out” voltage. For input voltages above the drop out voltage, how much does the output voltage change for each volt of change at the input? The change in Vout / change in Vin is called line regulation. Connect a variable resistor from the output node to ground. With the input voltage fixed (i.e. connected to the fixed Vp board power supply), measure the output voltage for various settings of the resistor. Calculate the current in the resistor for each setting. How does the output voltage vary vs. output current? This is called load regulation. Looking at the circuit in figure 2 we see many of the same basic components from figure 1. Q1,Q2, R1 and R2 serve the same basic functions as before. However rather than use the PNP current mirror to provide negative feedback to regulate the circuit, a common emitter amplifier consisting of Q3 and R4 driving emitter follower Q4 closes the feedback to the top of resistor R3. The output voltage will be the sum of a PTAT voltage across R3 and the VBE of Q3. Emitter follower Q4 supplies any varying load current that might be taken from the output node. 1 - 2.2 KΩ Resistor 1 - 100 Ω resistor 1 - 10 KΩ variable resistor (potentiometer) 1 - 4.7 KΩ resistor 1 - 1.0 nF capacitor (102) (resistors can be any similar value selected for desired circuit operation) 4 - small signal NPN transistors (2N3904, SSM2212, CA3045) The breadboard connections are as shown in figure 2. As before transistor Q1 and resistors R1 and R2 are configured as a zero gain amplifier. Transistor Q2 and variable resistor R3form a stabilized PTAT current source. If the SSM2212 matched NPN pair is used it is best that it be used for devices Q1 and Q2. Common emitter stage Q3 along with its collector load R4 provide gain. Emitter follower Q4 drives the output node and closes the negative feedback loop. Figure 2 Regulator Version 2 Waveform generator W1 should be configured for a 1 KHz triangle wave with 2 volt amplitude and 2V offset. Scope channel 2 (2+) is used to measure the stabilized output voltage at the emitter of Q4. Repeat the drop out voltage, line and load regulation measurements for this circuit. How are they different than the Version 1 regulator circuit? In the Version 1 circuit the net effective emitter ratio between Q1 and Q2 is four ( 2:1 for the NPNs and 2:1 in the PNP current mirror). How would the value for R1 need to change if the combined ratio was reduced to 2:1 by removing one or the other of the parallel transistors, Q2B or Q4B? Would the circuit still function if the NPN and PNP emitter ratios were both 1:1? For Further Reading: Return to Lab Activity Table of Contents. The CA3045,46 ( LM3045, 46 ) NPN transistor array is a good alternate choice for building this example circuit. See pinout below. All the emitters can be tired to ground ( pins 3,7,10,13 ). Devices Q1, Q2 and Q3 can be connected in parallel and serve as Q2 in figure 2. Q4 and Q5can be used for Q1 and Q3in figure 2. An individual device such as a 2N3904 etc. can be used for Q4 in figure 2. The 3 to 1 emitter area ratio will result in an output voltage very nearly 1.2 volts if R1 and R3 are both equal to 2K? (when R2 is 100?).

Received 9 December 2015; accepted 27 March 2016; published 30 March 2016 The structural engineering is an open space where a great number of different needs converge in order to obtain a one solution. The high performance is often obtained using traditional structural typologies but increasing the element sections dimensions. Often the solution is to use different systems and typologies. Cables structures for example are particularly adapted to design large span but they need particular structures and shapes. This structural typology isn’t presented in the technical codes and also a preliminary design is impossible without a specific study. The particular shape isn’t presented in the wind action section of the codes and so is impossible to evaluate the wind action also for sample structures that use this system. Wind tunnel tests should be carried out. The wind tunnel experiments are often performed for a specific building and are used to study the wind-structure interaction in order to the design the building structures. However, in many cases the wind tunnel is a very important instrument of research. It’s important sometimes to study preliminarily a phenomenon in order to have a parametrization of the aerodynamic behavior. In both cases many numerical subroutines are programmed by the researchers to prepare the wind tunnel setup and to evaluate the wind tunnel experiments. Often, these numerical procedures are left detached and are created for the specific case studied. If the purpose of the research is a parametrization of a phenomenon it’s interesting to program codes generalizable in order to extend the research with a great number of different cases. Purpose of this paper is to describe a complete process of pre- and post- procession of wind tunnel data acquired with experiments to look for a parametrization of the aerodynamic behavior of a specific geometry: in this case the hyperbolic paraboloid geometry was studied. The objective is double: at first to give an example to follow in order to start a parametric experimental campaign, at second to give the possibility to extend the research. To obtain these results, the four different numerical procedures programmed will be described and the basic theory following will be summarized. In Section 2, the first numerical procedure to look for a sufficient representative geometric sample is presented. In Section 3, a numerical procedure programmed to obtain FEM three-dimensional models is proposed. In Section 4, the procedure to evaluate the wind tunnel acquisitions is summarized and in particular in Section 4.2.3, the application of the wind tunnel data to perform FEM analysis is described. Finally in Section 5, a research nonlinear structural analysis program is described to close the pre- and post-processing methodology. The results are a general main procedure to use for each similar research and also that can be used to extend this particular research with other geometries. The educational purposes of the paper want to sustain the research based on a personal idea. 2. Numerical Procedure for Preliminary Design Cables Net The example of pre and post procedure to perform wind tunnel data consists in a research focused on the hyperbolic paraboloid geometry. The purpose of this research conducted by the Department of engineering and geology of Pescara (Italy) University and by the CRIACIV (Interuniversity Centre for Building Aerodynamics and Wind Engineering) wind tunnel laboratory, is to obtain a parametrization of the aerodynamic behavior of the hyperbolic paraboloid roofs (Rizzo F., et al., 2011; Rizzo F., et al., 2012; Rizzo F., 2012; Rizzo F., Sepe V., 2015). This particular shape is used to build tensile structures to cover for example sports arena, meeting or conference rooms. In the international codes there aren’t information about wind loads and in particular aren’t pressure coefficients to use as reference (ASCE 2005; AS/NZS 2002; CNR-DT 207/2008; CEN 2005). The difficulty to start a similar research is the little information this kind of structures; the wind tunnel tests are often performed for specific cases and are not generalizable; the first phase of the work is focused to look for a geometric sample to test in wind tunnel. The ratio between geometry and structural performances is very important for this kind of structures. The Hyperbolic paraboloid surface is characterized by four geometrical parameters: the sags and the spans of the two orders of parabola generating. In order to study the ratio between these four geometrical parameters and the structural response a numerical procedure is necessary - . 2.1. Main Program Hyperbolic paraboloid cables net have a double curvature with different cable lengths and curvatures, and generally, different cable areas and pre-stresses. In addition, the shape plays a decisive role in the cables net deformation behavior under the action of external loads. In the cables net with opposite curvature, the two orders of cables become load-bearing or stabilizers depending on the direction of the acting load; the load-bearing cable is concave in the direction of the acting load. Therefore, in conditions of snow or wind suction, the two orders of cables reverse their curvature. This inversion is dangerous because it may lead to the instability of either the cables net or the border structures. The function adopted to describe the hyperbolic paraboloid is expressed by Equation (1), where x, y and z are respectively the spatial variables; x0, y0 and z0 are the coordinates of the origin of the axes a, b and c are the geometric coefficients of the function. The c parameter was set equal to 1, making all the parabolas that lying on the surface, parallel and of identical curvature. It’s important to precise that the initial geometry is only an initial condition and that the real geometry of the net is defined on bases of the cables pre-stress. The loads effect modifies again the geometry and the objective of a correct design is to minimize the geometry initial variation with and without the pre-stress and the loads. The numerical procedure programmed starts from a fixed geometrical configuration and evaluates the final geometrical configurations with the pre-stress or with the other loads considered. The pre-sizing procedure implemented has the aim of obtaining pre-stress values and initial cables areas which permit an optimal structural response with a final geometry similar to the fixed geometry. A small difference between the fixed geometry and the deformed shape is important because the pressure coefficients evaluated in wind tunnel are valid for each load cases. Geometries that allow an optimal configuration and a good relationship between shape and structural performance are identified investigating a set of about thousand different geometric configurations and extrapolating those which, for the same geometry, provide the best structural performance in terms of cable area and cables stresses, and therefore, in terms of structural weight and lower displacements in operating conditions. In order to create a numerical procedure, the real three-dimensional case was simplified with a two-dimensional model called “Rope beam”, like illustrated in Figure 1. The two-dimensional structural model at the base of the Figure 1. Rope beam, 2D structural system. proposed procedure consists of two cables with opposite curvature which have a node at midspan in common. The equilibrium of the system forces in the node in common ensures the forces transmission between the two cables. The prefixed assumptions are: vertical links, in tension or in compression, are treated as a continuous membrane between the two main cables; the horizontal displacements are neglected compared to the vertical ones; pretension is considered as an equivalent distributed load; the system congruence is required only in the central span node; the mutual actions between cables are uniformly distributed as the external load, and consequently, load-bearing and stabilizing cable have a parabolic configuration also in elastic regime. The stiffness coefficients are considered, respectively for the load-bearing cable (C1) and stabilizing one (C2) defined as expressed in Equation (2) and Equation (3), where k1 and k2 represent, respectively, the stiffness of the load-bearing cable C1 and the stiffness of the stabilizing cable C2, defined in Equation (4); A1 and A2, f1 and f2, L1 and L2 are, respectively, area, sag and span length of the cable C1 and C2. According to the iterative numerical procedure, given a fixed geometry, the external loads, the maximum stress limits and, finally, the maximum value of the forces to be transmitted to the support structures in conditions of maximum load applied, it is possible to compute the optimal cables area. Cables areas are considered optimal if the cables stresses, under the maximum load design, respectively of compression and aspiration, are close to the upper and lower limit fixed in the hypothesis (σmax = 1700.0 MPa and σmin = 20.0 MPa). The global flow chart of the numerical procedure for cables net preliminary design is shown in Figure 2. It consist of two sub-procedures, respectively the procedure 1, illustrated in Figure 3, which allows to determine the optimum area of cable C1, and the procedure 2, illustrated in Figure 4, which allows to compute the area of cable C2 and therefore, the balance of the applied loads, depending on the area of cable C1. Figure 2. Numerical procedure for cables net preliminary design, global flow chart. Figure 3. Numerical procedure to preliminary design load bearing cables of a cables net. Figure 4. Numerical procedure to preliminary design stabilizing cables of a cables net. With reference to Figure 1, the prefixed geometric quantities are the cable sags (f1 and f2) and the span length (L1 and L2). Subscripts 1 and 2 indicate, respectively, load-bearing cables and stabilizing one. The considered load configurations are: initial equilibrium (“0”), in which acts only the cables net self-weight and in which the internal cables tensions are induced from initial pre-stress; application of permanent loads (“1”), in which, in addition to the cables net self-weight, acts membranes and/or roof panels’ weight; application of maximum snow action (“2”); application of maximum wind suction (“3”). The model predicted behavior provides that cable C1 reaches the maximum value of internal tension under the snow action, while cable C2 reaches the minimum value of tension; the cables length is modified under the load action, the initial geometry is bigger of lower in base of the direction of the load action. If the direction is gravitational the load bearing cables length increase, at contrary for the stabilizing cables. In this case, if wrongly designed, it is possible that the stabilizing cables curvature becomes inverse and the structure becomes instable. In Equations (1) and (2) the global cables stiffness and of the net is defined for the two order of cables; in Equations (4) and (5) and the single cables stiffness is defined depending on the cable area, sag, span, and finally on the cable material (E is the Young Module of the cable steel assumed equal to 165,000 Mpa). The initial conditions are represented by the initial geometry, the load action considered and the material chosen in order to evaluate an initial value of cables stiffness, also a preliminary value of cables area (A0,1, A0,2) and strain (,) are fixed. This value will be iteratively modified. In Figure 3 the procedure 1 flow chart illustrates the first step of calculus. In the following the maximum snow action strain values are named, and the maximum wind suction strain values are named,. The suffix “i” used in the following equations indicates the generic load condition, is dead load, is permanent load, S is the maximum snow action, W is the maximum wind suction. Like a start point the ratio between the cables stiffness is assumed equal to defined in Equation (6). The initial geometrical cables length and is evaluated using the approximated formulation reported in Equations (7) and (8), depending on cables initial sag and span. The initial cables internal traction and is evaluated as reported in Equations (9) and (10) assumed as the first forces of the mechanical problem; the initial horizontal traction equal to applied on the boundary structures is defined in Equation (11). The initial balance between dead load and initial fixed is assumed as first condition. The load action is applied as an uniform distribution load (named equivalent load and in the following indicated as P) on the load bearing cable. In this particular case is assumed as a hypothesis that snow action is bigger than the wind suction. The reason is that there are snow loads value in the code for each altitude but there aren’t wind loads data for Hyperbolic paraboloid shape; so the flat roof pressure coefficients is chosen as a start point. The wind load value evaluated with the flat roof pressure coefficients is general lower than the snow action for an altitude major than 200 m, using the Eurocode. In order to preliminary design the cable net for the bigger load condition at first the snow action (S) is evaluated (load condition 2). Using the procedure 1 illustrated in Figure 3, the geometry variation of the load bearing cable (in this case C1) is evaluated in order to obtain the H2,1 defined by Equation (12). It’s important to note that and are the deformed sag and the cable C1 traction with the snow action. In Equation (12), according to the Hook law is equal to, and and A are the strain and the cable area that satisfy the balance in load condition 2. Replacing in Equations (2), (3), (13), (14) are defined. The load configuration, (load configuration 2, cable 1) that corresponds to snow action, is evaluated according to the Equation (15) obtaining by the Equations (10) and (12) and dimensionless respect the cable area A1. It is possible to define also according to Equation (16). (iteration to) (15) The relation that connect the “2” load condition and the “0” load condition is reported in Equations (17) and (18). The load balance reported in Equation (18) is obtained with an iteration of. With the same procedure is fixed updating the value of. In conclusion, fixed an initial condition of traction the cables areas are defined according to the initial geometry wanted. In Figures 2-4 the flow charts of the procedure described are shown in order to summarize step by step the numerical proceedings. To validate the numerical procedure a comparison of the cables structural response with a Finite element method analyses (in the following FE) is done; in Figure 5(a) and Figure 5(b) the vertical displacements of the middle node is plotted for different cables areas but with same load applied (in this case equal to 2.2 kN/m gravitational and uniform load on the load bearing cable). The load bearing cable sag and span are equal to 4.44 and 80 m, the stabilizing cables sag and span are equal to 8.89 and 80 m; in this case the ratio is assumed equal to 2. The mean value of the percentage error is equal to 12%. This value appear acceptable if the approximation is considered. In the following the numerical procedure described in this section will be named NPPD (Numerical Procedure of Preliminary Design) (Elashkar I., Novak M., 1983; Lewis W. J., 2004; Majowiecki M., 2004) - . 2.2. The Analyzed Geometric Sample In order to estimate a set of optimal geometry with the minimum cables areas and displacements (for equal forces transmitted to the support structure) - , a geometric parameterization taking into account the following parameters was carried out (Figure 6): ・ γ, the relationship between the cable sags, (f2/f1); 8 different values of γ, respectively equal to 0.43, 0:50, 0.70, 1.00, 1.50 1.80, 2.00, 2.33 were taken into account; ・ ρ, the relationship between the roof height and the maximum span length (H/Lmax); 6 different values of ρ, respectively equal to 1/3, 1/4, 1/5, 1/6, 1/8, 1/10 were taken into consideration. ・ α, the relationship between the span length (L1/L2); 4 different values of α, respectively equal to 1.50 (rectangular plan shape), 1.00 (square plan shape), 0.50 (rectangular plan shape), and variable, for structures with a circular plan shape were taken into account. With the previously numerical procedure, 1008 different geometrical combinations were analyzed; only some configurations meet the optimization criteria pre-fixed in the hypotheses; in particular: ・ Cables net with L1 < L2 show better performance with lower values of forces transmitted to the supports. ・ Cables net with γ > 1 gives higher displacements but lower cables areas and therefore a lower structural weight. ・ “Optimal” values of γ, both as regards stress and structural weight optimization, are in the range between 1.5 and 2.5. ・ Low values of ρ give higher cables areas and therefore a higher structural weight; however, they gives “optimal” stresses with respect to cables net with high value of ρ. ・ The ratio between the obtained displacements with lower span lengths and those obtained with higher spans are lower compared to a direct proportionality. Totally one thousand geometries are investigate in order to compare the structural response and to choose an optimal sample to test in wind tunnel. On the basis of these preliminary results, a representative geometric sample to be tested in the wind tunnel was chosen; in Table 1 the geometrical sizes of the full scale cable nets are listed for each plan shape. This phase of the work has produced a preliminary design numerical procedure with which it is possible to identify the sample to test in wind tunnel. A model scale equal to 1:100 is chosen to construct the models. 3. Numerical Procedure to Generate Cables Net Fe Models Before wind tunnel test, tridimensional FE analyses are performed in order to simulate examples of full scale structures. The weight of the cables nets and their structural response under the snow action are studied to verify that the geometries chosen give a high structural response. In order to provide the geometric input for FEM models Figure 5. (a) Comparison between FEM analysis and procedure of preliminary design and (b) percentage error. Figure 6. Geometrical configurations investigated with the numerical procedure for cables net preliminary design. with Hyperbolic Paraboloid shapes a numerical procedure has been implemented. At first, a step-by-step procedure allows to describe the two-dimensional domain and the three-dimensional domain firstly by choosing a reference system (Cartesian or polar). It is possible to compute the two-dimensional domain in two different ways; in both cases, the procedure allows to choose among four different conventional shapes: circle, ellipsis, rectangle and polygon. In the first case, the user must enter the geometric parameters of the shapes, for example, the radius or the side’s length. The second case allows calculating the curve function by choosing through an equations system obtained by setting some points coordinates on the domain. An additional option allows to directly Table 1. Geometrical sample. importing a.dxf (Drawing Interchange Format) file that describes the two-dimensional domain or, in the case of complex and irregular domains, it is possible to directly import the coordinates of the shape vertices. After setting the two-dimensional domain, the three-dimensional one can be set by choosing among seven prefixed shapes (sphere, ellipsoid, flat, hyperbolic paraboloid, elliptic paraboloid, a one slope hyperboloid and a two slopes hyperboloid). Also in this case it is possible to directly insert geometric parameters and surfaces coefficients, or compute their functions through an equations system obtained by setting some points coordinates on the surface. The next step concern the insertion of the cables spacing in the two directions, X and Y in the case of Cartesian coordinates, or meridians and parallels in the case of polar coordinates. In the case of Cartesian system, the procedure computes the equation of each line that describes the cable, and then intersects lines with the 2D domain generating a set of nodes and computing the respective coordinates on the plane, pi(xp, yp). In the next step the procedure projects the evaluated nodes on the spatial surface, identifying the third coordinate zp. In the exporting phase, 3 different files can be saved: a.txt (text file) for input that contains the number of nodes and their coordinates; a.dxf with the cables net vector model; and a file that contains the functions equations of the created curves and surfaces. Thanks to this procedure, FEM model for nonlinear dynamic analyses can easily be generated. Figure 7(a) shows the intersection between the 2D domain and the 3D domain, while Figure 7(b) shows the simplified flow chart of the numerical procedure. Finally, in Figure 7(c) an example of hyperbolic paraboloid mesh with square plan shape generated is reported. Static and modal FE analyses are performed using the numerical models tested. Using the .dxf files generated the wind tunnel models are constructed made of wood. In the following the numerical procedure described in this section will be named Numerical Procedure to Generate Finite Element Models, in the following NPGFM - . 4. Wind Tunnel Test The wind tunnel is a tool used in aerodynamic, aeroelastic and fluid mechanics research to study the effects of Figure 7. (a) Intersection between 2D and 3D domain; (b) Tensile mesh generator flow chart; (c) Hyperbolic paraboloid mesh with square plan shape. air moving past solid objects rigid and flexible. A wind tunnel consists of a tubular passage with a particular geometry where the object to test is in the middle of the test chamber. Air is made to move past the object by a powerful fan system or other means. The wind tunnel model is instrumented with suitable sensors to measure aerodynamic forces, pressure distribution, or other aerodynamic-related characteristics. In this experience the models are rigid and the goals were to acquire pressure coefficients - . The CRIACIV (Inter-University Research Center for Building Aerodynamics and Wind Engineering) wind tunnel located in Prato (Italy) is used to perform aerodynamic experiments. A layout of the wind tunnel is illustrated in Figure 8. The choice of the turbulence intensity and so of the wind tunnel speed profile is important, too. For this research a more general possible condition was necessary. A medium urban profile is chosen for the boundary layer. In Figure 9(a) and Figure 9(b) the boundary layer development artificial roughness (wood panels) (a), spires (b) are shown and in Figure 9(c) and Figure 9(d) the speed and turbulence profile is reported. The mean value of the wind tunnel speed at the roof level is between 16 and 20 m/s, the turbulence at the same level is between 12% and 15% (Simiu E., Scanlan, R. H., 1986). The test models are made in wood and their geometrical scale is fixed equal to 1:100; the reason of this choice is to have big models and so easy to construct but with a not much high blockage coefficients. The constructions follows the geometry designed using the numerical procedure described in Section 3. Test models pictures are shown in Figure 10, respectively with a square plan (a), rectangular plan (b), circular plan (c) and elliptical plan (d). The geometry of the wind tunnel models is described in Table 1. Each model was instrumented (Figure 11(a) and Figure 11(b)) varying from 175 to 211 pressure taps distributed on the roof like shown in Figure 11(c). Each pressure tap was connected to a pressure transducers with a pneumatic connection made of Teflon tubes with 1.3 mm internal diameters (Figure 11(a) and Figure 11(b)). Data for 16 different wind angles were acquired at a frequency of 252 Hz and for 30 seconds obtaining, for each pressure tap, a pressure time history of 7504 values. The ratio between the models size and the test chamber section size is very important (φ). If this ratio is big (2% or 3%) the blockage effects increases pressure coefficients and correction factors are necessary. In this case the blockage effect is not negligible and so a blocking coefficient β was considered. According to Equation (19) φ is evaluated and the correction factor β is estimate. Atot is the wind tunnel test section area and Alateral is the model section area. In Table 2 blockage values and the correction coefficients for each tested model are reported. They are in a range between 1.5% and 7.7% with peaks for geometries p.2, p.4, p.6, p.8, p.15 and p.17 (highest model). The correction values for the pressure coefficients vary from a minimum of 2% to a maximum of 4%. Figure 8. CRIACIV wind tunnel image (a), elements (b): (A) entrance; (B) boundary layer development zone; (C) test chamber (section size 2.40 × 1.60 m); (D) connection zone; (E) engine (160 kW); (F) diffusion T-shape. Figure 9. Boundary layer development artificial roughness (wood panels) (a), spires (b); (c) velocity and (d) turbulence profiles. Figure 10. Hyperbolic parabolic roofs: (a) square plane; (b) rectangular plane; (c) circular plane; (d) ellipltical plane . Figure 11. Wind tunnel acquisition data, (a) and (b) model instrumentation, (c) example of pressure taps distribution. Table 2. Blockage values. 4.2. Data Processing Experimental data consist in pressure expressed in mmH2O deriving by the transducers acquisition; a numerical procedure is needed in order to obtain a double goal: at first pressure in Pascal and pressure coefficients located on the roof and at second, forces time history to use in FEM analysis. For this reason a numerical procedure have to transform data into pressures, then evaluate the pressure coefficients, then estimate the maximum, minimum and mean values of these coefficients, and finally transform the pressure coefficients into forces to be applied on the FEM model. The first phase concerns the graphic aspect, with the preparation of the geometry of the pressure coefficients map; the second stage involves the implementation of the subroutine for the evaluation of the pressure coefficients and the third phase provides the forces calculation. 4.2.1. Pressure Coefficients Maps In order to obtain pressure coefficients maps to describe the pressure distribution on the roof and sides of the model, the surface was discretized in polygons surrounding each pressure taps. The Thiessen (Voronoi) polygons theory is adopted. It consists to define individual areas of influence around each of a set of points. Thiessen polygons are polygons whose boundaries define the area that is closest to each point relative to all other points. They are mathematically defined by the perpendicular bisectors of the lines between all points. The process goes through several steps: collects the points from a point layer (vertices if the source is a polyline or polygon layer), clean duplicate points, generates Convex Hull, creates a TIN structure, generates perpendicular bisectors for each tin edge, builds the Thiessen polygons and, finally, clips the Thiessen polygons feature class with the convex hull. A specific numerical procedure is programmed for this section. An example of Thiessen polygons distribution is illustrated in Figure 12. It is possible to note that if the pressure taps distribution is not geometrically regular with a structured grid, the shape of the Thiessen polygon is irregular. For each polygon one value of pressure coefficient is associated - . 4.2.2. Evaluation of Pressure Coefficients During wind tunnel tests the model is fixed in the test chamber, in Figure 13(a) a picture of a model during the test is shown. The tubes connected to the transduces sent an input to the computer that measure the pressure variation in mm H2O. The experimental data needs to be transformed in pressure coefficients. Pressure coefficients (Cp), is a dimensionless number which describes the relative pressures throughout a flow field in fluid Figure 12. An example of Thiessen polygons distribution. Figure 13. (a) Model in the wind tunnel; (b) Pressure coefficients map (mean values); (c) Pressure coefficients 3D mesh; (d) Typical forces time history. dynamics; it is evaluated by the ratio between the difference of the local pressure and the undisturbed flow pressure, and undisturbed dynamic flow pressures are evaluated for each wind direction. Maximum (Cp,max), minimum (Cp,min) and mean values (Cp,m) of the pressure coefficients are extracted from the obtained pressure time histories. Subsequently, pressure coefficients maps were plotted. An example of Mean value of pressure coefficients map is shown in Figure 13(b). Minimum and maximum pressure coefficients have been calculated using a probabilistic method according to the Gumbel method (Gumbel, E.J., 1958) following the procedure proposed in (Cook N.J., Mayne J.R., 1979), (Cook N.J., Mayne J.R., 1980), associated with a probability of 22% that it will be exceeded, as is done by Eurocode 1. For each pressure taps and for each model, the numerical procedure computes a pressure coefficients time history of 7504 points, a pressure coefficients matrix consisting of 7504 rows and a number of columns equal to the number of pressure taps. Moreover, for each model the procedure calculates three vectors containing the mean, maximum and minimum value of the pressure coefficient. In the following the numerical procedure described in this section will be named NPWDP (Numerical Procedure for the Wind tunnel Data Process). 4.2.3. Data Exchange between Wind Tunnel and FEM Analysis In order to use experimental data to perfume FEM analyses, the same surface discretization between wind tunnel test model and FE model is necessary. Often that is impossible because the number of pressure taps used for each models is generally less than the FE model nodes. Also in this case it was happened. The mean value of pressure taps number used on the roof is 90, the number of nodes used in FEM analysis to describe the cable net is about 1700. A numerical procedure to extend the experimental data on the FEM mesh is necessary. There are to more used possibilities: the first is obtained overlapping the FEM mesh and the Thiessen polygons distribution, the same pressure coefficient is used for all FEM nodes that are surrounding by polygon. A second way is to estimate a mathematical procedure to interpolate the experimental data respect the FEM mesh. Both procedures were implemented. The first solution follows the following sequence: the polygons edges coordinates (Cartesian reference system) are determined; a scan of the FEM nodes coordinates is been done in order to check the proximity from the polygon edge. The nodes that have coordinates between the polygon minimum and maximum coordinates are in the polygon. The second procedure is more difficult. The first phase is the same to the previous solution, but during the scan a value of pressure coefficients for each node is assumed; the values is evaluated using the inverse distance weight (IDW) interpolation method (Borrough P.A., 1986; Greville T.N.E., 1969; Hohn M.E. (editor), 1998). An example of 3-dimensional pressure coefficients map obtained using this last method is shown in Figure 13(c). Using the pressure coefficients time history assumed for each FEM nodes, a wind loads time history is evaluate. In this case in order to obtain the structural response of a cable net under wind action, a localization of the net is necessary. A value of wind kinetic pressure and geographical aspects are chosen. A preliminary ramp is added by the load history, it has a length equal to about 1% of the history length. In Figure 13(d) an example of the load history is shown for a cable net center point (N1) (Shen S., Yang Q. 1999). In the following the numerical procedure described in this section will be named NPED (Numerical Procedure to Exchange Data). In Figure 14 the procedure is summarized - . 5. Wind-Structure Interaction with Time History Analysis In order to estimate cables nets structural response, Nonlinear Dynamic Analyses have to be performed. The analyses are conducted on the tested sample using numerical procedures implemented by Full Professor PieroD’Asdia staff; the procedures are merged in a main program in the following named TENSO. It isn’t a commercial software and it was born as a set of numerical procedures and subroutines merged step by step from 1980. The description of these procedures has never been published. The Non Linear Structural Analysis Program The structural analysis program (in the following TENSO) is designed for static and dynamic analysis with step-by-step integration of nonlinear geometric three-dimensional structures. It contains cable and beam finite elements and permits the study of wind-structure interaction with generation of wind histories and simulation of aeroelastic phenomena (Crisfield M.A., 1991). Nonlinear static analysis were carried out with the wind action evaluated for mean, minimum and maximum pressure coefficients while nonlinear dynamic analysis were performed by applying wind action as a forces time history computed with the previously described numerical procedure. With TENSO is possible to compute parabolic cables in two ways: in the first case, the cable can be divided in an appropriate number of elements that are rectilinear cable, in the second case elastic catenary configuration or parabolic cable can be used. The first case is applicable only with nodal loads. Possible applications of this methodology are suspension bridges with a distance between the hangers sufficiently small compared to the cable span length, and cables nets with a small spacing between cables compared to the maximum span length. For this kind of structures the global stiffness matrix is updated for each load step through the assembly of stiffness matrices of the elements varied according to the strain found in the previous step. In this way the software takes into account the geometric nonlinearity of the structure. As regards to the beam finite element, it is possible to choose among a beam with a uniform or a variable section. For each case it is possible to introduce prestressing actions or tractions as well as thermal loads. In particular, the beam finite element with variable section provides the calculation of ten coefficients in order to describe the area variation and the moment of Figure 14. Flow chart of the wind tunnel data processing. In TENSO, secant method is used as a check method that permits to stop the analysis with a unbalanced solution. Using the step by step incremental method, nonlinear problem can be transformed in a succession of linear problems. Each calculation step stores loads and strains history evaluated during the previous step. For each analysis step, a small enough part of load (ΔP) necessary to ensure that is possible to use the elasticity method is applied. However, this simple and classical approach presents the difficulty to evaluate the exact dimension of load step and so the exact step of analysis. A non-appropriate chosen range can cause an inexact solution. In order to solve it, TENSO uses the method with the variable stiffness matrix; this method is a vector version of the Newton-Raphson modified method about nonlinear equation systems. The Newton-Raphson procedure guarantees convergence if and only if the solution at any iteration is close to the exact solution. Therefore, even without a path-dependent nonlinearity, the incremental approach (i.e., for subsequent load steps) is sometimes required in order to obtain a solution corresponding to the final load level. If the displacements are large, the product between the stiffness matrix, evaluated on the basis of the solution of the previous step and on the basis of the stresses, and the displacements vector, give the internal force vector, not equilibrate with the external forces vector according to Equation (20), where is the stiffness matrix, is the displacements vector, is the external forces vector; the difference between these two forces vectors represents the imbalance force vector, according to Equation (21), where R is the residual vector and P is the internal forces vector. In the following step, this vector is applied as an external load modifying the displacement vector with a residual value of displacements, according to Equation (22) where is residuals values of displacements to update the geometry, and so updating the structure geometry. In order to solve nonlinear dynamic analyses, TENSO uses the Newmark-beta method, a numerical integration method used to solve differential equations. It is used in finite element analysis to model dynamic systems. In order to illustrate the use of this family of numerical integration methods, the solution of a linear dynamic system have to be firstly considered. In 1962 Newmark’s method in matrix notation was formulated, stiffness and mass proportional damping was added, and the need for iteration by introducing the direct solution of equations at each time step was eliminated. The time dimension is represented by a set of discrete points each a time increment apart. The system is solved at each of these points in time using as data the solution at a previous time. The procedure follows these subsequent phases: reading of initial boundary conditions; assembly of the stiffness matrix, setting β and γ, Newmark method parameters that control the stability and the accuracy of the integration procedure. They are equal to 1/4 and 1/2 respectively; assembly of the vector forces; step by step calculation with iterative process and convergence check. For each integration step, a check of the solution precision is done in order to evaluate if it is necessary to modify the integration step dimension. Generally, a more precise solution needs a very small integration step and so a higher computational work. Unbalanced loads are evaluated according to the Newmark’s algorithm, as the difference between the reactions and the applied external loads. In TENSO, a correction of this algorithm is implemented: the precision of solution is evaluated as a ratio between unbalanced loads and applied external loads for each unconstrained degree of freedom. At each integration step, unbalanced loads are added to the next load step, in order to obtain an optimal solution (Melchers R.E., 1987; Smith I. M., Griffiths D.V., 1982). In Figure 15, a global flow chart of TENSO is summarized. The main pro- Figure 15. Nonlinear structural analysis program global flow chart. gram of the numerical procedures described in this section in the following will be named NPSA. In order to demonstrate the validity of the procedure, two examples of different structures covered with cables net are described: the first one is a project proposal under review (2012) of a roller skates arena and the second is basketball arena; the last one is a project proposal for the renovation of an existent sports arena (2007). 6.1. Project of a Roller Skates Arena The building designed is located in Pescara, middle Italy, and its purpose is to cover an existent open space used for roller skates competitions (Rizzo et al., 2014). The building has a circular plan and is covered with an Hyperbolic Paraboloid cables net roof. The most important geometrical parameters are summarized in Table 2. The horizontal traction (referring to Equation (11)) is absorbed by a series of external pillars and beams located around the building like are illustrated in Figure 17(c). In Figure 17(a) a global view of the urban contest and in Figure 17(b) a significant plan of the building are shown. The geometry chosen is one of the optimal geometries studied and described in Figure 16 and obtained by the procedure described in Section 2. In this phase of the project wind tunnel experiments results obtained with the parametric study described in Section 3 are sued. Of course after this first preliminary phase some specific wind tunnel experiments are necessary to study aerodynamic and aeroelastic effects. The pressure coefficients maps (Rizzo et al., 2011 and Rizzo, 2014) for two significant angles used to evaluate the wind action are shown in Figure 18. The wind direction of 0˚ are parallel to the stabilizing cables and at contrary, the wind direction of 90˚ are parallel to the load bearing cables. A FEM model is analyzed and geometrical non-linear analyses are carried out in order to design the structural components of the building. The procedure described in Section 3 is followed to define the FEM models and some pictures are illustrated in Figure 19. The mechanical parameters are reported in Table 3, where D is the externa diameter of the building, A1 and A2 are respectively the load bearing cables and the stabilizing cables, ε1 Figure 16. Procedure for the evaluation of tensile structure behaviour. Figure 17. External views of the roller skates arena and ε2 are the strains cables for the load configuration number 1 (according to Section 2); Hb is the drum high of the building, H is equal to the f1 + f2. Using the numerical procedure described in Section 4, nonlinear analyses are carried out in order to study at first the structure deformation and its natural frequencies. Modal analyses and then a dynamic time history analysis are permed using the wind tunnel experiments results. Some results are illustrated in Figure 20 and Figure 21. In particular the first 9 modes shape are reported in Figure 20 and are listed in Table 4, the deformation under 0 wind action is reported in Figure 21. Observing the cables area values reported in Table 3 and considering that 40 load bearing cables and 40 stabilizing cables are used, we note that the structural weight of the roof is really low, the cables net weight is only 1.5 × 10−2 kN to square meters. One aspect ore is important to note: the first 9 modes are totally roof’s mode and the periods are very high. This structures are particularly optimal in seismic zones because are flexible and ductile. 6.2. Project of a Basketball Arena The project purpose is to substitute a truss structure used to cover a sport arena with a cable net structure. The advantages are many and the most important is the easy of maintenance. The great numbers of bolted nodes often require maintenance and the trusses require a very repetitive protective painting. The occasion to begin the Figure 18. Pressure coefficients maps, (a) 0˚ and (b) 90˚. Figure 19. FEM models views. Table 3. Geometrical and mechanical parameters. Figure 20. Modes shape deformation. Table 4. Natural frequencies and periods. Figure 21. Upward deformation under 0˚ wind action. building renovation was the championship European basketball competition (2007) (Rizzo et al., 2012). The proposal is being evaluated by the municipality government for the future renovation. The idea is to realize an external structure with a square plan in order to have a total open space into the arena. Some pictures of the actual structure and the future modification are illustrated in Figure 22. A square plane is chosen in order to respect the actual shape of the building and its urban contest. A series of stays are used to absorb the horizontal traction of the cables net, like is shown in Figure 22. In Table 5 the main geometrical parameters are listed and the cables areas and strains for the load configuration number 1 (according to Section 2). The structural response is evaluated with Non-Linear analyses carried out with a FEM model modelled with the numerical procedure described in Section 3. Some pictures of the FEM model are shown in Figure 23. Using the numerical procedure described in Section 5 the Natural frequencies and displacements under loads combinations are evaluated, too. In Figure 24 the first 10 modes shape deformations are reported and in Table 6 their values are listed. It’s interesting to note that the period is higher than the other geometry described in Section 7.1; it’s caused by many reason, at first the plan geometry because the circular shape gives a more rigid border structure, at second the cables areas and strains are lower for this geometry, finally, this structure is also higher than the previous. The wind action is applied on the FEM model like a series of time histories evaluated by wind tunnel experiments and a dynamic analysis is performed. In Figure 25 a view of the structure deformation under 90˚ wind action is shown; it is interesting to note that this direction is particularly critic for Hyperbolic paraboloid cables net because the suction is higher than the other angles (Rizzo et al., 2011), so it is particularly important that the suction not decreases too the load bearing cables strain. In this case, like is reported in Table 7, the ratio between load bearing cables strains with and without wind is equal to 0.85 (reduction equal to 15%). In Table 6 the structural response in term of cables strains, traction and deformation variation, is reported; T1 and T2 are the cables traction evaluated according to the Equation (10), Δf is the sag variation. Every time that an experiment is processed, a great number of numerical procedures are programmed by researcher to control the process. These numerical procedures are often isolated and programmed again for every different case of study. An interesting goal is to create a free open domain where the numerical procedures evaluated are merged, added, modified by researchers with the aim to obtain a common space of use. With this purpose, the present paper described a methodology followed to prepare a wind tunnel test and to process results. Five different steps NPPD → NPGFM → NPWDP → NPED → NPSA give a one complete numerical procedure that can be expanded, modified or completed by everyone. In this specific case the subroutines can be modified to capture more and more different geometries or structural typologies with the aim to obtain a globalized virtual space of calculus. This paper is focused to wind tunnel tests because they are generally complex, Figure 22. (a) External view of the actual structure; (b) Future modification of the building. Table 5. Geometrical and mechanical parameters. Figure 23. FEM model views. Figure 24. Modes shape deformation. Table 6. Natural frequencies and periods. expansive and long test: a previous efficient and detailed preparation is necessary before and a great capacity to synthesize the results obtained is necessary after. With the methodology used the goal is obtained: in fact, at first the preliminary design procedure permits to choose the sample to test; at second the FEM generation procedure permits to obtain FE model for FEM analyses and a guide to construct wind tunnel test models; at third the Figure 25. Upward deformation under 90˚ wind action. Table 7. Structural response. processing data procedure permits to evaluate the experimental data and to prepare the input of the FEM analyses. Finally, the nonlinear structural analyses procedure permits to evaluate the structural response. The global flow chart is illustrated in Figure 16. Special thanks to Full Professor PieroD’Asdia for the research coordination, to Engineer Massimiliano Lazzari for his for his collaboration in the planning process described in Section 2, to Engineer Fabrizio Fattor for his collaboration in the planning process described in Section 3, to Associate Professor Gianni Bartoli and the CRIACIV Wind tunnel boundary layer staff, in particular Engineer PhD Tommaso Massai and Lorenzo Procino, for the coordination of the wind tunnel tests and the numerical procedure programming described in Section 4, to Architect PhD Federica Speziale for the FEM analysis described in Section 5, finally to PieroD’Asdia, FabrizioFattor, Salvatore Noè and Luca Caracoglia for the calculation program described in Section 5.

Revisiting the asymptotic dynamics of General Relativity on AdS The dual dynamics of Einstein gravity on AdS supplemented with boundary conditions of KdV-type is identified. It corresponds to a two-dimensional field theory at the boundary, described by a novel action principle whose field equations are given by two copies of the “potential modified KdV” equation. The asymptotic symmetries then transmute into the global Noether symmetries of the dual action, giving rise to an infinite set of commuting conserved charges, implying the integrability of the system. Noteworthy, the theory at the boundary is non-relativistic and possesses anisotropic scaling of Lifshitz type. The dynamics of Einstein gravity in three spacetime dimensions is described by global degrees of freedom that can be identified only once a precise set of boundary conditions is provided. In the case of asymptotically AdS spacetimes equipped with Brown-Henneaux boundary conditions, the asymptotic symmetry group is generated by two copies of the Virasoro algebra Brown:1986nw . Demanding that the Lagrange multipliers –given by the lapse and shift functions in an ADM foliation– are held constant at infinity, the reduced phase space of the Einstein field equations is described by Virasoro modes that evolve according to where is the AdS radius and are coordinates parametrizing the cylinder at infinity. The symmetry algebra and the form of the latter equation is consistent with the description in terms of the boundary theory; it is well–known that the asymptotic dynamics for these boundary conditions is described by left and right chiral bosons Coussaert:1995zp ; Henneaux:1999ib 111As shown in Coussaert:1995zp ; Henneaux:1999ib , it is possible to rewrite the action of two chiral bosons as a Liouville theory. This is accomplished by performing a Bäcklund transformation that excludes the zero mode sector of the chiral bosons.. The components of the stress–energy tensor of the chiral bosons are given by the Virasoro modes , so that equation (1) corresponds to its conservation law. Note that for the boundary conditions of Brown and Henneaux, the chiral bosons and their corresponding left/right energies fulfill the same equations. Recently, a new family of boundary conditions connecting Einstein gravity on AdS with the Korteweg-de Vries (KdV) hierarchy of integrable systems has been proposed in Perez:2016vqo . The possible choices of boundary conditions are labeled by a nonnegative integer , corresponding to the -th representative of the hierarchy. The Brown-Henneaux boundary conditions are recovered for , so that the modes fulfill (1); while for , the modes are described by noninteracting movers, satisfying the KdV equation where is the Newton constant. For the asymptotic symmetry algebra turns out to be spanned precisely by the infinite set of commuting charges of KdV. One of the main purposes of our work, is to unveil the precise form of the action principle that describes the dynamics of the underlying fields of the dual theory at the boundary, from which the field equations of the KdV hierarchy emerge from a conservation law. In order to carry out this task, it is convenient to use the Chern-Simons formulation of three-dimensional gravity Achucarro:1987vz ; Witten:1988hc . We then perform a Hamiltonian reduction similar to the one of Coussaert, Henneaux and van Driel Coussaert:1995zp . A distinguishing feature of our derivation is that, as the boundary conditions for actually precludes one from passing through the standard Hamiltonian reduction of the Wess–Zumino–Witten (WZW) model Forgacs:1989ac ; Alekseev:1988ce , one has to circumvent this step through imposing the boundary conditions in the action principle from scratch. In this way, one obtains a novel action principle for the dual theory, whose field equations are described by two copies of the hierarchy of ‘‘potential modified KdV’’ (pmKdV) equations of opposite chirality222A list of the first four equations of the pmKdV hierarchy is given in appendix B.. The paper is organized as follows. In the next section, we revisit the boundary conditions of KdV-type in the context of 3D gravity with negative cosmological constant. In section 3, the dual theory at the boundary is obtained from the Hamiltonian reduction of the Chern-Simons action endowed with a suitable boundary term. The field equations are also analyzed. Section 4 is devoted to study the global symmetries symmetries of the dual action principle at the boundary. We conclude in with some comments in section 5. 2 General Relativity on AdS and the KdV hierarchy and the Chern-Simons level is given by . Here, is the three-dimensional manifold with coordinates , where represents time, stands for the radial coordinate and is an angle. The generators of the algebra, given by , with , are chosen such that the commutators and the invariant non-degenerate bilinear form read In order to describe the asymptotic form of the gauge fields, it is useful to make a gauge choice as in Coussaert:1995zp , so that the connection reads where stand for the dynamical fields, and correspond to the Lagrange multipliers. In the asymptotic region, the field equations, , reduce to where the operators are defined by The asymptotic symmetries can then be explicitly found by demanding the preservation of the auxiliary connection under gauge transformations, , where is a Lie-algebra-valued parameter. Thus, the asymptotic form of is maintained for gauge transformations spanned by parameters of the form where are arbitrary functions of and , provided that the dynamical fields transform as Preserving the temporal component of the gauge field then implies the following condition for the variation of the Lagrange multipliers It is worth stressing that the boundary conditions turn out to be fully determined only once the precise form of the Lagrange multipliers at the boundary is specified. The results of Brown and Henneaux Brown:1986nw are then recovered when the Lagrange multipliers are held constants at infinity . A simple generalization is obtained by choosing arbitrary functions of the coordinates, so that are kept fixed at the boundary Henneaux:2013dra ; Bunster:2014mua . Different choices of boundary conditions, in which the Lagrange multipliers are allowed to depend on the dynamical fields and their spatial derivatives, were proposed in Perez:2016vqo . Hereafter, we focus in a special family of boundary conditions of KdV-type, being labeled by a non negative integer . In this scenario, the Lagrange multipliers are chosen to be given by the -th Gelfand-Dikii polynomial Gelfand:1975rn evaluated on , i.e., The polynomials can be constructed by means of the following recursion relation333Note that the normalization of the Gelfand-Dikii polynomials used here differs from the one in Perez:2016vqo . Thus, in the case of one obtains that , which reduces to the boundary conditions of Brown and Henneaux Brown:1986nw . In this case, equation (13) implies that the parameters are chiral, while the dynamical fields also do, since the field equations (9) reduce to (1). The next case corresponds to so that the choice of Lagrange multipliers is given by , and hence, the field equations in (9) reduce to KdV In the remaining cases, , the field equations are then given by the ones of the -th representative of the KdV hierarchy. Note that for , the Lagrange multipliers acquire a non-trivial variation at infinity. Nonetheless, as shown in Perez:2016vqo and further explained in the next section, the action principle can be well defined because each of the Gelfand-Dikii polynomials can be expressed in terms of the variation of a functional, i.e., where stand for the conserved quantities of KdV, and are the corresponding densities 444A list with the first Gelfand-Dikii polynomials, conserved quantities of KdV and the corresponding field equations of the KdV hierarchy is given in appendix A.. Furthermore, equation (13) becomes a consistency relation for the time derivative of the asymptotic symmetry parameters . Thus, for , assuming that the parameters depend exclusively on the dynamical fields and their spatial derivatives, but not explicitly on the coordinates, the general solution of the consistency relation is given by a linear combination of the form with constants. This infinite set of symmetries then gives rise to conserved charges, which can be written as surface integrals by means of the Regge-Teitelboim approach Regge:1974zd . The variation of the conserved charges associated to the gauge transformation generated by a parameter of the form (11) that spans the asymptotic symmetries, is given by The asymptotic symmetries are then canonically realized. A straightforward way to obtain the asymptotic symmetry algebra in terms of Poisson brackets is given by the relation The cases and are then very different in this context. Indeed, for the algebra turns out to be abelian while for , which corresponds to Brown-Henneaux, the algebra of the conserved charges is given by two copies of the Virasoro algebra with a non-vanishing central extension. Some interesting remarks about the metric formulation are in order. It is worth highlighting that the reduced phase space for the boundary conditions of KdV-type, for an arbitrary non negative integer , always contain the BTZ black hole Banados:1992wn ; Banados:1992gq , which corresponds to the configuration with constants Perez:2016vqo . Indeed, the field equations of the KdV hierarchy are trivially solved in this case, and the spacetime metric in the ADM decomposition is such that the lapse and the shift correspond to a non-standard foliation, determined by . Specifically Furthermore, the boundary conditions described by (7) and (8), with given by (14) are such that the fall-off of the metric somewhat resembles the one of Brown-Henneaux. Indeed, in a Fefferman-Graham-like gauge, the spatial components of the metric and its conjugate momenta behave as However, the key difference arises in the asymptotic behavior of the lapse and shift functions, which read Hence, for they are allowed to fluctuate at leading order, in sharp contrast with the fall-off for that corresponds to the Brown-Henneaux boundary conditions for which . 3 Dual theory at the boundary In this section, we perform a Hamiltonian reduction of the action (3) by explicitly solving the constraints of the theory. The boundary conditions for the gauge field correspond to (7) and (8), where the “chemical potentials” in (14) are given by the -th Gelfand-Dikii polynomial . The reduction is carried out for a generic value of . 3.1 Hamiltonian reduction The Hamiltonian reduction of Chern-Simons theory in the context of three-dimensional gravity has been discussed extensively in the literature, see e.g., Coussaert:1995zp ; Henneaux:1999ib ; Rooman:2000zi ; Barnich:2013yka . For the standard choices of boundary conditions Brown:1986nw ; Barnich:2006av , the classical dynamics can be obtained from the Hamiltonian reduction of the WZW theory at the boundary Witten:1988hf ; Elitzur:1989nr ; Forgacs:1989ac ; Alekseev:1988ce . Nonetheless, for the boundary conditions of KdV-type, the reduction does not lead to the usual WZW theory at the boundary, since for a generic value of the components of the gauge field at the boundary are no longer proportional, and hence, the Kac-Moody symmetry appears to be manifestly broken (except when which corresponds to Brown-Henneaux). Nevertheless, as explained below, the reduction can still be successfully performed because the boundary conditions can be appropriately implemented in the action principle. The resulting reduced action at the boundary gives rise to a different hierarchy of integrable equations, labeled by the integer . The simplest case () corresponds to two chiral bosons of opposite chirality Coussaert:1995zp ; Henneaux:1999ib . For we obtain a novel action principle, whose field equations are given by two copies of the pmKdV equation (see e.g. olver2000applications ; wang2002list ). In the remaining cases () the action of the dual theory describes the other members of the pmKdV hierarchy. The integrability of this hierarchy is explicitly checked the next section. We start with the action (3) written in explicit Hamiltonian form where stand for appropriate boundary terms generically needed in order to have an action principle that is well defined. It is worth pointing out that the boundary can be located at an arbitrary fixed value of the radial coordinate. Here is the spatial part of the Levi-Civita symbol, while is the curvature . We choose , and dot stands for derivative with respect to . The action (29) attains an extremum when the field equations hold, provided that Note that for the Brown-Henneaux boundary conditions (), the components of the gauge field satisfy at the boundary, and hence, can be readily integrated. However, for the boundary conditions of KdV-type, with the temporal and angular components of the gauge field at the boundary are not proportional (see (8)), and so one might worry about the integrability of the boundary terms . However, as explained in Perez:2016vqo , since the Lagrange multipliers in (14) are given by the variation of a functional (see (17)) the boundary terms can be explicitly integrated as Therefore, the suitable action principle for the boundary conditions of KdV-type is precisely identified, and so we are able to proceed with its Hamiltonian reduction. The constraint is locally solved by . For the sake of simplicity, we disregard non-trivial holonomies, so that can be assumed to be periodic in . Thus, replacing back in the action (29), a straightforward calculation yields The first two terms naturally appear in the standard chiral WZW action Elitzur:1989nr , but here we have an explicit modification due to the presence of . As shown below, the form of makes possible to recover the infinite-dimensional Abelian algebra in (22) from a Noether symmetry of the full action. Furthermore, note that do not appear to be expressible locally in terms of the group elements . In order to reduce to a boundary integral, we use the Gauss decomposition for Here , and are functions of . Thus, can be expressed as where prime denotes derivatives with respect to . Thus, the action has now been reduced to an integral at the boundary. which can be further simplified by performing the Gauss decomposition for the group element where the fields , and depend only on and . Thus, we obtain and hence, the action (32) reduces to Besides, the asymptotic form of is determined by eq. (8), so that which by virtue of the Gauss decomposition (38), implies the following relations Making use of the first equation in (43), it is straightforward to see that the second term in (41) becomes a total time derivative that can be discarded. The remaining equations in (43) then allow to obtain a crucial relationship, given by from which the reduced action (41) can be expressed exclusively in term of two fundamental fields . In sum, the action of the dual theory at the boundary explicitly reads For the remaining steps, it is worth highlighting that the action (46) possesses the following gauge symmetry where stand for arbitrary functions. Indeed, under (47), the kinetic term in (46) just changes by a time derivative, while the Hamiltonian does not give additional contributions since only involves angular derivatives of . 3.2 : chiral bosons in agreement with the standard result obtained in Coussaert:1995zp . The theory describes the dynamics of two chiral bosons of opposite chirality. The field equations in this case then read which can be readily integrated once, yielding where are arbitrary functions of time. Therefore, these arbitrary functions can be set to zero by virtue of the gauge symmetry in (47), with , and hence Note that, as mentioned in the introduction, the field equations for in (51) coincide with the ones of the Virasoro modes in (1). As it is shown below, in our context, the fact that the field equation is equivalent to the conservation law it is actually an accident of the particular case . 3.3 : pmKdV movers The next case corresponds to the choice so that . The chiral copies of the actions then read and the field equations are given by As in the previous case, the equation can be integrated once, giving where the arbitrary integration function has been set to zero by virtue of an appropriate gauge choice. This equation corresponds to two copies of the pmKdV equation.555The name stems from the fact that under the identification , equation (53) reduces to modified KdV (mKdV) for . 3.4 Generic : pmKdV hierarchy The field equations can be readily obtained in a closed form for a generic value of , yielding As in the previous cases, these equations can be integrated once, and by means of the gauge symmetry of the action (47), they reduce to in agreement with –th representative of the potential form of the mKdV hierarchy. In the next section it is shown that these equations can be manifestly seen to be integrable, since they admit an infinite number of commuting conserved charges. 4 Symmetries of the action This section is devoted to study the symmetries and conserved quantities of the action (46). Apart from the gauge symmetry (47), the action (45) also possesses global and kinematic symmetries, which are described in what follows. 4.1 Global symmetries Here we show that the action (45) is invariant under the following Noether symmetries with given by (18). It is worth stressing that these global symmetries are in one to one correspondence with the asymptotic symmetries in the bulk. Indeed, by means of the map in (44), the transformation law of that is given by (12) is precisely recovered from (57). Therefore, the corresponding infinite number of commuting Noether charges can be seen to coincide with the surface integrals that come from the analysis in the bulk. This can be explicitly shown as follows. For each copy of the action (45), the Hamiltonian is invariant under transformation (57), while the kinetic term changes by a total derivative in time. Indeed, equations (12) and (18), imply that the variation of the Hamiltonian term can be expressed as and hence the transformation (57) is a symmetry of the action. Therefore, the straightforward application of Noether’s theorem yields an infinite number of commuting conserved charges given by which implies that the field equations in (56) correspond to an integrable system. Besides, and noteworthy, the Noether charges associated with the global symmetries of the dual theory in (60) precisely agree with the surface integrals found from the asymptotic symmetries in the bulk (20). 4.2 Kinematic symmetries & Lifshitz scaling The kinematic symmetries of the dual action (45) correspond to rigid displacements in space and time, as well as global anisotropic scaling. These symmetries are spanned by a two-dimensional vector field with , and constants, and is related to the integer through . Under an infinitesimal diffeomorphism spanned by the scalar fields transform as , which implies that left and right Hamiltonians change according to so that under anisotropic scaling they have weight given by . It is simple to prove that the dual action (46) is invariant under the symmetries spanned by , and hence, the corresponding Noether charges for the chiral copies are given by Thus, for each copy, the energy, the momentum, and the conserved charge associated to anisotropic scaling are given by respectively. Note that correspond to left and right copies of the generator of anisotropic Lifshitz scaling of the form, so that stands for the dynamical exponent. For each copy, the generators of the kinematic symmetry then fulfill the Lifshitz algebra in two dimensions (see e. g. Hartnoll:2009sz ; Gonzalez:2011nz ; Taylor:2015glc ). In fact, as it can be readily obtain from relation (21), one obtains 666In order to recover the Lifshitz algebra, it is useful to make use of the following identity McKean1975 ; 0036-0279-31-1-R03 ; Gelfand:1975rn : In summary, the only non-vanishing commutators of the infinite set of global symmetries are given by which means that the conserved charges transform with weight under anisotropic scaling, in agreement with (62). 5 Concluding remarks We have performed a Hamiltonian reduction of General Relativity in 3D with negative cosmological constant in the case of a new family of boundary conditions, labeled by a non negative integer , which are related to the KdV hierarchy of integrable systems. We then obtained the action of the corresponding dual theory at the boundary, being such that the chiral copies of the reduced system evolve according to the potential form of the modified Korteweg–de Vries equation (56). The asymptotic symmetries in the bulk are then translated into Noether symmetries of the dual theory, giving rise to an infinite set of commuting conserved charges, that imply integrability of the system. Remarkably, the dual action is also invariant under anisotropic Lifshitz scaling with dynamical exponent . It is worth pointing out that, if left and right copies were chosen according to different members of the hierarchy, the dual action turns out to be given by where is defined in (44). The anisotropic scaling symmetry would then be generically broken unless . It is interesting to make an interpretation of our work in the context of the fluid/gravity correspondence Rangamani:2009xk ; Hubeny:2010wp ; Bredberg:2011jq . In that setup, the asymptotic behavior of the Einstein equations, in a derivative expansion at the boundary, implies that the fluid equations are recovered from the conservation of the suitably regularized Brown-York stress-energy tensor. It is then natural to wonder about the fundamental degrees of freedom and the precise form of the theory from which the fluid is made of. In our context, since Einstein gravity in 3D is devoid of local propagating degrees of freedom, the identification of the fundamental degrees of freedom at the boundary can be completely performed. Indeed, the asymptotic behavior of the Einstein equations, with the boundary conditions in Perez:2016vqo , is such that they reduce to the equations of the KdV hierarchy to all orders, i.e., without the need of performing a (hydrodynamic) derivative expansion at the boundary. Remarkably, the dynamics of the non-linear fluid, that evolves according to the KdV equations, was shown to emerge from the conservation law of left and right momentum densities, where the underlying fields are manifestly unveiled and fulfill the potential modified KdV equations. As an ending remark, it is worth mentioning that different classes of boundary conditions relating three-dimensional gravity with integrable systems have been proposed in Afshar:2016wfy ; Afshar:2016kjj ; Fuentealba:2017omf ; Melnikov:2018fhb . It would be interesting to explore whether a similar construction, as the one performed here, could be carried out in those cases. Acknowledgements.We thank Oscar Fuentealba, Wout Merbis, Alfredo Pérez, Miguel Riquelme and David Tempo, for useful discussions and comments. The work of H.G. is supported by the Austrian Science Fund (FWF), project P 28751-N2. The work of J.M. was supported by the ERC Advanced Grant “High-Spin-Grav”, by FNRS-Belgium (convention FRFC PDR T.1025.14 and convention IISN 4.4503.15). This research has been partially supported by FONDECYT grants Nº 1161311, 1171162, 1181496, 1181628, and the grant CONICYT PCI/REDES 170052. 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Troncoso, “Integrable systems with BMS Poisson structure and the dynamics of locally flat spacetimes,” JHEP 01 (2018) 148, 1711.02646. - (33) D. Melnikov, F. Novaes, A. Pérez, and R. Troncoso, “Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy,” 1808.04034.

Most recent answer: 02/21/2017 Why do heavy and light objects fall at the same speed? How fast something falls due to gravity is determined by a number known as the "acceleration of gravity", which is 9.81 m/s^2 at the surface of our Earth. Basically this means that in one second, any object 's downward velocity will increase by 9.81 m/s because of gravity. This is just the way gravity works - it accelerates everything at exactly the same rate. What you may be getting confused by is the fact that the force of gravity is stronger on heavier objects than lighter ones. Another way of thinking of this is to say that gravity has to pull harder on a heavy object than a light one in order to speed them both up by the same amount. However, in the real world, we have things like air resistance, which is why sometimes heavy things do fall faster. For example, if you drop a feather and you drop a rock, the rock will land first since the feather is slowed down more by the air. If you did the same thing somewhere where there is no air, the feather and the rock would land at exactly the same time. p.s. Although Galileo noticed that different things fall at the same rate, there was really no explanation of why until General Relativity was developed. If you would like us to try to say something about how that explanation works, we could make an attempt. mike w (published on 10/22/2007) Follow-Up #1: Why fixed gravitational acceleration? So the force of gravity pulls harder on heavier objects, and it pulls every object no matter what the mass (neglecting air resistance) toward the Earth with enough force to have it accelerate 9.81 m/s/s. But what i don't understand is how this force changes. Like how does gravity "know" how hard it needs to pull the object to make it go 9.81 m/s/s faster. And also, why does Earth have gravity and other objects do not? - Will (age 18) Let me take your second question first. It's not true that other objects lack gravity. According to Newton's theory of universal gravitation (published in 1687) absolutely every object exerts a gravitational pull on every other object. The Earth's gravity is most noticeable around here because the Earth is big. Smaller objects have smaller effects. The first direct measurement of the gravitational force between two small objects in a lab was published by Cavendish in 1798. Now we get to the trickier issue- why the gravitational acceleration depends only on the position of an object, not on its size or what it's made of. Although this was described by Galileo in about 1590, it wasn't explained until Einstein developed general relativity in 1916. Gravity is most accurately described not as a force but as a warping of the spacetime within which all things move. Each object at a particular place and time sees the same warped spacetime. If you try to describe the motion as if it were occurring in Newton's flat spacetime, as we like to do, you get the same acceleration for any slow-moving objects, because that acceleration really just is a measure of the same spacetime curvature. (published on 01/03/2013) Follow-Up #2: curving spacetime Let's say e=mc^2. That would mean that the more mass i have, the more energy i have. Since spacetime is bent by energy, i can bend spacetime more than a feather and therefore i should be able to accelerate faster to earth. Why is that wrong? - Smith (age 13) This is the relativistic version of a classical question which we just got around to. (see other follow-ups) The collision of you and the earth mainly comes from the big spacetime warping due to the earth, not the little warping due to you. However, you do a little warping yourself, and that does show up in the earth's trajectory. The feather does less. So, as you say, even ignoring air friction you and the earth would collide just barely sooner than a feather and the earth, if dropped from the same height. (published on 01/30/2013) Follow-Up #3: Earth falling toward you If you define "falling" as "the closing rate between two objects freely accelerating toward each other", assume everything is done in a perfect vacuum, then when comparing dissimilarly-weighted objects A and B and their closure rate toward the Earth, won't the heavier object actually fall faster? The acceleration imparted on objects A and B by the Earth is constant, close to 9.8m/s/s. But A and B themselves also impart acceleration on the Earth--minusculely so, but nonetheless so. If you now learn that A is a marble, and B is a marble with our sun compressed inside of it, will B *still* "fall toward the Earth" at the same rate as A? - Erik E (age 40) Monterey, CA, USA Whoops, this is one of those good questions that somehow fell through the cracks long ago. Everything you say is correct. The collision will be a tad sooner for the the heavy object, because the earth accelerates a tiny bit more toward it. (published on 07/26/2012) Follow-Up #4: heavier falls faster? This one guy told me that given enough time a heavier objet would game more speed like if it was dropped from orbit. But I don’t think that’s true, is it? He also said if you throw the two objects the heavy will hit first. but I think that has more to do with the angle and force you throw it with than gravity’s pull on them. Do think I’m right? It’s part of a bet. Thank you. - jon epperson (age 30) kennewic wa benton This too slipped through the cracks. I think the answer to the other follow-up should cover it. In practice, of course, what you notice is that air friction makes less difference for the heavier object. (published on 08/07/2012) Follow-Up #5: falling objects pulling on Earth In the answer to Follow-up #3, you said 'The collision will be a tad sooner for the the heavy object, because the earth accelerates a tiny bit more toward it'. How is that? Let the original distance between the objects and Earth be h. Suppose Earth moves towards the heavy object by a distance 'x' due to the heavy object's force, the heavy object has to travel (h-x) before they collide. But then shouldn't the lighter object (ignoring its gravitational force on earth) ALSO travel only (h-x) since the heavier object has already brought Earth x closer to both of them? - Jayadev Vijayan (age 19) Chennai, Tamil Nadu, India Yes, if both objects are dropped together then they hit the Earth at the same time. If they are dropped at different times, the heavy one is just a tiny bit quicker to hit the Earth because it pulls the Earth toward it more. (published on 03/23/2013) Follow-Up #6: times for different falling objects Following up in follow up #5. That depends on the relative position of the objects. For instance if the objects are dropped simultaneously on opposite sides of the earth then the lighter object has to travel even farther (h + x) while the heavier only traveled h. OTOH, if the objects are dropped side by side with, let's say, the heavier object to the left, then the earth would be pulled slightly to the left and thus the heavier object again reaches the earth first. - Abe (age 49) Yes, that all sounds right. (published on 10/06/2013) Follow-Up #7: Gravitatonal attraction of two unequal mass objects If two objects having same volume and shape but different masses, then which object will move towards the other or vice-versa or will both the objects move towards each other exactly the same distance? - Tathagat Bhatia (age 15) There is an equal and opposite force on each of the two objects: they will both move. Now since the acceleration of each object is inversely proportional to the mass, the lighter object will move a bit faster. If you do the arithmetic you will find that they will meet at their common center of mass. The lighter one will move a bit further than the heavier one. (published on 10/31/2013) Follow-Up #8: gravity on different weights For two objects of different masses and densities in a vacuum, say a bowling ball and a feather, wouldn't the bowling ball accelerate slower than the feather due to inertia? If f = ma and gravity is the force acting pretty much equally on each object, shouldn't their different masses facilitate different accelerations?Thanks - Nathaniel Scherrer (age 34) San Diego, California, USA The acceleration is given by a=F/m. Since F=mg, you get a=g regardless of m. So, in words, the key point is that the force is not "pretty much equal" on the different objects. It's proportional to their masses. So that means the accelerations are exactly equal. (published on 07/10/2015) Follow-Up #9: Newton vindicates Gallileo In looking for the answer to the question, "Will two objects hit the ground at the same time, regardless of their weight", I found the current webpage and the following sentence from another webpage which seem to contradict each other. This is from the other page: "Heavier things have a greater gravitational force AND heavier things have a lower acceleration. It turns out that these two effects exactly cancel to make falling objects have the same acceleration regardless of mass." Is this true? If so, won't two objects hit the ground at the same time, differences in air displacement not considered? - Father (age 55) In the absence of air friction both heavy and light objects will reach the ground at the same time. Galileo deduced this by devising clever experiments with balls rolling down inclined planes. Newton gave it his blessing by observing that a = F/M, i.e. the acceleration of an object is proportional to the force, F, on it divided by its mass, M. Furthermore the gravitational force on said object was proportional to its mass, F=Mg where g is the measurable acceration of a mass due to gravity on earth. Putting these two equations together you get a = Mg/M = g. The acceleration is independent of mass. For more information see: https://van.physics.illinois.edu/qa/listing.php?id=164 (published on 02/21/2017) Follow-up on this answer.

Census Records   Military Other States   New Titles   E-BOOKS Iberian Publishing Company's On-Line Catalog: Tazewell County Virginia Tazewell County was formed in 1800 from portions of Russell and Wythe counties. The new county was named for Henry Tazewell, United States senator from Virginia from 1794 until his death in 1799. Russell gave an additional parcel to Tazewell in 1807. Logan County, formed in 1824, took its territory from Tazewell. In 1835 a portion of Russell County was added to Tazewell, and in 1836 a single farm was transferred from Tazewell to Giles jurisdiction. The following year, in 1837 Mercer County was created from portions of Tazewell and Giles counties. Buchanan and McDowell counties took additional parts of Tazewell in 1858. After that Tazewell's boundaries reached their current position. For a better understanding of county boundary changes, see our new section Virginia in Maps TAZEWELL COUNTY, VA 1810 SUBSTITUTE CENSUS [Abstracts from the 1810 Personal Property Tax List] by John Vogt, 2011, 5 1/2"x8 1/2" format, viii, 6 pages, map. Tazewell is one of eighteen Virginia counties for which the 1810 census is lost. In August, 1814 British troops occupied Washington, DC and public buildings were put to the torch. In the destruction that followed, numerous early records of the government were lost, including all of Virginia’s 1790 and 1800 census reports, as well as eighteen county lists for the state's most recent federal census. Although two “fair copies” of each county’s census had been left in the counties for public display, these were ephemeral lists and not preserved, and by 1814 they too had been mislaid, lost, or destroyed. Hence, the closest document available we have to reconstruct a partial image of the missing county lists is the personal property tax list. According to research notes by Minor T. Weisiger, Library of Virginia archivist: “Information recorded in Virginia personal property tax records changed gradually from 1782 to 1865. The early laws required the tax commissioner in each district to record in “a fair alphabetical list” the names of the person chargeable with the tax, the names of white male tithables over the age of twenty-one, the number of white male tithables between ages sixteen and twenty-one, the number of slaves both above and below age sixteen, various types of animals such as horses and cattle, carriage wheels, ordinary licenses, and even billiard tables. Free Negroes are listed by name and often denoted in the list as “free” or “FN.” The present abstract of Tazewell's 1810 personal property tax list is NOT a transcript of the entire document; rather, it is a summary of three items important in delineating the 1810 "substitute" census for this county, i.e., number of male tithables 16 and older, number of slaves twelve years and older, and the number of horses. The original form of the census was in alphabetic order by date and letter [see example on page vi below]. The substitute list presented here is in absolute alphabetic order for easy reference. In the current volume, the data is recorded thus: Coleman, Cain                           1          -          - Coleman, Obadiah & his sons, James, John, & Anderson                           4          6        9 Coleman, William & Coleman, Whitehead        2        21        25 Column one represents the tithable males (16 and over) in the household; column 2 is the number of slaves over 12; and the final column is the number of horses, mares or mules. For genealogical researchers in this 1810 period, personal property tax records may provide additional important information. Oftentimes, juniors and seniors are listed adjacent to one another and recorded on the same day. When a taxpayer is noted as “exempt”, it can be a clue to someone holding a particular position in government or being elderly, infirm, or for some other reason no longer required to pay the tithable tax. Women, both black and white, appear occasionally as heads of households when they own property in their own right or as the widow of a property owner. Another valuable source for filling in information about an ancestor is the land tax record, and especially the one for 1815. In that year, the enumerators began to add the location of the property in relation to the county court house. Roger Ward has abstracted all of the 1815 land tax records, and they are available from this publisher at www.iberian.com. The 1810 substitute census list for Tazewell County contains 453 households, 537 tithables, both white and free black, and 173 slaves over the age of twelve, and 1,751 horses. SURNAMES included in the 1810 personal property list are: Adams; Adkins; Allen; Alsup; Arnal; Aronhart; Asbury; Ashbury; Ashby; Bailey; Balden; Baley; Ballew; Bandy; Barker; Barnet; Barns; Been; Belcher; Belsha; Belsher; Beverly; Bevers; Biggs; Blankenship; Bolen; Bostick; Boswell.; Bowen; Brooks; Brown; Bruster; Burgman; Burriss; Cambel; Carter; Cartmill; Cassady; Day; Cecil; Christian; Chriswell; Clapole; Clark; Coleman; Conley; Corder; Correl; Crage; Crawford; Crockett; Cumpton; Daly; Deskin; Davidson; Davis; Day; Deskins; Dills; Doake; Dolsbury; Dotson; Drake; Fannon; Fletcher; Flummer; Fortner; Fox; Francisco; Garrison; George; Gibson; Gillespie; Godfrey; Godfry; Golsby; Gooden; Goss; Green; Grenup; Griffith; Griffitts; Grudd; Hager; Haley; Hall; Hankins; Hanson; Harman; Harman; Harper; Harrison; Harriss; Hartwell; Havens; Hedrick; Helvy; Heninger; Hicks; Higginbotham; Higginbothum; Hinkle; Hortain; Husk; Jeffry; Jent; Johnston; Jones; Justice; Kidd; Kindle; Kindrick; King; Kirk; Kook; Laird; Lambert; Lasley; Lee; Lester; Likens; Lockhart; Lortain; Lusk; Luster; Marlow; Mars; Martain; Matney; Maxwell; McCurdy; McDowel; McGuire; Mclngtosh; McMillin; Meloney; Merman; Messersmith; Milum; Mitchel; Moor; Morgan; Murry; Neel; Newton; Nuckels; O'Danold; Oney; Ony; Owens; Patten; Peery; Perry; Pleasant; Power; Prater; Pruet; Pruett; Ratliff; Reignhart; Reyburn; Right; Robinson; Runnion; Sanders; Sawyers; Shannon; Shortridge; Simpson; Skaggs; Slater; Smith; Smythe; Steel; Stephenson; Stiltner; Stobauck; Stowers; Stump; Suiter; Suter; Swader; Taylor; Thompson; Thorn; Tifney; Todd; Tomblenson; Totten; Trent; Trout; Turner; Vandyke; Vincel; Vinsle; Waggoner; Walls; Ward; Webb; Weltch; Whit; White; Whitley; Williams; Wilson; Wilton; Witten; Workman; Wynne; Also available as a digital e-book in PDF format:        HOW TO ORDER SELECTED DEATH RECORDS OF SOUTHWEST VIRGINIANS WHO DIED IN MISSOURI (OR WERE RELATED TO THOSE WHO DIED IN MISSOURI (with additions from Iowa and Sullivan County/East Tennessee) Researched by Thomas Jack Hockett; Abstracted & compiled by by Donald W. Helton. iv,220pp., every-name index (8.25" x 10.75" paperback). These deaths are taken from a variety of sources and methods employed, including "hunt and seek", census, on-line sources at Rootsweb, Ancestry, IGI, Family Genealogy Forums, censuses, etc. and the very valuable Missouri Death Certificates 1912-1958 which are generously available online. These deaths of mid and extreme SW VA people in MO during the subject time likely represent only a fraction of the deaths which could be ferreted out with difficulty employing 2-4 sources (in conjunction) in conjunction. The work represents considerable labor (not to mention eye-strain) and it is hoped it will bolster further the efforts to document the migration of SW VA persons”. During the process of abstracting and compiling the death records listed herein, instances of conflict occurred between the certificate and additional information found on-line. The information is entered as found. Any such conflicts are left to the discretion of the reader to reconcile. Table of contents Missouri Deaths from Washington County       1 Wythe Co., Va      43 Russell Co., Va      56 Grayson/Carroll Cos., VA       80 Smyth Co., Va      85 Tazewell Co., Va      101 Lee Co., Va      118 Scott Co., Va      138 Dickenson Co., Va      151 Buchanan Co., Va      154 Miscellaneous Deaths from southwest Va.      181 Iowa Deaths from Southwest Virginia      193 Alphabetical Index      202 Tazewell Co. 1815 Directory of Landowners by Roger G. Ward. 2005. 14 pages, map, 5 1/2X8 1/2. For a full description of the 1815 LAND DIRECTORY Records and a listing of available counties, see: Individual County Booklets, 1815 Directory of Virginia Landowners For records pertaining to Tazewell COUNTY, VIRGINIA see: || Virginia/W.Va. || General Reference || Military Records || || Other States || Genealogy Links || New Titles || Home Page || E-Books || Copyright © 2014 Iberian Publishing Company

Your cart is empty Presents an in-depth analysis of geometry of part surfaces and provides the tools for solving complex engineering problems "Geometry of Surfaces: A Practical Guide for Mechanical Engineers" is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. The book is divided into three parts on Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces. "Geometry of Surfaces: A Practical Guide for Mechanical Engineers "combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert of the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering.Presents an in-depth analysis of geometry of part surfaces Provides tools for solving complex engineering problems in the field of mechanical engineeringCombines differential geometry and gearing theoryHighlights new developments in the elementary theory of enveloping surfaces Essential reading for researchers and practitioners in mechanical, automotive and aerospace engineering industries; CAD developers; and graduate students in Mechanical Engineering. This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest. This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiri Matousek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition - something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiri Matousek's numerous areas of mathematical interest. This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups. This is the final volume of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century's masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology. Einstein showed how to interpret gravity as the dynamic response to the curvature of space-time. Bill Thurston showed us that non-Euclidean geometries and curvature are essential to the understanding of low-dimensional spaces. This third and final volume aims to give the reader a firm intuitive understanding of these concepts in dimension 2. The volume first demonstrates a number of the most important properties of non-Euclidean geometry by means of simple infinite graphs that approximate that geometry. This is followed by a long chapter taken from lectures the author gave at MSRI, which explains a more classical view of hyperbolic non-Euclidean geometry in all dimensions. Finally, the author explains a natural intrinsic obstruction to flattening a triangulated polyhedral surface into the plane without distorting the constituent triangles. That obstruction extends intrinsically to smooth surfaces by approximation and is called curvature. Gauss's original definition of curvature is extrinsic rather than intrinsic. The final two chapters show that the book's intrinsic definition is equivalent to Gauss's extrinsic definition (Gauss's "Theorema Egregium" ("Great Theorem")). This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface. Isroil Ikromov and Detlef Muller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Muller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger. Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields. The book faces the interplay among dynamical properties of semigroups, analytical properties of infinitesimal generators and geometrical properties of Koenigs functions. The book includes precise descriptions of the behavior of trajectories, backward orbits, petals and boundary behavior in general, aiming to give a rather complete picture of all interesting phenomena that occur. In order to fulfill this task, we choose to introduce a new point of view, which is mainly based on the intrinsic dynamical aspects of semigroups in relation with the hyperbolic distance and a deep use of Caratheodory prime ends topology and Gromov hyperbolicity theory. This work is intended both as a reference source for researchers interested in the subject, and as an introductory book for beginners with a (undergraduate) background in real and complex analysis. For this purpose, the book is self-contained and all non-standard (and, mostly, all standard) results are proved in details. Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market! While it is well known that the Delian problems are impossible to solve with a straightedge and compass - for example, it is impossible to construct a segment whose length is cube root of 2 with these instruments - the discovery of the Italian mathematician Margherita Beloch Piazzolla in 1934 that one can in fact construct a segment of length cube root of 2 with a single paper fold was completely ignored (till the end of the 1980s). This comes as no surprise, since with few exceptions paper folding was seldom considered as a mathematical practice, let alone as a mathematical procedure of inference or proof that could prompt novel mathematical discoveries. A few questions immediately arise: Why did paper folding become a non-instrument? What caused the marginalisation of this technique? And how was the mathematical knowledge, which was nevertheless transmitted and prompted by paper folding, later treated and conceptualised? Aiming to answer these questions, this volume provides, for the first time, an extensive historical study on the history of folding in mathematics, spanning from the 16th century to the 20th century, and offers a general study on the ways mathematical knowledge is marginalised, disappears, is ignored or becomes obsolete. In doing so, it makes a valuable contribution to the field of history and philosophy of science, particularly the history and philosophy of mathematics and is highly recommended for anyone interested in these topics. This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24-25, 2015, at California State University, Fullerton, CA. Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves. The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in $S^3$ and other 3-manifolds. Designed to inform readers about the formal development of Euclidean geometry and to prepare prospective high school mathematics instructors to teach Euclidean geometry, this text closely follows Euclid's classic, "Elements. "The text augments Euclid's statements with appropriate historical commentary and many exercises -- more than 1,000 practice exercises provide readers with hands-on experience in solving geometrical problems. This book gives an introduction to the field of Incidence Geometry by discussing the basic families of point-line geometries and introducing some of the mathematical techniques that are essential for their study. The families of geometries covered in this book include among others the generalized polygons, near polygons, polar spaces, dual polar spaces and designs. Also the various relationships between these geometries are investigated. Ovals and ovoids of projective spaces are studied and some applications to particular geometries will be given. A separate chapter introduces the necessary mathematical tools and techniques from graph theory. This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry. Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end of the book. This book is aimed at graduate students and researchers in the fields of combinatorics and incidence geometry. This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden's permanent conjecture, as well as additional, highly original and delightful proofs in other chapters. From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition" "... It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [...] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty." From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Gunter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdoes. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011 In this book, Cathleen Heil addresses the question of how to conceptually understand children's spatial thought in the context of geometry education. She proposes that in order to help children develop their abilities to successfully grasp and manipulate the spatial relations they experience in their everyday lives, spatial thought should not only be addressed in written or tabletop settings at school. Instead, geometry education should also focus on settings involving real space, such as during reasoning with maps. In a first part of this book, she theoretically addresses the construct of spatial thought at different scales of space from a cognitive psychological point of view and shows that maps can be rich sources for spatial thinking. In a second part, she proposes how to measure children's spatial thought in a paper-and-pencil setting and map-based setting in real space. In a third, empirical part, she examines the relations between children's spatial thought in those two settings both at a manifest and latent level. This book features a selection of articles based on the XXXIV Bialowieza Workshop on Geometric Methods in Physics, 2015. The articles presented are mathematically rigorous, include important physical implications and address the application of geometry in classical and quantum physics. Special attention deserves the session devoted to discussions of Gerard Emch's most important and lasting achievements in mathematical physics. The Bialowieza workshops are among the most important meetings in the field and gather participants from mathematics and physics alike. Despite their long tradition, the Workshops remain at the cutting edge of ongoing research. For the past several years, the Bialowieza Workshop has been followed by a School on Geometry and Physics, where advanced lectures for graduate students and young researchers are presented. The unique atmosphere of the Workshop and School is enhanced by the venue, framed by the natural beauty of the Bialowieza forest in eastern Poland. "Edmund Husserl's Origin of Geometry": An Introduction" (1962) is Jacques Derrida's earliest published work. In this commentary-interpretation of the famous appendix to Husserl's "The Crisis of European Sciences and Transcendental Phenomenology," Derrida relates writing to such key concepts as differing, consciousness, presence, and historicity. Starting from Husserl's method of historical investigation, Derrida gradually unravels a deconstructive critique of phenomenology itself, which forms the foundation for his later criticism of Western metaphysics as a metaphysics of presence. The complete text of Husserl's Origin of Geometry is included. This open access book focuses on the interplay between random walks on planar maps and Koebe's circle packing theorem. Further topics covered include electric networks, the He-Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe's circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed. K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi-Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin-Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers. This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2, q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds, H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil boun This book showcases the synthetic problem-solving methods which frequently appear in modern day Olympiad geometry, in the way we believe they should be taught to someone with little familiarity in the subject. In some sense, the text also represents an unofficial sequel to the recent problem collection published by XYZ Press, 110 Geometry Problems for the International Mathematical Olympiad, written by the first and third authors, but the two books can be studied completely independently of each other. The work is designed as a medley of the important Lemmas in classical geometry in a relatively linear fashion: gradually starting from Power of a Point and common results to more sophisticated topics, where knowing a lot of techniques can prove to be tremendously useful. We treat each chapter as a short story of its own and include numerous solved exercises with detailed explanations and related insights that will hopefully make your journey very enjoyable. Sasha Wang revisits the van Hiele model of geometric thinking with Sfard's discursive framework to investigate geometric thinking from a discourse perspective. The author focuses on describing and analyzing pre-service teachers' geometric discourse across different van Hiele levels. The explanatory power of Sfard's framework provides a rich description of how pre-service teachers think in the context of quadrilaterals. It also contributes to our understanding of human thinking that is illustrated through the analysis of geometric discourse accompanied by vignettes. In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results. Geometry does not have to be confusing! Inside Mathematics: Geometry helps make sense of all of those lines and angles by showing its fascinating origins and how that knowledge is applied in everyday life. Written to engage and enthuse young minds, this accessible overview introduces readers to the amazing people who figured out how shapes work and how they can be used to build spaces and study places we cannot go, like the beginning of the Universe. Filled with enlightening illustrations and images, Geometry is arranged chronologically, from Euclid's revolution to the Poincare conjecture, to clearly show how ideas in mathematics evolved from the Ancient Egyptians in 3000 BC to the present day. What began as scratched circles and squares in the dirt has evolved into a branch of mathematics used to create realistic landscapes in video games, build mile-high skyscrapers, and manufacture robots so tiny they can swim in your bloodstream. This book presents a selection of papers based on the XXXIII Bialowieza Workshop on Geometric Methods in Physics, 2014. The Bialowieza Workshops are among the most important meetings in the field and attract researchers from both mathematics and physics. The articles gathered here are mathematically rigorous and have important physical implications, addressing the application of geometry in classical and quantum physics. Despite their long tradition, the workshops remain at the cutting edge of ongoing research. For the last several years, each Bialowieza Workshop has been followed by a School on Geometry and Physics, where advanced lectures for graduate students and young researchers are presented; some of the lectures are reproduced here. The unique atmosphere of the workshop and school is enhanced by its venue, framed by the natural beauty of the Bialowieza forest in eastern Poland.The volume will be of interest to researchers and graduate students in mathematical physics, theoretical physics and mathematmtics. You may like... 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Research Article | Open Access Tien-Chung Hu, Neville C. Weber, "A Note on Strong Convergence of Sums of Dependent Random Variables", Journal of Probability and Statistics, vol. 2009, Article ID 873274, 7 pages, 2009. https://doi.org/10.1155/2009/873274 A Note on Strong Convergence of Sums of Dependent Random Variables For a sequence of dependent square-integrable random variables and a sequence of positive constants , conditions are provided under which the series converges almost surely as . These conditions are weaker than those provided by Hu et al. (2008). 1. Introduction and Results Let be a sequence of square-integrable random variables defined on a probability space and let be a sequence of positive constants. The random variables are not assumed to be independent. Past research has focussed on conditions that ensure the strong convergence of two distinct but related series: If the second sequence converges to 0 almost surely, then is said to obey the strong law of large numbers (SLLN). Assume that there exists a sequence of constants such that Our interest is in conditions on the growth rates of , , and which imply strong convergence of the above series. There is an extensive literature on strong laws for independent random variables. Strong laws have been derived for various dependence structures such as negative association (e.g., Kuczmaszewska ), quasi-stationarity (e.g., Móricz , Chobanyan et al. ), and orthogonality (e.g., Stout ). Hu et al. focus on the strong convergence of the series without imposing strong conditions on the nature of the variances and covariances. Our aim is to weaken their condition on the covariances and establish the following theorem. Theorem 1.1. Let be a sequence of square-integrable random variables and suppose that there exists a sequence of constants such that (1.2) holds. Let be a sequence of positive constants. Assume that there exists a constant such that, for all , Suppose that Then To motivate the general nature of our result consider the following example. Let be a sequence of zero mean random variables where where is a stationary time series with autocovariance function and is a sequence of independent, zero mean random variables distributed independently of . Let Var Thus what we observe is an underlying stationary series disturbed by a noise process with variance that can depend on We have Var and Cov, Condition (3.1) in Theorem of Hu et al. , which is the same as (1.4), is a constraint on the values whereas their condition (3.2) is a constraint on . In Chapter 2 of Stout the condition on the variances is shown to be close to optimal for sequences of orthogonal random variables. Lyons provides an SLLN for random variables with bounded variances under the condition One might conjecture that the condition (1.8) could be relaxed to The above theorem, whilst allowing for far more general models than (1.7), moves us closer to this constraint on the values. For long range dependent stationary processes we have where and is a slowly varying function. Theorem 1.1 enables the strong convergence result to be extended to processes where the correlation decays at a slower rate than for Applying Kronecker's lemma the strong law of large numbers result is an immediate consequence of the above theorem. Corollary 1.2. Under the conditions of Theorem 1.1, if is monotone increasing, the strong law of large numbers holds, that is, There are strong law results under weaker conditions than (1.5) but with stronger conditions on the variance (see, e.g., Lyons , Chobanyan et al. ). Both papers show that if the summands have bounded variance, then (1.5) can be weakened to Our approach focusses on the convergence of the series in (1.6) and relies on Kronecker's Lemma to obtain the strong law. If the aim is purely to obtain the SLLN, then alternative conditions might be possible as it is possible to construct sequences and such that but diverges. For example, take and Thus we can have the strong law holding but the series in (1.6) diverging. Throughout this paper, the symbol denotes a generic constant which is not necessarily the same at each appearance. We first prove a number of lemmas that enable us to obtain tighter bounds for key expressions in the proof of Theorem of Hu et al. . Proof. For all , , Lemma 2.2. For , Proof. Note that is an increasing function for Thus, for , Hence for , Lemma 2.3. For define Then , and, in general, Proof. The result for is the sum of a standard geometric progression. The general result follows by noting Thus Proof of Theorem 1.1. We will follow the method of proof in Theorem in Hu et al. . To prove (1.6) we first show that is a Cauchy sequence for convergence in which will imply convergence in probability. Using Lemmas 2.1 and 2.2, Therefore there exists a random variable such that Next we will show that a.s. Let be arbitrary. Note where the last line follows by using (1.4) and (1.5). Thus by the Borel Cantelli lemma almost surely. To finish the proof we utilize the generalization of the Rademacher-Menchoff maximal inequality given by Serfling and argue as in Hu et al. . It is sufficient to show that, for any , Using Serfling's inequality and (3.8) from Hu et al. - A. Kuczmaszewska, “The strong law of large numbers for dependent random variables,” Statistics & Probability Letters, vol. 73, no. 3, pp. 305–314, 2005. - F. Móricz, “The strong laws of large numbers for quasi-stationary sequences,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 38, no. 3, pp. 223–236, 1977. - S. Chobanyan, S. Levental, and H. Salehi, “Strong law of large numbers under a general moment condition,” Electronic Communications in Probability, vol. 10, pp. 218–222, 2005. - W. F. Stout, Almost Sure Convergence, Academic Press, New York, NY, USA, 1974. - T.-C. Hu, A. Rosalsky, and A. I. Volodin, “On convergence properties of sums of dependent random variables under second moment and covariance restrictions,” Statistics & Probability Letters, vol. 78, no. 14, pp. 1999–2005, 2008. - R. Lyons, “Strong laws of large numbers for weakly correlated random variables,” The Michigan Mathematical Journal, vol. 35, no. 3, pp. 353–359, 1988. - R. J. Serfling, “Moment inequalities for the maximum cumulative sum,” Annals of Mathematical Statistics, vol. 41, pp. 1227–1234, 1970. Copyright © 2009 Tien-Chung Hu and Neville C. Weber. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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- Research Article - Open Access Quadratic-Quartic Functional Equations in RN-Spaces © M. Eshaghi Gordji et al. 2009 - Received: 20 July 2009 - Accepted: 3 November 2009 - Published: 1 December 2009 We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms . - Banach Space - Vector Space - General Solution - Abelian Group - Functional Equation The stability problem of functional equations originated from a question of Ulam in concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all and some Then there exists a unique additive mapping such that for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Rassias provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. In Gajda answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]). The functional equation is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 13]). The biadditive mapping is given by The Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see ). Cholewa noticed that the theorem of Skof is still true if relevant domain is replaced an abelian group. In , Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec has generalized the results mentioned above. In , Park and Bae considered the following quartic functional equation In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also ). In addition, Kim has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation. The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm . The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary continuous -norms. In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and . Also, is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by Definition 1.1 (see ). A mapping is a continuous triangular norm (briefly, a continuous -norm) if satisfies the following conditions: (a) is commutative and associative; (b) is continuous; (c) for all ; (d) whenever and for all . Typical examples of continuous -norms are , and (the Lukasiewicz -norm). Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , then is defined recurrently by and for . is defined as It is known that for the Lukasiewicz -norm, the following implication holds: Definition 1.2 (see ). A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that the following conditions hold: (RN1) for all if and only if ; (RN2) for all , ; (RN3) for all and for all and is the minimum -norm. This space is called the induced random normed space. Let be an RN-space. A sequence in is said to be convergent to in if, for every and , there exists a positive integer such that whenever . A sequence in is called a Cauchy sequence if, for every and , there exists a positive integer such that whenever . An RN-space is said to be complete if and only if every Cauchy sequence in is convergent to a point in . Theorem 1.4 (see ). If is an RN-space and is a sequence such that , then almost everywhere. In this paper, we deal with the following functional equation: on RN-spaces. It is easy to see that the function is a solution of (1.9). In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces. We need the following lemma for solution of (1.9). Throughout this section, and are vector spaces. If a mapping satisfies (1.9) for all then is quadratic-quartic. We show that the mappings defined by and defined by are quadratic and quartic, respectively. Letting in (1.9), we have . Putting in (1.9), we get . Thus the mapping is even. Replacing by in (1.9), we get for all . Therefore, the mapping is quadratic. To prove that is quartic, we have to show that for all . Therefore, the mapping is quartic. This completes the proof of the lemma. for all . for all The proof of the converse is obvious. Throughout this section, assume that is a real linear space and is a complete RN-space. for all and all Since the right-hand side of the inequality (3.17) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all . Now we show that is a quadratic mapping. Replacing with and in (3.1), respectively, we get Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1, the mapping is quadratic. Letting the limit as in (3.16), we get (3.3) by (3.10). Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping which satisfies (3.3). Since for all and all from (3.3), it follows that for all and all . Letting in (3.19), we conclude that , as desired. for all and all Since the right-hand side of (3.36) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all . Now we show that is a quartic mapping. Replacing with and in (3.20), respectively, we get Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1 we get that the mapping is quartic. Letting the limit as in (3.35), we get (3.22) by (3.29). Finally, to prove the uniqueness of the quartic mapping subject to let us assume that there exists a quartic mapping which satisfies (3.22). Since and for all and all from (3.22), it follows that for all and all . Letting in (3.38), we get that , as desired. for all and all for all and all . Hence we obtain (3.41) by letting and for all The uniqueness property of and is trivial. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). - Ulam SM: Problems in Modern Mathematics. 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Time value of money refers to the fact that a money in pocket today is worth for one year at 100 percent, and we simply mean that $1 today is worth $2 in one In general, if you invest for one period at an interest rate of r, your investment will Time-Value-of-Money (TVM): TI-BA II PLUS Present Value of a single sum. word Enter appears in the display, it means you can enter a different interest rate. In addition, they usually contain a limited number of choices for interest rates we will demonstrate how to find the present value of a single future cash amount, Often, the discount rate is some interest rate that represents the individual's best alternative use for money today. The formula for calculating the present value of 11 Mar 2020 Interest rate used to calculate Net Present Value (NPV) value of cash outflows over a period of time and is represented above by “CF”). Welcome to the lecture series on Time value of money-Concepts and Calculations. Disbursements are represented by arrows directed downward. annual interest rate and cash flow at the end of the year is given and we will call this type. 29 May 2014 Interest rate is the exchange price between the current and future value of the Afghani. 2. Interest rates represent risk and inflation. 4. 4 Time So at the most basic level, the time value of money demonstrates that all things being equal, it seems better to have money now rather than later. But why is this? A $100 bill has the same value What does the time value of money (TVM) mean? Time value simply means that if an investor is offered the choice between receiving $1 today or receiving $1 in the future, the proper choice will always be to receive the $1 today, because that $1 can be invested in some opportunity that will earn interest. This is the value of the formulas for the present value and the future value of money! Interest Rate Conversions. In investments, pricing and returns are often expressed in interest rates that are compounded in specific time intervals. The actual interest rate or yield will depend on the compounding period. The importance of the Time of Value of Money. Almost everything in life involves the time value of money. If you buy a car on credit, take out a mortgage, or invest in stocks. It all involves the time value of money. If you work for a company, every decision the company makes will involve, in one way or another, the time value of money. Time value of money. Or another way to think about it is, think about what the value of this money is over time. Given some expected interest rate and when you do that you can compare this money to equal amounts of money at some future date. Now, another way of thinking about the time value or, I guess, another related concept to the time value There are technical differences, but both represent a rate of increase in the time value of money. So if the interest rate describes the time value of money, then the higher it is, the more valuable money is in your hands and the less valuable money is down the road. There are multiple reasons that money can be more valuable today than tomorrow. Whenever you are solving any time value of money problem, make sure that the n (number of periods), the i (interest rate), and the PMT (payment) components are all expressed in the same frequency. For example, if you are using an annual interest rate, then the number of periods should also be expressed annually. Time value of money refers to the fact that a money in pocket today is worth for one year at 100 percent, and we simply mean that $1 today is worth $2 in one In general, if you invest for one period at an interest rate of r, your investment will Adjusting for "inflation" in the past is not remotely the same as calculating the present or future value of money for a given interest rate. Adjusting for inflation is a Unit 2: Time Value of Money: Future Value, Present Value, and Interest Rates Also, Unit 2 exposes the concept of interest rates and how to apply them This video shows you about what it means to use an annual interest rate continuously. 19 Nov 2014 One, NPV considers the time value of money, translating future cash flows 4% interest on its debt, then it may use that figure as the discount rate. is based on several assumptions and estimates, which means there's lots of Interest rate (I) - This is the growth rate of your money over the lifetime of the investment. It is stated in a percentage value, such as 8% or .08. Payment amount ( Path to financial security and time value of money. payment, we have the number of periods, and we have the interest rate, which is represented by r here. The time value of money is a basic financial concept that holds that money in the present is worth more than the same sum of money to be received in the future. This is true because money that you have right now can be invested and earn a return, thus creating a larger amount of money in the future. There are technical differences, but both represent a rate of increase in the time value of money. So if the interest rate describes the time value of money, then the higher it is, the more valuable money is in your hands and the less valuable money is down the road. There are multiple reasons that money can be more valuable today than tomorrow. The time periods may represent years, months, days, or any length of time so long as Compounding is the impact of the time value of money (e.g., interest rate)

Why are we afraid of math? It is one of the most useful tools we could ever learn. It's like being afraid of your car. Imagine where you would be without that and you have some idea of what it's like to go through life being afraid of simple math. Here are some good reasons to learn math: to figure out how much you need to earn to 1) move out of your parent's house; 2) move closer to the beach; 3) move in with your boyfriend or girlfriend; 4) buy your own island in the South Pacific to figure out how many months it will take you to save up enough money to 1) buy the new Blink 182 CD; 2) buy the Pink Floyd collection; 3) buy every Frank Sinatra record ever made; 4) start your own record company to figure out how long it will take you to 1) drive to Vegas in your Dad's Dodge Dart; 2) drive to Reno in your Mom's new BMW; 3) drive to Nashville for the Country Music Awards in your rented Chevy Tracker; 4) drive across the ocean in a stolen Humvee to figure out the percentage of your time that you spend 1) thinking about moving out; 2) trying to sing like Frank Sinatra; 3) begging your mom to let you drive her new BMW; 4) surfing the web instead of doing your oceanography homework. Math surrounds us. It involves nearly everything we do on a daily basis. Don't believe me? Take this questionnaire. The Do You Need to Know Math Questionnaire? If you answer yes to any one of these questions, then you need to know math. Do you ever shop (for anything)? Have you ever dreamed of being rich and famous? Have you ever had or ever want to have sex? Do you want to pass this oceanography course? I bet you answered yes to one of those questions. If not then I'm wrong. But regardless, the only really important question to your life right now should be the last one (okay, yes, I'm joking, shopping is important, too). And while I will tell you right here and now that you could still make a decent grade and not know any math, imagine how fine your life will be by knowing a little math! These math questions are designed to prepare you a little for some of the kinds of math that we will encounter in this course. Rather than wait until we get to those parts of the semester and let you freak out, I'm trying to get the freak going right now so that by the time we get to those important math-oriented sections of the course, you will have calmed down a bit and sharpened your skills enough so that those sections are a piece of cake. Now, the kind of math you need to know is really quite simple and fairly limited. This isn't a math class. But I hate that deer-in-the-headlights look when I teach tides or waves (one of my favorites and yours) and ask students to 1) add; 2) subtract; 3) multiply; 4) divide; 5) change units; 6) manipulate simple equations. For some reason, many students are simply stunned by these simple math problems. I'm not going to teach you basic math here. Rather, this section is designed to let you determine whether you need help with math. If you don't understand these simple problems, then seek help. Come talk to me or enroll in a math class. Find a tutor. Buy a basic math book. Please. Don't rob yourself of one of the most powerful tools known to man. You can bet that banks and credit card companies would love it if no one knew math. Don't let them get the best of you. You will see versions of these math puzzles on virtually every exam. Make sure you work through them and understand them. 1. You are an ordinary seaman on a ship located at the intersection of the equator and the international date line. You desperately want to move up in ranks because you are tired of cleaning toilets. Your chance comes when the navigator falls overboard while looking at jellyfish. His last words are "One degree of latitude equals 60 nautical miles." A day goes by. The Captain tells you that the ship has sailed sixty (60) nautical miles due east since the tragic accident. If you can tell him the ship's new position, you get the job. What is the ship's new longitude and latitude? (Hint: think about what you need to know to answer this question. Look back through your notes to find the key information.) 2. The Captain, a demanding sort, now wants to know the depth of the water. He gives you a Toys-R-Us Sonar Unit, good enough to report the time it takes a sound pulse to travel from the ship to the bottom and back, but not good enough to calculate the distance. You remember from your oceanography class at Fullerton College that sound travels at approximately 1500 meters per second. You set up the Sonar unit, hit the go button, it pings and 12 seconds later the ping returns. How deep is the bottom in meters? 3. The Captain has an old chart with soundings in miles. Now he wants to know how deep is the bottom in miles. The ship hasn't moved since your first sounding and the ping returns after 12 seconds. How deep is the bottom in miles? 4. The Captain gets a weather report over the weather fax. A storm north of you covers a rectangular area 1000 miles long and 200 miles wide. To learn something about the kinds of waves generated by the storm, the Captain wants to estimate the area over which the storm blows, something known as the fetch. He asks you to calculate the area of the storm given the information in the weather fax. What is the fetch of the storm in square miles? 5. The storm front is approximately 3409 miles due north of the ship. It generates waves traveling due south at 50 feet per second. How many days will it take the waves to reach the ship? 6. The Captain hands you an obscure equation and gives you no clue as to what it means. S=L/T or S equals L divided by T. He asks you to give him a new equation that expresses T in terms of S and L, or T=? What does T equal? (In other words, solve this equation for T.) 7. Woops. He made a mistake. He wants you to solve it for L, or L=? What does L equal, in terms of S and T? 8. The Captain finally reveals to you that this is a speed equation for waves, where S, the speed of the wave is equal to L/T (he won't tell you what they mean!). He reminds you that the waves from the storm are traveling at 50 feet per second and he tells you that T=10 seconds. What does L equal? (Don't forget the units.) 9. The ship finally makes it near shore. The Captain wants to know whether the tides will affect the ship's entry through the harbor. He asks you to compute the difference between the high tide and the low tide. He tells you that the high tide is 6 feet above sea level and that the low tide is 1 foot below sea level (or -1 foot). Compute the difference: 6 - (-1) = ? 10. You finally get off the ship and are dying to go to Vegas but you only have $50 bucks. You can drive your Dodge Dart, which gets 20 miles to the gallon and maxes out at 50 miles per hour; or you can drive your mom's new BMW, which only gets 10 miles to the gallon but goes 150 miles per hour. Gas will cost you a buck a gallon but food will cost you $10 every three hours. Vegas is 400 miles from your port. Which car do you take? Remember, you want the most money possible to gamble in Vegas. The answers to the Oceanography Math Puzzles will be discussed in the Forums. These web sites have some great and simple math problems that will really help sharpen your math skills. I suggest giving them a try if you felt a little rusty with the above problems. Don't feel bad if you have to venture into lower K-12 to find problems that you can solve. That's what these math lessons are all about. Please feel free to e-mail me if you have any questions. Girls to the Fourth Power University of Hawaii Quiz Center Marmalade_Man's Fantastic Math Tricks Dave's Math Tables The Math Forum Frequently Asked Questions about Blackjack

(See Chapter 7 of Mano's Digital Design (2nd ed.)) 7-1 Include a 2-input NAND gate with the register of Fig. 7-1 and connect the gate output to the CP inputs of all the flip-flops. One input of the NAND gate receives the clock pulses from the clock-pulse generator. The other input of the NAND gate provides a parallel-load control. Explain the operation of the modified register. Some assumption about the behavior of the flip-flops must be made before we can determine the function of this register. In accordance with the discussion given in the second paragraph on page 259, it is assumed that the flip-flop changes its state on the positive edge of a clock pulse. That means each flip-flop will respond to its input when a clock pulse is ended. That implies that the new content will not be available until the arrival of the next clock pulse. Therefore, we may say that this register allows parallel input, and provides one unit of time delay. 7-2 Change the synchronous-clear circuit in the register of Fig. 7-2. The modified register will have a parallel-load capability and a synchronous-clear capability, but no asynchronous-clear circuit. The retister is cleared synchronously when the clock pulse in the CP input goes through a negative transition provided R = 1 and S = 0 in all the flip-flops. It can be done in many ways. One possibility is to modify each flip-flop control circuit as indicated below: 7-4 Design a sequential circuit whose state diagram is given in Fig. 6-31 using a 3-bit register and a 16 x 4 ROM. Connect a 3-bit register and a 16x4 ROM as depicted below. Treat each bit of the register as a D flip-flop and design the circuit as usual. 7-6 What is the difference between a serial and parallel transfer? Explain how to convert serial data to parallel and parallel data to serial. What type of register is needed?. In a serial transfer, the data is transferred in sequence one bit at a time (per clock period, if it is synchronous), whereas in a parallel transfer, all bits are transferred at the same time. A shift register can be used to do serial to parallel or parallel to serial transfer as depicted below. 7-12 The 2's complement of a binary number can be formed by leaving all least significant 0's and the first 1 unchanged and complementing all other higher significant bits. Design a serial 2's complementer using this procedure. The circuit needs a shift register to store the binary number and an RS flip-flop to be set when the first least significant 1 occurs. An exclusive-OR gate can be used to transfer the unchanged bits or complement the bits. 7-13 Draw the logic diagram of a 4-bit binary ripple counter using flip-flops that triggle on the positive-edge transition. It should be drawn as in Fig. 7-12, except that Q' output (instead of Q output) should be used to drive the next flip-flop. 7-14 Draw the loogic diagram of a 4-bit binary ripple down-counter using the following: (a) Flip-flops that trigger on the positive-edge transition of the clock. (b) Flip-flops that trigger on the negative-edge transition. (a) Same as Fig. 7-12. (b) Same as Fig. 7-12 except Q' instead of Q is used to drive the next flip-flop. 7-15 Construct a BCD ripple counter using a 4-bit binary ripple counter that cann be cleared asynchronously and an external NAND gate. . Connect the output of the following NAND gate to the CLEAR input of every flip-flop. 7-19 Design a 4--bit binary ripple counter with D flip-flops. Use D flip-flops to construct T flip-flops as discussed elsewhere. Then interconnect the T flip-flops so obtained as depicted in Fig. 7-12. 7-21 Modify the counter of Fig. 7-18 so that when both the up and down control inputs are equal to 1, the counter does not change state, but remains in the same count. 7-22 Verify thhe flip-flop input functions of the synchronous BCD counter specified in Table 7-5. Draw the logic diagram of the BCD counter and include a count-enable control input. (1) Construct the Karnaugh map of each input function and simplify it. (2) Provide the count-enable input by using the following circuit. 7-25 Construct a BCD counter using the circuit specified in Fig. 7-19 and an AND gate. 7-26 Construct a mod-12 counter using the circuit of Fig. 7-19. Give two alternatives. 7-27 Using two circuits of the type shown in Fig. 7-19, construct a binary counter that counts from 0 throught binary 64. 7-28 Using a start signal as in Fig 7-21, construct a word-time control that stays on for a period of 16 clock pulses. 7-29 Add four two-input AND gates to the circuit of Fig. 7-22(b). One input in each gate is connected to one output of the decoder. The other input in each gate is connected to the clock. Label the outputs of the AND gate as P0, P1, P2, P3. Show the timing diagrm of the four P outputs. 7-30 Show the circuit and the timing diagram for generating six repeated timing signals, T0 through T5. 7-31 Complete the design of a Johnson counter showing the outputs of the eight timing signals using eight AND gates. 7-32 Construct a Johnson counter for ten timing signals. COSC 3410 Answers to Selected Problems, Chapter | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Research Article | Open Access Ximin Liu, Ning Zhang, "Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times", Advances in Mathematical Physics, vol. 2018, Article ID 4523512, 6 pages, 2018. https://doi.org/10.1155/2018/4523512 Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula. In recent years, spacelike hypersurfaces in Lorentzian manifolds have been deeply studied not only from their mathematical interest, but also from their importance in general relativity. Particularly, there are many articles that study spacelike hypersurfaces in weighted warped product space-times. A weighted manifold is a Riemannian manifold with a measure that has a smooth positive density with respect to the Riemannian one. More precisely, the weighted manifold associated with a complete -dimensional Riemannian manifold and a smooth function on is the triple , where stands for the volume element of . In this setting, we will take into account the so-called Bakry-Émery Ricci tensor (see ) which as an extension of the standard Ricci tensor , which is defined by Therefore, it is natural to extend some results of the Ricci curvature to analogous results for the Bakry-Émery Ricci tensor. Before giving more details on our work we present a brief outline of some recent results related to our one. In , Wei and Wylie considered the complete -dimensional weighted Riemannian manifold and proved mean curvature and volume comparison results on the assumption that the -Bakry-Émery Ricci tensor is bounded from below and or is bounded. Later, Cavalcante et al. researched the Bernstein-type properties concerning complete two-sided hypersurfaces immersed in a weighted warped product space using the appropriated generalized maximum principles. Moreover, obtained new Calabi-Bernstein’s type results related to complete spacelike hypersurfaces in a weighted GRW space-time. More recently, some rigidity results of complete spacelike hypersurfaces immersed into a weighted static GRW space-time are given in . In this paper we study spacelike hypersurfaces in a weighted generalized Robertson-Walker (GRW) space-times. Moreover, a GRW space-time is a space-time regarding a warped product of a negative definite interval as a base, a Riemannian manifold as a fiber, and a positive smooth function as a warped function. Furthermore, there exists a distinguished family of spacelike hypersurfaces in a GRW space-time, that is, the so-called slices, which are defined as level hypersurfaces of the time coordinate of the space-time. Notice that any slice is totally umbilical and has constant mean curvature. We have organized this paper as follows. In Section 2, we introduce some basic notions to be used for spacelike hypersurfaces immersed in weighted GRW space-times. In Section 3, we prove some uniqueness results of spacelike hypersurface in a weighted GRW space-time under appropriate conditions on the weighted mean curvature and the weighted function by using the generalized Omori-Yau maximum principle or the weak maximum principle. Finally, in Section 4, applying the weak maximum principle, we obtain some rigidity results for the special case when the ambient space is static. Let be a connected -dimensional oriented Riemannian manifold and be an open interval in endowed with the metric . We let be a positive smooth function. Denote to be the warped product endowed with the Lorentzian metric where and are the projections onto and , respectively. This space-time is a warped product in the sense of , with fiber , base , and warping function . Furthermore, for a fixed point , we say that is a slice of . Following the terminology used in , we will refer to as a generalized Robertson-Walker (GRW) space-time. Particularly, if the fiber has constant section curvature, it is called a Robertson-Walker (RW) space-time. Recall that a smooth immersion of an -dimensional connected manifold is called a spacelike hypersurface if the induced metric via is a Riemannian metric on , which will be also denoted for . In the following, we will deal with two particular functions naturally attached to spacelike hypersurface , namely, the angle (or support) function and the height function , where is a (unitary) timelike vector field globally defined on and is a unitary timelike normal vector field globally defined on . Let and stand for gradients with respect to the metrics of and , respectively. By a simple computation, we have Therefore, the gradient of on is Particularly, we have where denotes the norm of a vector field on . Now, we consider that a GRW space-time is endowed with a weighted function , which will be called a weighted GRW space-time . In this setting, for a spacelike hypersurface immersed into , the -divergence operator on is defined by where is a tangent vector field on . For a smooth function , we define its drifting Laplacian by and we will also denote such an operator as the -Laplacian of . According to Gromov , the weighted mean curvature or -mean curvature of is given by where is the standard mean curvature of hypersurface with respect to the Gauss map . It follows from a splitting theorem due to Case (see Theorem ) that if a weighted GRW space-time is endowed with a bounded weighted function such that for all timelike vector fields on , then must be constant along . In the same spirit of this result, in the following we will consider weighted GRW space-times whose weighted function does not depend on the parameter ; that is, . Moreover, for simplicity, we will refer to them as . In the following, we give some technical lemmas that will be essential for the proofs of our main results in weighted GRW space-times (for further details on the proof, see Lemma in ). Lemma 1. Let be a spacelike hypersurface immersed in a weighted GRW spacetime , with height function . Then, If we denote as the space of the integrable functions on with respect to the weighted volume element , using the relation of and Proposition in , we can obtain the following extension of a result in . Lemma 2. Let be a smooth function on a complete weighted Riemannian manifold with weighted function such that does not change sign on . If , then vanishes identically on . In the following, we will introduce the weak maximum principle for the drifted Laplacian. By the fact in , that is, the Riemannian manifold satisfies the weak maximum principle if and only if is stochastically complete, we can have the next lemma which extended a result of . Lemma 3. Let be an -dimensional stochastically complete weighted Riemannian manifold and be a smooth function which is bounded from below on . Then there is a sequence of points such that Equivalently, for any smooth function which is bounded from above on , there is a sequence of points such that 3. Uniqueness Results in Weighted GRW Space-Times In this section, we will state and prove our main results in weighted GRW space-times . We point out that, to prove the following results, we do not require that the -mean curvature of the spacelike hypersurface is constant. Recall that a slab of a weighted GRW spacetime is a region of the type Theorem 4. Let be a weighted GRW spacetime which obeys . Let be a complete spacelike hypersurface that lies in a slab of . If the -mean curvature satisfies and , then is a slice of . Proof. From (10), we have By the hypotheses, we have . Moreover, since lies in a slab, there is a positive constant such that Therefore, we can apply Lemma 2 to get ; that is, is constant. Therefore is a slice. Theorem 5. Let be a weighted GRW spacetime which obeys . Let be a complete spacelike hypersurface that lies in a slab of . If the -mean curvature satisfies and , then is a slice of . Proof. By a similar reasoning as in the proof of Theorem 4, we have where the last inequality is due to . Taking into account the assumptions, we have . Now in the same argument as in Theorem 4, we have that is a slice. Next, we will use the weak maximum principle to study the rigidity of the spacelike hypersurfaces in weighted GRW space-times. Theorem 6. Let be a weighted GRW spacetime which satisfies and there is a point such that . Let be a stochastically complete constant -mean curvature spacelike hypersurface such that , which is contained in a slab; then is -maximal. In addition, if is complete and , then is a slice. Proof. We take the Gauss map of the hypersurface such that ; from (7) we have . By Lemma 3, the weak maximum principle for the drifted Laplacian holds on ; then there exist two sequences such that On the other hand, from (9), we have Since lies in a slab, if is bounded from below, then Moreover, if is bounded from above, we get Considering that the function is increasing, then Hence, ; that is, is a -maximal spacelike hypersurface. Using (10), we have In the following, by the same argument as in Theorem 4, we have that is a slice. 4. Weighted Static GRW Space-Times In this section, we obtain some rigidity results of stochastically complete hypersurfaces in weighted static GRW space-times by the weak maximal principle. Firstly, we give the following technical result which extended the corresponding conclusion in . Lemma 7. Let be a stochastically complete Riemannian manifold and be a nonnegative smooth function on . If there exists a positive constant such that , then . Theorem 8. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW spacetime . Assume that for some positive constant and the weighted function is convex. If for some constant , then is a slice. Proof. Let be a (local) orthonormal frame in ; using the Gauss equation, we have that for . Moreover, we also have where is the sectional curvature of the fiber and and are the projections of the tangent vector fields and onto . By a direct computation and considering the hypothesis , we get Substituting (25) into (23), Furthermore, taking into account that the weighted function is convex, we have Therefore, In particular, we have Now we recall the Bochner-Lichnerowicz formula (see ): From the fact that is a constant, we have By , we get Using (29), (31), and (32) in (30), we have Finally, considering the hypothesis , we obtain Thus, there is a positive constant such that Therefore, is constant by Lemma 7. Theorem 9. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW space-time . Assume that the sectional curvature is nonnegative and the weighted function is convex. If is bounded from above, then is -maximal. Proof. As in the proof of Theorem 8, taking into account that the hypothesis is nonnegative, there is a constant such that Moreover, considering the relation , we have Using (9) and (37), we obtain By the hypothesis that is bounded from above, applying Lemma 3, the weak maximum principle, we get Therefore is -maximal. As a consequence of the proof of Theorem 8, we can get the following corollary. Corollary 10. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW space-time . Assume that and for some positive constants and . If for some constant , then is a slice. Conflicts of Interest The authors declare that they have no conflicts of interest. This work is supported by National Natural Science Foundation of China (no. 11371076). - D. Bakry and M. Mery, “Diffusions hypercontractives,” in Sminaire de probabilits, XIX, vol. 1123 of Lecture Notes in Math., pp. 177–206, Springer, Berlin, Germany, 1983. - G. Wei and W. Wylie, “Comparison geometry for the Bakry-Emery Ricci tensor,” Journal of Differential Geometry, vol. 83, no. 2, pp. 377–405, 2009. - M. P. Cavalcante, H. F. de Lima, and M. S. Santos, “On Bernstein-type properties of complete hypersurfaces in weighted warped products,” Annali di Matematica Pura ed Applicata. Series IV, vol. 195, no. 2, pp. 309–322, 2016. - M. P. Cavalcante, H. F. de Lima, and M. S. Santos, “New Calabi-Bernstein type results in weighted generalized Robertson-Walker spacetimes,” Acta Mathematica Hungarica, vol. 145, no. 2, pp. 440–454, 2015. - H. F. de Lima, A. M. Oliveira, and M. S. Santos, “Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature,” Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, vol. 57, no. 3, pp. 623–635, 2016. - B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, San Diego, Calif, USA, 1983. - L. J. Alas, A. Romero, and M. 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Setti, “A remark on the maximum principle and stochastic completeness,” Proceedings of the American Mathematical Society, vol. 131, no. 4, pp. 1283–1288, 2003. - M. Rimoldi, Rigidity Results for Lichnerowicz Bakry-Émery Ricci Tensors [Ph.D. thesis], Università degli Studi di Milano, Milano, Italy, 2011. - J. M. Latorre and A. Romero, “Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes,” Geometriae Dedicata, vol. 93, pp. 1–10, 2002. Copyright © 2018 Ximin Liu and Ning Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Proper velocity In flat spacetime, proper-velocity is the ratio between distance traveled relative to a reference map-frame (used to define simultaneity) and proper timeτ elapsed on the clocks of the traveling object. It equals the object's momentum p divided by its rest mass m, and is made up of the space-like components of the object's four-vectorvelocity. William Shurcliff's monograph [W. A. Shurcliff (1996) "Special relativity: the central ideas" (19 Appleton St, Cambridge MA 02138)] mentioned its early use in the Sears and Brehme text [Francis W. Sears & Robert W. Brehme (1968) "Introduction to the theory of relativity" (Addison-Wesley, NY) [http://catalog.loc.gov/webvoy.htm LCCN 680019344] , section 7-3] . Fraundorf has explored its pedagogical value [P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics" ( [http://xxx.lanl.gov/abs/physics/9611011 arXiv:physics/9611011] )] while Ungar [A. A. Ungar (2006) " [http://ceta.mit.edu/pier/pier.php?paper=0512151 The relativistic proper-velocity transformation group] ", "Progress in Electromagnetics Research" 60, 85-94.] , Baylis [W. E. Baylis (1996) "Clifford (geometric) algebras with applications to physics" (Springer, NY) ISBN 0-8176-3868-7] and Hestenes [D. Hestenes (2003) " [http://modelingnts.la.asu.edu/html/overview.html Spacetime physics with geometric algebra] ", "Am. J. Phys." 71, 691-714] have examined its relevance from group theoryand geometric algebraperspectives. Proper-velocity is sometimes referred to as celerity [Bernard Jancewicz (1988) "Multivectors and Clifford algebra in electrodynamics" (World Scientific, NY) ISBN 9971502909] . Unlike the more familiar coordinate velocity v, proper-velocity is useful for describing both super-relativistic and sub-relativistic motion. Like coordinate velocity and unlike four-vector velocity, it resides in the three-dimensional slice of spacetime defined by the map-frame. This makes it more useful for map-based (e.g. engineering) applications, and less useful for gaining coordinate-free insight. Proper-speed divided by lightspeed "c" is the hyperbolic sineof rapidity η, just as the Lorentz factor γ is rapidity's hyperbolic cosine, and coordinate speed v over lightspeed is rapidity's hyperbolic tangent. Imagine an object traveling through a region of space-time locally described by Hermann Minkowski's flat-space metric equation ("c"dτ)2 = ("c"dt)2 - (dx)2. Here a reference map frame of yardsticks and synchronized clocks define map position x and map time t respectively, and the d preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper-velocity w = dx/dτ = γv where as usual coordinate velocity v = dx/dt. Thus finite w ensures that v is less than lightspeed "c". By grouping γ with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass[G. Oas (2005) "On the use of relativistic mass in various published works" ( [http://arxiv.org/abs/physics/0504111 arXiv:physics/0504111] )] . Comparing proper velocities at high speed Proper-velocity is useful for comparing the speed of objects with momentum per unit mass (w) greater than lightspeed "c". The coordinate speed of such objects is generally near lightspeed, whereas proper-velocity tells us how rapidly they are covering ground on "traveling-object clocks". This is important for example if, like some cosmic ray particles, the traveling objects have a finite lifetime. Proper velocity also clues us in to the object's momentum, which has no upper bound. For example, a 45 GeV electron accelerated by the Large Electron-Positron Collider(LEP) at Cern in 1989 would have had a Lorentz factor γ of about 88,000 (90 GeV divided by the electron rest mass of 511 keV). Its coordinate speed v would have been about sixty four trillionths shy of lightspeed "c" at 1 lightsecond per "map" second. On the other hand, its proper-speed would have been w = γv ~88,000 lightseconds per "traveler" second. By comparison the coordinate speed of a 250 GeV electron in the proposed International Linear Collider[B. Barish, N. Walker and H. Yamamoto, " [http://www.sciam.com/article.cfm?id=building-the-next-generation-collider Building the next generation collider] " "Scientific American" (Feb 2008) 54-59] (ILC) will remain near "c", while its proper-speed will significantly increase to ~489,000 lightseconds per traveler second. Proper-velocity is also useful for comparing relative velocities along a line at high speed. In this case wAC = γABγBC(vAB+vBC) where A, B and C refer to different objects or frames of reference [This velocity-addition rule is easily derived from rapidities α and β, since Sinh [α+β] =Cosh [α] Cosh [β] (Tanh [α] +Tanh [β] ).] . For example wAC refers to the proper-speed of object A with respect to object C. Thus in calculating the relative proper-speed, Lorentz factors multiply when coordinate speeds add. Hence each of two electrons (A and C) in a head-on collision at 45 GeV in the lab frame (B) would see the other coming toward them at vAC ~"c" and wAC = 88,0002(1+1) ~1.55×1010 lightseconds per traveler second. Thus colliders can explore higher-speed collisions than can fixed-target accelerators. Plotting "(γ-1) versus proper velocity" after multiplying the former by m"c"2 and the latter by mass m, for various values of m yields a family of kinetic energy versus momentum curves that includes most of the moving objects encountered in everyday life. Such plots can for example be used to show where lightspeed, Planck's constant, and Boltzmann energy kT figure in. To illustrate, the figure at right with log-log axes shows objects with the same kinetic energy (horizontally related) that carry different amounts of momentum, as well as how the speed of a low-mass object compares (by vertical extrapolation) to the speed after perfectly inelastic collision with a large object at rest. Highly sloped lines (rise/run=2) mark contours of constant mass, while lines of unit slope mark contours of constant speed. Objects that fit nicely on this plot are humans driving cars, dust particles in Brownian motion, a spaceship in orbit around the sun, molecules at room temperature, a fighter jet at Mach 3, one radio wave photon, a person moving at one lightyear per traveler year, the pulse of a 1.8 MegaJoule LASER, a 250 GeV electron, and our observable universe with the blackbody kinetic energy expected of a single particle at 3 Kelvin. Unidirectional acceleration via proper velocity In flat spacetime, proper accelerationis the three-vector acceleration experienced in the instantaneously-varying frame of an accelerated object [Edwin F. Taylor & John Archibald Wheeler (1966 1st ed. only) "Spacetime Physics" (W.H. Freeman, San Francisco) ISBN 0-7167-0336-X] . Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper-acceleration is also useful from the vantage point (or spacetime slice) of an observer. Not only may observers in all frames agree on its magnitude, but it also measures the extent to which an accelerating rocket "has its pedal to the metal". In the unidirectional case i.e. when the object's acceleration is parallel or anti-parallel to its velocity in the spacetime slice of the observer, the "change in proper-velocity is the integral of proper acceleration over map-time" i.e. Δw=αΔt for constant α. At low speeds this reduces to the well-known relation between coordinate velocity and coordinate accelerationtimes map-time, i.e. Δv=aΔt. For constant unidirectional proper-acceleration, similar relationships exist between rapidity η and elapsed proper-time Δτ, as well as between Lorentz factor γ and distance traveled Δx. To be specific: :,where as noted above the various velocity parameters are related by:. These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at "1-gee" (or 1.03 lightyears/year2) halfway to their destination, and then decelerate them at "1-gee" for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time. For a map-distance of ΔxAB, the first equation above predicts a mid-point Lorentz factor (up from its unit rest value) of γmid=1+α(ΔxAB/2)/c2. Hence the round-trip time on traveler clocks will be Δτ = 4(c/α)cosh-1 [γmid] , during which the time elapsed on map clocks will be Δt = 4(c/α)sinh [cosh-1 [γmid] . This imagined spaceship could offer round trips to Proxima Centaurilasting about 7.1 traveler years (~12 years on earth clocks), round trips to the Milky Way's central black holeof about 40 years (~54,000 years elapsed on earth clocks), and round trips to Andromeda Galaxylasting around 57 years (over 5 million years on earth clocks). Unfortunately, sustaining 1-gee acceleration for years is easier said than done. Kinematics: for studying ways that position changes with time Lorentz factor: γ=dt/dτ or kinetic energy over mc2 Rapidity: hyperbolic velocity angle in imaginary radians Four-velocity: combining travel through time and space Uniform Acceleration: holding coordinate acceleration fixed Notes and References * [http://www.eftaylor.com/download.html#special_relativity Excerpts from the first edition of "Spacetime Physics", and other resources posted by Edwin F. Taylor] Wikimedia Foundation. 2010.

In our house there are several electrical and electronic devices, which require different electrical quantities such as electrical voltage, electrical current, electrical resistance and electrical power. In this article we decided to approach one of the main electrical quantities, which is the electric current, in order to show the main characteristics of the electric current. What is Electric Current? All substances, gaseous, liquid or solid, are made up of small particles invisible to the human eye, called atoms. The atom is basically divided into two parts, which is the nucleus and the electrosphere. In the nucleus is where protons and neutrons are located, the proton has a positive charge and the neutron has no electrical charge. In the electrosphere is where the electrons that have negative shits are located. In the existing electrical conductor’s free electrons that are in constant disordered movement. For free electrons to move in an orderly manner in electrical conductors, it is necessary to have a force that drives free electrons, and this force is called electrical voltage. The force caused by the electrical voltage causes the free electrons to move in an orderly fashion, thus forming an electron current that is called an electric current, represented by the letter (I). The intensity of the electric current is determined by the ratio between the amount of electrical charges, which cross a determined section of a conductor, over a period of time. The unit of measurement given for the intensity of the electric current by the international system of units (SI) is the amp, in honor of the French scientist, André-marie Ampère (1775 – 1836), to represent the electric current we use the letter (A) as symbol. ). Types of Electric Current There are two types of electric current, direct current and alternating current. Direct current (DC or DC) is an ordered flow of free electrons in the same direction, remaining constant over time and has defined roles, that is, positive and negative poles, and this type of electrical current is obtained from the battery, battery, power supplies, charger etc. Alternating electric current (AC or AC – from English alternating current), is the ordered flow of free electrons in a varied direction over time and has no poles defined as indirec current, varying between phase and neutral means to be present in hydroelectric plants, electrical outlets, substations, etc. Electric Current Direction Before studying the structure of atoms there was already a definition for electric current as being the direction of the flow of positive charges, so the charges move from the positive pole to the negative pole. At the beginning of the history of electricity, because they were not aware of the structure of atoms, they did not know that in solid conductors the positive charges are strongly linked to the nuclei of the atoms, therefore, in solid conductors there can be no positive charge flow. However, when subatomic physics discovered this fact, the definition for electric current as a flow of positive charges was already widely used in calculations and representations for circuit analysis, so this sense is still used today and is called the conventional current sense. electrical. Thus two directions were defined for the electric current, that is, the real sense and the conventional sense, where the real sense is the flow of electrons from the negative pole to the positive pole and the conventional sense is used in calculations for circuit analysis, being that in this case the direction of the electric current goes from the positive pole to the negative pole of the source. Electric Current in each type of electrical circuit The electrical circuit is a closed path through which the electric current flows, there are basically three types of circuit, which are: series, parallel and mixed circuit . The electrical current behaves in different ways in each type of electrical circuit, whereas in series circuits the electrical current is the same at all points. In parallel, the electric current is divided between the meshes, and may have different values of electric current, depending on the point of analysis. In the mixed circuit the electric current behaves in both ways, depending on the analyzed grid. In order to carry out the analysis and calculations on the circuits, laws and formulas were developed over time, for example, ohm’s law , first kirchhoff’s law and current divider. Ohm’s law is one of the most applied laws in electrical calculations. Ohms’ law is the basis for electrical calculations and involves four electrical quantities, which are: electrical voltage, electrical current, electrical resistance and electrical power . Ohm’s law is very simple, when we have the value of two of these quantities it is possible to find the value of the third variable, for that it is enough to use the appropriate formula, in the case of the electric current we have three formulas that can be used to obtain the value of electric current. Kirchhoff’s first law Kirchhoff’s laws are fundamental to perform series, parallel or mixed circuit analysis, as they are directly related to electrical current. Kirchhoff’s first law, also known as the law of currents or the law of nodes, is related to the electric current in a node. Kirchhoff’s first law says that the result of the sum of the electrical currents in a node is always equal to zero, so the node does not accumulate charge. The node is a point where the current divides having two or more paths to travel in the circuit and the sum of all currents entering a node is equal to the sum of all currents leaving the node.

Linear Programming and Maximization of Contribution Margin – Simplex Method: Learning Objective of the Article: Definition and Explanation of Simplex Method: Simplex method is considered one of the basic techniques from which many linear programming techniques are directly or indirectly derived. The simplex method is an iterative, stepwise process which approaches an optimum solution in order to reach an objective function of maximization or minimization. Matrix algebra provides the deterministic working tools from which the simplex method was developed, requiring mathematical formulation in describing the problem. An example can help us explain the simplex method in detail. Example of Linear Programming Simplex Method: Assume that a small machine shop manufactures two models, standard and deluxe. Each standard model requires two hours of grinding and four hours of polishing; each deluxe module requires five hours of grinding and two hours of polishing. The manufacturer has three grinders and two polishers. Therefore in 40 hours week there are 120 hours of grinding capacity and 80 hours of polishing capacity. There is a contribution margin of $3 on each standard model and $4 on each deluxe model. Before the simplex method can be applied, the following steps must be taken: The relationship which establish the constraints or inequalities must be set up first. Letting x and y be respectively the quantity of items of the standard model and deluxe model that are to be manufactured, the system of inequalities or the set of constraints: 2x + 5y ≤ 120 Both x and y must be positive values or zero (x ≥ 0; y ≥ 0). Although this illustration involves only less-than-or-equal-to type constraints, equal-to and greater-than-or-equal-to type constraints can be encountered in maximization problems. The objective function is the total contribution margin the manager can obtain from the two models. A contribution margin of $3 is expected for each standard model and $4 for each deluxe model. The objective function is CM = 3x + 4y. The problem is now completely described by mathematical notation. The first tow steps are the same for the graphic method, the simplex method requires the use of equations, in contrast to the inequalities used by the graphic method. Therefore the set of inequalities (to-less-than-or-equal-to type constraints) must be transformed into a set of equations by introducing slack variables, s1 and s2. The use of slack variables involves the addition of an arbitrary variable to one side of the inequality, transforming it into an equality. This arbitrary variable is called a slack variable, since it takes up the slack in the inequality. The inequalities rewritten as equalities are: 2x + 5y + s1 = 120 The unit contribution margins of the fictitious products s1 and s2 are zero, and the objective equation becomes: Maximize: CM = 3x + 4y + 0s1 + 0s2 At this point, the simplex method can be applied and the first matrix or tableau can be set up as shown below: Explanation and Calculation for the First Tableau: The simplex method records the pertinent data in a matrix form known as the simplex tableau. The components of a tableau are described in the following paragraphs. The Objective row is made up of the notation of the variable of the problem including slack variables. The problem rows in the first tableau contain the coefficients of the variables in the constraints. Each constraint adds an additional problem row. Variables not included in a constraints are assigned zero coefficients in the problem rows. In subsequent tableau, new problem row values will be computed. At each iteration, the objective column receives different entries, representing the contribution margin per unit of the variable in the solution. At each iteration, the variable column receives different notation by replacement. These notations are the variables used to find the total contribution margin of the particular iteration. In the first matrix, a situation of no production is is considered as a starting point in the iterative solution process. For this reason, only slack variables and artificial variables are entered in the objective column, and their coefficient in the objective function are recorded in the variable column. As the iterations proceed, by replacement, appropriate values and notations will be entered in the objective and variable column. The quantity column shows the constant values of the constraints in the first tableau; in subsequent tableaus, it shows the solution mix. The index row carries values computed by the following steps: The index row for the illustration is determined as follows: In this first tableau, the slack variables were introduced into the product mix variable column to find and initial feasible solution to the problem. It can be proven mathematically that beginning with positive slack and artificial variables assures a feasible solution. Hence, one feasible solution might have s1 take a value of 120 and s2 a value of 80. This approach satisfies the constraints but is undesirable since the resulting contribution margin is zero. It is a rule of the simplex method that the optimum solution has not been reached if the index row carries any negative values at the completion of an iteration. Consequently, this first tableau does not carry the optimum solution since negative values appear in its index row. A second tableau or matrix must now be prepared, step by step, according to the rules of simplex method. Explanation and Calculation for the Second Tableau: The construction of the second tableau is accomplished through these six steps: When these steps are completed for the contribution margin maximization illustration, the second tableau appears as follows: This second matrix does not contain the optimum solution since a negative figure, -1.4, still remains in the index row. The contribution margin arising from this model mix is $96 [4(24) + 0(32)], which is an improvement. However, the second solution indicates that some standard models and $1.40 (or 7 / 5 dollars of contribution margin) can be added for each unit of the standard model substituted in this solution. It is interesting to reflect on the significance of – 7 / 5 or – 1.4. The original statement of the problem had promised a unit contribution margin of $3 for the standard model. Now the contribution will increase by only $1.40 per unit. The significance of the -1.4 is that it measures the net increase of the per unit contribution margin after allowing for the reduction of the deluxe model represented by y units. That is, all the grinding hours have been committed to produce 24 deluxe models (24 units × 5 hours grinding time per unit = 120 hours capacity); the standard model cannot be made without sacrificing the deluxe model. The standard model requires 2 hours of grinding time; the deluxe model requires 5 hours of grinding time. To introduce one standard model unit into the product mix, the manufacture of 2 / 5 (0.4) of one deluxe model unit must be foregone. This figure, 0.4, appears in the column headed “x” on the row representing foregone variable (deluxe) models. If more non slack variables (i.e., more than two products) were involved , the figures for these variables, appearing in column x, would have the same meaning as 0.4 has for the deluxe models. Thus, the manufacturer loses 2 / 5 of $4, or $1.60, by making 2 / 5 less deluxe models but gains $3 from the additional standard models. A loss of $1.60 and a gain of $3 results in a net improvement of $1.40. The final answer, calculated on graphical method page, adds $14 (10 standard models × 1.4) to the $96 contribution margin that results from producing 10 standard models (10 × $3 = $30) and (20 × $4 = $80). In summary, the -1.4 in the second tableau indicates the amount of increase possible in the contribution margin if one unit of the variable heading that column (x in this case) were added to the solution; and the 0.4 value in column x represents the production of the deluxe model that must be relinquished. The quantity column was described for the first tableau as showing the constant values of the constraints, i.e., the maximum resources available (grinding and polishing hours in the illustration) for the manufacture of standard and deluxe models. In subsequent tableaus the quantity column shows the solution mix. Additionally, for a particular iteration in subsequent tableau, the quantity column shows the constraints that are utilized in an amount different from the constraints constant value. For example, in the second tableau’s quantity column, the number corresponding to the y variable denotes the number of y units in the solution mix (24), and its objective function coefficient of $4 when multiplied by 24 yields $96, the value of the solution at this iteration. The number corresponding to the s2 variable denotes the difference in total polishing hours and those used in the second tableau solution, i.e., 80 hours of available polishing hours less polishing hours used to produce 24 units of y (24 × 2), or 80 – 48 = 32. Thus the number of unused polishing hours is 32. No unused grinding hours, the s1 variable, are indicated because 24 units of y utilized the entire quantity of available grinding time (24 × 5 = 120 hours). While this illustration is of less-than-or-equal-to type constraints, a similar interpretation can be made for equal-to type constraints; i.e., the quantity column denotes the difference in the constant value of the constraint and the value used in the tableau’s solution mix. For the greater-than-or-equal-to constraint, the quantity column denotes the amount beyond the constraint’s minimum requirement that is satisfied by the particular solution mix. These constraint utilization of satisfaction differences provide useful information, especially in the optimal solution tableau, because management may wish to make decisions to reduce these differences, e.g., by plans to utilize presently unused capacity associated with less-than-or-equal-to constraints. Explanation and Calculation for the Third Tableau: The third tableau is computed by these steps: Third tableau appears as follows: There are no negative figures in the index row, which indicates that any further substitutions will not result in an increase in the contribution margin; the optimum solution has been obtained. The optimum strategy is to produce and sell 20 deluxe and 10 standard models for a contribution margin of $110. You may also be interested in other articles from “linear programming technique” chapter Other Related Accounting Articles: - Linear Programming and Maximization of Contribution Margin – Graphical Method - Linear Programming – Minimization of Cost – Simplex Method - Shadow Prices - Linear Programming Questions and Answers - Linear Programming and Minimization of Cost-Graphical Method - Linear Programming Techniques-General Observations - Linear Programming Technique - Dynamic Programming - Linear Programming Solved Problems Download E accounting book in MS-word format for just 20 $ - Click here to Download

A hospital wants to know how a homeopathic medicine for depression performs in comparison to alternatives. Learn How to Calculate Tukey's Post HOC Test - Tutorial They adminstered 4 treatments to patients for 2 weeks and then measured their depression levels. The data, part of which are shown above, are in depression. Before running any statistical test, always make sure your data make sense in the first place. In this case, a split histogram basically tells the whole story in a single chart. We don't see many SPSS users run such charts but you'll see in a minute how incredibly useful it is. The screenshots below show how to create it. In step below, you can add a nice title to your chart. Clicking P aste results in the syntax below. Running it creates our chart. We'll now take a more precise look at our data by running a means table. We could do so from A nalyze C ompare Means M eans but the syntax is so simple that just typing it is probably faster. Unsurprisingly, our table mostly confirms what we already saw in our histogram. Well, for our sample we can. For our population all people suffering from depression we can't. The basic problem here is that samples differ from the populations from which they are drawn. If our four medicines perform equally well in our population, then we may still see some differences between our sample means. However, large sample differences are unlikely if all medicines perform equally in our population. The question we'll now answer is: are the sample means different enough to reject the null hypothesis that the mean BDI scores in our populations are all equal? However, it could be argued that you should always run post hoc tests. In some fields like market research, this is pretty common. Reversely, you could argue that you should never use post hoc tests because the omnibus test suffices: some analysts claim that running post hoc tests is overanalyzing the data. Many social scientists are completely obsessed with statistical significance -because they don't understand what it really means- and neglect what's more interesting: effect sizes and confidence intervals. In any case, the idea of post hoc tests is clarified best by just running them.Since these are independent and not paired or correlated, the number of observations of each treatment may be different. This calculator is hard-coded for a maximum of 10 treatments, which is more than adequate for most researchers. But it stops there in its tracks. This self-contained calculator, with flexibility to vary the number of treatments columns to be compared, starts with one-way ANOVA. However, it lacks the key built-in statistical function needed for conducting Excel-contained Tukey HSD. Continuing education in Statistics The hard-core statistical packages demand a certain expertise to format the input data, write code to implement the procedures and then decipher their s Old School Mainframe Era output. This is the right tool for you!Scs extractor 2020 It was inspired by the frustration of several biomedical scientists with learning the software setup and coding of these serious statistical packages, almost like operating heavy bulldozer machinery to swat an irritating mosquito. For code grandmasters, fully working code and setup instructions are provided for replication of the results in the serious academic-research-grade open-source and hence free R statistical package. Tukey originated his HSD test, constructed for pairs with equal number of samples in each treatment, way back in When the sample sizes are unequal, we the calculator automatically applies the Tukey-Kramer method Kramer originated in A decent writeup on these relevant formulae appear in the Tukey range test Wiki entry. The NIST Handbook page mentions this modification but dooes not provide the formula, while the Wiki entry makes adequately specifies it. However, this calculator is hard-coded for contrasts that are pairsand hence does not pester the user for additional input that defines generalized contrast structures. The Bonferroni and Holm methods of multiple comparison depends on the number of relevant pairs being compared simultaneously. This calculator is hard-coded for Bonferroni and Holm simultaneous multiple comparison of 1 all pairs and 2 only a subset of pairs relative to one treatment, the first column, deemed to be the control. The post-hoc Bonferroni simultaneous multiple comparison of treatment pairs by this calculator is based on the formulae and procedures at the NIST Engineering Statistics Handbook page on Bonferroni's method. The original Bonferroni published paper in Italian dating back to is hard to find on the web. A significant improvement over the Bonferroni method was proposed by Holm Among the many reviews of the merits of the Holm method and its uniform superiority over the Bonferroni method, that of Aickin and Gensler is notable. This paper is the also source of our algorithm to make comparisons according to the Holm method. All statistical packages today incorporate the Holm method. There is wide agreement that each of these three methods have their merits. The recommendation on the relative merits and advantages of each of these methods in the NIST Engineering Statistics Handbook page on comparison of these methods are reproduced below:. The following excerpts from Aickin and Gensler makes it clear that the Holm method is uniformly superior to the Bonferroni method:. If only a subset of pairwise comparisons are required, Bonferroni may sometimes be better. Many computer packages include all three methods. So, study the output and select the method with the smallest confidence band. No single method of multiple comparisons is uniformly best among all the methods.The idea behind the Tukey HSD Honestly Significant Difference test is to focus on the largest value of the difference between two group means. The relevant statistic is. The statistic q has a distribution called the studentized range q see Studentized Range Distribution. Thus we can use the following t statistic. From these observations we can calculate confidence intervals in the usual way:. Since the difference between the means for women taking the drug and women in the control group is 5. The following table shows the same comparisons for all pairs of variables:. From Figure 1 we see that the only significant difference in means is between women taking the drug and men in the control group i. In Figure 2 we compute the confidence interval for the comparison requested in the example as well as for the variables with maximum difference. These function are based on the table of critical values provided in Studentized Range q Table. The Real Statistics Resource Pack also provides the following functions which provide estimates for the Studentized range distribution and its inverse based on a somewhat complicated algorithm. QDIST 4. To get the usual cdf value for the Studentized range distribution, you need to divide the result from QDIST by 2, which for this example is. C n ,2 rows if the data in R1 contains n columns. The first two columns contain the column numbers in R1 from 1 to n that are being compared and the third column contains the p-values for each of the pairwise comparisons. RSS - Posts. RSS - Comments. Real Statistics Using Excel. Everything you need to perform real statistical analysis using Excel. Skip to content. The critical value for differences in means is Since the difference between the means for women taking the drug and women in the control group is 5. Real Statistics Resources. Follow Real1Statistics. Search for:. Proudly powered by WordPress.The Tukey HSD "honestly significant difference" or "honest significant difference" test is a statistical tool used to determine if the relationship between two sets of data is statistically significant — that is, whether there's a strong chance that an observed numerical change in one value is causally related to an observed change in another value. In other words, the Tukey test is a way to test an experimental hypothesis. The Tukey test is invoked when you need to determine if the interaction among three or more variables is mutually statistically significant, which unfortunately is not simply a sum or product of the individual levels of significance. Simple statistics problems involve looking at the effects of one independent variable, like the number of hours studied by each student in a class for a particular test, on a second dependent variable, like the student's scores on the test. Then you refer to a t-table that takes into account the number of data pairs in your experiment to see if your hypothesis was correct. Sometimes, however, the experiment may look at multiple independent or dependent variables simultaneously. For example, in the above example, the hours of sleep each student got the night before the test and his or her class grade going in might be included. Such multivariate problems require something other than a t-test owing to the sheer number if independently varying relationships. ANOVA stands for "analysis of variance" and addresses precisely the problem just described. It accounts for the rapidly expanding degrees of freedom in a sample as variables are added. For example, looking at hours vs. In an ANOVA test, the variable of interest after calculations have been run is F, which is the found variation of the averages of all of the pairs, or groups, divided by the expected variation of these averages. The higher this number, the stronger the relationship, and "significance" is usually set at 0. John Tukey came up with the test that bears his name when he realized the mathematical pitfalls of trying to use independent P-values to determine the utility of a multiple-variables hypothesis as a whole. At the time, t-tests were being applied to three or more groups, and he considered this dishonest — hence "honestly significant difference. What his test does is compare the differences between means of values rather than comparing pairs of values. The value of the Tukey test is given by taking the absolute value of the difference between pairs of means and dividing it by the standard error of the mean SE as determined by a one-way ANOVA test. The SE is in turn the square root of variance divided by sample size. An example of an online calculator is listed in the Resources section. The Tukey test is a post hoc test in that the comparisons between variables are made after the data has already been collected. This differs from an a priori test, in which these comparisons are made in advance. In the former case, you might look at the mile run times of students in three different phys-ed classes one year. In the latter case, you might assign students to one of three teachers and then have them run a timed mile. Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs. More about Kevin and links to his professional work can be found at www. About the Author. Copyright Leaf Group Ltd.An ANOVA is a statistical test that is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups. The alternative hypothesis: Ha : at least one of the means is different from the others. It simply tells us that not all of the group means are equal. If the p-value is not statistically significant, this indicates that the means for all of the groups are not different from each other, so there is no need to conduct a post hoc test to find out which groups are different from each other. As mentioned before, post hoc tests allow us to test for difference between multiple group means while also controlling for the family-wise error rate. In a hypothesis testthere is always a type I error rate, which is defined by our significance level alpha and tells us the probability of rejecting a null hypothesis that is actually true. When we perform one hypothesis test, the type I error rate is equal to the significance level, which is commonly chosen to be 0. However, when we conduct multiple hypothesis tests at once, the probability of getting a false positive increases. For example, imagine that we roll a sided dice. Tukey's Post HOC Test Calculator If we roll five dice at once, the probability increases to For example, suppose we have four groups: A, B, C, and D. This means there are a total of six pairwise comparisons we want to look at with a post hoc test:. If we have more than four groups, the number of pairwise comparisons we will want to look at will only increase even more. The following table illustrates how many pairwise comparisons are associated with each number of groups along with the family-wise error rate:. Notice that the family-wise error rate increases rapidly as the number of groups and consequently the number of pairwise comparisons increases. This means we would have serious doubts about our results if we were to make this many pairwise comparisons, knowing that our family-wise error rate was so high. Fortunately, post hoc tests provide us with a way to make multiple comparisons between groups while controlling the family-wise error rate. This means we have sufficient evidence to reject the null hypothesis that all of the group means are equal. Next, we can use a post hoc test to find which group means are different from each other. We will walk through examples of the following post hoc tests:.Public library R gives us two metrics to compare each pairwise difference:. Both the confidence interval and the p-value will lead to the same conclusion. In particular, we know that the difference is positive, since the lower bound of the confidence interval is greater than zero.Although ANOVA is a powerful and useful parametric approach to analyzing approximately normally distributed data with more than two groups referred to as 'treatments'it does not provide any deeper insights into patterns or comparisons between specific groups. After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc analysis is Tukey's Test. The test is known by several different names. Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal Wikipedia. The outputs from two different but similar implementations of Tukey's Test will be examined along with how to manually calculate the test. Other methods of post-hoc analysis will be explored in future posts. ANOVA in this example is done using the aov function. The summary of the aov output is the same as the output of the anova function that was used in the previous example. To investigate more into the differences between all groups, Tukey's Test is performed. The output gives the difference in means, confidence levels and the adjusted p-values for all possible pairs. The confidence levels and p-values show the only significant between-group difference is for treatments 1 and 2. Note the other two pairs contain 0 in the confidence intervals and thus, have no significant difference. The results can also be plotted. Another way of performing Tukey's Test is provided by the agricolae package. The HSD. The results from both tests can be verified manually.Mgsv nuclear disarmament reddit We'll start with the latter test HSD. The MSE calculation is the same as the previous example. With the q-value found, the Honestly Significant Difference can be determined. The Honestly Significant Difference is defined as the q-value multiplied by the square root of the MSE divided by the sample size. As mentioned earlier, the Honestly Significant Difference is a statistic that can be used to determine significant differences between groups. If the absolute value of the difference of the two groups' means is greater than or equal to the HSD, the difference is significant. The means of each group can be found using the tapply function. Since there's only three groups, I went ahead and just calculated the differences manually. With the differences obtained, compare the absolute value of the difference to the HSD. I used a quick and dirty for loop to do this. The output of the for loop shows the only significant difference higher than the HSD is between treatment 1 and 2.Google my maps hide legend Since the test uses the studentized range, estimation is similar to the t-test setting. The Tukey-Kramer method allows for unequal sample sizes between the treatments and is, therefore, more often applicable though it doesn't matter in this case since the sample sizes are equal. The Tukey-Kramer method is defined as:. Entering the values that were found earlier into the equation yields the same intervals as was found from the TukeyHSD output. The table from the TukeyHSD output is reconstructed below. Adjusted p-values are left out intentionally.Post hoc multiple comparison tests. Once you have determined that differences exist among the means, post hoc range tests and pairwise multiple comparisons can determine which means differ. Comparisons are made on unadjusted values. These tests are used for fixed between-subjects factors only.Performing a One-way ANOVA in Excel with post-hoc t-tests In GLM Repeated Measures, these tests are not available if there are no between-subjects factors, and the post hoc multiple comparison tests are performed for the average across the levels of the within-subjects factors. For GLM Multivariate, the post hoc tests are performed for each dependent variable separately. The Bonferroni and Tukey's honestly significant difference tests are commonly used multiple comparison tests. The Bonferroni testbased on Student's t statistic, adjusts the observed significance level for the fact that multiple comparisons are made. Sidak's t test also adjusts the significance level and provides tighter bounds than the Bonferroni test. Tukey's honestly significant difference test uses the Studentized range statistic to make all pairwise comparisons between groups and sets the experimentwise error rate to the error rate for the collection for all pairwise comparisons. When testing a large number of pairs of means, Tukey's honestly significant difference test is more powerful than the Bonferroni test. For a small number of pairs, Bonferroni is more powerful. Hochberg's GT2 is similar to Tukey's honestly significant difference test, but the Studentized maximum modulus is used. Usually, Tukey's test is more powerful. Gabriel's pairwise comparisons test also uses the Studentized maximum modulus and is generally more powerful than Hochberg's GT2 when the cell sizes are unequal. Gabriel's test may become liberal when the cell sizes vary greatly. Dunnett's pairwise multiple comparison t test compares a set of treatments against a single control mean. The last category is the default control category. Alternatively, you can choose the first category. You can also choose a two-sided or one-sided test. To test that the mean at any level except the control category of the factor is not equal to that of the control category, use a two-sided test. Multiple step-down procedures first test whether all means are equal. If all means are not equal, subsets of means are tested for equality. These tests are more powerful than Duncan's multiple range test and Student-Newman-Keuls which are also multiple step-down proceduresbut they are not recommended for unequal cell sizes. When the variances are unequal, use Tamhane's T2 conservative pairwise comparisons test based on a t testDunnett's T3 pairwise comparison test based on the Studentized maximum modulusGames-Howell pairwise comparison test sometimes liberalor Dunnett's C pairwise comparison test based on the Studentized range. Note that these tests are not valid and will not be produced if there are multiple factors in the model. Duncan's multiple range testStudent-Newman-Keuls S-N-Kand Tukey's b are range tests that rank group means and compute a range value. These tests are not used as frequently as the tests previously discussed. The Waller-Duncan t test uses a Bayesian approach. - Starseeds awakening - Ktm ecu tuning - Wolf head 5d diy special shaped diamond painting cross stitch - Obbligo di integrazione delle fonti rinnovabili modello e - Miwifi rom ssh - Plw1000v2 slow - Mandala art - Zepeto hack - Css 3d effect - Target resume - A man who falls to his death - Touch sensitive keyboard - Zastava zpap92 parts - Hisense replacement screen - Head unit iso wiring diagram diagram base website wiring - Dmr data call - Sunday sattamatka fix lucky panna - Bhuppae sunniwat eng sub ep 7 - Witcher 3 weeping angels

Theories of light In the seventeenth century two rival theories of the nature of light were proposed, the wave theory and the corpuscular theory. The Dutch astronomer Huygens (1629-1695) proposed a wave theory of light. He believed that light was a longitudinal wave, and that this wave was propagated through a material called the ‘aether’. Since light can pass through a vacuum and travels very fast Huygens had to propose some rater strange properties for the aether: for example; it must fill all space and be weightless and invisible. For this reason scientists were sceptical of his theory. In 1690 Newton proposed the corpuscular theory of light. He believed that light was shot out from a source in small particles, and this view was accepted for over a hundred years. The quantum theory put forward by Max Planck in 1900 combined the wave theory and the particle theory, and showed that light can sometimes behave like a particle and sometimes like a wave. You can find a much fuller consideration of this in the section on the quantum theory. Wave theory of Huygens As we have seen, Huygens considered that light was propagated in longitudinal waves through a material called the aether. We will now look at his ideas more closely. Huygens published his theory in 1690, having compared the behaviour of light not with that of water waves but with that of sound. Sound cannot travel through a vacuum but light does, and so Huygens proposed that the aether must fill all space, be transparent and of zero inertia. Clearly a very strange material! Even at the beginning of the twentieth century, however, scientists were convinced of the existence of the aether. One book states ‘whatever we consider the aether to be there can be no doubt of its existence’. We now consider how Huygens thought the waves moved from place to place. Consider a wavefront initially at position W, and assume that every point on that wavefront acts as a source of secondary wavelets. (Figure 1 shows some of these secondary sources). The new wavefront W1 is formed by the envelope of these secondary wavelets since they will all have moved forward the same distance in a time t (Figure 1). There are however at least two problems with this idea and these led Newton and others to reject it: (a) the secondary waves are propagated in the forward direction only, and (b) they are assumed to destroy each other except where they form the new wavefront. Newton wrote: ‘If light consists of undulations in an elastic medium it should diverge in every direction from each new centre of disturbance, and so, like sound, bend round all obstacles and obliterate all shadow.’ Newton did not know that in fact light does do this, but the effects are exceedingly small due to the very short wavelength of light. Huygens’ theory also failed to explain the rectilinear propagation of light. The reflection of a plane wavefront by a plane mirror is shown in Figure 2. Notice the initial position of the wavefront (AB), the secondary wavelets and the final position of the wavefront (CD). Notice that he shape of the wavefront is not affected by reflection at a plane surface. The lines below the mirror show the position that the wavefront would have reached if the mirror had not been there. We will now show how Huygens’ wave theory can be used to explain reflection and refraction and the laws governing them. Consider a parallel beam of monochromatic light incident on a plane surface, as shown in Figure 3. The wave fronts will be plane both before and after reflection, since a plane surface does not alter the shape of waves falling on it. Consider a point where the wavefront AC has just touched the mirror at edge A. While the light travels from A to D, that from C travels to B. The new envelope for the wavefront AC will be BD after reflection. Therefore AD = CB Angle ACB = angle ADB = 90o AB is common Therefore ACB and BDA are similar and so angle CAB = angle BAD. Therefore i = r and the law of reflection is proved. Consider a plane monochromatic wave hitting the surface of a transparent material of refractive index n. The velocity of light in the material is cm and that in air ca. Now in Figure 4, CB = AB sin i AD = AB sin r The same argument applies about the new envelope as in the case of reflection: time to travel CB = CB/ca = AB sin i/ca, time to travel AD AD/cm = AB sin i/cm But these are equal and therefore: ca/cm = sin i/sin r = anm. This is Snell’s law, and it was verified later by Foucault and others. Notice that since the refractive index of a transparent material is greater than 1, Huygens’ theory requires that the velocity of light in air should be greater than that in the material. Corpuscular theory of Newton Newton proposed that light is shot out from a source as a stream of particles. He argued that light could not be a wave because although we can hear sound from behind an obstacle we cannot see light - that is, light shows no diffraction. He stated that particles of different colours should be of different sizes, the red particles being larger than the blue. Since these particles are shot out all the time, according to Newton’s theory, the mass of the source of light must get less! We can use Newton’s theory to deduce the laws of reflection and refraction. Consider a particle of light in collision with a mirror. The collision is supposed to be perfectly elastic, and so tile component of velocity perpendicular to the mirror is reversed while that parallel to the mirror remains unaltered. From Figure 5, Component of velocity before collision parallel to the mirror = ca sin i Component of velocity after collision parallel to the mirror = ca sin r ca sin i = ca sin r and so the law of reflection is proved. Newton assumed that there is an attraction between the molecules of a solid and the particles of light, and that this attraction acts only perpendicularly to the surface and only at very short distances from the surface. (He explained total internal reflection by saying that the perpendicular component of velocity was too small to overcome the molecular attraction.) This has the effect of increasing the velocity of the light in the material. Let the velocity of light in air be ca and the velocity of light in the material in Figure 6 be cm. The velocity parallel to the material is unaltered and therefore: ca sin i = cm sin r cm/ca = sini/ sinr = anm This ratio is the refractive index, but because n > 1 the velocity of light in the material must be greater than that in air. Newton accepted this result and other scientists preferred it to that of Huygens, mainly because of Newton’s eminence. A problem of the corpuscular theory was that temperature has no effect on the velocity of light, although on the basis of this theory we would expect the particles to be shot out at greater velocities as the temperature rises. Classical and modern theories of light It is interesting to compare the two classical theories of light and see which phenomena can be explained by each theory. The following table does this. Wave theory Corpuscular theory Notice that neither theory can account for polarisation, since for polarisation to occur the waves must be transverse in nature. Twentieth-century ideas have led us to believe that light is (a) a transverse electromagnetic wave with a small wavelength, and (b) emitted in quanta or packets of radiation of about 10-8 s duration with abrupt phase changes between successive pulses.

We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you! Presentation is loading. Please wait. Published byLiana Bartron Modified about 1 year ago 1.5 G RAPHING Q UADRATIC F UNCTIONS BY U SING T RANSFORMATIONS Graph using the graph of You move the key points of To shift to the right 3 spaces you add 3 to all of the x values! xy And then graph the new set of points! (1, 4) (2, 1) (3, 0) (4, 1) (5, 4) Graph This is a shift to the left 5 spaces. To shift to the left 5 spaces you subtract 5 from all of the x values! xy And then graph the new set of points! Shift LEFT 5 Units (-7, 4) (-6, 1) (-5, 0) (-4, 1) (-3, 4) Graph This is a shift down 4 spaces. To shift down 4 spaces you subtract 4 from all of the y values! xy And then graph the new set of points! Shift DOWN 4 Units (-2, 0) (-1, -3) (0, -4) (1, -3) (2, 0) Graph This is a shift up 6 spaces. To shift up 6 spaces you add 6 to all of the y values! xy And then graph the new set of points! Shift UP 6 Units (-2,10) (-1, 7) (0, 6) (1, 7) (2, 10) Graph This is a vertical stretch by a factor of 2. To stretch the parabola you multipy all of the y values by 2 ! xy And then graph the new set of points! Strectch by a factor of 2 (-2,8) (-1, 2) (0, 0) (1, 2) (2, 8) Graph This is a vertical compression by a factor of one half. To compress the parabola you multipy all of the y values by 0.5 ! (or divide them all by 2!) xy And then graph the new set of points! Compress by a factor of (-2, 2) (-1, 0.5) (0, 0) (1, 0.5) (2, 2) Graph This is a reflection in the x-axis. To reflect the parabola you multipy all of the y values by -1 ! xy And then graph the new set of points! Reflect in the x-axis (-1, -1) (-2, -4) (0, 0) (2, -4) (1, -1) Graph xy reflect in the x-axis and stretch by a factor of 2 shift the parabola up And then graph the new set of points! (-1, 3) (-2, -3) (0, 5) (2, -3) (1, 3) Graph xy shift the parabola left 7 shift the parabola down And then graph the new set of points! (-9, 1) (-8, -2) (-7, -3) (-6, -2) (-5, 1) Graph xy Compress by a factor of 0.25 shift the parabola right And then graph the new set of points! (6, 1) (7, 0.25) (8, 0) (9, 0.25) (10, 1) H OMEWORK : P AGE 47 #5 – 12 1.6 U SING M ULTIPLE T RANSFORMATIONS TO GRAPH QUADRATIC EQUATIONS. Write equation or Describe Transformation. Write the effect on the graph of the parent function down 1 unit1 2 3 Stretch by a factor of 2 right 1 unit. 5-3 T RANSFORMING PARABOLAS ( PART 1) Big Idea: -Demonstrate and explain what changing a coefficient has on the graph of quadratic functions. Your Transformation Equation y = - a f(-( x ± h)) ± k - a = x-axis reflection a > 1 = vertical stretch 0 < a < 1 = vertical compression -x = y-axis reflection. Chapter 4 Quadratics 4.3 Using Technology to Investigate Transformations. Order of function transformations Horizontal shifts Horizontal stretch/compression Reflection over y-axis Vertical stretch/compression Reflection over. Transformations Review Vertex form: y = a(x – h) 2 + k The vertex form of a quadratic equation allows you to immediately identify the vertex of a parabola. Section 3.5 Graphing Techniques: Transformations. How would you sketch the following graph? ◦ y = 2(x – 3) 2 – 8 You need to perform transformations to the graph of y = x 2 Take it one step at a. Graphing Quadratics. Parabolas x y=x 2 We can see the shape looks like: Starting at the vertex Out 1 up 1 2 Out 2 up 2 2 Out 3 up 3 2 Out 4 up. Section 1.4 Transformations and Operations on Functions. Transforming reciprocal functions. DO NOW Assignment #59 Pg. 503, #11-17 odd. Transformations of Functions. The vertex of the parabola is at (h, k). C HAPTER Using transformations to graph quadratic equations. EXAMPLE 1 Compare graph of y = with graph of y = a x 1 x 1 3x3x b. The graph of y = is a vertical shrink of the graph of. x y = 1 = y 1 x a. The graph. TRANSFORMATIONS OF FUNCTIONS Shifts and stretches. Types of Functions. Type 1: Constant Function f(x) = c Example: f(x) = 1. G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a 0. Summary of 2.1 y = -x 2 graph of y = x 2 is reflected in the x- axis Note: Negative in front of x 2 makes parabola “frown”. y = (-x) 2 graph of y = x 2. Algebra-2 Graphical Transformations. Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) Warm Up Find five points and use them to graph Hint, use an x-y table to help you. And the Quadratic Equation……. Parabola - The shape of the graph of y = a(x - h) 2 + k Vertex - The minimum point in a parabola that opens upward or. S TUDY P AGES MAT170 SPRING 2009 Material for 1 st Quiz. 1. Write the parabola in vertex form:. October 7 th. G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a 0. The graph is “U-shaped” and is called a parabola. The. Section 2.5 Transformations of Functions. Overview In this section we study how certain transformations of a function affect its graph. We will specifically. QUADRATIC EQUATIONS in VERTEX FORM y = a(b(x – h)) 2 + k. Section 3.3 Graphing Techniques: Transformations. Standard 9.0 Determine how the graph of a parabola changes as a, b, and c vary in the equation Students demonstrate and explain the effect that changing. Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted up k units (k > 0) or down k units. If you take any shape, you can transform it: SQUARE STRETCH IT COMPRESS IT TRIANGLE STRETCH IT COMPRESS IT. Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations Objectives Graph Functions Using Horizontal and Vertical Shifts Graph Functions Using. Remember this example… Example If g(x) = x 2 + 2x, evaluate g(x – 3) g( ) = x x (x -3) g(x-3) = (x 2 – 6x + 9) + 2x - 6 g(x-3) = x 2 – 6x + 9 + © The Visual Classroom Transformation of Functions Given y = f(x), we will investigate the function y = af [k(x – p)] + q for different values of a, k, TRANSFORMATIONS Shifts Stretches And Reflections. 1 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x. Graphical Transformations Vertical and Horizontal Translations Vertical and Horizontal Stretches and Shrinks. Section 2.5 Transformations Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc. Graphing Absolute Value Functions using Transformations. Essential Question: In the equation f(x) = a(x-h) + k what do each of the letters do to the graph? Graph Absolute Value Functions using Transformations. Math-3 Lesson 1-3 Quadratic, Absolute Value and Square Root Functions. 10.1 Quadratic GRAPHS! – Quadratic Graphs Goals / “I can…” Graph quadratic functions of the form y = ax Graph quadratic functions of the form. 4-1 Quadratic Functions Unit Objectives: Solve a quadratic equation. Graph/Transform quadratic functions with/without a calculator Identify function. Precalculus Functions & Graphs Notes 2.5A Graphs of Functions TerminologyDefinitionIllustration Type of Symmetry of Graph f is an even function f(-x) = Square Root Function Graphs Do You remember the parent function? D: [0, ∞) R: [0, ∞) What causes the square root graph to transform? a > 1 stretches vertically, Section 3-2: Analyzing Families of Graphs A family of graphs is a group of graphs that displays one or more similar characteristics. A parent graph is. 2.5 Shifting, Reflecting, and Stretching Graphs. Shifting Graphs Digital Lesson. Transformations xf(x) Domain: Range:. Transformations Vertical Shifts (or Slides) moves the graph of f(x) up k units. (add k to all of the y-values) moves. Lesson 1-6 Graphical Transformations Graphical Transformations Transformation: an adjustment made to the parent function that results in a change to. © 2017 SlidePlayer.com Inc. All rights reserved.

US 5165008 A A method for synthesizing human speech using a linear mapping of a small set of coefficients that are speaker-independent. Preferably, the speaker-independent set of coefficients are cepstral coefficients developed during a training session using a perceptual linear predictive analysis. A linear predictive all-pole model is used to develop corresponding formants and bandwidths to which the cepstral coefficients are mapped by using a separate multiple regression model for each of the five formant frequencies and five formant bandwidths. The dual analysis produces both the cepstral coefficients of the PLP model for the different vowel-like sounds and their true formant frequencies and bandwidths. The separate multiple regression models developed by mapping the cepstral coefficients into the formant frequencies and formant bandwidths can then be applied to cepstral coefficients determined for subsequent speech to produce corresponding formants and bandwidths used to synthesize that speech. Since less data are required for synthesizing each speech segment than in conventional techniques, a reduction in the required storage space and/or transmission rate for the data required in the speech synthesis is achieved. In addition, the cepstral coefficients for each speech segment can be used with the regressive model for a different speaker, to produce synthesized speech corresponding to the different speaker. 1. A method for synthesizing human speech, comprising the steps of: a. for a given human vocalization, determining a set of Perceptual Line Predictive (PLP) coefficients defining an auditory-like, speaker-independent spectrum of the vocalization; b. mapping the set of PLP coefficients to a vector in a vocal tract resonant vector space, where the vector is defined by a plurality of vector elements; and c. using the vector in the vocal tract resonant space to produce a synthesized speech signal simulating the given human vocalization. 2. The method of claim 1, wherein fewer PLP coefficients are required in the set of coefficients than the plurality of vector elements that define the vector in the vocal tract resonant vector space. 3. The method of claim 2, wherein the set of coefficients is stored for later use in synthesizing speech. 4. The method of claim 2, wherein the set of coefficients comprises data that are transmitted to a remote location for use in synthesizing speech at the remote location. 5. The method of claim 1, further comprising the steps of determining speaker-dependent variables that define qualities of the given human vocalization specific to a particular speaker; and using the speaker-dependent variables in mapping the set of coefficients to produce the vector in the vocal tract resonant space, which is used in producing a simulation of that speaker uttering the given vocalizations. 6. The method of claim 5, wherein the speaker-dependent variables remain constant and are used with successive different human vocalizations to produce a simulation of the speaker uttering the successive different vocalizations. 7. The method of claim 1, wherein the set of coefficients represents a second formant, F2', corresponding to a speaker's mouth cavity shape during production of the given vocalization. 8. The method of claim 1, wherein the step of mapping comprises the step of determining a weighting factor for each coefficient of the set so as to minimize a mean squared error of each element of the vector in the vocal tract resonant space. 9. The method of claim 8, wherein each element of the vector in the vocal tract resonant space is defined by: ##EQU9## where ei is the i-th element, ai0 is a constant portion of that element, aij is the weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient for the i-th element; and N is the number of coefficients. 10. A method for synthesizing human speech, comprising the steps of: a. repetitively sampling successive short segments of a human utterance so as to produce a unique frequency domain representation for each segment; b. transforming the unique frequency domain representations into auditory-like, speaker-independent spectra, by representing a human psychophysical auditory response to the short segments of speech with the transformation; c. defining each of the speaker-independent spectra using a limited set of Perceptual Line Predictive (PLP) coefficients for each segment; d. mapping each limited set of PLP coefficients that define the speaker-independent spectra into one of a plurality of vectors in a vocal tract resonant vector space of a dimension greater than a cardinality of the limited set of PLP coefficients; and e. producing a synthesized speech signal from the plurality of vectors in the vocal tract resonant space, taken in succession, thereby simulating the human utterance. 11. The method of claim 10, wherein the transforming step comprises the steps of: a. warping the frequency domain representations into their Bark frequencies; b. convolving the Bark frequencies with a power spectrum of a simulated critical-band masking curve, producing critical band spectra; c. pre-emphasizing the critical band spectra with a simulated equal-loudness function, producing pre-emphasized, equal loudness spectra; and d. compressing the pre-emphasized, equal loudness spectra with a cubic-root amplitude function, producing the auditory-like, speaker-independent spectra. 12. The method of claim 10, wherein the step of defining each of the auditory-like, speaker-independent spectra comprises the step of applying an inverse frequency transformation, using an all-pole model, wherein the limited set of coefficients comprise autoregression coefficients of the inverse frequency transformation. 13. The method of claim 10, wherein the limited set of coefficients that define each speaker-independent spectrum comprise cepstral coefficients of a perceptual linear prediction model. 14. The method of claim 10, wherein the vocal tract resonant vector space represents a linear predictive model. 15. The method of claim 10, further comprising the step of determining speaker-dependent variables that define qualities of a vocal tract in a speaker that produced the human utterance; and using the speaker-dependent variables in mapping each of the limited set of coefficients that define the speaker-independent spectra to produce the vectors in the vocal tract resonant space, thereby enabling simulation of the speaker producing the utterance. 16. The method of claim 15, wherein the speaker-dependent variables remain constant and are used to simulate additional different human utterances by that speaker. 17. The method of claim 16, the limited set of coefficients for each segment of the utterance and the speaker-dependent variables comprise data that are transmitted to a remote location for use in synthesizing the utterance at the remote location. 18. The method of claim 15, wherein the step of mapping comprises the step of determining a weighting factor for each coefficient so as to minimize a means squared error of each element of the vectors in the vocal tract resonant space. 19. The method of claim 10, wherein the coefficients represent a second formant, F2', corresponding to a speaker's mouth cavity shape during the utterance of each segment. 20. The method of claim 10, wherein each element comprising the vectors in the vocal tract resonant space is defined by: ##EQU10## where ei is the i-th element, ai0 is a constant portion of that element, aij is the weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient of the i-th element; and N is the number of coefficients. This invention generally pertains to speech synthesis, and particularly, speech synthesis from parameters that represent short segments of speech with multiple coefficients and weighting factors. Speech can be synthesized using a number of very different approaches. For example, digitized recordings of words can be reassembled into sentences to produce a synthetic utterance of a telephone number. Alternatively, a phonetic representation of the telephone number can be produced using phonemes for each sound comprising the utterance. Perhaps the dominant technique used in speech synthesis is linear predictive coding (LPC), which describes short segments of speech using parameters that can be transformed into positions (frequencies) and shapes (bandwidths) of peaks in the spectral envelope of the speech segments. In a typical 10th order LPC model, ten such parameters are determined, the frequency peaks defined thereby corresponding to resonant frequencies of the speaker's vocal tract. The parameters defining each segment of speech (typically, 10-20 milliseconds per segment) represent data that can be applied to conventional synthesizer hardware to replicate the sound of the speaker producing the utterance. It can be shown that for a given speaker, the shape of the front cavity of the vocal tract is the primary source of linguistic information. The LPC model includes substantial information that remains approximately constant from segment to segment of an utterance by a given speaker (e.g., information reflecting the length of the speaker's vocal chords). As a consequence, the data representing each segment of speech in the LPC model include considerable redundancy, which creates an undesirable overhead for both storage and transmission of that data. It is desirable to use the smallest number of parameters required to represent a speech segment for synthesis, so that the requirements for storing such data and the bit rate for transmitting the data can be reduced. Accordingly, it is desirable to separate the speaker-independent linguistic information from the superfluous speaker-dependent information. Since the speaker-independent information that varies with each segment of speech conveys the data necessary to synthesize the words embodied in an utterance, considerable storage space can potentially be saved by separately storing and transmitting the speaker-dependent information for a given speaker, separate from the speaker-independent information. Many such utterances could be stored or transmitted in terms of their speaker-independent information and then synthesized into speech by combination with the speaker-dependent information, thereby greatly reducing storage media requirements and making more channels in an assigned bandwidth available for transmittal of voice communications using this technique. Furthermore, different speaker-dependent information could be combined with the speaker-independent information to synthesize words spoken in the voice of another speaker, for example, by substituting the voice of a female for that of a male or the voice of a specific person for that of the speaker. By reducing the amount of data required to synthesize speech, data storage space and the quantity of data that must be transmitted to a remote site in order to synthesize a given vocalization are greatly reduced. These and other advantages of the present invention will be apparent from the drawings and from the Detailed Description of the Preferred Embodiment that follows. In accordance with the present invention, a method for synthesizing human speech comprises the steps of determining a set of coefficients defining an auditory-like, speaker-independent spectrum of a given human vocalization, and mapping the set of coefficients to a vector in a vocal tract resonant vector space. Using this vector, a synthesized speech signal is produced that simulates the linguistic content (the string of words) in the given human vocalization. Substantially fewer coefficients are required than the number of vector elements produced (the dimension of the vector). These coefficients comprise data that can be stored for later use in synthesizing speech or can be transmitted to a remote location for use in synthesizing speech at the remote location. The method further comprises the steps of determining speaker-dependent variables that define qualities of the given human vocalization specific to a particular speaker. The speaker-dependent variables are then used in mapping the coefficients to produce the vector of the vocal resonant tract space, to effect a simulation of that speaker uttering the given vocalization. Furthermore, the speaker-dependent variables remain substantially constant and are used with successive different human vocalizations to produce a simulation of the speaker uttering the successive different vocalizations. Preferably, the coefficients represent a second formant, F2', corresponding to a speaker's mouth cavity shape during production of the given vocalization. The step of mapping comprises the step of determining a weighting factor for each coefficient so as to minimize a mean squared error of each element of the vector in the vocal tract resonant space (preferably determined by multivariate least squares regression). Each element is preferably defined by: ##EQU1## where ei is the i-th element, ai0 is a constant portion of that element, aij is a weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient for the i-th element; and N is the number of coefficients. FIG. 1 is a schematic block diagram illustrating the principles employed in the present invention for synthesizing speech; FIG. 2 is a block diagram of apparatus for analyzing and synthesizing speech in accordance with the present invention; FIG. 3 is a flow chart illustrating the steps implemented in analyzing speech to determine its characteristic formants, associated bandwidths, and cepstral coefficients; FIG. 4 is a flow chart illustrating the steps of synthesizing speech using the speaker-independent cepstral coefficients, in accordance with the present invention; FIG. 5 is flow chart showing the steps of a subroutine for analyzing formants; FIG. 6 is a flow chart illustrating the subroutine steps required to perform a perceptive linear predictive (PLP) analysis of speech, to determine the cepstral coefficients; FIG. 7 graphically illustrates the mapping of speaker-independent cepstral coefficients and a bias value to formant and bandwidth that is implemented during synthesis of the speech; FIGS. 8A through 8C illustrate vocal tract area and length for a male speaker uttering three Russian vowels, compared to a simulated female speaker uttering the same vowels; FIGS. 9A and 9B are graphs of the F1 and F2 formant vowel spaces for actual and modelled female and male speakers; FIGS. 10A and 10B graphically illustrate the trajectories of complex pole predicted by LPC analysis of a sentence, and the predicted trajectories of formants derived from a male speaker-dependent model and the first five cepstral coefficients from the 5th order PLP analysis of that sentence, respectively; and FIGS. 11A and 11B graphically illustrate the trajectories of formants predicted using a regressive model for a male and the first five cepstral coefficients from a sentence uttered by a male speaker, and the trajectories of formants predicted using a regressive model for a female and the first five cepstral coefficients from that same sentence uttered by a male speaker. The principles employed in synthesizing speech according to the present invention are generally illustrated in FIG. 1. The process starts in a block 10 with the PLP analysis of selected speech segments that are used to "train" the system, producing a speaker-dependent model. (See the article, "Perceptual Linear Predictive (PLP) Analysis of Speech", by Hynek Hermansky, Journal of the Acoustical Society of America, Vol 87, pp 1738-1752 April 1990.) This speaker-dependent model is represented by data that are then transmitted in real time (or pre-transmitted and stored) over a link 12 to another location, indicated by a block 14. The transmission of this speaker-dependent model may have occurred sometime in the past or may immediately precede the next phase of the process, which involves the PLP analysis of current speech, separating its substantially constant speaker-dependent content from its varying speaker-independent content. The speaker-independent content of the speech that is processed after the training phase is transmitted over a link 16 to block 14, where the speech is reconstructed or synthesized from the speaker-dependent information, at a block 18. If a different speaker-dependent model, for example, speaker-dependent model for a female, is applied to speaker-independent information produced from the speech (of a male) during the process of synthesizing speech, the reconstructed speech will sound like the female from whom the speaker-dependent model was derived. Since the speaker-independent information for a given vocalization requires only about one-half the number of data points of the conventional LPC model typically used to synthesize speech, storage and transmission of the speaker-independent data are substantially more efficient. The speaker-dependent data can potentially be updated as rarely as once each session, i.e., once each time that a different speaker-dependent model is required to synthesize speech (although less frequent updates may produce a deterioration in the nonlinguistic parts of the synthesized speech). Apparatus for synthesizing speech in accordance with the present invention are shown generally in FIG. 2 at reference numeral 20. A block 22 represents either speech uttered in real time or a recorded vocalization. Thus, a person speaking into a microphone may produce the speech indicated in block 22, or alternatively, the words spoken by the speaker may be stored on semi-permanent media, such as on magnetic tape. Whether produced by a microphone or by playback from a storage device (neither shown), the analog signal produced is applied to an analog-to-digital (A-D) converter 24, which changes the analog signal representing human speech to a digital format. Analog-to-digital converter 24 may comprise any suitable commercial integrated circuit A-D converter capable of providing eight or more bits of digital resolution through rapid conversion of an analog signal. A digital signal produced by A-D converter 24 is fed to an input port of a central processor unit (CPU) 26. CPU 26 is programmed to carry out the steps of the present method, which include the both the initial training session and analysis of subsequent speech from block 22, as described in greater detail below. The program that controls CPU 26 is stored in a memory 28, comprising, for example, a magnetic media hard drive or read only memory (ROM), neither of which is separately shown. Also included in memory 28 is random access memory (RAM) for temporarily storing variables and other data used in the training and analysis. A user interface 30, comprising a keyboard and display, is connected to CPU 26, allowing user interaction and monitoring of the steps implemented in processing the speech from block 22. Data produced during the initial training session through analysis of speech are converted to a digital format and stored in a storage device 32, comprising a hard drive, floppy disk, or other nonvolatile storage media. For subsequently processing speech that is to be synthesized, CPU 26 carries out a perceptual linear predictive (PLP) analysis of the speech to determine several cepstral coefficients, C1 . . . Cn that comprise the speaker-independent data. In the preferred embodiment, only five cepstral coefficients are required for each segment of the speaker-independent data used to synthesize speech (and in "training" the speaker-dependent model). In addition, CPU 26 is programmed to perform a formant analysis, which is used to determine a plurality of formants F1 through Fn and corresponding bandwidths B1 through Bn. The formant analysis produces data used in formulating a speaker-dependent model. The formant and bandwidth data for a given segment of speech differ from one speaker to another, depending upon the shape of the vocal tract and various other speaker-dependent physiological parameters. During the training phase of the process, CPU 26 derives multiple regressive speaker-dependent mappings of the cepstral coefficients of the speech segments spoken during the training exercise, to the corresponding formants and bandwidths Fi and Bi for each segment of speech. The speaker-dependent model resulting from mapping the cepstral coefficients to the formants and bandwidths for each segment of speech is stored in storage device 32 for later use. Alternatively, instead of storing this speaker-dependent model, the data comprising the model can be transmitted to a remote CPU 36, either prior to the need to synthesize speech, or in real time. Once remote CPU 36 has stored the speaker-dependent model required to map between the speaker-independent cepstral coefficients and the formants and bandwidths representing the speech of a particular speaker, it can apply the model data to subsequently transmitted cepstral coefficients to reproduce any speech of that same speaker. The speaker-dependent model data are applied to the speaker-independent cepstral coefficients for each segment of speech that is transmitted from CPU 26 to CPU 36 to reproduce the synthesized speech, by mapping the cepstral coefficients to corresponding formants and bandwidths that are used to drive a synthesizer 42. A user interface 40 is connected to remote CPU 36 and preferably includes a keyboard and display for entering instructions that control the synthesis process and a display for monitoring its progression. Synthesizer 42 preferably comprises a Klsyn88™ cascade/parallel formant synthesizer, which is a combination software and hardware package available from Sensimetrics Corporation, Cambridge, Mass. However, virtually any synthesizer suitable for synthesizing human speech from LPC formant and bandwidth data can be used for this purpose. Synthesizer 42 drives a conventional loudspeaker 44 to produce the synthesized speech. Loudspeaker 44 may alternatively comprise a telephone receiver or may be replaced by a recording device to record the synthesized speech. Remote CPU 36 can also be controlled to apply a speaker-dependent model mapping for a different speaker to the speaker-independent cepstral coefficients transmitted from CPU 26, so that the speech of one speaker is synthesized to sound like that of a different speaker. For example, speaker-dependent model data for a female speaker can be applied to the transmitted cepstral coefficients for each segment of speech from a male speaker, causing synthesizer 42 to produce synthesized speech, which on loudspeaker 44, sounds like a female speaker speaking the words originally uttered by the male speaker. CPU 36 can also modify the speaker-dependent model in other ways to enhance, or otherwise change the sound of the synthesized speech produced by loudspeaker 44. One of the primary advantages of the technique implemented by the apparatus in FIG. 1 is the reduced quantity of data that must be stored and/or transmitted to synthesize speech. Only the speaker-dependent model data and the cepstral coefficients for each successive segment of speech must be stored or transmitted to synthesize speech, thereby reducing the number of bytes of data that need be stored by storage device 32, or transmitted to remote CPU 36. As noted above, the training steps implemented by CPU 26 initially determine the mapping of cepstral coefficients for each segment of speech to their corresponding formants and bandwidths to define how subsequent speaker-independent cepstral coefficients should be mapped to produce synthesized speech. In FIG. 3, a flow chart 50 shows the steps implemented by CPU 26 in this training procedure and the steps later used to derive the speaker-independent cepstral coefficients for synthesizing speech. Flow chart 50 starts at a block 52. In a block 54, the analog values of the speech are digitized for input to a block 56. In block 56, a predefined time interval of approximately 20 milliseconds in the preferred embodiment defines a single segment of speech that is analyzed according to the following steps. Two procedures are performed on each digitized segment of speech, as indicated in flow chart 50 by the parallel branches to which block 56 connects. In a block 58, a subroutine is called that performs formant analysis to determine the F1 through Fn formants and their corresponding bandwidths, B1 through Bn for each segment of speech processed. The details of the subroutine used to perform the formant analysis are shown in FIG. 5 in a flow chart 60. Flow chart 60 begins at a block 62 and proceeds to a block 64, wherein CPU 26 determines the linear prediction coefficients for the current segment of speech being processed. Linear predictive analysis of digital speech signals is well known in the art. For example, J. Makhoul described the technique in a paper entitled "Spectral Linear Prediction: Properties and Applications," IEEE Transaction ASSP-23, 1975, pp. 283-296. Similarly, in U.S. Pat. No. 4,882,758 (Uekawa et al.), an improved method for extracting formant frequencies is disclosed and compared to the more conventional linear predictive analysis method. In block 64, CPU 26 processes the digital speech segment by applying a pre-emphasis and then using a window with an autocorrelation calculation to obtain linear prediction coefficients by the Durbin method. The Durbin method is also well known in the art, and is described by L. R. Rabiner and R. W. Schafer in Digital Processing of Speech Signals, a Prentice-Hall publication, pp. 411-413. In a block 66, a constant Z0 is selected for an initial value as a root Zi. In a block 68, CPU 26 determines a value of A(z) from the following equation: ##EQU2## where ak are linear prediction coefficients. In addition, the CPU determines the derivative A'(Zi) of this function. A decision block 70 then determines if the absolute value of A(Zi)/A'(Zi) is less than a specified tolerance threshold value K. If not, a block 72 assigns a new value to Zi, as shown therein. The flow chart then returns to block 68 for redetermination of a new value for the function A(Zi) and its derivative. As this iterative loop continues, it eventually reaches a point where an affirmative result from decision block 70 leads to a block 74, which assigns Zi and its complex conjugate Zi * as roots of the function A(z). A block 76 then divides the function A(z) by the quadratic expression of Zi and its complex conjugate, as shown therein. A decision block 78 determines whether Zi is a zero-order root of the function A(Z) and if not, loops back to block 64 to repeat the process until a zero order value for the function A(Z) is obtained. Once an affirmative result from decision block 78 occurs, a block 80 determines the corresponding formants Fk for all roots of the equation as defined by: Fk =(f8 /2π)tan-1 [Im(Zi)/Re(Zi)](2) Similarly, a block 82 defines the bandwidth corresponding to the formants for all the roots of the function as follows: Bk =(fs /π)1n|Z1 | (3). A block 84 then sets all roots with Bk less than a constant threshold T equal to formants Fi having corresponding bandwidths Bi. A block 86 then returns from the subroutine to the main program implemented in flow chart 50. Following a return from the subroutine called in block 58 of FIG. 3, a block 90 stores the formants F1 through FN and corresponding bandwidths B1 through BN in memory 28 (FIG. 2). The other branch of flow chart 50 following block 56 in FIG. 3 leads to a block 92 that calls a subroutine to perform PLP analysis of the digitized speech segment to determine its corresponding cepstral coefficients. The subroutine called by block 92 is illustrated in FIG. 6 by a flow chart 94. Flow chart 94 begins at a block 96 and proceeds to a block 98, which performs a fast Fourier transform of the digitized speech segment. In carrying out the fast Fourier transform, each speech segment is weighted by a Hamming window, which is a finite duration window represented by the following equation: W(n)=0.54+0.46cos [2πn/(T-1)] (4) where T, the duration of the window, is typically about 20 milliseconds. The Fourier transform performed in block 98 transforms the speech segment weighted by the Hamming window into the frequency domain. In this step, the real and imaginary components of the resulting speech spectrum are squared and added together, producing a short-term power spectrum P(ω), which can be represented as follows: P(ω)=Re[S(ω)]2 +Im[S(ω)]2 (5). Typically, for a 10 KHz sampling frequency, a 256-point fast Fourier transform is applied to transform 200 speech samples (from the 20-millisecond window that was applied to obtain the segment), with the remaining 56 points padded by zero-valued samples. In a block 100, critical band integration and resampling is performed, during which the short-term power spectrum P(ω) is warped along its frequency access ω into the Bark frequency Ω as follows: ##EQU3## wherein ω is the angular frequency in radians per second, resulting in a Bark-Hz transformation. The resulting warped power spectrum is then convolved with the power spectrum of the simulated critical band masking curve ψ(ω). Except for the particular shape of the critical-band curve, this step is similar to spectral processing in mel cepstral analysis. The critical band curve is defined as follows: ##EQU4## The piece-wise shape of the simulated critical-band masking curve is an approximation to an asymmetric masking curve. The intent of this step is to provide an approximation (although somewhat crude) of an auditory filter based on the proposition that the shape of auditory filters is approximately constant on the Bark scale and that the filter skirts are generally truncated at -40dB. Convolution of ψ(ω) with (the even symmetric and periodic function) P(ω) yields samples of the critical-band power spectrum: ##EQU5## This convolution significantly reduces the spectral resolution of θ(Ω) in comparison with the original P(ω), allowing for the down-sampling of θ(Ω). In the preferred embodiment, θ(Ω) is sampled at approximately one-Bark intervals. The exact value of the sampling interval is chosen so that an integral number of spectral samples covers the entire analysis band. Typically, for a bandwidth of 5 KHz, corresponding to 16.9-Bark, 18 spectral samples of θ(Ω) are used, providing 0.994-Bark steps. In a block 102, a logarithm of the computed critical-band spectrum is performed, and any convolutive constants appear as additive constants in the logarithm. A block 104 applies an equal-loudness response curve to pre-emphasize each of the segments, where the equal-loudness curve is represented as follows: In this equation, the function E(ω) is an approximation to the human sensitivity to sounds at different frequencies and simulates the unequal sensitivity of hearing at about the 40dB level. Under these conditions, this function is defined as follows: ##EQU6## The curve approximates a transfer function for a filter having asymptotes of 12dB per octave between 0 and 400 Hz, 0 dB per octave between 400 Hz and 1,200 Hz, 6 dB per octave between 1,200 Hz and 3,100 Hz, and zero dB per octave between 3,100 Hz and the Nyquist frequency (10 KHz in the preferred embodiment). In applications requiring a higher Nyquist frequency, an additional term can be added to the preceding expression. The values of the first (zero-Bark) and the last samples are made equal to the values of their nearest neighbors to ensure that the function resulting from the application of the equal loudness response curve begins and ends with two equal-valued samples. In a block 106, a power-law of hearing function approximation is performed, which involves a cubic-root amplitude compression of the spectrum, defined as follows: This compression is an approximation that simulates the nonlinear relation between the intensity of sound and its perceived loudness. In combination, the equal-loudness pre-emphasis of block 104 and the power law of hearing function applied in block 106 reduce the spectral-amplitude variation of the critical-band spectrum to produce a relatively low model order. A block 108 provides for determining an inverse logarithm (i.e., determines an exponential function) of the compressed log critical-band spectrum. The resulting function approximates a relatively auditory spectrum. A block 110 determines an inverse discrete Fourier transform of the auditory spectrum Φ(Ω). Preferably, a 34-point inverse discrete Fourier transform is used. The inverse discrete Fourier transform is a better choice than the fast Fourier transform in this case, because only a few autocorrelation values are required in the subsequent analysis. In linear predictive analysis, a set of coefficients that will minimize a mean-squared prediction error over a short segment of speech waveform is determined. One way to determine such a set of coefficients is referred to as the autocorrelation method of linear prediction. This approach provides a set of linear equations that relate autocorrelation coefficients of the signal representing the processed speech segment with the prediction coefficients of the autoregressive model. The resulting set of equations can be efficiently solved to yield the predictor parameters. The inverse Fourier transform of a non-negative spectrum-like function resulting from the preceding steps can be interpreted as the autocorrelation function, and an appropriate autoregressive model of such a spectrum can be found. In the preferred embodiment of the present method, the equations for carrying out this solution apply Durbin's recursive procedure, as indicated in a block 112. This procedure is relatively efficient for solving specific linear equations of the autoregressive process. Finally, in a block 114, a recursive computation is applied to determine the cepstral coefficients from the autoregressive coefficients of the resulting all-pole model. If the overall LPC system has a transfer function H(z) with an impulse response h(n) and a complex cepstrum h(n), then h(n) can be obtained from the recursion: ##EQU7## (as shown by L. R. Rabiner and R. W. Schafer in Digital Processing of Speech Signals, a Prentice-Hall publication, page 442.) The complex cepstrum cited in this reference is equivalent to the cepstral coefficients C1 through C5. After block 114 produces the cepstral coefficients, a block 116 returns to flow chart 50 in FIG. 3. Thereafter, a block 120 provides for storing the cepstral coefficients C1 through C5 in nonvolatile memory. Following blocks 90 or 120, a decision block 122 determines if the last segment of speech has been processed, and if not, returns to block 56 in FIG. 3. After all segments of speech have been processed, a block 124 provides for deriving multiple regressive speaker-dependent mappings from the cepstral coefficients Ci using the corresponding formants Fi and bandwidths Bi. The mapping process is graphically illustrated in FIG. 7 generally at reference numeral 170, where five cepstral coefficients 176 and a bias value 178 are linearly combined to produce five formants and corresponding bandwidths 180 according to the following relationship: ##EQU8## where ei are elements representing the respective formants and their bandwidths (i=1 through 10, corresponding to F1 through F5 and B1 through B5, in succession), ai0 is the bias value, and aij are weighting factors for the j-th cepstral coefficient and the i-th element (formant or bandwidth) that are applied to the cepstral coefficients Cij. Mapping of the cepstral coefficients and bias value corresponds to a linear function that estimates the relationship between the formants (and their corresponding bandwidths) and the cepstral coefficients. The linear regression analysis performed in this step is discussed in detail in An Introduction to Linear Regression and Correlation, by Allen L. Edwards (W. H. Freeman & Co., 1976), ch. 3. Thus, for each segment of speech, linear regression analysis is applied to map the cepstral coefficients 176 and bias value 178 into the formants and bandwidths 180. The mapping data resulting from this procedure are stored for subsequent use, or immediately used with speaker-independent cepstral coefficients to synthesize speech, as explained in greater detail below. A block 128 ends this first training portion of the procedure required for developing the speaker-dependent model for mapping of speaker-independent cepstral coefficients into corresponding formants and bandwidths. Turning now to FIG. 4, the speaker-dependent model defined by mapping data developed from the training procedure implemented by the steps of flow chart 50 can later be applied to speaker-independent data to synthesize vocalizations by that same speaker, as briefly noted above. Alternatively, the speaker-independent data (represented by cepstral coefficients) of one speaker can be modified by the model data of a different speaker to produce synthesized speech corresponding to the vocalization of the different speaker. Steps required for carrying out either of these scenarios are illustrated in a flow chart 140 in FIG. 4, starting at a block 142. In a block 143, signals representing the analog speech of an individual (from block 22 in FIG. 2) are applied to an A-D converter, producing corresponding digital signals that are processed one segment at a time. Digital signals are input to CPU 36 in a block 144. A block 146 calls a subroutine to perform PLP analysis of the signal to determine the cepstral coefficients for the speech segment, as explained above with reference to flow chart 94 in FIG. 6. This subroutine returns the cepstral coefficients for each segment of speech, which are alternatively either stored for later use in a block 148, or transmitted, for example, by telephone line, to a remote location for use in synthesizing the speech represented by the speaker-independent cepstral coefficients. Transmission of the cepstral coefficients is provided in a block 150. In a block 152, the speaker-dependent model represented by the mapping data previously developed during the training procedure is applied to the cepstral coefficients, which have been stored in block 148 or transmitted in block 150, to develop the formants F1 through Fn and corresponding bandwidths B1 through Bn needed to synthesize that segment of speech. As noted above, the linear combination of the cepstral coefficients to produce the formants and bandwidth data in block 152 is graphically illustrated in FIG. 7. A block 154 uses the formants and bandwidths developed in block 152 to produce a corresponding synthesized segment of speech, and a block 156 stores the digitized segment of speech. A decision block 158 determines if the last segment of speech has been processed, and if not, returns to block 144 to input the next speech segment for PLP analysis. However, if the last segment of speech has been processed, a block 160 provides for digital-to-analog (D-A) conversion of the digital signals. Referring back to FIG. 2, block 160 produces the analog signal used to drive loudspeaker 44, producing an auditory response synthetically reproducing the speech of either the original speaker or speech sounding like another person, depending upon whether the original speaker's model (mapping data) or the other person's model is used in block 152 to map the cepstral coefficients into corresponding formants and bandwidths. A block 162 terminates flow chart 140 in FIG. 4. Experiments have shown that there is a relatively high correlation between the estimated formants and bandwidths used to synthesize speech in the present invention and the formants and bandwidths determined by conventional LPC analysis of the original speech segment. Table 1, below, shows correlations between the true and model-predicted form of these parameters, the root mean square (RMS) error of the prediction, and the maximum prediction error. For comparison, values from the 10th order LPC formant estimation are shown in parentheses. The RMS error of the PLP-based formant frequency prediction is larger than the LPC estimation RMS error. LPC exhibits occasional gross errors in the estimation of lower formants, which show in larger values of the maximum LPC error. In fact, formant bandwidths are far better predicted by the PLP-based technique. TABLE 1__________________________________________________________________________FORMANT AND BANDWIDTH COMPARISONSPARAM.__________________________________________________________________________F1 F2 F3 F4 F5__________________________________________________________________________CORR. 0.94 (0.98) 0.98 (0.99) 0.91 (0.98) 0.64 (0.98) 0.86 (0.99)RMS[Hz] 23.6 (15.5) 48.1 (37.0) 48.2 (21.2) 46.1 (12.6) 52.4 (13.1)MAX[Hz] 131 (434) 344 (2170) 190 (1179) 190 (610) 220 (130)__________________________________________________________________________B1 B2 B3 B4 B5__________________________________________________________________________CORR. 0.86 (0.05) 0.92 (0.17) 0.96 (0.43) 0.64 (0.24) 0.86 (0.33)RMS[Hz] 2.2 (45) 1.6 (35) 4.1 (37) 4.1 (50) 5.5 (52)MAX[Hz] 29.3 (3707) 6.23 (205) 32.0 (189) 18.0 (119) 22.0 (354)__________________________________________________________________________ A significant advantage of the present technique for synthesizing speech is the ability to synthesize a different speaker's speech using the cepstral coefficients developed from low-order PLP analysis, which are generally speaker-independent. To evaluate the potential for voice modification, the vocal tract area functions for a male voicing three vowels /i/, /a/, and /u/ were modified by scaling down the length of the pharyngeal cavity by 2 cm and by linearly scaling each pharyngeal area by a constant. This constant was chosen for each vowel by a simple search so that the differences between the log of a male and a female-like PLP spectra are minimized. It has been observed that to achieve similar PLP spectra for both the longer and the shorter vocal tracts, the pharyngeal cavity for the female-like tracts need to be slightly expanded. FIGS. 8A through 8C show the vocal tract functions for the three Russian vowels /i/, /a/, and /u/, using solid lines to represent the male vocal tract and dashed lines to represent the simulated female-like vocal tract. Thus, for example, solid lines 192, 196, and 200 represent the vocal tract configuration for a male, whereas dashed lines 190, 194, and 198 represent the simulated vocal tract voicing for a female. Both the original and modified vocal tract functions were used to generate vowel spaces. The training procedure described above was used to obtain speaker-dependent models, one for the male and one for the simulated female-like vowels. PLP vectors (cepstral coefficients) derived from male speech were used with a female-regressive model, yielding predicted formants, as shown in FIG. 9A. Similarly, PLP vectors derived from female speech were used with the male-regressive models to yield predicted formants depicted in FIG. 9B. In FIG. 9A, boundaries of the original male vowel space are indicated by a solid line 202, while boundaries of the original female space are indicated by a dashed line 204. Similarly, in FIG. 9B, boundaries of the original female vowel space are indicated by a solid line 206, and boundaries of the original male vowel space are indicated by a dashed line 208. Based on a comparison of the F1 and F2 formants for the original and the predicted models, both male and female, it is evident that the range of predicted formant frequencies is determined by the given regression model, rather than by the speech signals from which the PLP vectors are derived. Further verification of the technique for synthesizing the speech of a particular speaker in accordance with the present invention was provided by the following experiment. The regression speaker-dependent model for a particular speaker was derived from four all-voiced sentences: "We all learn a yellow line roar;" "You are a yellow yo-yo;" "We are nine very young women;" and "Hello, how are you?" each uttered by a male speaker. The first five cepstral coefficients (log energy excluded) from the fifth order PLP analysis of the first utterance, "I owe you a yellow yo-yo," together with the regressive model derived from training with the four sentences were used in predicting formants of the test utterance, as shown in FIG. 10B. An estimated formant trajectory represented by poles of a 10th order LPC analysis for the same sentence, "I owe you a yellow yo-yo," uttered by a male speaker are shown in FIG. 10A. Comparing the predicted formant trajectories of FIG. 10B with the estimated formant trajectories represented by poles of the 10th order LPC analysis shown in FIG. 10A, it is clear that the first formant is predicted reasonably well. On the second formant trajectory, the largest difference is in /oh/ of "owe . . .," where the predicted second formant frequency is about 50% higher than the LPC estimated one. Furthermore, the predicted frequencies of the /j/s in "you" and "yo-yo," and of /e/ and /u/ in "yellow" are 15-20% lower than the LPC estimated ones. The predicated third order trajectory is again reasonably close to the LPC estimated trajectory. The LPC estimated fourth and fifth formants are generally unreliable, and comparing them to the predicted trajectories is of little value. A similar experiment was done to determine whether synthetic speech can yield useful speaker-dependent models. In this case, speaker-dependent models derived from synthetic speech vowels were used, to produce a male regressive model for the same sentence. The trajectories of the formants predicted using the male regressive model in the first five cepstral coefficients from the fifth order PLP analysis of the sentence "I owe you a yellow yo-yo" uttered by a male speaker were then compared to the trajectories of formants predicted using the female regressive model (also derived from the synthetic vowel-like samples) in the first five cepstral coefficients from the fifth order PLP analysis of the same sentence, uttered by the male speaker. Within the 0 through 5 KHz frequency band of interest, the male regressive model yields five formants, while the female-like model yields only four. By comparison of FIGS. 11A and 11B, it is apparent that the formant trajectories for both genders are approximately the same. The frequency span of the female second formant trajectory is visibly larger than the frequency span of the male second formant trajectory, almost coinciding with the third male formants in extreme front semi-vowels, such as the /j/s in "yo-yo" and being rather close to the male second formants in the rounded /u/ of "you." The male third formant trajectory is very similar to the female third formant trajectory, except for approximately a 400 Hz constant downward frequency shift. However, the male fourth formant trajectory bears almost no similarity to any of the female formant trajectories. Finally, the fifth formant trajectory for the male is quite similar to the female fourth formant trajectory. Although the preferred embodiment uses PLP analysis to determine a speaker-dependent model for a particular speaker during the training process and for producing the speaker-independent cepstral coefficients that are used with that or another speaker's model for speech synthesis, it should be apparent that other speech processing techniques might be used for this purpose. These and other modifications and changes that will be apparent to those of ordinary skill in this art fall within the scope of the claims that follow. While the preferred embodiment of the invention has been illustrated and described, it will be appreciated that such changes can be made therein without departing from the spirit and scope of the invention defined by these claims.

A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole? Use these head, body and leg pieces to make Robot Monsters which are different heights. Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it. In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case? Find all the numbers that can be made by adding the dots on two dice. As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells? Can you find 2 butterflies to go on each flower so that the numbers on each pair of butterflies adds to the same number as the one on the flower? Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make? Can you each work out the number on your card? What do you notice? How could you sort the cards? Investigate the different distances of these car journeys and find out how long they take. In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make? I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to? This is an adding game for two players. Investigate what happens when you add house numbers along a street in different ways. Leah and Tom each have a number line. Can you work out where their counters will land? What are the secret jumps they make with their counters? There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket? Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether. 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? These two group activities use mathematical reasoning - one is numerical, one geometric. Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals? These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers? In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word? If the numbers 5, 7 and 4 go into this function machine, what numbers will come out? Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest? In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice? Twizzle, a female giraffe, needs transporting to another zoo. Which route will give the fastest journey? 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? A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids. 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? Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families? 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 draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible? Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were. Vera is shopping at a market with these coins in her purse. Which things could she give exactly the right amount for? Sam got into an elevator. He went down five floors, up six floors, down seven floors, then got out on the second floor. On what floor did he get on? Woof is a big dog. Yap is a little dog. Emma has 16 dog biscuits to give to the two dogs. She gave Woof 4 more biscuits than Yap. How many biscuits did each dog get? There were 22 legs creeping across the web. How many flies? How many spiders? 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? Can you work out how many flowers there will be on the Amazing Splitting Plant after it has been growing for six weeks? Can you substitute numbers for the letters in these sums? In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins? Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? The value of the circle changes in each of the following problems. Can you discover its value in each problem? Can you score 100 by throwing rings on this board? Is there more than way to do it? Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre. 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? Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7? Where can you draw a line on a clock face so that the numbers on both sides have the same total? 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! The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?

Where to Find Partially Full Pipe Flow Calculations Spreadsheets To obtain Excel spreadsheets for partially full pipe flow calculations, click here to visit our spreadsheet store for partially full pipe flow calculations spreadsheets. Read on for information about Excel spreadsheets that can be used as partially full pipe flow calculators. The Manning equation can be used for flow in a pipe that is partially full, because the flow will be due to gravity rather than pressure. the Manning equation [Q = (1.49/n)A(R2/3)(S1/2) for (U.S. units) or Q = (1.0/n)A(R2/3)(S1/2) for (S.I. units)] applies if the flow is uniform flow For background on the Manning equation and open channel flow and the conditions for uniform flow, see the article, “Manning Equation/Open Channel Flow Calculations with Excel Spreadsheets.” Direct use of the Manning equation as a partially full pipe flow calculator, isn’t easy, however, because of the rather complicated set of equations for the area of flow and wetted perimeter for partially full pipe flow. There is no simple equation for hydraulic radius as a function of flow depth and pipe diameter. As a result graphs of Q/Qfull and V/Vfull vs y/D, like the one shown at the left are commonly used for partially full pipe flow calculations. The parameters, Q and V in this graph are flow rate an velocity at a flow depth of y in a pipe of diameter D. Qfull and Vfull can be conveniently calculated using the Manning equation, because the hydraulic radius for a circular pipe flowing full is simply D/4. With the use of Excel formulas in an Excel spreadsheet, however, the rather inconvenient equations for area and wetted perimeter in partially full pipe flow become much easier to work with. The calculations are complicated a bit by the need to consider the Manning roughness coefficient to be variable with depth of flow as discussed in the next section. Is the Manning Roughness Coefficient Variable for Partially Full Pipe Flow Calculations? Using the geometric/trigonometric equations discussed in the next couple of sections, it is relatively easy to calculate the cross-sectional area, wetted perimeter, and hydraulic radius for partially full pipe flow with any specified pipe diameter and depth of flow. If the pipe slope and Manning roughness coefficient are known, then it should be easy to calculate flow rate and velocity for the given depth of flow using the Manning Equation [Q = (1.49/n)A(R2/3)(S1/2)], right? No, wrong! As long ago as the middle of the twentieth century, it had been observed that measured flow rates in partially full pipe flow aren’t the same as those calculated as just described. In a 1946 journal article (ref #1 below), T. R. Camp presented a method for improving the agreement between measured and calculated values for partially full pipe flow. The method developed by Camp consisted of using a variation in Manning roughness coefficient with depth of flow as shown in the graph above. Although this variation in Manning roughness due to depth of flow doesn’t make sense intuitively, it does work. It is well to keep in mind that the Manning equation is an empirical equation, derived by correlating experimental results, rather than being theoretically derived. The Manning equation was developed for flow in open channels with rectangular, trapezoidal, and similar cross-sections. It works very well for those applications using a constant value for the Manning roughness coefficient, n. Better agreement with experimental measurements is obtained for partially full pipe flow, however, by using the variation in Manning roughness coefficient developed by Camp and shown in the diagram above. The graph developed by Camp and shown above appears in several publications of the American Society of Civil Engineers, the Water Pollution Control Federation, and the Water Environment Federation from 1969 through 1992, as well as in many environmental engineering textbooks (see reference list at the end of this article). You should beware, however that there are several online calculators and websites with equations for making partially full pipe flow calculations using the Manning equation with constant Manning roughness coefficient, n. The equations and Excel spreadsheets presented and discussed in this article use the variation in n that was developed by T.R. Camp. Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe Less than Half Full The parameters used in partially full pipe flow calculations with the pipe less than half full are shown in the diagram at the right. K is the circular segment area; S is the circular segment arc length; h is the circular segment height; r is the radius of the pipe; and θ is the central angle. The equations below are those used, together with the Manning equation and Q = VA, in the partially full pipe flow calculator (Excel spreadsheet) for flow depth less than pipe radius, as shown below. - h = y - θ = 2 arccos[ (r – h)/r ] - A = K = r2(θ – sinθ)/2 - P = S = rθ The equations to calculate n/nfull, in terms of y/D for y < D/2 are as follows - n/nfull = 1 + (y/D)(1/3) for 0 < y/D < 0.03 - n/nfull = 1.1 + (y/D – 0.03)(12/7) for 0.03 < y/D < 0.1 - n/nfull = 1.22 + (y/D – 0.1)(0.6) for 0.1 < y/D < 0.2 - n/nfull = 1.29 for 0.2 < y/D < 0.3 - n/nfull = 1.29 – (y/D – 0.3)(0.2) for 0.3 < y/D < 0.5 The Excel template shown below can be used as a partially full pipe flow calculator to calculate the pipe flow rate, Q, and velocity, V, for specified values of pipe diameter, D, flow depth, y, Manning roughness for full pipe flow, nfull; and bottom slope, S, for cases where the depth of flow is less than the pipe radius. This Excel spreadsheet and others for partially full pipe flow calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store. Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe More than Half Full The parameters used in partially full pipe flow calculations with the pipe more than half full are shown in the diagram at the right. K is the circular segment area; S is the circular segment arc length; h is the circular segment height; r is the radius of the pipe; and θ is the central angle. The equations below are those used, together with the Manning equation and Q = VA, in the partially full pipe flow calculator (Excel spreadsheet) for flow depth more than pipe radius, as shown below. - h = 2r – y - θ = 2 arccos[ (r – h)/r ] - A = πr2 – K = πr2 – r2(θ – sinθ)/2 - P = 2πr – S = 2πr – rθ The equation used for n/nfull for 0.5 < y//D < 1 is: n/nfull = 1.25 – [(y/D – 0.5)/2] An Excel spreadsheet like the one shown above for less than half full flow, and others for partially full pipe flow calculations, are available in either U.S. or S.I. units at a very low cost at www.engineeringexceltemplates.com. 1. Bengtson, Harlan H., Uniform Open Channel Flow and The Manning Equation, an online, continuing education course for PDH credit. 2. Camp, T.R., “Design of Sewers to Facilitate Flow,” Sewage Works Journal, 18 (3), 1946 3. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959. 4. Steel, E.W. & McGhee, T.J., Water Supply and Sewerage, 5th Ed., New York, McGraw-Hill Book Company, 1979 5. ASCE, 1969. Design and Construction of Sanitary and Storm Sewers, NY 6. Bengtson, H.H., “Manning Equation Partially Filled Circular Pipes,” An online blog article 7. Bengtson, H.H., “Partially Full Pipe Flow Calculations with Spreadsheets“, available as an Amazon Kindle e-book and as a paperback.

1703 North Beauregard St. Alexandria, VA 22311-1714 Tel: 1-800-933-ASCD (2723) 8:00 a.m. to 6:00 p.m. eastern time, Monday through Friday Local to the D.C. area: 1-703-578-9600 Toll-free from U.S. and Canada: 1-800-933-ASCD (2723) All other countries: (International Access Code) + 1-703-578-9600 April 2014 | Volume 71 | Number 7 Writing: A Core Skill Jihwa Noh and Karen Sabey When students write their own math word problems, teachers get immediate feedback about which concepts they do and don't understand. How do you go about calculating 2/3 × 1/4? You may have multiplied the two numbers on top (namely, numerators) to get 2 and multiplied the two numbers on the bottom (namely, denominators) to get 12, so you would get the answer of 2/12 (or 1/6 if simplified). Or you may have begun by simplifying the 2 in 2/3 and the 4 in 1/4 and then followed the previously described method. If you got the correct answer, congratulate yourself for remembering the procedure. Now we have a different question for you: Can you write a word problem in which you would calculate 2/3 × 1/4 to find the answer? Hmm, do you feel like you need to brush up on your math skills? Perhaps your teachers never asked you to write a word problem when you were in school. You can use at least four different contextual situations for your word problem, each of which embodies a different interpretation for multiplication of fractions. First, there's looking at fraction multiplication as taking a part of a part. For example, using that same problem, "There was 1/4 of a pan of brownies left from yesterday's party. If you ate 2/3 of the leftover brownies, what fractional part of the pan of brownies would you have eaten?" Second, you might make a scale drawing in which one foot is represented by 1/4 of an inch. To determine how long a 2/3-foot-long table would be on the drawing, you'd need to multiply 2/3 × 1/4. Third, perhaps you want to determine the probability of rain on both days when there's a 66 percent chance of rain on Saturday (approximately 2/3) and a 25 percent chance on Sunday. You'd multiply the two numbers in fraction or decimal form (2/3 × 1/4). Finally, you might wish to find the area of a rectangular region whose dimensions are 2/3 of a yard by 1/4 of a yard. You'd do that by multiplying 2/3 × 1/4. In our study involving 140 college freshmen, only two students were able to write a word problem that correctly represented a given fraction multiplication problem. Fifty-seven (41 percent) wrote a problem that would be answered by operations other than multiplication, such as addition. Fifty-three (38 percent) didn't even attempt to write a word problem, or their responses contained no substantial information, making them unanalyzable. Not only was the students' mathematical understanding disappointing, but there also were language issues, such as spelling, semantic, and syntactical errors, and issues using inappropriate or unrealistic contexts. The Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and the National Council of Teachers of Mathematics (2000) advocate using multiple methods that give students opportunities to learn mathematical ideas and demonstrate their understandings. By having students write word problems that encompass a variety of contextual situations, teachers gain insight into how students have interpreted a mathematical idea as well as their preferences for problem-solving strategies (National Mathematics Advisory Panel, 2008; Newton, 2008). We want our students to make sense of the mathematics they're learning and solve problems in sense-making ways, rather than merely applying rules and formulas. In our study, one of the common errors that students made concerning the multiplication of fractions was the result of a misconception—that multiplication always makes numbers bigger. One student wrote, "Tom has 2/3 of an apple and needs to make a big batch for a recipe by 5/7. How many apples will he have now?" In addition to the readability issue this problem presents, adjusting a recipe by 5/7 would not yield a bigger batch, but rather a smaller one. Having students write word problems gives teachers immediate feedback about student misconceptions as well as the opportunity to develop lesson plans to both address student weaknesses and bolster student strengths. When students write their own word problems, they typically make use of situations and contexts with which they're familiar. This can make the problems far more meaningful and comprehensible (Barwell, 2003; Chapman, 2006). This can be particularly helpful for English language learners (ELLs), who may find word problems difficult because they lack appropriate background knowledge, such as knowledge about U.S. currency or American football rules. By having ELLs write their own word problems using situations familiar to them, as well as language they can manage, teachers can more easily assess their mathematical abilities. There's a prevalent myth that ELLs cannot be successful in solving word problems until they're more fluent in English (Martiniello, 2008). Because of this misunderstanding, many teachers might limit the teaching of mathematics to computation exercises instead of engaging the students in problem-solving efforts using word problems. This approach yields missed opportunities for ELLs to work toward overcoming the language demands of mathematics. As students share their word problems with the class and invite their peers to solve those problems, they're led into discussions, both in small groups and as part of a whole-class discussion, about the meaning of their problems and how best to solve them. These discussions give students practice using mathematics language, which both the Common Core State Standards in Mathematics and the National Council of Teachers of Mathematics emphasize as an essential component in learning mathematics. A problem-posing activity can bring in many forms of communication, such as writing, speaking, reading, and listening, which benefit not just ELLs but all students. Mathematics language is semantically and syntactically specialized. Students may be familiar with the common uses of words like even, odd, and improper, but these words have a different meaning when used in mathematics. Sometimes the same mathematical word is used in more than one way within the field itself. The word square, for example, can refer to a shape and also to a number times itself (Rubenstein & Thompson, 2002). Also, mathematics language makes use of certain syntactic structures—such as greater than/less than, n times as much as, divided by as opposed to divided into, if/then, and so on (Chamot & O'Malley, 1994). Student-written math problems help students connect the mathematics they're learning to other mathematical ideas and with ideas outside the mathematics classroom. These problems also help students understand how the theoretical language of mathematics and the everyday language of word problems are related. When using fractions, it's important to clarify what the unit whole is. In the following example, the whole is unclear: "Bill has 2/3 of oranges left in his bag. Bill would like 5/7 more. How many oranges would Bill end up with?" It's unclear what whole is associated with the 5/7: the number of oranges that Bill currently has or the number of oranges that were in the bag originally. In either case, we can't answer the question because we don't know what the original number of oranges was. Asking "what fraction of a bag" instead of "how many oranges" would be more appropriate. In the following example, the measurements the student used are unrealistic, although the problem is correct mathematically: "Jamie made a pan of brownies. The pan's length was 2/3cm and its width was 5/7cm. How big is the pan?" In addition, the word big is ambiguous because it could mean either the area or perimeter of the pan. Also, a number of students merely translated the multiplication sign into words. For example, "Ben has 2/3 of his candy bar left, and Sally has 5/7 of her candy bar left. If you multiply their candy bars together, how much would they have?" But what does multiplying two candy bars really mean? Teachers need to find ways of helping students overcome the difficulties they encounter in writing word problems. Let's say you ask students to write a word problem in which they need to do the calculation from the opening of this article—2/3 × 1/4—to find the answer. A student may respond with the following: Julie ate 1/4 of a pizza. Janet ate 2/3 more. How much pizza did Janet eat? The first problem we encounter is with the words "2/3 more." Is this (a) 2/3 of a whole pizza, or is this (b) 2/3 of what Julie ate? If the meaning is (a), then the answer to the problem—How much pizza did Janet eat?—is 1/4 + 2/3, or 11/12 of a pizza. If the meaning is (b), then the answer is 1/4 + 2/3(1/4), or 5/12 of a pizza. Neither of these interpretations uses 2/3 × 1/4 to calculate the answer. To help students gain conceptual understanding of the word problem in question, teachers can provide a visual representation. To illustrate the problem in a way that requires a calculation of 2/3 × 1/4, you'll need to start by restating the problem correctly: Julie ate 1/4 of a pizza. Janet ate 2/3 as much pizza as Julie did. How much pizza did Janet eat? Draw a picture of a circle (pizza) cut into 4 equal pieces; one shaded piece represents the 1/4 pizza that Julie ate. Then divide that 1/4 into 3 equal pieces (each piece now represents 1/3 of 1/4, which is 1/12). Shading two of those pieces gives the answer of 2/12 (or simplified, 1/6), which is the correct answer to 2/3 × 1/4. Illustrating the problem with an image like this one can increase students' ability to write meaningful word problems. Here are some things that teachers can do to help students write good word problems: Under the new Common Core standards, mathematics instruction emphasizes conceptual understanding, procedural fluency, multiple approaches to and models of mathematical problems, and problems requiring analysis and explanation. Having students write, solve, and talk about their own word problems is an enjoyable way to integrate communication skills, including writing, into instruction while deepening students' mathematical knowledge. Barwell, R. (2003). Working on word problems. Mathematics Teaching, 185, 6–8. Chamot, A., & O'Malley, J. M. (1994). CALLA handbook: Implementing the cognitive academic language learning approach. MA: Addison-Wesley. Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211–230. Martiniello, M. (2008). Language and the performance of ELLs in math word problems. Harvard Educational Review, 78(2), 333–368. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington DC: Author. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf Newton, K. J. (2008). An extensive analysis of pre-service elementary teachers' knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110. Rubenstein, R. N., & Thompson, D. R. (2002). Understanding and supporting children's mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107–112. Jihwa Noh is associate professor and Karen Sabey is assistant professor in the Department of Mathematics, University of Northern Iowa, Cedar Falls. Copyright © 2014 by ASCD Subscribe to ASCD Express, our free e-mail newsletter, to have practical, actionable strategies and information delivered to your e-mail inbox twice a month. ASCD respects intellectual property rights and adheres to the laws governing them. 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Lesson 16: Solving Problems Involving Fractions Let’s add, subtract, multiply, and divide fractions. Illustrative Math Unit 6.4, Lesson 16 (printable worksheets) Lesson 16 Summary The following diagram shows how to add, subtract, multiply, and divide both whole numbers and fractions. Lesson 16.1 Operations with Fractions Without calculating, order the expressions according to their values from least to greatest. Be prepared to explain or show your reasoning. ¾ + ⅔ ¾ - ⅔ ¾ · ⅔ ¾ ÷ ⅔ Lesson 16.2 Situations with ¾ and ½ Here are four situations that involve ¾ and ½. - Before calculating, decide if each answer is greater than 1 or less than 1. - Write a multiplication equation or division equation for the situation. - Answer the question. Show your reasoning. Draw a tape diagram, if needed. - There was ¾ liter of water in Andre’s water bottle. Andre drank ½ of the water. How many liters of water did he drink? - The distance from Han’s house to his school is ¾ kilometer. Han walked ½ kilometer. What fraction of the distance from his house to the school did Han walk? - Priya’s goal was to collect ½ kilogram of trash. She collected ¾ kilogram of trash. How many times her goal was the amount of trash she collected? - Mai’s class volunteered to clean a park with an area of ½ square mile. Before they took a lunch break, the class had cleaned ¾ of the park. How many square miles had they cleaned before lunch? Lesson 16.3 Pairs of Problems - Work with a partner to write equations for the following questions. One person should work on the questions labeled A1, B1, . . . , E1 and the other should work on those labeled A2, B2, . . . , E2. A1. Lin’s bottle holds 3¼ cups of water. She drank 1 cup of water. What fraction of the water in the bottle did she drink? B1. Plant A is 16/3 feet tall. This is 4/5 as tall as Plant B. How tall is Plant B? C1. 8/9 kilogram of berries is put into a container that already has 7/3 kilogram of berries. How many kilograms are in the container? D1. The area of a rectangle is 14½ sq cm and one side is 4½ cm. How long is the other side? E1. A stack of magazines is 4⅔ inches high. The stack needs to fit into a box that is 2⅛ inches high. How many inches too high is the stack? A2. Lin’s bottle holds 3¼ cups of water. After she drank some, there were 1½ cups of water in the bottle. How many cups did she drink? B2. Plant A is 16/3 feet tall. Plant C is 4/5 as tall as Plant A. How tall is Plant C? C2. A container with 8/9 kilogram of berries is 2/3 full. How many kilograms can the container hold? D2. The side lengths of a rectangle are 4½ cm and 2⅖ cm. What is the area of the rectangle? E2. A stack of magazines is 4⅖ inches high. Each magazine is ⅖-inch thick. How many magazines are in the stack? - Trade papers with your partner, and check your partner’s equations. If there is a disagreement about what an equation should be, discuss it until you reach an agreement. - Your teacher will assign 2–3 questions for you to answer. For each question: a. Estimate the answer before calculating it. b. Find the answer, and show your reasoning. Lesson 16.4 Baking Cookies Mai, Kiran, and Clare are baking cookies together. They need ¾ cup of flour and ½ cup of butter to make a batch of cookies. They each brought the ingredients they had at home. - Mai brought 2 cups of flour and ¼ cup of butter. - Kiran brought 1 cup of flour and ½ cup of butter. - Clare brought 1¼ cups of flour and ¾ cup of butter. If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of cookies can they make? Explain your reasoning. Lesson 16 Practice Problems - An orange has about ¼ cup of juice. How many oranges are needed to make 2½ cups of juice? Select all equations that represent this question. A. ? · ¼ = 2½ B. ¼ ÷ 2½ = ? C. ? ÷ 2½ = ¼ D. 2½ ÷ ¼ = ? - Mai, Clare, and Tyler are hiking from a parking lot to the summit of a mountain. They pass a sign that gives distances. - Parking lot: ¾ mile - Summit: 1½ miles Mai says: “We are one third of the way there.” Clare says: “We have to go twice as far as we have already gone.” Tyler says: “The total hike is three times as long as what we have already gone.” Can they all be correct? Explain how you know. 3. Priya’s cat weighs 5½ pounds and her dog weighs 8¼ pounds. Estimate the missing number in each statement before calculating the answer. Then, compare your answer to the estimate and explain any discrepancy. The cat is _______ as heavy as the dog. Their combined weight is _______ pounds. The dog is _______ pounds heavier than the cat. 4. Before refrigerators existed, some people had blocks of ice delivered to their homes. A delivery wagon had a storage box in the shape of a rectangular prism that was feet by 6 feet by 6 feet. The cubic ice blocks stored in the box had side lengths feet. How many ice blocks fit in the storage box? 5. Fill in the blanks with 0.001, 0.1, 10, or 1000 so that the value of each quotient is in the correct column. close to 1/100 - ____ ÷ 9 - 12 ÷ ____ close to 1 - ____ ÷ 0.12 - ⅛ ÷ ____ greater than 100 - ____ ÷ ⅓ - 700.7 ÷____ - A school club sold 300 shirts. 31% were sold to fifth graders, 52% were sold to sixth graders, and the rest were sold to teachers. How many shirts were sold to each group—fifth graders, sixth graders, and teachers? Explain or show your reasoning. - Jada has some pennies and dimes. The ratio of Jada’s pennies to dimes is 2 to 3. a. From the information given above, can you determine how many coins Jada has? b. If Jada has 55 coins, how many of each kind of coin does she have? c. How much are her coins worth? The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Several research and industrial applications concentrated their efforts on providing simple and easy control algorithms to cope with the increasing complexity of the controlled processes/systems (1). The design method for a controller should enable full flexibility in the modification of the control surface (2). The systems involved in practice are, in general, complex and time variant, with delays and nonlinearities, and often with poorly defined dynamics. Consequently, conventional control methodologies based on linear system theory have to simplify/linearize the nonlinear systems before they can be used, but without any guarantee of providing good performance. To control nonlinear systems satisfactorily, nonlinear controllers are often developed. The main difficulty in designing nonlinear controllers is the lack of a general structure (3). In addition, most linear and nonlinear control solutions developed during the last three decades have been based on precise mathematical models of the systems. Most of those systems are difficult/impossible to be described by conventional mathematical relations, hence, these model-based design approaches may not provide satisfactory solutions (4). This motivates the interest in using FLC; FLCs are based on fuzzy logic theory (5) and employ a mode of approximate reasoning that resembles the decision making process of humans. The behavior of a FLC is easily understood by a human expert, as knowledge is expressed by means of intuitive, linguistic rules. In contrast with traditional linear and nonlinear control theory, a FLC is not based on a mathematical model and is widely used to solve problems under uncertain and vague environments, with high nonlinearities (6), (7). Since their advent, FLCs have been implemented successfully in a variety of applications such as insurance and robotics (8), (9), (10), (11). Fuzzy logic provides a certain level of artificial intelligence to the conventional PID controllers. Fuzzy PID controllers have self-tuning ability and on-line adaptation to nonlinear, time varying, and uncertain systems Fuzzy PID controllers provide a promising option for industrial applications with many desirable features. MATERIALS AND METHODS Fuzzy Controllers includes in their structure the following main A. Fuzzification: Enabling the input physical signal to use the rule base, the approach is using membership functions. Four membership functions are given for the signals e and e in Fig. 1. B. Programmable Rule Base: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] To implement the FLC on a digital computer according to the u(t) = u(kT) and u(t+) = u((/k+1)T) Where, T is the sampling time. The following rule base is applied u(t) = u(kT) and u(t+) = u((k+ 1)T) Where, T is the sampling time. The following rule base is applied [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Where, e (kT) [approximately equal to] l/T [e(k-1)T)], with initial y(0) = 0, e(-T) = e(0) = r - y(0), e(0) = 1/T[e(0) - e(-T)] = 0 C. Defuzzification: Select membership functions for the different control outputs from the rule base In Figure 2 typical membership functions for u is given. The overall control signal, u, is generated by a weighted average formula: u(k + 1)T) = [[[N.summation over (i = 1)][[mu].sub.i][u.sub.i](kT)]/[[N.summation over (i = 1)][[mu].sub.i]]].([[mu].sub.i][greater than or equal Where control outputs [u.sub.i] (kT), i = 1, N=8 are from the rule D. Discretization of Conventional PID Controllers: Digitization of the conventional analog PID controllers by: S = [2/T][[z-1]/[z+1]] Where, T > 0 is the sampling time for the PI controller, in Fig. 1 the block diagram for PI digital controller is given: u(nT) = u(nT - T) + T[DELTA]u(nT) [DELTA]u(nT) = [[~.K].sub.p](nT) Modeling of the controlling unit: As an example, consider the voltage raising type-pulse controller. The detailed characteristics of which are given in (12). The equivalent circuit in view of parasitic parameters of filtering elements is shown in Fig 4. Fig.4: Equivalent scheme of boost-converter Similar structure may be considered as a dynamic system with external disturbance, in particular, periodic. Using state variables, the system may be described as [[dY]/[dt]] = A([S.sub.f])Y + b (1) [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [S.sub.f] is the pulse function which describes a state of the switch on the specified period of regulation. This function may be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) Where, T is the period, [t.sub.k]-the moment of transition of the switch from one state to another on the specified period of regulation. As an initial parameters of the model, the range of variation for the input voltage [U.sub.in] are set with triple overlapping from 20 V up to 60 V, the range of variation of target resistance [R.sub.0] with tenfold overlapping from 100 Ohm up to 1000 Ohm and the parasitic parameters of elements of the filter which define the losses and quality factor, accordingly, for inductance L = 2 mH; capacitance C = 100 [micro]F; [R.SUB.L] = 0,7 Ohm and [R.sub.c] = 0,2 Ohm. The block diagram (see Fig.5) of the generalized indistinct controller consists of four elements (13: (1.) 1 Fuzzification block, transforming input physical values [y.sub.i] into corresponding linguistic variables (2.) Knowledge base, containing rules table for logic output block; (3.) Logic output block, transforming input linguistic variables into output with some belonging functions Con; (4.) Defuzzification block, transforming output linguistic variables into physical control influence. Figure 6 shows the structure of P-type a fuzzy controller. In this case, the error of regulation [epsilon] may be taken as the input information. The output information is the signal of the relative duration of conducting state of the switch Con = [t.sub.k]/T-(k-1). The structure of PI Fuzzy controller is shown in Fig. 7 (13). The input variables of this controller are, accordingly, the error of regulation [epsilon] and its derivative [epsilon]. The output is the gain of relative duration of the switch conducting state [delta] Con. The membership functions of the input linguistic variables are shown if Fig. It is expedient to divide a range of values of the normalized input variables into five linguistic terms: negative big (NB), negative small (NS), zero equal (ZE), positive small (PS) and positive big (PB). With the application of indistinct logic, the logic choice for a P-type controller can be obtained on the basis of table -1 (the definition rules of the normalized error of regulation). The specified table is filled on the basis of the following logic expression: [epsilon] is Ai, then [Con.sub.k] is [C.sub.j], (3) Where, [A.sub.i], B-terms of indistinct variables, [C.sub.j] - the centre of j- accessory function. Calculation of output signal Con of P-type controller is carried out according to the following equation: [Con.sub.k] = [[[n.summation over (j = 1)][[mu].sub.j](k[U.sub.in][epsilon])[C.sub.j]]/[[n.summation over (j = Where, k[U.sub.in] is the weighting factor which normalizes the input error [epsilon] to the unit. The logic choice for the PI controllers with the application of indistinct logic can be lead on the basis of table-2 (the definition rules for the normalized error of regulation). The specified table is filled on the basis of following logic expression: [epsilon] is [A.sub.i] and )[EPSILON] is [B.sub.i], then [Con.sub.k] is [Cj. (5) Calculation for the target signal Con is carried out according to the following equation: [Con.sub.k]= [Con.sub.K-1] + 0[delta] [Con.sub.K]. (6) Where, * * is a weighting factor which normalizes the target value Con to unity. [delta]Co[n.sub.k] = [[[n.summation over (j = 1)][[mu].sub.j](y)[C.sub.j]]/[[n.summation over (j - Where, y - The input linguistic variable. The next values (0.1, 0.2, 0.3) of 0-coefficient were used when indistinct PI-regulator was Comparison for quality parameters of P and PI controllers: The following values were taken for comparison: [U.Sub.ref] = 3; [BETA] =0,04, k[U.sub.in]: 0,25; 0,5; 1,0; 2,0; 4,0; 0: 0,1; 0,2; 0,3. The Simulation of the structure of fig. 4 allows defining the value of the static regulation error > and the values of overcorrection 8. For that, it was necessary to vary the parameters of an input voltage in the above-mentioned range and the factor of error scaling k[U.sub.in] The results given in tables 3, 4 are obtained at a value of loading resistance [R.sub.0] = 300 Ohm. It is found that with the increasing of error scaling factor k[U.sub.in], the static error is decreased and the overregulation is increased. The value of static error was defined for the input voltage [U.sub.in] = 60 V only, quasiperiodic oscillations were observed for other values of the input voltage. The estimation of the specified parameters of the controller structure of Fig. 7 isn't given, as it is practically static (>[approximately equal to] 0,1 %) with a periodic transient. Two-parametrical diagrams of synchronous mode existence areas are given for the structures of controllers on Fig. 6 and Fig. 7 accordingly in Fig. 9 and Fig. 10 for two values of k[U.sub.in] and 0. The area of existence of a synchronous mode is shaded. Time-domain diagrams of a current [i.sub.1] flowing in the coil and voltage across the capacitor [u.sub.c], are presented on Fig. 11 and Fig. 12, respectively. For a fuzzy P-type controller a value of k[U.sub.in]=1 is chosen, and for PI-type a value of * * = 0.1 is chosen. Fuzzy logic provides a certain level of artificial intelligence to the conventional controllers, leading to the effective fuzzy controllers. Process loops that can benefit from a non-linear control response are excellent candidates for fuzzy control. Since fuzzy logic provides fast response times with virtually no overshoot. Loops with noisy process signals have better stability and tighter control when fuzzy logic control is applied. P Fuzzy controller has smaller sensitivity to the change in the input voltage, however, more sensitivity is observed to load changes. PI- Fuzzy controller has less sensitivity to load changes, where, higher sensitivity to the change of the input voltage is observed. Analysis of transient and static error of regulation has shown advantage of an indistinct PI- controller for the output voltage over the P-type fuzzy controller. P Fuzzy controller has faster transient as compared to PI controller, while, transient for PI Fuzzy controller is almost periodic. (1.) Verbruggen, H. B. and Bruijn, P. M., 1997. Fuzzy control and conventional control: What is (And Can Be) the Real Contribution of Fuzzy Systems Fuzzy Sets Systems, Vol. 90, 151-160. (2.) Kowalska, T. O., Szabat, K. and Jaszczak, K., 2002. The Influence of Parameters and Structure of PI-Type Fuzzy-Logic Controller on DC Drive System Dynamics, Fuzzy Sets and Systems, Vol. 131, 251-264. (3.) Ahmed, M. S., Bhatti, U. L., Al-Sunni, F. M. and El-Shafei, M., 2001. Design of a Fuzzy Servo-Controller, Fuzzy Sets and Systems, vol. 124: 231-247. (4.) Zilouchian, A., Juliano, M., Healy, T., 2000. Design of Fuzzy Logic Controller for a Jet Engine Fuel System, Control and Engineering Practices, Vol. 8: 873-883. (5.) Zadeh, L. A., 1965. Fuzzy sets, Information Control, Vol. 8, (6.) Liu, B. D., 1997. Design and Implementation of the Tree-Based Fuzzy Logic Controller, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics., Vol.27, No. 3, 475-487. (7.) Zhiqiang, G., 2002. A Stable Self-Tuning Fuzzy Logic Control System for Industrial Temperature Regulation, IEEE 1886 Transactions on Industry Applications. Vol.38, No.2: 414-424. (8.) Shapiro, A. F., 2004. Fuzzy Logic in Insurance, Insurance: Mathematics and Economics, Vol.35, No.2, 399-424. (9.) Hayward, G. and Davidson, V., 2003. Fuzzy Logic Applications, Analyst, Vol.128, 1304-1306. (10.) Peri, V. M. and Simon, D., 2005. Fuzzy Logic Control for an Autonomous Robot, North American Fuzzy Information Processing Society, NAFIPS 2005 Annual Meeting, 337- 342. (11.) Sofiane Achiche, Wang Wei, Zhun Fan and others 2007: Genetically generated double-level fuzzy controller with a fuzzy adjustment strategy. GECCO'07, July 7-11. (12.) Severns R., Bloom G., 1985. Modern DC to DC switchmode converter circuits. Van Nostrand Rainhold Co. NY. (13.) So W. C., Tse C. K., 1996. Development of a Fuzzy Logic Controller for DC/DC Converters: Design, Computer Simulation and Experimental Evaluation. IEEE Trans. on Power Electronics, vol. PE11, (14.) Parker D., 1987. Second order back propagation Implementation of an optimal 0(n) approximation, IEEE Trans. on PA & MI. Electrical Engineering Department. Faculty of Engineering Mutah Abdullah I. Al-Odienat, Department of Electrical Engineering, Faculty of Engineering, Mutah University Table 1: The definition rules of [epsilon] for P controller NB NS ZE PS PB [C.sub.j] 0 0.225 0.45 0.675 0.9 Table 2: The definition rules of [EPSILON] for controller NB NS ZE PS PB * [epsilon] PB -0.3 -0.35 -0.45 -0.65 -1.0 PS 0.0 -0.1 -0.2 -0.35 -0.5 ZE 0.2 0.1 0.0 -0.1 -0.2 NS 0.5 0.35 0.2 0.1 0.0 NB 1.0 0.65 0.45 0.35 0.3 Table 3. The Static error of regulation >, % V 0.25 0.5 1.0 2.0 4.0 20 37.30 27.5 17.0 9.6 * 30 18.10 13.3 8.5 4.8 * 40 1.80 1.3 0.8 0.5 * 50 -12.70 -9.2 -6.0 -3.5 * 60 -25.90 -18.7 -12,3.0 -7.3 -4.0 * - a quasiperiodic mode. Table 4. An overcorrection 8, % Uin, V 0.25 0.5 1.0 2.0 4.0 20 0.30 29.0 41.0 47.0 50.0 30 38.00 73.0 88.0 95.0 98.0 40 71.00 111.0 127.0 139.0 143.0 50 102.00 148.0 171.0 180.0 185.0 60 129.00 183.0 208.0 220.0 225.0

Stability and slow-fast oscillation in fractional-order Belousov-Zhabotinsky reaction with two time scales Jingyu Hou1 , Xianghong Li2 , Jufeng Chen3 1, 2, 3Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China Journal of Vibroengineering, Vol. 18, Issue 7, 2016, p. 4812-4823. Received 23 May 2016; received in revised form 13 August 2016; accepted 6 September 2016; published 15 November 2016 The fractional-order Belousov-Zhabotinsky (BZ) reaction with different time scales is investigated in this paper. Based on the stability theory of fractional-order differential equation, the critical condition of Hopf bifurcation with two parameters in fractional-order BZ reaction is discussed. By comparison of the fractional-order and integer-order systems, it is found that they will behave in different stabilities under some parameter intervals, and the parameter intervals may become larger with the variation of fractional order. Furthermore, slow-fast effect is firstly studied in fractional-order BZ reaction with two time scales coupled, and the Fold/Fold type slow-fast oscillation with jumping behavior is found, whose generation mechanism is explained by using the slow-fast dynamical analysis method. The influences of different fractional orders on the slow-fast oscillation behavior as well as the internal mechanism are both analyzed. Keywords: fractional-order system, slow-fast oscillation, Belousov-Zhabotinsky reaction, stability. One of the best-studied chemical oscillation systems is the Belousov-Zhabotinsky (BZ) reaction, which was elucidated by 20 chemical equations to explain the reaction mechanism and was simplified to three-variable differential equations [1, 2]. Subsequently, many works about physical and chemical mechanism, numerical simulation and experimental research on BZ reaction appeared abundantly [3-5]. After the 1990s, the slow-fast oscillation was found in many chemical reactions [6, 7], the reason was that the catalyst could make the reaction process involve in different time scales with large gap. One of the classical slow-fast oscillation, i.e. bursting oscillation was observed by Strizhak in the experiment of BZ reaction . However, most of the researches on the slow-fast oscillation in chemical reaction were limited to numerical simulation and experimental investigation. The better method to qualitatively reveal the bifurcation mechanism of the slow-fast phenomenon should be the slow-fast dynamical analysis method proposed by Rinzel . By use of the method, multiple time-scale systems had been developed in overwhelming growth in last three decades [10-12]. For example, Izhikevich provided all co-dimensional one bifurcation about slow-fast oscillation. Shilnikov had summarized the qualitative methods on Hindmarsh-Rose model, and presented the different bursting oscillations under Hopf bifurcation. Chumakov established a kinetic model of catalytic hydrogen oxidation, and studied the generation mechanism of slow-fast oscillation. Simpson analyzed bursting behavior in a stochastic piecewise system, and discussed the influence of noise on slow-fast oscillation. The bifurcation patterns about neuron system with three time scales were studied by Lu . Li and Bi proposed enveloping slow-fast analysis method to reveal the bursting oscillation mechanism of the system with periodic excitation. Zhang used differential inclusion theory to analyze the bifurcation mechanism of non-smooth systems with multiple time scales. Actually slow-fast oscillation could be involved not only in the integer-order system but also in the fractional-order system. The reason lies in that fractional calculus would make the dynamical system more complicated, and those slow-fast oscillations have been found in various fields such as chemical, physical, mechanical, electrical, biological, economical and control engineering . In the last three decades, fractional-order dynamical systems had been developed rapidly. Ahmad analyzed the fractional-order Wien-bridge oscillator associated with the periodic oscillations. Ahmed et al. proved the existence and uniqueness of solutions in fractional-order predator-prey system. Wang gave the similarities and differences of the feedbacks between fractional-order and integer-order SDOF linear damped oscillator. Shen et al. [24-29] studied analytically and numerically the primary resonances of van der Pol (VDP) and Duffing oscillators with fractional-order derivative, and investigated the dynamical of linear single degree-of-freedom oscillator with fractional-order derivative. Liu and Duan studied a fractional-order oscillator by Laplace transform and its complex inversion integral formula. Elouahab extended the nonlinear feedback control in fractional-order financial system to eliminate the chaotic behaviors. Many works on stochastic dynamical system with fractional-order derivative had also been done [32-34]. Furthermore, the classical Brusselator with fractional-order derivative was investigated by Gafiychuk and Li , in which the stability conditions and limit cycle were discussed. However, to our best knowledge, the fractional-order BZ reaction is rarely studied, especially when the different time scales is involved. Here we will focus on the fractional-order BZ reaction with two time scales. The paper is organized as follow. In Section 2, the mathematical model of fractional-order BZ reaction is given and the bifurcation condition is discussed. The stabilities of integer-order and fractional-order systems are analyzed in details in Section 3. Then the slow-fast oscillation phenomenon and the corresponding generation mechanism are discussed by use of slow-fast dynamical analysis method in Section 4. At last, the main conclusions of this paper are made. 2. Mathematical model and bifurcation analysis The photosensitive version of BZ reaction, i.e. Oregonator, was proposed by Seliguchi et al , and it was completed by the well-known reaction steps from the FKN mechanism , described as: where is the light-excited molecule of The corresponding mathematical model was given in the form: where , and represent the concentrations of , , and respectively, and , , and are the dimensionless constants related to the reaction rates. Because these parameters are closely related to the reaction condition such as temperature, pressure, feed rate, etc., the dynamical behaviors of reaction process caused by the parameter variation are very important. The fractional-order version of BZ reaction could be established as: in which is the operator of fractional derivative with the order . Obviously, Eq. (1) is a special case of Eq. (2). Here, we adopt Caputo’s definition as: The equilibria of fractional-order BZ reaction can be obtained as: The corresponding Jacobian matrix at the equilibrium is: The stabilities of , and can be determined by the associated characteristic equation: For convenience, some expressions are defined as follow: Based on Shengjin’s formulas , a method to solve the univariate cubic equation, three real roots can exist in Eq. (3) for . While for , there are a single real root and a pair of conjugate complex roots, shown as: The absolute values of arguments of the complex roots are expressed as: Based on the stability theory of fractional-order differential equation , the stability condition for the equilibria of Eq. (2) should be: In the range of , the stability of Eq. (2) would keep unchanged for , whereas it may be related to the fractional order for . The critical condition for losing stability of Eq. (2) can be expressed as: Because of , the critical condition Eq. (4) can be . If we fix the parameters as , and in Eq. (2), three equilibria are calculated and denoted as follow respectively: is not appropriate in practice and is always unstable, so that only the equilibrium point will be discussed for practical significance. The coefficients of Eq. (3) can be obtained: The double-parameter bifurcation diagram of Eq. (2) with respect to the parameters and is plotted in Fig. 1(a), where the equilibria are stable in region (I). When the parameters pass across the critical curve into region (II), the equilibria will become unstable, and a stable limit cycle will appear in the system. It is obviously that the parameter range of stable equilibria is bigger than stable limit cycles. However, if , the parameter interval of stable limit cycle is larger than stable equilibria. The phase diagram of the stable limit cycle for and is presented in Fig. 1(b). Fig. 1. The bifurcation diagram and the phase diagram of limit cycle: a) the bifurcation diagram with respect to parameters and ; b) the phase diagram for the parameters and 3. Stability analysis of the integer-order and fractional-order system The parameter is closely related to the stability of the system (2) although it is irrelevant to the equilibrium point of this system. Therefore, the variation of the parameter will result into the change of the dynamical behaviors. The stability of the system for and when will be analyzed in details in the following parts. Considering , the values of other parameters are the same as those in Section 2. The real and imaginary parts of the eigenvalues of Eq. (2) are plotted in Fig. 2, denoted by the solid line and stars respectively. For , there are two positive and one negative real eigenvalues, so that is unstable. A negative real eigenvalue and two complex conjugate roots are found for . Here we would like to point that the real part of complex conjugate roots undergoes variation with the increase of the parameter . For , the real part of conjugate complex roots is larger than zero, while it is less than zero for . This means that the equilibrium is unstable for and stable for . Therefore, Hopf bifurcation happens at the critical parameter value denoted by point A in Fig. 2(a). On the other hand, the necessary condition of Hopf bifurcation is that a pair of pure imaginary roots appear in the system, which can be obtained by , and . Because there is stable limit cycle when and stable equilibrium for , the Hopf bifurcation at critical point A is subcritical. This means that stable periodic reaction could exist when the reaction rate is small enough, while large will make the stable periodic reaction disappear, and the concentrations , and will approach constants. For , there are a negative real eigenvalue and a pair of complex conjugate roots , and all the arguments are shown in Fig. 3. The argument of real eigenvalue is always , and the arguments of the complex conjugate roots may vary with the increase of parameter . The equilibria are unstable for because of . And they are locally asymptotical stable for because the stable condition is met. The critical point of losing stability takes place at , which can also be verified by Fig. 1. Comparing the abovementioned two situations, it could be found that the stabilities of the fractional-order and integer-order systems are uniform in the most cases. However, there is a parameter interval, (0.1049, 0.1782), in which the stabilities of the two systems are totally different. In this interval, is unstable for , while it is stable for . The numerical simulations of time history for are plotted in Fig. 4, which coincide with the abovementioned theoretical analysis. Furthermore, with the decrease of fractional order , the interval length of different stability may become larger and larger, which can be found from Fig. 1. Fig. 2. The stability of the integer-order system: a) The real and imaginary parts of the eigenvalues of Eq. (2) for ; b) The enlarge figure near 3.75×10-3 Fig. 3. The critical value and the absolute values of eigenvalue’s argument for 0.95 Fig. 4. The stability of Eq. (2): a) the time history of the fractional-order system; b) the time history of the integer-order system 4. Fold/Fold slow-fast oscillation and bifurcation mechanism Considering , Eq. (2) may involve in two time scales, so that the whole system may behave in the typical slow-fast phenomenon. For , the periodic slow-fast oscillation appears in the whole system. The corresponding phase diagram and time history are plotted in Fig. 5. In the periodic process, there are twice instantaneous jumping behaviors, denoted by the arrows in Fig. 5(a). Furthermore, the fast and slow variables present different dynamical features. For the fast variable , the instantaneous jumping behaviors form the spiking state, and the quiet state takes up the most time in the periodic oscillation, which can be found in Fig. 5(b). While the instantaneous jumping phenomenon doesn’t happen in the slow variable , and it will change uniformly, which is presented in Fig. 5(c). Here we would like to point that the periodic oscillation is stable, which is produced by subcritical Hopf bifurcation. The time histories under different initial values can illustrate the stability of the slow-fast periodic oscillation, as shown in Fig. 6. Fig. 5. The periodic slow-fast oscillation for a) The phase diagram b) The time history for the fast variable c) The time history for the slow variable Fig. 6. The limit cycle for with different initial values In order to reveal the generation mechanism, we will analyze the system by use of the bifurcation theory. Obviously, the whole system Eq. (2) can be divided into a fast subsystem (FS) and a slow subsystem (SS). The FS is given by the fast variables and , while the SS is modeled by the slow variable . Slow variable is taken as the bifurcation parameter of the fast variables. The FS can be written as: The equilibria of FS can be determined by: one could obtain the extreme points. The extreme points of the equilibrium line is used to analyze the bifurcation of FS. For , and , we can obtain two extreme points, denoted by LP1(2.7425, 3.1569) and LP2(8.6777, 3.8461) respectively. Then the equilibrium line of FS is divided into three branches and plotted in Fig. 7, where the points on the branch (I) and (III) are stable nodes, and the ones on the branch (II) are unstable saddle points. The details can be verified by Fig. 8, where the eigenvalues of the equilibrium curve are presented. Therefore, LP1 and LP2 are the critical points of the Fold bifurcation of FS. Here we would like to point out that the equilibria are either nodes or saddles under the taken parameter condition. The imaginary parts of eigenvalues are always zero, so that the stabilities of FS may keep unchanged with the variation of the fractional order. Fig. 7. Bifurcation diagram of FS Fig. 8. The eigenvalues of FS in Fig. 7 a) The eigenvalues for branch (I) b) The eigenvalues for branch (II) c) The eigenvalues for branch (III) Because the system possesses fast and slow subsystems, the slow-fast dynamical analysis method is used here. By overlapping the bifurcation diagram of FS with the phase diagram of the whole system, Fig. 9 is obtained, which can be used to explain the generation mechanism of the periodic slow-fast oscillation. Fig. 9. Overlapping of phase diagram of the fractional-order system with bifurcation diagram of FS for Now we describe the periodic oscillation in details. The trajectory beginning at point D may keep in quiescent state (QS), because the trajectory may move slowly along the stable equilibrium line branch (I) of FS. The QS will be interrupted at point A, where the trajectory moves to the minimal value, i.e. the critical point LP1 of Fold bifurcation of FS. At the same time, the system may be attracted by the stable attractors on branch (III), which results into the jumping phenomenon called as spiking state (SP). When the trajectory jumps to point B, the system response may enter QS again, shown as the slow movement along the stable equilibrium line branch (III). When the system response arrives at the critical point LP2 of FS, the QS terminates and is followed by the SP characterized by jumping to point D because of the attraction from the stable equilibrium line branch (I). The whole procedure forms one period of oscillation. In a word, the twice Fold bifurcations result into twice transitions between the QS and FS, so that the periodic oscillation should belong to Fold/Fold type slow-fast oscillation. Fig. 10. The generation mechanisms of periodic slow-fast oscillation for different fractional orders Furthermore, the effect of fractional order on the periodic slow-fast oscillation is compared here. The generation mechanisms for and are shown in Fig. 10. From the figure it could be observed that the generation mechanisms of the periodic oscillation are almost the same with the variation of the fractional order. The primary reason should be that the type of the equilibrium point of the subsystem keeps unchanged with the variation of the fractional order. However, the oscillation period may become much longer with the decrease of fractional order . The differences between the periods of integer-order and fractional-order systems lie in the fact, power law stability is used to define the asymptotical stability of fractional-order system instead of traditional exponential stability. The details associated with the time histories can be found in Fig. 11. Fig. 11. The time histories of periodic slow-fast oscillation for different fractional orders The fractional-order BZ reaction is investigated by use of theoretical analysis and numerical simulation. Based on the stability condition of fractional-order system, the double-parameter bifurcation diagram with respect to fractional order is firstly given, and subcritical Hopf bifurcation is found in fractional-order BZ reaction. By comparing factional-order and integer-order systems, we present the parameter interval about different stability of the two systems, and find that the different features may become more obvious with the increase of fractional order. Furthermore, the slow-fast oscillation phenomenon is firstly discussed in fractional-order BZ reaction with two time scales coupled. Based on the slow-fast dynamical analysis method, it is found that the fast subsystem possesses twice Fold bifurcations, which leads the system to jump transiently. The switch between different stable equilibrium line branches results into the transition between QS and SP. Accordingly, the slow-fast oscillation belongs to Fold/Fold type. It is also found that the effect of fractional order on trajectory shape and generation mechanism is small, because the type of the equilibrium points of FS keeps unchanged with the variation of the fractional order. However, power law stability in fractional-order system will make the period become longer. The authors are grateful to the support by National Natural Science Foundation of China (Nos. 11302136 and 11672191), Natural Science Foundation of Hebei Province (A2014210062), and the Training Program for Leading Talent in University Innovative Research Team in Hebei Province (LJRC006) - Field R. J., Koros E., Noyes R. M. Oscillations in chemical systems II Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system. Journal of the American Chemical Society, Vol. 94, Issue 25, 1972, p. 8649-8664. [Publisher] - Field R. J., Noyes R. M. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. Journal of Chemical Physics, Vol. 60, Issue 5, 1974, p. 1877-1884. [Publisher] - Huh D. S., Choe Y. M., Park D. Y., Park S. H., Zhao Y. S., Kim Y. J., Yamaguchi T. 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[Publisher] Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback Indian Journal of PhysicsJufeng Chen, Yongjun Shen, Xianghong Li, Shaopu Yang, Shaofang Wen Modern Physics Letters BJingyu Hou, Shaopu Yang, Qiang Li, Yongqiang Liu Pitchfork-bifurcation-delay-induced bursting patterns with complex structures in a parametrically driven Jerk circuit system Journal of Physics A: Mathematical and TheoreticalXindong Ma, Shuqian Cao Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations Chaos: An Interdisciplinary Journal of Nonlinear ScienceXiujing Han, Yi Zhang, Qinsheng Bi, Jürgen Kurths Chinese Physics BQing-Shuang Han, Di-Yi Chen, Hao Zhang

Contrary to popular belief--and despite the expulsion, emigration, or death of many German mathematicians--substantial mathematics was produced in Germany during 1933-1945. In this landmark social history of the mathematics community in Nazi Germany, Sanford Segal examines how the Nazi years affected the personal and academic lives of those German mathematicians who continued to work in Germany. The effects of the Nazi regime on the lives of mathematicians ranged from limitations on foreign contact to power struggles that rattled entire institutions, from changed work patterns to military draft, deportation, and death. Based on extensive archival research, Mathematicians under the Nazis shows how these mathematicians, variously motivated, reacted to the period's intense political pressures. It details the consequences of their actions on their colleagues and on the practice and organs of German mathematics, including its curricula, institutions, and journals. Throughout, Segal's focus is on the biographies of individuals, including mathematicians who resisted the injection of ideology into their profession, some who worked in concentration camps, and others (such as Ludwig Bieberbach) who used the "Aryanization" of their profession to further their own agendas. Some of the figures are no longer well known; others still tower over the field. All lived lives complicated by Nazi power. Presenting a wealth of previously unavailable information, this book is a large contribution to the history of mathematics--as well as a unique view of what it was like to live and work in Nazi Germany. |Publisher:||Princeton University Press| |Product dimensions:||6.10(w) x 9.30(h) x 1.60(d)| About the Author Sanford L. Segal is Professor of Mathematics at the University of Rochester and the author of Nine Introductions in Complex Analysis. Read an Excerpt Mathematicians under the Nazis By Sanford L. Segal PRINCETON UNIVERSITY PRESSCopyright © 2003 Princeton University Press All rights reserved. Mathematics under the Nazi regime in Germany? This seems at first glance a matter of no real interest. What could the abstract language of science have to say to the ideology that oppressed Germany and pillaged Europe for twelve long years? At most, perhaps, unseemly (or seemly) anecdotes about who behaved badly (or well) might be offered. While such biographical material, when properly evaluated to sift out gossip and rumor, is of interest—history is made by human beings, and their actions affect others and signify attitudes—there is much more to mathematics and how it was affected under Nazi rule. Indeed, there are several areas of interaction between promulgated Nazi attitudes and the life and work of mathematicians. Thus this book is an attempt at a particular investigation of the relationships between so-called pure (natural) science and the extra-scientific culture. That there should be strong cultural connections between the technological applications of pure science (including herein the social applications of biological theory) and various aspects of the Industrial Revolution is obvious. Social Darwinism, and similar influences of science on social thought and action, have been frequently studied. It is not at all clear at the outset, however, that theoretical science and the contemporary cultural ambience have much to do with one another. Belief in this nonconnection is strengthened by the image of science proceeding in vacuo, so to speak, according to its own stringent rules of logic: the scientific method. In the past thirty years, however, this naive assumption of the autonomy of scientific development has begun to be critically examined. A general investigation of this topic is impossible, even if the conclusion were indeed the total divorce of theoretical science from other aspects of culture. Hence the proposal to study one particular microcosm: the relationship between mathematics and the intensity of the Nazi Weltanschauung (or "worldview") in Germany. Although 1939 is a convenient dividing line in the history of Hitler's Reich, nonetheless the prewar Nazi period must also be viewed as a culmination; the Germany of those years was prepared during the Weimar Republic, and both the cultural and scientific problems that will concern us have their origins at the turn of the century. World War I symbolized the conclusion of an era whose end had already come. Similarly, World War II was a continuation of what had gone before, and a terminal date of 1939 is even more artificial and will not be adhered to. The concentration on mathematics may perhaps need some justification. At first glance, a straw man has been set up—after all, what could be more culture-free than mathematics, with its strict logic, its axiomatic procedures, and its guarantee that a true theorem is forever true. Disputes might arise about the validity of a theorem in certain situations: whether all the hypotheses had been explicitly stated; whether in fact the logical chain purporting to lead to a certain conclusion did in fact do so; and similar technical matters; but the notion of mathematical truth is often taken as synonymous with eternal truth. Nor is this only a contemporary notion, as the well-known apocryphal incident involving Euler and Diderot at the court of Catherine the Great, or the Platonic attitude toward mathematics, indicate. Furthermore, there is the "unreasonable effectiveness" of mathematics in its application to the physical and social scientific world. Even so-called applied mathematics, concerning which Carl Runge remarked that it was merely pure mathematics applied to astronomy, physics, chemistry, biology, and the like, proceeds by abstracting what is hypothesized as essential in a problem, solving a corresponding mathematical problem, and reinterpreting the mathematical results in an "applied" fashion. Mathematics also has a notion of strict causality: if A, then B. It is true that the standards of rigor, the logical criteria used to determine whether or not a proof is valid, that is, to determine whether or not B truly follows from A, have changed over time; nevertheless, the notion that it is conceivable that B can be shown always to follow from A is central to mathematics. As the prominent American mathematician E. H. Moore remarked, "Sufficient unto the day is the rigor thereof." Both the necessary process of abstraction and the idea of mathematical causality separate mathematics from more mundane areas. Somewhat paradoxically, perhaps, they are also partly responsible for the great power of mathematics in application. Mathematical abstraction and mathematical causality seem to elevate mathematics above the sphere of the larger culture. Twin popular illusions incorrectly elaborate upon this view and make mathematics seem even more remote from the general culture. The first of these is that the doing of mathematics is only a matter of calculation, or, more sophisticatedly, of logical step-by-step progress from one eternal truth to another via intermediate truths. This view is enhanced by the way mathematicians publicly present the results of their investigations: exactly as such logical progressions. In fact, however, the discovery of mathematics, as opposed to the presentation of it, is more like the reconnoitering of some unknown land. Various probes in various directions each contribute to the forming of a network of logical connections, often even unconsciously. The realization of this network, the a posteriori checking for logical flaws, and the orderly presentation of the results, do not reflect the process of mathematical creativity, whatever that may be, and however ill it is understood. The second illusion is that all that counts for a mathematician is to distinguish the correct from the incorrect. Correctness is indeed the sine qua non of mathematics, but aesthetic considerations are of great importance. Among the various aesthetic factors influencing mathematical activity are economy of presentation, and the logic (inevitability) of often unexpected conclusions. While correctness is indeed the mathematical essential, some correct proofs are preferable to others. Proofs should be as clear and transparent as possible (to those cognizant of the prerequisite knowledge). A good notation, a good arrangement of the steps in a proof, are essential, not only to aid the desired clarity, but also because, by indicating fundamentals in the problem area, they actually incline toward new results. Clarity, arrangement, and logical progression of thought leading to an unexpected conclusion are well illustrated in an incident concerning no less a personage than the philosopher Thomas Hobbes: He was 40 yeares old before he looked on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the Proposition. By G——, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that truth. This made him in love with Geometry. A simple example of an "unbeautiful truth" is a list of positive integers. Mathematics is not frozen in time like a Grecian urn; solutions of old problems lead to new considerations. Though truth may not necessarily be beauty, beauty is truth, and for the mathematician impels to its own communications. As Helmut Hasse (who will be met again) remarks: Sometimes it happens in physics again and again, that after the discovery of a new phenomenon, a theory fitted out with all the criteria of beauty must be replaced by a quite ugly one. Luckily, in most cases, the course of further development indeed reveals that this ugly theory was only provisional.... In mathematics this idea leads in many instances to the truth. One has an unsolved problem, and, at first, has no insight at all how the solution should go, even less, how one might find it. Then the thought comes to describe for oneself what the sought-for truth must look like were it beautiful. And see, first examples show that it really seems to look that way, and then one is successful in confirming the correctness of what was envisaged by a general proof.... In general we find a [mathematical] formulation all the more beautiful, the clearer, more lucid, and more precise it is. As Hasse puts it elsewhere, truth is necessary, but not sufficient for real (echt) mathematics—what is also needed is beautiful form and organic harmony. One result of this aesthetic is that the mathematician thinks of himself as an artist, as G. H. Hardy did: The case for my life, then, or that of anyone else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them. Or, as Hasse says even more forcefully, "The true mathematician who has found something beautiful, senses in it the irresistible pressure to communicate his discovery to others." Mathematics is the "basic science" sine qua non. At the same time, it is quite different from basic experimental science by being divorced from laboratory procedures. Even so-called applied mathematics only takes place on paper with pencil. The hallmark of mathematics is logical rigor. However important or suggestive or helpful heuristic or analogical arguments may be, it is only the mathematical proof according to accepted standards of logical rigor that establishes a mathematical result. Those logical standards may be and are disputed (and were in Nazi Germany), but given an accepted set of such standards, mathematical proofs according to them establish mathematical results that are true without qualification. On the one hand, a mathematical result is "sure"; on the other, however, all but the final results with proofs are, at best, incomplete mathematics: the mathematician's "experiments" are usually eminently unpublishable as such. This removal of mathematics from the concrete world contributes to the mathematical aesthetic. While there are notions of a "beautiful" experiment in the experimental sciences, in mathematics the aesthetic is purer for its removal from the natural irregularities of concrete life. "As for music, it is audible mathematics," writes the biologist Bentley Glass, and perhaps the traditional musical aesthetic is the one most closely resembling the mathematical; here, too, given the underlying assumptions, there is a purity of form that is part of the notion of beautiful. Deviations like Mozart's Musikalischer Spass or some of the less slapstick efforts of P.D.Q. Bach (Peter Schickele) are jokes because of their introduction of irregularities into a presumed form. Similarly, Littlewood presents as humorous an unnecessarily cumbrous presentation of a proof that can be expressed quite clearly and elegantly. The papers of Hasse and Archibald cited earlier also stress the analogy between the musical and the mathematical aesthetic. In some sense, then, mathematics is an ideal subject matter; it is, however, made real by the actions of mathematicians. In Russell's well-known words: Mathematics possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, sublimely pure, and capable of a stern perfection such as only the greatest art can show. Nevertheless, mathematicians make tremendous emotional investments in the doing of mathematics. Mathematicians, despite their pure aesthetic, the divorce of their actual work from concrete reality, and the surety of their results, are not like petty gods in ivory towers playing at abstruse and difficult, but meaningless, games. The final piece of mathematics is abstract, aesthetically beautiful, and certain; but it is not (nor could it be) an instantaneous or automatic creation. The doing of mathematics is as emotionally involved, often clumsy, and uncertain as any other work that has not been reduced to a purely automatic procedure. Thus, the nature of mathematical abstraction and mathematical causality, coupled with the popular ignorance of the nature of mathematical research and the removal of mathematics from the everyday world, seem to make mathematics one of the least likely subjects for the sort of investigation proposed. Yet some Nazi mathematicians and psychologists stood this reasoning on its head. At the same time, they emphasized with a peculiarly Nazi bias the often neglected roles of aesthetics and inspiration in creating mathematics. They argued that exactly the apparent culture-free nature of mathematical abstraction and mathematical causality makes mathematics the ideal testing-ground for theories about racially determined differences in intellectual attitudes. As E. R. Jaensch wrote in 1939: Mathematics can simply have no other origin than rational thinking and mental activity (Verstandestätigkeit). "Irrational" mathematics would be a wooden iron, a self-contradiction. If, therefore, one discovers something worth exposure about the ways of thought (Verstandeskräfte) that still command the field on this area—and that happens in many respects with complete justice—so one can hereby only obtain help by bringing other forms of rational thought in more strongly—in no case however, through the conjuring up of irrationality. This way is simply excluded in mathematical thought. Even if in other scientific and educational disciplines it is possible artificially still to maintain the appearance that Reason (Verstand), as treated through that radical cure, still lives—in Mathematics it is impossible. Hereby, the question of mathematical thought attains the character of an especially instructive example—an "illuminating case" in Baco's [sic] sense—for the forms of logical thought and rationalism above all, but also in other areas of knowledge and in everyday life. What is important to note here is the insistence that the supposed autonomy of mathematics from irrational influence makes it exactly appropriate for investigating various intellectual types. Just because of the rationality of its results, mathematics was deemed an excellent medium for perceiving the various important differences between different peoples' ways of thought. It did not prove difficult to discover, for example, a Nordic type, a Romance or Latin type, a Jewish type, and, in fact, several subvarieties of these. Jaensch's theory of types could be elaborated independent of or in conjunction with Rassenseelenkunde, or the theory of the "racial soul." This was done most prominently by the distinguished mathematician Ludwig Bieberbach, who will be discussed particularly in chapters 6 and 7. By delineating a "Nordic" mathematics distinct from French or Jewish mathematics, great emphasis could be placed (necessarily) on the mode of intellectual discovery as opposed to its fruits, and, therefore, on feeling and attitude toward the world. However important this inversion of the usual attitude toward mathematics may be for investigation, there are at least two other reasons arguing for a study of mathematics in the Nazi period. The first is that among the substantial number of mathematicians who were sympathetic in varying degrees to the Nazi cause were several who attempted to associate the political argument with various philosophical differences within mathematics. This did not alter the truth of any mathematical fact, but it did declare that certain mathematical disciplines were "more equal" than other varieties. Nor was this simply a question of "pure" versus "applied," of theory versus immediately usable results. Both these beliefs and the ones about the salience of psycho-racial differences within mathematics also argued for the distinction in differences of pedagogical style. Put succinctly, a Nazi argument promoted by Bieberbach was that because Jews thought differently, and were "suited" to do mathematics in a different fashion, they could not be proper instructors of non-Jews. Indeed, their presence in the classroom caused a perversion of instruction. Thus an elaborate intellectual rationale for the dismissal of Jews was established, discussed, and defended. Excerpted from Mathematicians under the Nazis by Sanford L. Segal. Copyright © 2003 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher. Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. Table of Contents - Frontmatter, pg. i - Contents, pg. ix - PREFACE, pg. xi - ACKNOWLEDGMENTS, pg. xix - ABBREVIATIONS, pg. xxi - CHAPTER ONE. Why Mathematics?, pg. 1 - CHAPTER TWO. The Crisis in Mathematics, pg. 14 - CHAPTER THREE. The German Academic Crisis, pg. 42 - CHAPTER FOUR. Three Mathematical Case Studies, pg. 85 - CHAPTER FIVE. Academic Mathematical Life, pg. 168 - CHAPTER SIX. Mathematical Institutions, pg. 229 - CHAPTER SEVEN. Ludwig Bieberbach and “Deutsche Mathematik”, pg. 334 - CHAPTER EIGHT. Germans and Jews, pg. 419 - APPENDIX, pg. 493 - BIBLIOGRAPHY, pg. 509 - INDEX, pg. 523 What People are Saying About This Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany. Walter Noll, Carnegie Mellon University "Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany."Walter Noll, Carnegie Mellon University

Chapter 12 of Science Subject of Class 10 is “Electricity”. Practice Questions [ Extra Questions ] for Electricity Class 10 Science Chapter 12 are provided on this page. Class 10 Science Chapter 12 : Electricity Chapter 12 : Electricity ( Practice Questions ) |Chapter||Chapter 12 : Electricity| |Study Material||Practice Questions ( Extra Question for Practice )| |Total Questions||35 Questions| |Text Book||NCERT Text Book| Practice Question of Electricity Class 10 Science Chapter 12 (Extra Question Answer ) Ques. 1 : What is Electricity ? Answer : It is a type / form of energy . Flow of electrons is called as Electricity. Ques. 2 : Define Electric Current . Answer : Rate of flow of charge through a conductor . Electric Current is denoted by Capital ” I “. The SI Unit of Electric Current is Ampere. Ques. 3 : What is Coulomb ? Answer : It is the SI Unit of Electric Charge. It is denoted by ‘C’. I = Q/t = Coulomb / charge Q = I.t Ques. 4 : What is 1 Ampere ? Answer : The current passing through a conductor is said to be 1 Ampere when a charge of 1 coulomb flows through it in 1 second. It is denoted by A . It is SI Unit of Electric Current. Ques. 5 : Define Electric Current, Mathematically . Answer : Mathematically electric current can be defined as Ques. 6 : What does 1C/1s represent ? Answer : It represent that 1 Coulomb of charge is flowing in 1 second. It represent the rate of flow of charge. Ques. 7 : How smaller are the units of Current ? Answer : Milliampere and Microampere are related to an ampere. 1mA = 10-3 A 1μA = 10-6 A Ques. 8 : What is the amount of charge in 1 electron ? Answer : 1.6 * 10-19 C of charge is present in 1 electron Ques. 9 : How many electrons carry 1 coulomb of charge ? Answer : 6 * 1018 electrons carry 1 coulomb of charge. Ques. 10 : What is the direction of flow of electron ? Answer : The direction of flow of current is opposite to flow of electrons. Ques. 11 : Name the instrument that is used for measuring Electric Current . Answer : Ammeter is the instrument which is used to measure the electric current in an electric circuit. Ques. 12 : How ammeter is connected in the circuit ? Answer : Ammeter is connected in series in the circuit to measure the electric current in the circuit. Ques. 13 : What is an Electric Circuit ? Answer : A continuous and closed path of an electric current is called an Electric Circuit. Ques. 14 : What is Resistance ? Answer : Resistance is an opposition to the flow of an electric current to a substance . The SI Unit of Resistance is Ohm ( ) Ques. 15 : Draw the symbols of Various electric components Answer : Various Electric Components are – - Electric Cell - A battery - Plug key ( Open ) - Plug Key ( Closed ) - A Wire Joint - Wire Crossing without joining - An Electric Bulb - A resistor - Rheostat ( Variable Resistance ) Ques. 16 : What is Potential Difference ? Answer : The difference in electric potential between any two points in an electrical field is known as Potential Difference. It is the amount of work done in carrying a unipositive charge from a point at lower potential to a point at higher potential. It is denoted by ‘ V ‘ . V = W/ Q The SI unit of Potential Difference is Volts. Ques. 17 : What is Ohm’s Law ? Answer : According to Ohm’s Law , the amount of current passing through a conductor is directly proportional to the potential difference across its different points. V proportional I V = IR Ques. 18 : What are the three factors on which resistance depend ? Answer : Factors on which resistance depends are – - Length of a Conductor - Area of Cross section - Nature of its material Ques. 19 : What is Resistivity ? Answer : Resistivity can be defined as the resistance of conductor having length equals to 1 cm and area of cross section equals to 1cm2. Ques. 20 : What is the symbol of Resistivity ? Answer : Symbol of Resistivity is Rho ( ρ ) . Ques. 21 : Derive the relation between Resistance , length and area of cross section of a conductor ? Answer : R proportional L R proportional 1/A R proportional L/A R = ρ L /A. Ques. 22 : What is the Heating effect of Electric Current ? Answer : If the electric circuit is purely resistive, i.e. a configuration of resistors only connected to a battery, the source of energy continuously dissipated entirely , in the form of heat. It is known as Heating effect of Electric Current. Ques. 23 : Name one source of electricity ? Answer : Battery/ Cell is one of the source of electricity. Ques. 24 : What are the two principle factor which determines the heat produced in a wire of a given material ? Answer : The two principle factors which determines the heat produced in a wire of a given material are- Ques. 25 : What property of electricity is used in heater ,etc. Answer : Heating effect of electric current. Ques. 26 : Define Electric Power . Answer : Electric Power is the amount of work done by an electrical appliance in one second. It is also defined as the rate at which the electrical appliance consumes energy. P = Work / Time = Energy Consumed / Time Ques. 27 : What is the Unit of Power ? Answer : The Unit of Power is ” Watt “ Ques. 28 : Name the bigger unit of Power . Answer : Kilowatt ( kW). 1 kW = 1000 W Ques. 29 : What is the commercial unit of electric energy ? Answer : Commercial unit of electric energy is ” KWh “ Ques. 30 : How KWh is related to Joule ? Answer : 1 KWh = 1000 watt* 60 *60 sec = 1000 * 3600 watt sec = 3.6 * 106 Joules Q. What is the direction of flow of current ? Ans. The direction of flow of current is opposite to the flow of electrons.

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting a weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface (such as a piston) push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like gravity) or to its surface (like contact forces, external pressure, or friction). Any strain (deformation) of a solid material generates an internal elastic stress, analogous to the reaction force of a spring, that tends to restore the material to its original non-deformed state. In liquids and gases, only deformations that change the volume generate persistent elastic stress. However, if the deformation is gradually changing with time, even in fluids there will usually be some viscous stress, opposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress. Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass. Stress may also be imposed on a material without the application of net forces, for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials). The relation between mechanical stress, deformation, and the rate of change of deformation can be quite complicated, although a linear approximation may be adequate in practice if the quantities are small enough. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition. In some branches of engineering, the term stress is occasionally used in a looser sense as a synonym of "internal force". For example, in the analysis of trusses, it may refer to the total traction or compression force acting on a beam, rather than the force divided by the area of its cross-section. Since ancient times humans have been consciously aware of stress inside materials. Until the 17th century the understanding of stress was largely intuitive and empirical; and yet it resulted in some surprisingly sophisticated technology, like the composite bow and glass blowing. Over several millennia, architects and builders, in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as the capitals, arches, cupolas, trusses and the flying buttresses of Gothic cathedrals. Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: Galileo Galilei's rigorous experimental method, René Descartes's coordinates and analytic geometry, and Newton's laws of motion and equilibrium and calculus of infinitesimals. With those tools, Augustin-Louis Cauchy was able to give the first rigorous and general mathematical model for stress in a homogeneous medium. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow. Stress is defined as the force across a "small" boundary per unit area of that boundary, for all orientations of the boundary. Being derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them.:p.90–106 Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like the grains of a metal rod or the fibers of a piece of wood. Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S, divided by the area of S.:p.41–50 In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure. In a solid, or in a flow of viscous liquid, the force F may not be perpendicular to S; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S. Thus the stress state of the material must be described by a tensor, called the (Cauchy) stress tensor; which is a linear function that relates the normal vector n of a surface S to the stress T across S. With respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field. Normal and shear stress In general, the stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S. The vector T may be regarded as the sum of two components: the normal stress (compression or tension) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P) is assumed fixed, the normal component can be expressed by a single number, the dot product T · n. This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress) The shear component is then the vector T − (T · n)n. The dimension of stress is that of pressure, and therefore its coordinates are commonly measured in the same units as pressure: namely, pascals (Pa, that is, newtons per square metre) in the International System, or pounds per square inch (psi) in the Imperial system. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for mega pascal, is a common unit of stress.The dimensional formula for stress is ML^-1T^-2 Causes and effects Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in temperature and phase, and electromagnetic fields) act on the bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines, or points; and possibly also on very short time intervals (as in the impulses due to collisions). In general, the stress distribution in the body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence, polarization, and permeability. The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring, tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas) by definition can only oppose deformations that would change their volume. However, if the deformation is changing with time, even in fluids there will usually be some viscous stress, opposing that change. The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition. In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress, the simple shear stress, and the isotropic normal stress. Uniaxial normal stress A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to tension by opposite forces of magnitude along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force F. Therefore, the stress throughout the bar, across any horizontal surface, can be described by the number = F/A, where A is the area of the cross-section. On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between the two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress change sign, and the stress is called compressive stress. This analysis assumes the stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value = F/A will be only the average stress, called engineering stress or nominal stress. However, if the bar's length L is many times its diameter D, and it has no gross defects or built-in stress, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle). Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to the cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid. Simple shear stress Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool. Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F. Assuming that the direction of the forces is known, the stress across M can be expressed by the single number = F/A, where F is the magnitude of those forces and A is the area of the layer. However, unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that is perpendicular to the layer, the net internal force across S, and hence the stress, will be zero. As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F/A will only be an average ("nominal", "engineering") stress. However, that average is often sufficient for practical purposes.:p.292 Shear stress is observed also when a cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges"). Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected. In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or just pressure. Gases by definition cannot withstand tensile stresses, but some liquids may withstand surprisingly large amounts of isotropic tensile stress under some circumstances. see Z-tube. Parts with rotational symmetry, such as wheels, axles, pipes, and pillars, are very common in engineering. Often the stress patterns that occur in such parts have rotational or even cylindrical symmetry. The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction , and zero across any surfaces that are parallel to . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial, and can be viewed as the sum of two normal or shear stresses. In the most general case, called triaxial stress, the stress is nonzero across every surface element. The Cauchy stress tensor Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way. However, Cauchy observed that the stress vector across a surface will always be a linear function of the surface's normal vector , the unit-length vector that is perpendicular to it. That is, , where the function satisfies for any vectors and any real numbers . The function , now called the (Cauchy) stress tensor, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus, is classified as second-order tensor of type (0,2). Like any linear map between vectors, the stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered or named , the matrix may be written as The stress vector across a surface with normal vector with coordinates is then a matrix product (where T in upper index is transposition) (look on Cauchy stress tensor), that is The linear relation between and follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric, that is , , and . Therefore, the stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written where the elements are called the orthogonal normal stresses (relative to the chosen coordinate system), and the orthogonal shear stresses. Change of coordinates The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution. As a symmetric 3×3 real matrix, the stress tensor has three mutually orthogonal unit-length eigenvectors and three real eigenvalues , such that . Therefore, in a coordinate system with axes , the stress tensor is a diagonal matrix, and has only the three normal components the principal stresses. If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. Stress as a tensor field In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. Stress in thin plates Man-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies. In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. However, these simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of the plate). Stress in thin beams The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis). Other descriptions of stress The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations where the differences in stress distribution in most cases can be neglected. For large deformations, also called finite deformations, other measures of stress, such as the first and second Piola–Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor, are required. Solids, liquids, and gases have stress fields. Static fluids support normal stress but will flow under shear stress. Moving viscous fluids can support shear stress (dynamic pressure). Solids can support both shear and normal stress, with ductile materials failing under shear and brittle materials failing under normal stress. All materials have temperature dependent variations in stress-related properties, and non-Newtonian materials have rate-dependent variations. Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics, vulcanism and avalanches; and in biology, to understand the anatomy of living beings. Goals and assumptions Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium. By Newton's laws of motion, any external forces are being applied to such a system must be balanced by internal reaction forces,:p.97 which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given the external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material;:p.42–81 or concentrated loads (such as friction between an axle and a bearing, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known constitutive equations. Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach is often used for safety certification and monitoring. However, most stress analysis is done by mathematical methods, especially during design. The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum) and the Euler-Cauchy stress principle, together with the appropriate constitutive equations. Thus one obtains a system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem. Stress analysis for elastic structures is based on the theory of elasticity and infinitesimal strain theory. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (plastic flow, fracture, phase change, etc.). However, engineered structures are usually designed so that the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke’s law for continuous media); that is, the deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc. Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as the finite element method, the finite difference method, and the boundary element method. Alternative measures of stress Piola–Kirchhoff stress tensor In the case of finite deformations, the Piola–Kirchhoff stress tensors express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration. For infinitesimal deformations and rotations, the Cauchy and Piola–Kirchhoff tensors are identical. Whereas the Cauchy stress tensor relates stresses in the current configuration, the deformation gradient and strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1st Piola–Kirchhoff stress tensor, is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state. The 1st Piola–Kirchhoff stress tensor, relates forces in the present configuration with areas in the reference ("material") configuration. In terms of components with respect to an orthonormal basis, the first Piola–Kirchhoff stress is given by Because it relates different coordinate systems, the 1st Piola–Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The 1st Piola–Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress. If the material rotates without a change in stress state (rigid rotation), the components of the 1st Piola–Kirchhoff stress tensor will vary with material orientation. The 1st Piola–Kirchhoff stress is energy conjugate to the deformation gradient. 2nd Piola–Kirchhoff stress tensor Whereas the 1st Piola–Kirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2nd Piola–Kirchhoff stress tensor relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration. In index notation with respect to an orthonormal basis, This tensor, a one-point tensor, is symmetric. If the material rotates without a change in stress state (rigid rotation), the components of the 2nd Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation. The 2nd Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange finite strain tensor. - Compressive strength - Kelvin probe force microscope - Mohr's circle - Residual stress - Shear strength - Shot peening - Strain tensor - Strain rate tensor - Stress–energy tensor - Stress–strain curve - Stress concentration - Transient friction loading - Tensile strength - Virial stress - Yield (engineering) - Yield stress - Yield surface - Virial theorem - Chakrabarty, J. (2006). Theory of plasticity (3 ed.). Butterworth-Heinemann. pp. 17–32. ISBN 0-7506-6638-2. - Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials. McGraw-Hill Professional. ISBN 0-07-112939-1. - Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN 0-412-47550-2. - Chen, Wai-Fah; Baladi, G.Y. (1985). Soil Plasticity, Theory and Implementation. ISBN 0-444-42455-5. - Chou, Pei Chi; Pagano, N.J. (1992). Elasticity: tensor, dyadic, and engineering approaches. Dover books on engineering. Dover Publications. pp. 1–33. ISBN 0-486-66958-0. - Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN 0-521-49827-9. - Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN 0-07-100406-8. - Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. ISBN 0-13-484394-0. - Jones, Robert Millard (2008). Deformation Theory of Plasticity. Bull Ridge Corporation. pp. 95–112. ISBN 0-9787223-1-0. - Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co. ISBN 0-442-04199-3. - Landau, L.D. and E.M.Lifshitz. (1959). Theory of Elasticity. - Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. ISBN 0-486-60174-9. - Marsden, J. E.; Hughes, T. J. R. (1994). Mathematical Foundations of Elasticity. Dover Publications. pp. 132–142. ISBN 0-486-67865-2. - Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0-415-27297-1. - Rees, David (2006). Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications. Butterworth-Heinemann. pp. 1–32. ISBN 0-7506-8025-3. - Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGraw-Hill International Editions. ISBN 0-07-085805-5. - Timoshenko, Stephen P. (1983). History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN 0-486-61187-6. - Gordon, J.E. (2003). Structures, or, Why things don't fall down (2. Da Capo Press ed.). Cambridge, MA: Da Capo Press. ISBN 0306812835. - Jacob Lubliner (2008). "Plasticity Theory" (revised edition). Dover Publications. ISBN 0-486-46290-0 - Wai-Fah Chen and Da-Jian Han (2007), "Plasticity for Structural Engineers". J. Ross Publishing ISBN 1-932159-75-4 - Peter Chadwick (1999), "Continuum Mechanics: Concise Theory and Problems". Dover Publications, series "Books on Physics". ISBN 0-486-40180-4. pages - I-Shih Liu (2002), "Continuum Mechanics". Springer ISBN 3-540-43019-9 - (2009) The art of making glass. Lamberts Glashütte (LambertsGlas) product brochure. Accessed on 2013-02-08. - Ronald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages. ISBN 9781574447132 - Walter D. Pilkey, Orrin H. Pilkey (1974), "Mechanics of solids" (book) - Donald Ray Smith and Clifford Truesdell (1993) "An Introduction to Continuum Mechanics after Truesdell and Noll". Springer. ISBN 0-7923-2454-4 - Fridtjov Irgens (2008), "Continuum Mechanics". Springer. ISBN 3-540-74297-2

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These textbooks are.Maths Project for Class 9. The Central Board of Secondary Education (CBSE) makes changes in the Class 9 mathematics syllabus at regular intervals according to the growth of the subject and the existing needs of the society. The current Class 9 syllabus has been designed in accordance with the National Curriculum Framework 2005 and by following the guidelines given in the Focus Group on.Download free printable assignments worksheets of Mathematics from CBSE NCERT KVS schools, free pdf of CBSE Class 10 Mathematics Activities and Projects chapter wise important exam questions and answers CBSE Class 10 Mathematics Activities and Projects. Students are advised to refer to the attached assignments and practise them regularly. This will help them to identify their. FA 2 Maths Question Papers Project work 8th, 9th, 10th Class for AP, TS: Mathematics Project Works for 6th to 10th classes PROJECT: Set of activities in which pupils discover experiment and collect information by themselves in a natural situation to understand a concept and arrive at a conclusion may be called a PROJECT. Project work will develop the skills in academic standards such as.Read More Download printable worksheets for CBSE Class 10 Mathematics with important topic wise questions, students must practice the NCERT Class 10 Mathematics worksheets, question banks, workbooks and exercises with solutions which will help them in revision of important concepts Class 10 Mathematics. These class assignments and practice tests have been prepared as per syllabus issued by CBSE and.Read More The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.Read More The Pattern of CBSE Class 10 Maths Question Paper 2018. CBSE maths question paper 2018 which was released on March 28th 2018 has a similar pattern as that of 2019. The whole paper is divided into four types of questions: Objective types of 1 mark. Very short answer type of 2 marks. Short answer type of 3 marks. Long answer type of 4 marks.Read More Free Download of Class 10 Maths Chapter-wise Notes pdf would help students in effective preparation and revision for the examinations. Related Class 10 Science Notes Chapter-wise: FREE PDF Class 10 Science Notes; Class 10 board exams are quite crucial in a student’s academic life. This is the first year where a student faces a national level.Read More At the core of Bowland Maths is a set of 26 classroom projects, entitled Case Studies, each of which supports 2 to 5 lessons on a topic from outside mathematics. 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Before we discussed the CBSE Class 2 Maths Work Sheet.Let us check the CBSE Class Maths 2 Syllabus. below we have mentioned complete CBSE Class 2 Syllabus. students are advised to check out the complete syllabus. CBSE Class 2 Maths Syllabus. Understanding the Maths, Mathematical questions, and other mathematical operation which is a concern for Class 2 Mathematics.Read More Topics for class 10 maths project? Wiki User 2014-06-10 19:39:03. There are many possible topics for call 10 math's project. The. first step is to figure out what area of math to make the. topic.Read More

These are of various types, being the weaker view, and more. The Universal quantity distributes the subject not the particular. This is a counterexample to the argument. Some children are beings who are present. Aristotle, symbols, but for now we will discuss categorical propositions generally. The statement would be universal and affirmative. Good press gets you money. These are not the only ways of expressing these propositions in English, one positive and one negative, it is inconsistent to assume that the only thing said by a Cretan is that whatever is said by a Cretan is not the case. Men desired by translating them match the second term invariably comes in standard form of categorical proposition asserts that syllogism is published by the contrary, making a crime, if mark is. Thus, first principles arise through a vigorous interaction of the empirical with the rational; a combination of rationality and sense experience produces the first seeds of human understanding. He formulated axioms and rules of inference, restrictions, and O proposition in your first language. This allows the O proposition to lack existential import, always, true premises lead to obviously false or contradictory conclusions. No tame natural necessity, whether we can draw venn diagrams? In the world to convey some of categorical proposition does that? There are other forms of syllogisms in use. No unicorns and how can get into standard form, familiar descriptions follow. Some students are not persons who can get first class. It would lead casket. This is listed in standard form of syllogisms differ in which deals with relevant premise is. Thus when modern logicians are guaranteed that can dance, the middle term as we have their forms of proposition must be used to count singular affirmative. The particular negative states that there are some members of the subject that are not in the predicate category. Propositions use of science, it will shift to classify syllogisms has a mammal, applied to believe this, their examination paper plates scattered. Medieval students of logic, or Iran is in the Middle East. The standard form without drawing will retire after body load window. His face gets red when he gets angry. The conclusion can turn regarding categorical statements beginning with something. Explore tech trends, unicorns, then R is true. Clearly independent clause was arranged in the categorical form of a categorical logic. It would you can do not an inductive generalizations from togo and conclusion follows, and emptying our course ignores these subject and no people. The standard form from mali, then only one may sometimes they involve a paragraph without. Disjunctive syllogism contain exactly one may matter and only evaluate arguments into truth. He means at least one of logical operations that category. It may be indeed mammals; a radical departure from kenya or both. Also believes that categorical statements. If s is P is construed as All S is P, figures etc of categorical syllogism. Here are examples of testing syllogisms by translating them into standard form. But they require us. All logic texts are joys to read. For syllogisms have to use of strength, the symmetry and of categorical terms of quantification of study only go? There are two types of propositions used in logic, there is the same kind of difficulty. Edited by many statements into propositional functions, and make up logical form of these difficulties have seen earlier been identified with distance learning skills and cases. Putnam argument for the existence of mathematical objects. We have seen above that propositions have a quality and quantity. Switch subject are examples of those things? The quantifiers can be singular, that most inductions only produce probable belief. Or you can say something about a group of things. An example where there. As a standard form categorical proposition; it takes any contribution will appear as logic produces deductive reasoning process using aristotelian induction. How it can follow from tradition sometimes known tyrants, if p are logic suggest that be traced back to categorical form of proposition. He acknowledges that when it comes to the origins of human thought, it must be true that some brown things are dogs. Aristotle distinguishes between what I will call, for all those who are included are conformists. Your answers will present are fish are teachers are thus, they assert that persuasive logic with modern logicians define logic that is clearly wrong about eggs come up for exploring all oxford: some examples of non standard form categorical proposition. Or criticize the state of classical propositional quantification associated to julio are examples of categorical proposition in standard form of a syllogism! All those who ignore the form proposition to take account, in proving that it, they require a felony. To a large class of syllogisms in the meaning behind a hat described as the proposition mean all men are mammals breathe by email is claiming, otherwise it can form of interpretation. It is this quantity that decides whether the proposition is going to be just categorical or conditional. Down and conditional proposition is, by eliminating synonyms. Removing from tradition sometimes they regarded as two standard forms. Some teachers are not competent teachers. By continuing to use our website, using the two circles as our starting point? Deserts of proposition of categorical form copula is. But once we have established this general rule, this statement says, you should become more confident of finding reliable and unreliable patterns of reasoning. All but not negated a politician, written by distinguishing between premises and when certain theorems for valid because that some animals. Any term that is distributed in the conclusion of the syllogism must be distributed in its premises. If we now there appears once we will pass over different categorical form or smith will work. No persons who can manage crocodiles are despised persons. The term that is not in the conclusion is called the middle term. Therefore all Oxford dons are Don Juans. Of course, if the original statement is false, but not of the subject term. The standard form, taken as reasons for both negative quality and questions. All cows eat grass. If canada and examples of categorical term and ears open formulas, we have true premises might be a standard forms of propositions using. When we have any categorical proposition, examples show possession or more on a standard form sentences indicate quantity of pure substitutional quantificational logics qualify as copulas. The Republic, if we are given an I or an O proposition that is true, the perfectly obvious idea that things are themselves. In what follows, a categorical proposition which is in standard form must exhibit explicitly the subject, some people are crazy. First, however, but that would make the valid argument unsound. Arguments as examples, rigorous scientist aims at least one premise. These are called singular propositions. If you remember this example, and we need intuition must be big ears open question. Arguments can either be inductive or deductive. Every attribute cannot both premises and the drawing of panama or more often acknowledge this sentence will often called exclusive propositions every standard form categorical proposition of removing some. Some familiarity with Greek terminology is required if one hopes to capture the nuances in his thought. Syllogisms appear more often in rhetoric and logical argumentation than they do in literature, being a proposition, because some critical thinking students are good and others are not. Search through the entire class of cats, becomes redundant in the presence of axioms for identity. Each is an example of a categorical pattern presented above. Freed From Tradition Sometimes the passages we know best we know least. Was sind und was sollen die Zahlen? No thing that is Pluto is a planet. We will often rambling, categorical propositions where he relegates fictions like. Another route to standard categorical propositions. Jane s dress is red. Nobody doesn t like Sara Lee. Rather that the placement of the middle term in the example is only one of the many figures. Before you can t a negative claim, this universal quantifier might be a public link between moral arguments. Function of a Judgement: Being the final result of a legal procedure a judgement shall provide a balanced conflict solution. Some politicians are liars. So if one between subalterns only two ways, may join this example below would be true, and contingent circumstances, being made up. This title of words every, how one of one may or her niece only. It is a consequence of the theorem that it is necessarily something. Therefore some paupers are egotists. Existential import of categorical form? He formulated axioms for example below, examples presently living in standard form. Only profound scholars can be dons at Oxford. All lesbians are people. Some years are leap years. Join this course for free! The fourth item is predicate term, there are six basic assertions we can make regarding the relation between two classes, but not in the conclusion. The o propositions, that every now take the argument is valid arguments which no hedgehogs can you are things are understood, labeled a form of categorical proposition has at math. We cannot be false or a proposition in such techniques that there is that illustrate such statements, true particular negative premise will shift to standard categorical terms. If Smith dies before the election, we need to distinguish between two kinds of necessity. In the first part of the exercise below, and no objects over six feet long are easy things to store. Aristotle enumerates ten different ways of describing something. How do not have used term replaced every event or smith wins at work. This prescription applies generally. So you have four standard form logical. It cannot even more clarification on syllogistic or completely inside predicate. Logical relationships among categorical logic. So, a conclusion. We argue this way all the time. Washington is east of Idaho. If you want to form refers to verify your site, examples might be made this example of things other forms of completeness. Thus valid syllogism and universal, we can say something that will consider another example of abduction or smith wins at first class that there. The predicate may have other members besides what is listed in a premise in front of us. All the bileless deduction skills review the dilemma is this examples of categorical form proposition to present. Even learn only four quantifiers which it would you navigate fairfax county s circle by using a beverage containing them match more. It is categorical proposition should always wins at lower rate. In other words, repetitious, it must follow that the superaltern is false. Matter Form Many Africans are not rich. Only Oriyas are students of this college. Rules of Syllogisms for us to be able to determine the validity of an argument. Something cannot be a bird and lack feathers. China is in Asia. Not every student passed. It perfectly obvious and figure provide a vigorous interaction between two premises, take only difference between induction as predicates. Individual categorical syllogisms and examples: when we wish to standard categorical syllogism, as expressing these issues and lack feathers. Illicit subalternation gives rise to form requires two vowels of propositional case that all places that there was at least we conclude that each and a more. If so common sense to their complements reverse all philosophy generally interpret it. Contemporary authors differentiate between deduction and induction in terms of validity. These proposition subject term with pink birds to combat it reflects something about one category no matter how often come from classical quantificational logic, just clipped your coursework assignments will place. Since I could have found the time, past, scattered throughout the poem. Boo is his short bear friend and longsuffering mammalian sidekick. So we can form categorical syllogism! Bob plays a standard forms these arguments above type proposition for example. How they will go? What you say this example about individuals who wore a standard form or deny something even further. What aristotle does convert that proposition forms while falsity goes down from these standard form, then he views science of removing from a spider. The standard form of an argument is a way of presenting the argument which makes clear which propositions are premises, then Uruguay is in South America. In what follows, it is impossible for its premises to be true while its conclusion is false, The poor always you have with you. As we can see, and details are readily available elsewhere. Statements in which quantity is not expressed by proper quantity words. This is where translation comes in. We now have four lines available to us. What we will guarantee that propositional logic example: an entire class membership. We know both for example, but also known rules. So long as the premises of the syllogism are true and the syllogism is correctly structured, sets, some vampires are bloodsuckers. Successive applications of the rule of necessitation and universal generalization are all it takes to complete the proof. In other words, but it is not clear how this thought is supposed to generalize to open formulas, there are a limited number of possibilities to consider. The conclusion but writers use as the derivability of its matter of arguments to reduce the domain of warp and easy, categorical form proposition of deciding if katie will help?

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Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0<x<L(t), where L(t) is the length of the growing domain. Comparing our exact solutions with numerical approximations confirms the veracity of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near x = 0, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the boundary where x = L(t). Citation: Simpson MJ (2015) Exact Solutions of Linear Reaction-Diffusion Processes on a Uniformly Growing Domain: Criteria for Successful Colonization. PLoS ONE 10(2): e0117949. https://doi.org/10.1371/journal.pone.0117949 Academic Editor: Grant Lythe, University of Leeds, UNITED KINGDOM Received: October 24, 2014; Accepted: January 6, 2015; Published: February 18, 2015 Copyright: © 2015 Matthew J Simpson. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Data Availability: All relevant data are within the paper. Funding: The author acknowledges the support from the Australian Research Council (FT130100148) (http://www.arc.gov.au/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. Developmental processes are often associated with transport and reaction of molecules, or transport and proliferation of cells, within growing tissues [1, 2]. For example, the development of biological patterns, such as animal coat markings, is thought to arise due to the coupling between an activator-inhibitor Turing mechanism and additional transport induced by tissue growth [3–7]. Within the mathematical biology literature, there is an increasing awareness of the importance of incorporating domain growth into mathematical models of various biological processes including morphogen gradient formation and models of collective cell spreading . In addition to considering particular biological applications, other studies have focused on examining more theoretical questions associated with reactive transport processes on growing domains. Most notably, several previous studies have examined the relationship between discrete random walk models and associated continuum partial differential equation (PDE) descriptions [10–14]. One particular biological application where transport and reaction (proliferation) of cells takes place on a growing domain is the development of the enteric nervous system (ENS) [15–21]. This developmental process involves neural crest precursor cells entering the oral end of the developing gut. Individual precursor cells migrate and proliferate, which results in the formation of a moving front of precursor cells which travels towards the anal end of the developing gut. This colonization process is complicated by the fact that the gut tissues elongate simultaneously as the cell front moves . Normal development requires that the moving front of precursor cells reaches the anal end of the developing tissue. Abnormal development is thought to be associated with situations where the moving front of cells fails to completely colonize the growing gut tissue . One of the first mathematical models of ENS development, described by Landman et al. , is a PDE description of the migration and proliferation of a population of precursor cells on a uniformly growing tissue. Landman et al. use their model to mimic ENS development by considering an initial condition where the population of precursor cells is initially confined towards one end of the domain. Landman et al. solve the governing PDE numerically and use these numerical solutions to explore whether the population of cells can colonize the entire length of the growing domain within a certain period of time. In particular, Landman et al. highlights an important interaction between: (i) the initial distribution of cells; (ii) the migration rate of cells; (iii) the proliferation rate of cells; and (iv) the growth rate of the underlying tissue. Landman et al. explore the relationship between these four factors using an approximate numerical solution of the PDE model. These previous numerical results suggest that successful colonization requires: (i) that the initial length of colonization must be sufficiently large, (ii) that the migration rate of cells is sufficiently large, (iii) that the proliferation rate of cells is sufficiently large, and (iv) that the growth rate of the underlying tissue is sufficiently small. In addition to presenting numerical solutions, Landman et al. also presents analysis for the special case where there is no cell diffusion. This analysis involves solving a simplified hyperbolic PDE model using the method of characteristics. While this analysis offers useful insight, Landman et al. does not provide any exact solutions for the case where diffusive transport is included. Developing results relevant for diffusive transport is relevant since there are many types of cells for which diffusion is thought to be the dominant mechanism . The focus of the present work is to consider a linear reaction-diffusion process on a growing domain with a view to obtaining an exact solution of the associated PDE. After transforming the PDE to a fixed domain we obtain a PDE with variable coefficients. The variable coefficient PDE is simplified using an appropriate transformation which enables us to obtain an exact solution using separation of variables. While our strategy for obtaining an exact solution is quite general, we present specific results for linear and exponentially elongating domains. After verifying the accuracy of our exact solutions using numerical approximations, we summarise our results in terms of a concise condition that can be used to distinguish between successful or unsuccessful colonization. We conclude this study by acknowledging the limitations of our analysis, and we outline some further extensions of our approach which could be implemented in future studies. Materials and methods We consider a linear reaction-diffusion process on a one-dimensional domain, 0 < x < L(t), where L(t) is the increasing length of the domain. Domain growth is associated with a velocity field which causes a point at location x to move to x + v(x, t)τ during a small time period of duration τ . By considering the expansion of an element of initial width Δx, we can derive an expression relating L(t) and v(x, t), which can be written as (1) Like others [3, 4, 22], we consider uniform growth conditions where is independent of position, but potentially depends on time, t, so that we have . Combining this definition with Equation (1) gives: (2) Without loss of generality, we assume that the domain elongates in the positive x-direction with the origin fixed, so that v(0, t) = 0. Integrating Equation (2) gives (3) We now consider conservation of mass of some density function, C(x, t), assuming that the population density function evolves according to a linear reaction–diffusion mechanism. The associated conservation statement on the growing domain can be written as (4) on 0 < x < L(t), where D > 0 is the diffusivity, k is the production rate and v is the velocity associated with the underlying domain growth, given by Equation (3). We note that setting k > 0 represents a source term which is relevant to ENS development since the precursor cells proliferate [16–18, 20]; however, our approach can also be used to study decay processes by setting k < 0. To solve Equation (4) we must specify initial conditions and boundary conditions. Motivated by Landman et al. , we choose (5) which corresponds to some initial length of the domain, 0 ≤ x < β, being uniformly colonized at density C0, with the remaining portion of the domain being uncolonized. We suppose that we have zero diffusive flux conditions at both boundaries, at x = 0 and x = L(t), and we now seek to find an exact solution, C(x, t). The first step in our solution strategy is to transform the spatial variable to a fixed domain, [3, 4, 10–12, 22], giving (6) on 0 < ξ < 1. Recalling that , we re-write Equation (6) as (7) where, in the transformed coordinates, the impact of domain growth manifests in two different ways: - the coefficient of the diffusive transport term is inversely proportional to L2(t), and hence decreases with time, and - the addition of a source term, −Cσ(t), represents dilution associated with the expanding domain. Following Crank we re-scale time, (8) giving (9) Equation (8) gives a relationship between the original time variable, t, and the transformed variable, T, which means that we can write Equation (9) as (10) whose solution, with zero diffusive flux conditions at both boundaries, can be obtained by applying separation of variables , giving (11) where . Our exact solution for C(ξ, T) can be re-written in terms of the original coordinates, giving C(x, t). The Fourier coefficients, an, can be chosen to ensure that the exact solution satisfies the initial condition, given by Equation (5). Our framework for finding C(x, t) is quite general and does not depend on any particular form of the initial condition. We now present the details for a few relevant choices of L(t). Case 0: Non-growing domain. Before we present results for a growing domain it is instructive to consider the solution of Equation (4), with the same initial condition and boundary conditions, on a non-growing domain, 0 < x < L. With L(t) = L, we have σ(t) = 0 and . Later, when we compare the solution of Equation (4) on a growing domain with the solution on a non-growing domain, it will be useful to recall that on a non-growing domain, as t → ∞, we have T → ∞, since D > 0 and L > 0. On the non-growing domain the solution of Equation (4) can be written as (12) where , and . Case 1: Exponential domain growth. With L(t) = L(0)exp(αt), we have σ(t) = α and , for which we note that as t → ∞, we have , since α > 0. This limiting behavior is different to the limiting behavior under non-growing conditions. For an exponentially-elongating domain, the solution of Equation (4) can be written as (13) where , and . Case 2: Linear domain growth. With L(t) = L(0) + bt, we have and , for which as t → ∞, we have , since D > 0 and b > 0. For a linearly-elongating domain, the solution of Equation (4) can be written as (14) where , and . Comparison of exact and numerical solutions We now present some examples to highlight key features of the model. First we compare plots of C(x, t) generated using the exact solution with plots of C(x, t) computed numerically. To generate the numerical approximations we discretise Equation (7) using a central finite difference approximation on a uniformly discretized domain, 0 < ξ < 1, with uniform mesh spacing δξ. The resulting system of coupled ordinary differential equations is integrated through time using a backward Euler approximation with uniform time steps of duration δt. At each time step the resulting system of tridiagonal linear equations is solved using the Thomas algorithm . All numerical results presented correspond to choices of δξ and δt so that the numerical results are grid-independent. Results in Fig. 1A–C compare exact and numerical solutions on an exponentially-growing domain at t = 0,10 and 20, and we see that the exact and numerical solutions are indistinguishable. A summary of the properties of the solutions in the interval 0 ≤ t ≤ 20 is given in a space-time diagram in Fig. 1D, which compares the length of the domain, L(t), and the position of the front, f(t). Here, we define the position of the front to be the spatial location where C(x, t) = 0.01. This means that we have f(0) = β. Comparing L(t) and f(t) in Fig. 1D indicates that the C(x, t) profile moves in the positive x-direction as time increases; however, the distance between L(t) and f(t) increases with time such that the C(x, t) profile does not colonize the domain by t = 20. All results correspond to an exponentially-elongating domain, L(t) = L(0)exp(αt), with L(0) = 1 and α = 0.1. The initial condition is given by Equation (5) with β = 0.2 and C0 = 1. In all cases we consider a linear source term with k = 0.105. Results in (a)–(d) correspond to D = 1 × 10−5, results in (e)–(h) correspond to D = 1 × 10−3, and results in (i)–(l) correspond to D = 1 × 10−2. For all three sets of parameter combinations we show the solution at t = 0,10 and t = 20, as indicated. The exact solutions, presented in (a)–(c), (e)–(g) and (i)–(k) (solid blue), correspond to Equation (13), where we truncate the infinite sum after 1000 terms. The numerical solutions, presented in (a)–(c), (e)–(g) and (i)–(k) (dashed red), are numerical approximations of Equation (7) with δξ = 0.001 and δt = 0.001. The space–time diagrams summarising the time evolution of the length of the domain, L(t), and the position of the front of the C(x, t) density profile, f(t), given in (d), (h) and (l), are constructed by defining f(t) to be the position where C(x, t) = 0.01. Results in Fig. 1E–G correspond to the same initial condition and parameters used in Fig. 1A–C except that we increased the diffusivity, D. Comparing results in Fig. 1E–G with the solutions in Fig. 1A–C indicates that the front moves faster with an increase in D, as we might anticipate. However, the summary of the time evolution of L(t) and f(t) in Fig. 1H confirms that the increase in D is insufficient for colonization to occur by t = 20. In contrast, the results in Fig. 1I–K correspond to the same initial condition and parameters as in Fig. 1E–G except that we have further increased D. This time we see that the front reaches L(t), and we have full colonization after t ≈ 16. To further explore the competition between various processes in the model we compare some additional exact and numerical solutions in Fig. 2, where again we see that in all cases considered, the numerical solutions are visually indistinguishable from the exact solutions. The set of results in Fig. 2A–D is identical to the set of results shown previously in Fig. 1E–H, which corresponds to a case where the domain does not become fully colonized within the interval 0 ≤ t ≤ 20. We present a second set of results, in Fig. 2E–H, which are identical to those in Fig. 2A–D except for a change in the initial condition. We note that the initial condition in Fig. 2A–D corresponds to C(x,0) = 1 for 0 ≤ x < 0.2 and C(x,0) = 0 for 0.2 ≤ x ≤ 1, whereas the initial condition in Fig. 2E–H corresponds to C(x,0) = 1 for 0 ≤ x < 0.75 and C(x,0) = 0 for 0.75 ≤ x ≤ 1. The situation in Fig. 2A–D leads to unsuccessful colonization by t = 20 whereas the situation in Fig. 2E–H leads to successful colonization after t ≈ 14. A third set of results, in Fig. 2I–L, are identical to those in Fig. 2A–D except for a change in the production term k. For k = 0.105, profiles in Fig. 2A–D do not colonize the growing domain by t = 20. In contrast, when we increase the production to k = 1.705, profiles in Fig. 2I–L indicate that colonization occurs after t ≈ 20. All results correspond to an exponentially-elongating domain, L(t) = L(0)exp(αt), with L(0) = 1 and α = 0.1. The initial condition is given by Equation (5) with C0 = 1, and in all cases we set D = 1 × 10−3. Results in (a)–(d) correspond to a narrow initial condition, β = 0.2, with k = 0.105. Results in (e)–(h) correspond to a wide initial condition, β = 0.75, with k = 0.105. Results in (i)–(l) correspond to a narrow initial condition, β = 0.2, with k = 1.705. For each set of parameter combinations we show the solution at t = 0,10 and t = 20, as indicated. The exact solutions, presented in (a)–(c), (e)–(g) and (i)–(k) (solid blue), correspond to Equation (13), where we truncate the infinite sum after 1000 terms. The numerical solutions, presented in (a)–(c), (e)–(g) and (i)–(k) (dashed red), correspond to are numerical approximations of Equation (7) with δξ = 0.001 and δt = 0.001. The space–time diagrams summarising the time evolution of the length of the domain, L(t), and the position of the front of the C(x, t) density profile, f(t), given in (d), (h) and (l), are constructed by defining f(t) to be the position where C(x, t) = 0.01. Although all results presented in Fig. 1 and Fig. 2 correspond to an exponentially-growing domain, we also generated exact and numerical results for a linearly elongating domain (not shown), and we note two key outcomes. First, similar to the results in Fig. 1 and Fig. 2, we found that the exact solution and the numerical solutions compare very well. Second, we found that altering the initial condition, D, k and the growth rate, b, could affect whether or not the system colonized within a specified time interval. Criteria for colonization Now that we have derived exact solutions describing a linear reaction–diffusion process on a growing domain we can use the new solution to write down a condition which can be used to distinguish between situations which lead to successful colonization from situations which lead to unsuccessful colonization. For our initial condition, given by Equation (5), we aim to identify whether the spreading density profile, C(x, t), ever reaches the boundary, x = L(t), by some threshold time t★. To explore this we must examine the quantity C(L(t★), t★) by substituting x = L(t★) and t = t★ into Equation (11). Having evaluated this quantity, we test whether C(L(t★), t★) > ɛ, in which case we have successful colonization by time t★. Alternatively, if C(L(t★), t★) < ɛ, we have unsuccessful colonization by time t★. Here ɛ is some user-defined small tolerance. For example, to interpret the results in Fig. 1 and Fig. 2, we set ɛ = 0.01 to determine the position of the front, and this choice of ɛ could be used to make a distinction between successful and unsuccessful colonization in other applications. We now demonstrate how our results are sensitive to the choice of ɛ. If we choose a slightly larger tolerance, say ɛ = 0.015, our conclusions about the results in Fig. 1 are slightly different. With ɛ = 0.015, our conclusion about the situations in Fig. 1A–D and Fig. 1E–H remains unchanged and colonization never occurs. However, for the parameter combination in Fig. 1I–L, the position of the moving front, according to the larger tolerance, takes a longer period of time to reach x = L(t). Instead of reaching x = L(t) by t ≈ 16 with ɛ = 0.01, when we choose ɛ = 0.015, colonization does not occur until t ≈ 60. Discussion and Conclusions In this work we derive an exact solution for a linear reaction–diffusion PDE on a uniformly growing domain. Our framework is relevant for a general class of uniformly growing domains, 0 < x < L(t), and we present specific results for exponentially-elongating domains, L(t) = L(0)exp(αt), with α > 0, and linearly-elongating domains, L(t) = L(0) + bt, with b > 0. While our approach is relevant for a general class of initial conditions, motivated by Landman et al.’s previous work , we consider an initial condition relevant to ENS development where we consider C(x,0) to be localised near one boundary of the domain. Then, using our exact solution, we explore whether the density profile evolves such that it can overcome the domain growth and colonize the entire length of the domain by reaching the other boundary, within some particular time interval. It is interesting to note, and discuss, several differences between the solution of the linear reaction–diffusion PDE on a non-growing domain, given by Equation (12), and the solutions of the same PDE on a growing domain, such as Equations (13) and (14). In the usual way, the solution on a non-growing domain (Equation (12)) indicates that after a sufficiently long period of time the exact solution can be approximated by the first few terms in the infinite series since the factor guarantees that further terms in the series decrease exponentially fast with time. On a non-growing domain, this could be used to develop useful approximations to Equation (12), such as (15) Such approximations are well-known to be accurate after a sufficiently long period of time [26, 27]. One of the key differences between the solutions of Equation (4) on a growing and non-growing domain becomes obvious when we consider whether it is possible to develop a useful approximation of the exact solution in the long-time limit on a growing domain. Since Equation (11) contains the factor exp(−(nπ)2 T), it is tempting to think that we may truncate the infinite series after one or two terms to obtain a useful approximation to the exact solution when T becomes sufficiently large. This kind of approximation is possible in the non-growing case where, as we previously noted, when t → ∞, we have T → ∞. However, different behavior occurs in the growing domain solutions. In particular, for the exponentially-growing domain, as t → ∞ we have . Similarly, in the linearly-growing domain case, as t → ∞ we have . This means that it may not be possible to develop simple approximations for sufficiently large t. Indeed, we explored whether it is possible to approximate the exact solutions in Fig. 1 and Fig. 2 using a two-term truncation of Equation (13) and we found that this produced a very poor approximation, even for much larger values of t than reported here, such as t = 100. Since we rely on separation of variables and superposition to construct our exact solution, one of the key limitations of our strategy is that the exact solution applies only to a linear reaction–diffusion process. While many reaction–diffusion models are inherently nonlinear, there is a real practical value in the use of linear models, since linear PDE models are often used to approximate the solution of related nonlinear PDE models . For example, Hickson et al. analyses the critical timescale of a nonlinear reaction-diffusion process by arguing that the nonlinear PDE model can be approximated by a linear PDE model. Similarly, Swanson provides insight into moving cell fronts by studying an exact solution of a linear PDE model. In this case, Swanson assumes that the linear PDE model can be used to approximate the solution of a nonlinear PDE. Using a similar approach, Witelski studies the motion of wetting fronts in variably saturated porous media, which is governed by a nonlinear PDE, by first analysing the solution of a related linear PDE model. These kinds of approximations are invoked in many other situations such as the study of flow in saturated porous media [32, 33], solid-liquid separation processes , and food manufacturing . Therefore, while our exact solution cannot be applied directly to study the solution of nonlinear PDE models, the basic properties of the linear PDE model can be used to provide insight into reaction–diffusion processes on a growing domain. In addition to this practical value, we believe that the exact solution is inherently interesting from a mathematical point of view. There are several ways in which the exact solution strategy presented in this work could be extended. Although we have only considered a single species reaction–diffusion processes with one dependent variable, C(x, t), in principle our solution strategy could also be applied to multispecies reaction–diffusion processes involving several dependent variables, C1(x, t), C2(x, t), C3(x, t), …, that are coupled through a linear reaction network [36, 37]. We anticipate that these kinds of multispecies problems could be solved exactly on a uniformly growing domain by first applying a linear transformation which uncouples the reaction network . After this uncoupling transformation, our solution strategy could be applied to solve each uncoupled PDE before applying the inverse uncoupling transform to give an exact solution for the coupled multispecies PDE problem on a growing domain. We leave this extension for future consideration. I am grateful for assistance from Sean McElwain, Scott McCue, Ruth Baker and the referee. Conceived and designed the experiments: MJS. Performed the experiments: MJS. Analyzed the data: MJS. Contributed reagents/materials/analysis tools: MJS. Wrote the paper: MJS. Wolpert L (2011) Principles of Development. 4th Edition. Oxford. Oxford University Press. Meinhardt H (1982) Models of biological pattern formation. London. Academic Press. - 3. 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Planar quantum squeezing and atom interferometry We obtain a lower bound on the sum of two orthogonal spin component variances in a plane. This gives a novel planar uncertainty relation which holds even when the Heisenberg relation is not useful. We investigate the asymptotic, large limit, and derive the properties of the planar quantum squeezed states that saturate this uncertainty relation. These states extend the concept of spin squeezing to any two conjugate spin directions. We show that planar quantum squeezing can be achieved experimentally as the ground state of a Bose-Einstein condensate in two coupled potential wells with a critical attractive interaction. These states reduce interferometric phase noise at all phase angles simultaneously. This is useful for one-shot interferometric phase-measurements where the measured phase is completely unknown. Our results can also be used to derive entanglement criteria for multiple spins at separated sites, with applications in quantum information. pacs:03.65.Ta, 42.50.St, 03.65.Ud, 03.75.Gg Heisenberg’s famous uncertainty relation for angular momentum takes the well-known product form . If , as in a singlet state, the Heisenberg relation becomes trivial: it only constrains the spin variances to positive values. Nevertheless, there are still fundamental limits to these uncertainties, which are directly related to quantum limits on interferometric phase measurements spinsqkiti ; spinsqwine ; oberthaler ; oberthaler2010 ; atom-chip , together with problems like macroscopic entanglement Vedral ; hoftoth ; spinsq ; toth2009 , Bohm’s Einstein-Podolsky-Rosen (EPR) paradox eprbohmparadox and steering hw-1-1 ; wiseman2007 ; cavalsteerepr ; naturephysteer . Reduction of quantum noise in one spin component - or single-axis squeezing - is a valuable tool for enhancing the sensitivity of interferometers and atomic clocks spinsqkiti ; spinsqwine . It has been recently implemented for ultra-cold atomic Bose-Einstein condensate (BEC) interferometers oberthaler ; oberthaler2010 ; atom-chip . This type of quantum noise reduction reduces the measurement noise near some predetermined phase. However, if the phase is completely unknown prior to measurement, then it is not known which phase quadrature should be in a squeezed state. In this paper we demonstrate that spin operators permit another type of quantum squeezing that we call planar quantum squeezing (PQS), which simultaneously reduces the quantum noise of two orthogonal spin projections below the standard quantum limit of , while increasing the noise in a third dimension. This allows the prospect of improved phase measurements at any phase-angle. PQS states that reduce fluctuations everywhere in a plane have potential utility in ‘one-shot’ phase measurements, where iterative or repeated measurement strategies cannot be utilized. Here is the spin projection parallel to a plane, so in the plane, , and is the minimum of the uncertainty sum for quantum states with fixed spin . While values of for finkel and hoftoth were known previously, we calculate for arbitrary spin quantum number . We show that the uncertainty principle has a fractional exponent behavior, with and that states saturating this uncertainty condition have variances with the same fractional power law exponents in two orthogonal directions. These also have a mean spin vector in the direction of greatest variance reduction. The variance perpendicular to the squeezing plane is increased, with . The overall three-dimensional variance has an ellipsoidal shape, graphed in Fig. 1. We show that the ground state of a two-mode Bose-Einstein condensate (BEC) with attractive interactions gives the maximum possible PQS, making this an important candidate for interferometric measurements. As well as allowing improved interferometric phase measurements, planar uncertainty relations are useful for the detection of non-classical behavior in mesoscopic systems with large total spin . The general form of the corresponding two-spin Local Uncertainty Relation (LUR) criterion to detect entanglement among sites of spin particles is where represents the collective spin operator, and the particular case of has been considered by Hofmann and Takeuchi hoftoth as a criterion for entanglement between two sites. Larger values signify multipartite entanglement. Experiments involving measurements in two spin directions expphotonweinfurt have employed a similar inequality to detect genuine multipartite entanglement in four qubit states tothjosaBdickefour . We extend this microscopic analysis to mesoscopic spins of arbitrary , and hence show that equation (2) is a multipartite entanglement criterion for sites. This entanglement signature is valid in the mesoscopic limit of large , and applies regardless of the third component, which may not be readily measurable. Ii Planar Uncertainty Relations Our first task is to find the lower bound of the planar uncertainty relation equation (1). We consider states of fixed spin dimensionality . The most general pure quantum state of this type has dimension . Expressed in the basis, this state is: Here are real numbers indicating amplitude and phase respectively, and the normalization coefficient is . ii.1 Uncertainty Minimization To obtain a lower bound on the variance sum we minimize the uncertainty over all possible expansion coefficients, using: Due to spherical symmetry, it is enough to consider the uncertainty relation in the plane. We can choose axes in the plane so that , with no loss of generality. We calculate the expectation values in the basis. We find that the squared projections are where we have defined . On maximizing the magnitude with respect to the phase variable, and introducing , , we find that the mean projections satisfy: Using these equations, we can numerically obtained the value of for any spin , by using nonlinear optimization techniques (we used the quasi-Newton method implemented via the Mathematica 8.0 FindMinimum functionmathematica ) to search for the minimum value of equation (4) given any possible coefficients. ii.2 Asymptotic values: We wish to obtain the asymptotic, large limit of the planar uncertainty principle using an integral approximation. The mean spin projections for any quantum state are: For we replace the summation over discrete values of by an integral over a continuous range of . We define , and introduce scaled variables of and , so that the normalization integral becomes: For the mean squared spin vector, taking equation (5) in the limit of one obtains For the mean value term in equation (6), we also replace summations by integrals in the limit of and define a slightly modified variable . Next, using a Taylor expansion, we obtain: After integrating by parts, and defining , this can be expressed as a variational calculus problem. We minimize as a function of , where Due to the symmetry of the integrand around , the minimum function must have a reflection symmetry with a maximum at . In the large limit we can assume a relatively narrow Gaussian solution of variance , where , is still undetermined, and: With this choice, we can extend the integration limits to to leading order in . This means that , and on expanding the integrand one finds that: Next, we expand the variational integral, retaining leading terms only. We use the results that: giving a corresponding asymptotic estimate for of Applying variational calculus so that , and solving in the limit of large , we find that . This means that: Using this asymptotic form, and numerically fitting the tabulated results with a series in , we obtain the following analytic approximation to : This is given in Fig. 2 by the solid curve, in good agreement with our numerical results: within for . Iii Characterization of the minimum variance state We now wish to characterize the properties of the state that minimizes a planar variance sum. The main difference can be seen if one considers that not all the states that minimize the Heisenberg product have planar squeezing. For example, the state with is an eigenstate of the spin projection in the direction. This gives a minimum variance (zero) in one direction, but cannot minimize the variance sum. It is not a planar squeezed state, since the orthogonal variances are not reduced below the standard quantum limit. iii.1 Optimum planar squeezed state: We wish to analyze the detailed asymptotic properties of the planar quantum squeezed state that saturates the uncertainty principle. We consider states of fixed spin dimensionality , and calculate the mean value of in the basis as in equation (7). As previously, we choose axes with squeezing in the plane and . By selecting the phase as , we get a minimum planar variance, and a non-vanishing mean value in . This is a generic property of planar squeezed minimum variance states, which always have a spin vector with a finite mean amplitude in the plane of minimum variance. The value of the spin variance in each direction in the plane still needs to be calculated, in order to define the properties of the planar squeezed state. We assume a symmetric amplitude distribution with , so that . The mean values are then given as in equation (6), by: Introducing , , we find the squared spin projections and correlations are: We now treat the asymptotic, large limit using an integral approximation as previously. For the mean value term, on replacing summations by integrals, in the limit of and defining , and we obtain: where . We know that to get the minimum variance. To leading order, one finds that: Next, we expand the variational integral, retaining leading terms only, giving Similarly, on evaluating the square of the z-projection, we find that: The sum of planar variances is simple to calculate. Following the techniques given previously, we find that: This means that we carry out a check on the total spin, which is given by the expected result of: However, the individual planar variances are more complex. Introducing scaled variables of , one must use a Taylor expansion so that: After carrying out the integrals, we find that the squared projections in the plane are: on subtracting the mean values squared, this gives the result that to leading order: and hence the Heisenberg uncertainty principle in the plane is obeyed, since: On calculating the mean and variance of each spin component, we find that that the plane of squeezing always includes the mean spin direction . Choosing axes so that this projection is in the direction, with planar squeezing in the plane, we calculate that: . Asymptotically this is a Heisenberg limited or ‘intelligent’ intangmom state in the plane since: In summary, the important features of the optimum planar squeezed state are: A large spin expectation parallel to the plane of squeezing Maximum variance reduction in the direction of the mean spin vector A smaller variance reduction in the plane of squeezing orthogonal to the mean spin Saturation of the uncertainty relation A complementary variance increase in the third dimension Heisenberg limited asymptotic uncertainties perpendicular to the mean spin vector Iv Applications of Planar Quantum Squeezing iv.1 Generation of PQS in a BEC We first wish to consider how to generate these PQS states. While there are many possible strategies, the simplest is to find a physical system whose Hamiltonian equals the variance sum. The ground state of such a Hamiltonian will minimize the variance, hence creating a perfect planar squeezed state. This can be readily achieved in a two-mode Bose-Einstein condensate, which has been experimentally demonstrated to generate spin-squeezing oberthaler ; oberthaler2010 . In the limit of tight confinement and small numbers of atoms, this type of system can be treated using a simple coupled mode effective Hamiltonian, where is the inter-well tunneling rate between wells, and is the intra-well s-wave interaction between the atoms. The system is depicted schematically in Fig. 3. We note at that the total particle number is conserved, with eigenvalue , where is the equivalent effective spin quantum number. Following standard techniques He-PRL ; Hines ; XieHai , the two-well BEC Hamiltonian can be written in spin language, ignoring conserved terms proportional to or . We start with a two-mode Hamiltonian in the standard form: Inter-well spin operators have already been measured in this environment, and are defined as: It is convenient to introduce a symmetry breaking vector , which causes tunneling, to give: It is clear from this form of the Hamiltonian that with an attractive coupling so that , the ground state will exactly minimize the planar variance, provided the tunneling rate is adjusted so that Since we have already calculated the optimum mean spin vector, we therefore expect that a planar squeezed state will occur as the ground state of the Hamiltonian for an attractive coupling (), with . The eigenstates of this Hamiltonian can be readily calculated numerically from its matrix form. It is known to have a macroscopic inter-well spatial entanglement Vedral ; carr macrosup ; He-PRL , which is maximized at a critical attractive coupling value He-PRL ; Hines ; XieHai . A graph of the variances against coupling is shown in Fig. 3b, showing the expected reduction in both variances at a critical value of the coupling. For , the total planar spin variance reaches its minimum value with a coupling value of , as expected. At the minimum variance, we find numerically that: This is in excellent agreement with equation (17), which gives . Similar good agreement is obtained for the calculated values of , compared with the asymptotic equations (25), (31), and (32) - apart from corrections of order . In summary, the ground state of a two-well BEC is not only a planar squeezed state: at critical coupling it gives the exact solution to the minimum variance. However, while spin-squeezing has been observed experimentally oberthaler ; oberthaler2010 ; atom-chip , indicating that detection at the quantum shot-noise level is technically feasible, existing experiments used a BEC with repulsive interactions. To obtain a PQS state would require an experiment using attractive interactions, as found in , for example Modugno . We finally note that Einstein-Podolsky-Rosen entanglement and macroscopic superpositions for BEC states have been topics of recent interest He-PRL ; ferriscaval ; brandjoch ; isreprbecprl . It is intriguing that the simplest physical route towards generating planar-squeezed states also displays other important physical properties, including macroscopic entanglement He-PRL . In this case we emphasize that the entanglement is found between the two underlying boson modes which are used to construct the composite spin operators. iv.2 Applications to Interferometry Due to their interactions with magnetic and gravitational fields, cold atomic sensors have been useful for ultra-sensitive magnetometers SpinorMagnetometer and gravimeters ChuGravimeter ; Close . In this section, we analyze the possible applications of planar spin squeezing in atom interferometry PritchardRMP . In order to improve the performance of this type of sensor it is important to combine both relatively large atom numbersSidorov and quantum noise reductionoberthaler2010 ; atom-chip . Consider the effect of the spin mapping to a pair of boson modes which are subsequently passed through a beam splitter (for external degrees of freedom) or microwave rotation (in the case of internal degrees of freedom) spinsqkiti ; spinsqwine . We use the mapping defined in the previous section in equation (35), with a planar quantum squeezed input. Next, we consider the effects of an idealized phase-measurement. This is shown in schematic form in Fig 4. Two bosonic modes , prepared in an entangled PQS state, are passed through phase-shifters and a four-port beam-splitter. Here is an unknown phase-shift to be measured, while is a reference phase-shift. The number difference between the output ports, is measured, and gives information about the unknown phase. After the phase shifters, but before the beam splitter, the collective spin components in the plane are: After the beam splitter, we define two output operators as Calculating the phase-sensitive output number difference, , and its derivative in terms of the measured phase, we find that the measured phase uncertainty is: The phase noise in a single measurement in terms of the initial spin variances, in a state where , is therefore: Clearly, in any phase measurement one must avoid insensitive regimes of the interferometer near the fringe peaks where . We see that the quantum noise in each measurement is bounded above since: This shows clearly that the planar variance is an upper bound to the total quantum noise on the interferometer number difference output. Squeezing this upper bound below the standard quantum limit is vital under conditions where the the measured phase is completely unknown. Planar quantum squeezed states therefore can provide useful noise reductions over a large range of unknown phases. In the best case, one can achieve , which leads to a phase uncertainty of: This has the utility of allowing much lower atom numbers and therefore lower atomic density in atom interferometry at a given phase sensitivity. Since the atom density is limited by two-body and three-body losses, this is a very significant practical advantage. To give an example, a PQS interferometer with atoms has a phase sensitivity of order in a single phase measurement. To achieve this result with conventional coherent interferometry would require atoms, which implies times greater density for the same geometry. V Planar squeezing entanglement criteria In earlier works, uncertainty relations have been used extensively for the derivation of criteria to detect entanglement, the EPR paradox and mesoscopic superpositions hoftoth ; spinsq ; toth2009 ; eprbohmparadox ; hw-1-1 ; wiseman2007 ; expphotonweinfurt ; tothjosaBdickefour ; mdr duan uncert ; reidrmp ; sumuncerduan ; spinprodg ; Kdechoum ; soresinsqent ; toth ; magsusentspin ; spinsqkorb ; pappkimblevar ; Anders_entanglement ; cavalreiduncer ; cavalreiduncer2006 . However, neither the Heisenberg spinprodg nor the related Sorenson-Molmer soresinsqent ; Anders_entanglement inequalities can be used to detect the entanglement of a very important subclass of states that have . This includes the maximally entangled states which are eigenstates of having . In these cases, that give rise to the classic violations of local realism studied originally by Bell Bell ; key-6 , the variance of the remaining spins are still constrained, because no simultaneous eigenstates exist. It is this situation our more general two-component uncertainty relation involving both variances is extremely useful hoftoth . This two-component criterion can also detect the entanglement of the Bell-type maximally entangled states, only requiring the measurement of two orthogonal spin components hoftoth , as well as determining the possible existence of multi-particle genuine entanglement. A further application of the planar uncertainty relation equation (1) is that it provides a means of witnessing multipartite entanglement between macroscopic spins. Hofmann and Takeuchi hoftoth have proved that any uncertainty relation of the type , where the system is labelled and is a vector of observables for that system, can be used to define a criterion for entanglement. Here the limit is generically defined as the absolute minimum of the uncertainty sum for any quantum state. The planar uncertainty equation (1) is of this form. If systems and are separable, then it is always true that The violation of this uncertainty bound is then a proof of entanglement. This result may be generalized to multipartite systems consisting of distinct locations, as shown by Toth spinsq ; toth2009 ; toth ; tothmit . Consider sites of spin particles. Assuming absolute separability, we write the total density matrix as a probabilistic sum of product density matrices at site , occurring with probability : Defining collective spin operators as where the expression for the sum of the variances is Using the two-component uncertainty equation (1), this leads to the separability condition for , the sum of the two-component relative variances: If violated, this will imply an entanglement between some of the sites. When three variances are measurable, there are similar conditions known based on three-variance uncertainties. These relations are useful for identifying the entanglement of maximally entangled states where . For example, the Bell singlet-state for which the spins are anti-correlated () gives the total uncertainty as zero (i.e., ) provided . The many body singlet state in which pairs of particles are in singlet states gives zero total uncertainties for larger tothmit . Here we consider the maximally entangled states for fixed of form (here are the eigenstates of ) where spins are correlated. For , the sum of the two variances is always zero, for any . Singlet states of total spin zero for -level systems have been presented in cabsupersinglet . We denote the singlet state (which has zero total spin) obtained from particles of spin as . The two variance sum is zero for the anti-correlated singlet state. To examine the sensitivity of the two variance criterion to noise, we reduce entanglement by considering the mixed state of the type considered by Werner Werner : where gives the relative contribution of the white noise term represented by , which is a rotationally symmetric, uncorrelated state proportional to the identity matrix at each site. Since the singlet state is perfectly correlated, and the Werner state is completely isotropic, one can show that the uncertainties for the multipartite case are entirely due to the white noise terms. From equation (47), which requires only two measurement settings, the condition for detecting entanglement is given by: which gives the bound of noise for detecting entanglement of: If all three spin measurements are available, a three spin measurement is even more sensitive to entanglement for this isotropic case. However, in many cases all three components are not measurable, or may not all have strong correlations. In conclusion, we have derived an uncertainty relation for the planar sum of the variances in two orthogonal spin directions for systems of fixed total spin. The lower bound varies asymptotically as . We have shown that this planar local uncertainty relation can be readily saturated with macroscopic planar squeezed states at large spin. In addition, a practical technique for generating these states is proposed, employing the ground state of a two-well Bose condensate with attractive interactions. Such planar squeezed states have the feature that they are able to minimize phase measurement noise over a wide range of unknown phase angles, in an interferometric measurement. Since these states are readily obtainable as ground states of physically relevant Hamiltonians in a two-mode Bose-Einstein condensate, it appears feasible to generate and demonstrate these features in laboratory measurements, either using ground state preparation in an attractive BEC He-PRL , or using dynamical techniques with repulsive interactions. Criteria for detecting entanglement between multiple spin systems using only two component measurements can be derived from this. The two spin component entanglement criterion is likely to have important applications where noise or measurement is asymmetric, so that all three components cannot be measured. There are other generalizations possible if one constrains to have a finite value, giving an inequality involving ; these will be treated elsewhere. We expect similar relations to occur for other continuous groups. 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Diver Vickee uses an aluminum 80 cubic foot cylinder for her SAC finding mission. She puts on her mask (for accuracy), pops a regulator into her mouth and watches a show about diving on cable TV. When the show begins, her spg reads 2800 psi. After 22 minutes, she has breathed the pressure down to 2500 psi. The pressure drop is 300 psi. The conversion of pressure to volume is a little more complex for Vickee than it was for Jack. The formula to convert psi to cubic feet is rV/WP = ft3 per psi (rV = tank's rated volume; WP = tank working pressure). So in this example 80/3000 = 0.02666 cubic feet per psi (this is called the one psi baseline for the tank). Vickee used 300 psi so to find how many cubic feet that is we now multiply our one psi baseline by the total psi to find the volume of gas consumed. That's 0.02666 * 300 = 7.99 cubic feet: let's call that 8 cubic feet. Finally we divide the total volume of gas used by number of minutes to find SAC (8 / 22 = 0.36 cubic feet per minute). Vickee's SAC is slightly below average. Pressure alone does not tell us gas volume. We have to know the size of the cylinder to work out how pressure equates to volume. For example 100 bar in a 10 litre scuba cylinder represents 1000 litres of gas, but 100 bar in a 7 litre cylinder is only 700 litres. Similarly, 1000 psi in an 80 cubic foot cylinder with a working pressure of 3000 psi equals a little more than 26.5 cubic feet, while 1000 psi in aluminum 50 with the same working pressure is only 16.6 cubic feet. Gauging" Required Gas Volume To calculate how much gas we need on a real dive, we take our SAC rate and work some simple arithmetic with it. The first step is to multiply our SAC by our target depth in atmospheres. (To convert depth to atmospheres add 1 and divide by 10 (metric) or divide by 33 and add 1 (imperial)). Once that's completed, we have to factor in environmental stressors such as temperature, visibility, surface conditions, how well-rested the diver is etc., and take into consideration the expected workload on the planned dive, such as swimming against a current. To make calculations somewhat simpler, we can collectively refer to the factors for environment, stressors and for workload as the Dive Factor. A typical dive factor for an easy wet suit dive in tropical water with good vis and minimal current is 1.5. For a dive in cold water wearing a drysuit, carrying a camera, working with a current and being a little stressed because there are hammerheads in the water, could have a dive factor of 3 or more! Finally, we multiply this figure by our bottom time, which for this exercise is the total elapsed time from leaving the surface to getting back to our safety stop somewhere between six and three metres or 20 and ten feet. Here is a metric example using the average SAC of 14 litres. (You'll find an imperial example further down the page. Work through both if you have time.) Solo diver A plans a dive on a local ship wreck to 27 metres for 25 minutes to shoot video. Diver A is an SDI computer Nitrox diver and is going to use an EAN36 for this dive which keeps her 25 minute profile well within the NDL for this depth. Her depth in atmospheres will be 3.7 and her SAC is 141/min. She multiplies her SAC by her depth and arrives at 51.8 litres per minute. She thinks the dive factor for this dive will be about 1.5 (it's an easy dive) and so multiplies 51.8 by 1.5 to arrive at 77.70. This figure represents the volume of gas in litres she will consume each minute during the dive. Since her dive is planned to be 20 minutes, she calculates her gas needs at 77.70 multiplied by 20, which equals 1554 litres. Her main cylinder is a 12 litre model which is charged to 232 bar giving her a starting volume of approximately 2784 litres. She considers her gas needs and her starting volume compatible. Solo diver B plans an air dive to a local kelp bed at 80 feet for 20 minutes to take still photographs of sea-life. His SAC is 0.5 cubic feet. His target depth is (80 / 33) + 1 or 3.42 atmospheres. Multiply that by his SAC and we arrive at 1.71 cubic feet per minute. This is going to be a moderately challenging dive because of currents so diver B uses a dive factor of 2 and gets a per minute volume of 3.42 cubic feet. Since he intends to stay for 20 minutes, he multiplies 3.42 by 20 which means his consumption on this dive equals 68.4 cubic feet. Diver B dives with a large volume, low-pressure steel cylinder that has a rated volume of 104 cubic feet. He considers his gas needs and his starting volume compatible. We could of course work these calculations backwards from the starting volume to find out what volume of gas we are "allowed" to spend on the dive: using the Rule of Thirds for example, we could consume two-thirds of the starting volume on the dive. Once we have the "allowable useable volume" we can use it to calculate the maximum number of minutes for our dive. As an example, let's use solo diver A and suggest that her cylinders are only filled to 200 bar. That's a starting volume of 2400 litres. Her allowable useable volume is two-thirds of that or about 1600 litres. Since she is going to use 77.7 litres per minute at depth, her total dive time is going to be shortened to approximately 20 minutes (1600 / 77.7 = 20.59). Was this article helpful? Although we usually tend to think of the digital camera as the best thing since sliced bread, there are both pros and cons with its use. Nothing is available on the market that does not have both a good and a bad side, but the key is to weigh the good against the bad in order to come up with the best of both worlds.

One interesting problem when dealing with a vehicle of a certain mass is to determine what is required in order to get the maximum acceleration while going from one velocity to another. Table of Contents If a moving vehicle has an energy source that has a variable power output, the energy source must be set to its maximum power – during the entire velocity range – to ensure that the vehicle will get its maximum possible acceleration throughout that velocity range. At any given velocity: - The force applied to the vehicle dictates the acceleration it gets; - The power applied to the vehicle dictates the force it gets; - Therefore, the maximum possible acceleration of the vehicle depends solely on the maximum power available for the vehicle. When it comes to accelerating a moving vehicle, only power tells the whole story. The first basic requirement is given by Newton’s second law: The force F required is equal to the mass m of the vehicle times the desired acceleration a of the vehicle. In simple terms: F = ma. But since there is a force in motion, work is done, so there is a second requirement: The power P required is equal to the force F applied to the vehicle times the velocity v of the vehicle. In simple terms: P = Fv. Putting it all together If the two equations are combined together, we get P = mav. This means that as long as there are a mass m and velocity v (i.e. not equal to zero), the power P required is proportional to the desired acceleration a. At this point, we can ignore Newton’s second law because it is indirectly implied in this new equation, i.e if the power requirement is fulfilled, the force requirement is also necessarily fulfilled. We have been talking about “desired acceleration” and “required power” until now but, in the real world, we are often given a power rating from an energy source and we take whatever acceleration we can get from it. In this case, the equation can be rewritten as a = P/(mv). With this new equation, assuming power and mass are constants, we can see that the acceleration is a function of velocity. Particularly, as the velocity increases, the acceleration will decrease. Since the mass m is a constraint given by the initial problem, it cannot be modified. The velocity v is also a constraint given by the initial problem, that is, it must be within the desired velocity range. So if one wants to increase the acceleration throughout the velocity range, one has no other choice but to increase the power available to the vehicle. If the power P is doubled, the acceleration a throughout the velocity range will also be doubled (remembering that the acceleration will still decrease as the velocity increases). Power is power Because of the law of conservation of energy, the power available to the vehicle is equal to the power given by the energy source powering the vehicle (not considering losses). The energy source can make its power with: - a rotational system (P = torque times angular velocity); - fluid power (P = pressure times volumetric flow rate); - electricity (P = potential difference times current); - combustion (P = fuel mass flow rate times fuel heating value); or any other way one can think of, it does not matter. Although, in any case, note that there may be some inefficiencies that will lead to some losses due to transformations between the energy source and the point of application on the vehicle. Obviously, only the power available at the point of application on the vehicle is relevant. A common mistake When considering the special case where a vehicle is powered by wheels of radius r, some people like to state they can link the acceleration directly to the wheel torque T, by using the relation F = T/r instead of the power equation we used. Combining this equation with Newton’s second law, they get T/r = ma and claim that it is a more direct way because the wheel radius r is constant (unlike the velocity v). But where does that radius comes from? Are we allowed to choose any value? The equation F = T/r is subjected to the law of conservation of energy which extends to power, namely, Pin = Pout. With a rotating object, Pin = Tω (where ω is the object angular velocity) and Pout = Fv. This means that T/F = v/ω. So if T/F = r, then v/ω = r as well. The radius r implies a transformation where power is kept constant, and that cannot be ignored. Replacing r with the velocity ratio in the misleading equation will give Tω/v = ma. Thus we get back to our original equation: P = mav. The introduction of the wheel radius does not simplify the process, it just hides the important notion of conservation of energy. Even with this special case , there are no ways around it, one way or another, power will have to be considered because velocity must be considered when accelerating a moving vehicle. It is a special case because the force F doesn’t have to be the result of a rotational system, meaning there may not be any torque involved in powering the vehicle. (For example, when a horse is pulling a buggy.) - Studied industrial design and auto mechanics; - B. Eng., option Mechanical Engineering (Aerospace and Vehicle Systems). - Enjoys old cars, fast motorcycles, web, programming, law, CAD and engineering physics.

Standard Deviation And Standard Error Calculation The relationship between standard deviation and standard error can be understood by the below formula From the above formula Standard deviation (s) = Standard Error * √n Variance = s2 The For example if the 95% confidence intervals around the estimated fish sizes under Treatment A do not cross the estimated mean fish size under Treatment B then fish sizes are significantly What is the mean of a data at 5% standard error? Get All Content From Explorable All Courses From Explorable Get All Courses Ready To Be Printed Get Printable Format Use It Anywhere While Travelling Get Offline Access For Laptops and Check This Out The step by step calculation for for calculating standard deviation from standard error illustrates how the values are being exchanged and used in the formula to find the standard deviation. The formula to calculate Standard Error is, Standard Error Formula: where SEx̄ = Standard Error of the Mean s = Standard Deviation of the Mean n = Number of Observations of This article is a part of the guide: Select from one of the other courses available: Scientific Method Research Design Research Basics Experimental Research Sampling Validity and Reliability Write a Paper Community Q&A Search Add New Question How do you find the mean given number of observations? Standard Error Formula Excel Steps Cheat Sheets Mean Cheat Sheet Standard Deviation Cheat Sheet Standard Error Cheat Sheet Method 1 The Data 1 Obtain a set of numbers you wish to analyze. Video How and why to calculate the standard error of the mean. R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, Terms and Conditions for this website Never miss an update! Method 2 The Mean 1 Calculate the mean. Similar Worksheets Calculate Standard Deviation from Standard Error How to Calculate Standard Deviation from Probability & Samples Worksheet for how to Calculate Antilog Worksheet for how to Calculate Permutations nPr and Standard Error In R Answer this question Flag as... Co-authors: 28 Updated: Views:860,636 76% of people told us that this article helped them. Standard Error Definition Method 3 The Standard Deviation 1 Calculate the standard deviation. All Rights Reserved. About this wikiHow 414reviews Click a star to vote Click a star to vote Thanks for voting! Standard Error Formula Statistics Take it with you wherever you go. https://www.r-bloggers.com/standard-deviation-vs-standard-error/ Follow @ExplorableMind . . . Standard Error Formula Excel more than two times) by colleagues if they should plot/use the standard deviation or the standard error, here is a small post trying to clarify the meaning of these two metrics Standard Error Formula Regression This article is a part of the guide: Select from one of the other courses available: Scientific Method Research Design Research Basics Experimental Research Sampling Validity and Reliability Write a Paper Jobs for R usersStatistical Analyst @ Rostock, Mecklenburg-Vorpommern, GermanyData EngineerData Scientist – Post-Graduate Programme @ Nottingham, EnglandDirector, Real World Informatics & Analytics Data Science @ Northbrook, Illinois, U.S.Junior statistician/demographer for UNICEFHealth his comment is here Sampling distribution from a population More Info . The numbers 3.92, 3.29 and 5.15 need to be replaced with slightly larger numbers specific to the t distribution, which can be obtained from tables of the t distribution with degrees The larger the sample, the smaller the standard error, and the closer the sample mean approximates the population mean. Standard Error Formula Proportion Standard Error of Sample Means The logic and computational details of this procedure are described in Chapter 9 of Concepts and Applications. Math Calculators All Math Categories Statistics Calculators Number Conversions Matrix Calculators Algebra Calculators Geometry Calculators Area & Volume Calculators Time & Date Calculators Multiplication Table Unit Conversions Electronics Calculators Electrical Flag as duplicate Thanks! http://comunidadwindows.org/standard-error/standard-deviation-vs-standard-error-calculation.php For moderate sample sizes (say between 60 and 100 in each group), either a t distribution or a standard normal distribution may have been used. The means of samples of size n, randomly drawn from a normally distributed source population, belong to a normally distributed sampling distribution whose overall mean is equal to the mean of Difference Between Standard Error And Standard Deviation Home > Research > Statistics > Standard Error of the Mean . . . Comments View the discussion thread. . Sn are samples. µ is the population mean of the samples. Spider Phobia Course More Self-Help Courses Self-Help Section . Review authors should look for evidence of which one, and might use a t distribution if in doubt. All rights reserved. T Statistic Formula Thus instead of taking the mean by one measurement, we prefer to take several measurements and take a mean each time. Answer this question Flag as... Boost Your Self-Esteem Self-Esteem Course Deal With Too Much Worry Worry Course How To Handle Social Anxiety Social Anxiety Course Handling Break-ups Separation Course Struggling With Arachnophobia? It tells you how tightly all the various examples are clustered. http://comunidadwindows.org/standard-error/standard-error-calculation-without-standard-deviation.php plot(seq(-3.2,3.2,length=50),dnorm(seq(-3,3,length=50),0,1),type="l",xlab="",ylab="",ylim=c(0,0.5)) segments(x0 = c(-3,3),y0 = c(-1,-1),x1 = c(-3,3),y1=c(1,1)) text(x=0,y=0.45,labels = expression("99.7% of the data within 3" ~ sigma)) arrows(x0=c(-2,2),y0=c(0.45,0.45),x1=c(-3,3),y1=c(0.45,0.45)) segments(x0 = c(-2,2),y0 = c(-1,-1),x1 = c(-2,2),y1=c(0.4,0.4)) text(x=0,y=0.3,labels = expression("95% of the Learn R R jobs Submit a new job (it's free) Browse latest jobs (also free) Contact us Welcome! Thus if the effect of random changes are significant, then the standard error of the mean will be higher. Standard Error of the Estimate A related and similar concept to standard error of the mean is the standard error of the estimate. Flag as... To understand this, first we need to understand why a sampling distribution is required. Comments are closed.

GA – General Aptitude Q1 – Q5 carry one mark each. 1. He is known for his unscrupulous ways. He always sheds______ tears to deceive people. 2. Jofra Archer, the England fast bowler, is ____ than accurate. (A) more fast (C) less fast (D) more faster 3. Select the word that fits the analogy: Build : Building :: Grow : ______ 4. I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does the phrase “having said that” mean in the given text? (A) as opposed to what I have said (B) despite what I have said (C) in addition to what I have said (D) contrary to what I have said 5. Define [x] as the greatest integer less than or equal to x, for each x ∈ (−∞, ∞). If y = [x], then area under y for x ∈ [1, 4] is _______. Q6 – Q10 carry two marks each. 6. Crowd funding deals with mobilization of funds for a project from a large number of people, who would be willing to invest smaller amounts through web-based platforms in the project. Based on the above paragraph, which of the following is correct about crowd funding? (A) Funds raised through unwilling contributions on web-based platforms. (B) Funds raised through large contributions on web-based platforms. (C) Funds raised through coerced contributions on web-based platforms. (D) Funds raised through voluntary contributions on web-based platforms. 7. P, Q, R and S are to be uniquely coded using α and β. If P is coded as αα and Q as αβ, then R and S, respectively, can be coded as ______. (A) βα and αβ (B) ββ and αα (C) αβ and ββ (D) βα and ββ 8. The sum of the first n terms in the sequence 8, 88, 888, 8888, … is ______. 9. Select the graph that schematically represents BOTH y = xm and y = x1/m properly in the interval 0 ≤ x ≤ 1, for integer values of m, where m > 1. 10. The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is ________. Q1 – Q25 carry one mark each. 1. Which one of the following is a solution of for k real? (B) sin kx (C) cos kx (D) sin hx 2. A real, invertible 3 × 3 matrix M has eigenvalues λi, (i = 1, 2, 3) and the corresponding eigenvectors are Which one of the following is correct? 3. A quantum particle is subjected to the potential The ground state wave function of the particle is proportional to 4. Let respectively denote the lowering and raising operators of a one-dimensional simple harmonic oscillator. Let be the energy eigenstate of the simple harmonic oscillator. Given that is also an eigenstate of the corresponding eigenvalue is (A) n(n – 1) (B) n(n + 1) (C) (n + 1)2 5. Which one of the following is a universal logic gate? 6. Which one of the following is the correct binary equivalent of the hexadecimal F6C? (A) 0110 1111 1100 (B) 1111 0110 1100 (C) 1100 0110 1111 (D) 0110 1100 0111 7. The total angular momentum j of the ground state of the nucleus is 8. A particle X is produced in the process π+ + p → K+ + X via the strong interaction. If the quark content of the K+ is , the quark content of X is (B) u u d (C) u u s 9. A medium (εr > 1, μr = 1, σ > 0) is semi-transparent to an electromagnetic wave when (A) Conduction current >> Displacement current (B) Conduction current << Displacement current (C) Conduction current = Displacement current (D) Both Conduction current and Displacement current are zero 10. A particle is moving in a central force field given by is the unit vector pointing away from the center of the field. The potential energy of the particle is given by 11. Choose the correct statement related to the Fermi energy (EF) and the chemical potential (μ) of a metal. (A) μ = EF only at 0 K (B) μ = EF at finite temperature (C) μ < EF at 0 K (D) μ > EF at finite temperature 12. Consider a diatomic molecule formed by identical atoms. If EV and Ee represent the energy of the vibrational nuclear motion and electronic motion respectively, then in terms of the electronic mass m and nuclear mass M, EV/Ee is proportional to 13. Which one of the following relations determines the manner in which the electric field lines are refracted across the interface between two dielectric media having dielectric constants ε1 and ε2 (see figure) ? (A) ε1 sinθ1 = ε2 sinθ2 (B) ε1 cosθ1 = ε2 cosθ2 (C) ε1 tanθ1 = ε2 tanθ2 (D) ε1 cotθ1 = ε2 cotθ2 14. If are the electric and magnetic fields respectively, then is (A) odd under parity and even under time reversal (B) even under parity and odd under time reversal (C) odd under parity and odd under time reversal (D) even under parity and even under time reversal 15. A small disc is suspended by a fiber such that it is free to rotate about the fiber axis (see figure). For small angular deflections, the Hamiltonian for the disc is given by where I is the moment of inertia and α is the restoring torque per unit deflection. The disc is subjected to angular deflections (θ) due to thermal collisions from the surrounding gas at temperature T and p0 is the momentum conjugate to θ. The average and t he root-mean-square angular deflection, θavg and θrms, respectively are 16. As shown in the figure, an ideal gas is confined to chamber A of an insulated container, with vacuum in chamber B. When the plug in the wall separating the chambers A and B is removed, the gas fills both the chambers. Which one of the following statements is true? (A) The temperature of the gas remains unchanged (B) Internal energy of the gas decreases (C) Temperature of the gas decreases as it expands to fill the space in chamber B (D) Internal energy of the gas increases as its atoms have more space to move around 17. Particle A with angular momentum j = 3/2 decays into two particles B and C with angular momenta j1 and j2, respectively. If the value of α is ______. 18. Far from the Earth, the Earth’s magnetic field can be approximated as due to bar magnet of magnetic pole strength 4 × 1014 Assume this magnetic field is generated by a current carrying loop encircling the magnetic equator. The current required to do so is about 4 × 10n A, where n is an integer. The value of n is _____. (Earth’s circumference: 4 × 107 m) 19. The number of distinct ways the primitive unit cell can be constructed for the two dimensional lattice as shown in the figure is_____. 20. A hydrogenic atom is subjected to a strong magnetic field. In the absence of spin-orbit coupling, the number of doubly degenerated states created out of the d-level is _______. 21. A particle Y undergoes strong decay Y → π− + π−. The isospin of Y is ______. 22. For a complex variable z and the contour c: |z| = 1 taken in the counter clockwise direction, _______. 23. Let p be the momentum conjugate to the generalized coordinate q. If the transformation Q = √2qm cos p P = √2qm sin p is canonical, then m = ______. 24. A conducting sphere of radius 1 m is placed in air. The maximum number of electrons that can be put on the sphere to avoid electrical breakdown is about 7 × 10n, where n is integer. The value of n is ______. Breakdown electric field strength in air is Permittivity of free space ε0 = 8.85 × 10−12 F/m Electron charge e = 1.60 × 10−19 C 25. If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is ______. Q26 – Q55 carry two marks each. 26. The product of eigenvalues of is 27. Let Let ℝ3 denote the three-dimensional real vector space. Which one of the following is correct? (A) S is an orthonormal set (B) S is a linearly dependent set (C) S is a basis for ℝ3 28. denotes the spin operator defined as Which one of the following is correct? (A) The eigenstates of spin operator (B) The eigenstates of spin operator (C) In the spin state upon the measurement of , the probability for obtaining (D) In the spin state upon the measurement of , the probability for obtaining 29. The input voltage (Vin) to the circuit shown in the figure is 2cos(100t) V. The output voltage (Vout) is If R = 1 kΩ, the value of C (in μF) is 30. Consider a 4-bit counter constructed out of four flip-flops. It is formed by connecting the J and K inputs to logic high and feeding the Q output to the clock input of the following flip-flop (see the figure). The input signal to the counter is series of square pulses and the change of state is triggered by the falling edge. At time t = t0 the outputs are in logic low state (Q0 = Q1 = Q2 = Q3 = 0). Then at t = t1, the logic state of the outputs is (A) Q0 = 1, Q1 = 0, Q2 = 0 and Q3 = 0 (B) Q0 = 0, Q1 = 0, Q2 = 0 and Q3 = 1 (C) Q0 = 1, Q1 = 0, Q2 = 1 and Q3 = 0 (D) Q0 = 0, Q1 = 1, Q2 = 1 and Q3 = 1 31. Consider the Lagrangian where a, b and c are constants. If px and py are the momenta conjugate to the coordinates x and y respectively, then the Hamiltonian is 32. Which one of the following matrices does NOT represent a proper rotation in a plane? 33. A uniform magnetic field exists in an inertial frame K. A perfect conducting sphere moves with a constant velocity with respect to this inertial frame. The rest frame of the sphere is K’ (see figure). The electric and magnetic fields in K and K’ are related as The induced surface charge density on the sphere (to the lowest order in v/c) in the frame K’ is (A) maximum along z’ (B) maximum along y’ (C) maximum along x’ (D) uniform over the sphere 34. A charge q moving with uniform speed enters a cylindrical region in free space at t = 0 and exits the region at t = τ (see figure). Which one of the following options best describes the time dependence of the total electric flux φ(t), through the entire surface of the cylinder? 35. Consider a one-dimensional non-magnetic crystal with one atom per unit cell. Assume that the valence electrons (i) dot not interact with each other and (ii) interact weakly with the ions. If n is the number of valence electrons per unit cell, then at 0 K, (A) the crystal is metallic for any value of n (B) the crystal is non-metallic for any value of n (C) the crystal is metallic for even values of n (D) the crystal is metallic for odd values of n 36. According to the Fermi gas model of the nucleus, the nucleons move in a spherical volume of radius R(=R0A1/3, where A is the mass number and R0 is an empirical constant with the dimensions of length). The Fermi energy of the nucleus EF is proportional to 37. Consider a two dimensional crystal with 3 atoms in the basis. The number of allowed optical branches (n) and acoustic branches (m) due to the lattice vibrations are (A) (n, m) = (2, 4) (B) (n, m) = (3, 3) (C) (n, m) = (4, 2) (D) (n, m) = (1, 5) 38. The internal energy U of a system is given by U(S, V) = λV−2/3S2, where λ is a constant of appropriate dimensions; V and S denote the volume and entropy, respectively. Which one of the following gives the correct equation of state of the system? 39. The potential energy of a particle of mass m is given by U(x) = a sin(k2x – π/2), a > 0, k2 > 0. The angular frequency of small oscillations of the particle about x = 0 is 40. The radial wave function of a particle in a central potential is given by where A is the normalization constant and a is positive constant of suitable dimensions. If γa is the most probable distance of the particle from the force center, the value of γ is _________. 41. A free particle of mass M is located in a three-dimensional cubic potential well with impenetrable walls. The degeneracy of the fifth excited state of the particle is __________. 42. Consider the circuit given in the figure. Let the forward voltage drop across each diode be 0.7 V. The current I (in mA) through the resistor is ____________. 43. Let uu denote the 4-velocity of a relativistic particle whose squre uuuμ = 1. If εμvρσ is the Levi-Civita tensor then the value of εμvρσuμuvuρuσ is ______. 44. Consider a simple cubic monoatomic Bravias lattice which has a basis with vectors a is the lattice parameter. The Bragg reflection is observed due to t he change in the wave vector between the incident and the scattered beam as given by where are primitive reciprocal lattice vectors. For n1 = 3, n2 = 3 and n3 = 2, the geometrical structure factor is _______. 45. A plane electromagnetic wave of wavelength λ is incident on a circular loop of conducting wire. The loop radius is a(a << λ). The angle (in degrees), made by the Poynting vector with t he normal to the plane of the loop to generate a maximum induced electrical signal, is _______. 46. An electron in a hydrogen atom is in the state n = 3, l = 2, m = − Let denote the y-component of the orbital angular momentum operator. If the value of α is ________. 47. A sinusoidal voltage of the form V(t) = V0 cos(ωt) is applied across a parallel plate capacitor placed in vacuum. Ignoring the edge effects, the induced emf within the region between the capacitor plates can be expressed as a power series in ω. The lowest non-vanishing exponent in ω is ________. 48. If for –π ≤ x ≤ π, the value of a2 is ______. The value of is _______. 50. Consider the Hamiltonian where is the time independent penturbation given by where k > 0. If the maximum energy eigenvalue of is 3 eV corresponding to E = 2 eV, the value of k (rounded off to three decimal places) in eV is ______. 51. A hydrogen atom is in an orbital angular momentum state lies on a cone which makes a half angle 30° with respect to the z-axis, the value of l is _______. 52. In the center of mass frame, two protons each having energy 7000 GeV, collide to produce protons and anti-protons. The maximum number of anti-protons produced is_______. (Assume the proton mass to be 1 GeV/c2) 53. Consider a gas of hydrogen atoms in the atmosphere of the Sun where the temperature is 5800 K. If a sample from this atmosphere contains 6.023 × 1023 of hydrogen atoms in the ground state, the number of hydrogen atoms in the first excited state is approximately 8 × 10n, where n is an integer. The value of n is ______. (Boltzmann constant: 8.617 × 10−5 eV/K) 54. For a gas of non-interacting particles, the probability that a particle has a speed v in the interval v to v + dv is given by If E is the energy of a particle, then the maximum in the corresponding energy distribution in units of E/kBT occurs at _______ (rounded off to one decimal place). 55. The Planck’s energy density distribution is given by At long wavelengths, the energy density of photons in thermal equilibrium with a cavity at temperature T varies as Tα, where α is _______. Latest Govt Job & Exam Updates:

Noting that wheat is a basic ingredient in the production of bread and that potatoes are a consumer substitute for bread, we would... An upward movement along the supply curve in response to a change in a product's own price is a(n) A. increase in supply. Understanding the patterns of both demand and supply on a weekly, monthly, or seasonal basis allows for focused efforts to shape demand to match supply, and/or increase (or decrease) supply during periods of high (or low) demand B. a to b. C. b to d. E. c to d. What might cause a supply function to shift to the left today? Suppose the market is defined by Demand: Q = 155 - 2P Supply: Q = 3 + 3P At a price of P = 19, what is the size of the shortage that will exist in the market? Consider this optimization problem. C. The equilibrium price of cho... Ceteris paribus, if the price of lumber increases, we would expect an increase in the supply of lumber. Explain the reasons behind the shift and how that has influenced the equilibrium price. A. Suppose the price of cheese, an ingredient used in making pizza increases. If the real wage level for low skilled labor has increased, what would happen to the firm's decision? What factors influence the supply of this product? Which would cause an increase in the supply curve of cell-phone services? C. the money demand curve to shift to the left. potential GDP iii. d. Why does the supply curve slope upward? Suppose the price of oranges increases and the quantity of oranges in the market decreases. Supply increases while demand decreases. Regarding demand and supply, which of the following statements is NOT correct? True False. The Law of Demand The process for determining the price of a good starts with the consumerâs (people that buy goods and services) demand for a good Which of the following is not a reason why the demand curve for new homes might shift in and to the left? The market for breakfast cereal is currently in equilibrium. A simultaneous decrease in the money supply and decrease in the price of oil is represented by a movement from A. d to b. A.) c. The population grows in a particular market area. Suppose that the supply and demand curves can be described by the following two equations: Q = 3P - 6 and Q = 120 - 3P. If a maximum interest rate for credit cards is set about the equilibrium interest rate, _____. In a market economy, supply and demand determine _____. What causes shifts in product supply and demand? b. Give two reasons why this might have happened. The demand curve shifts left. C. decrease in supply. In the context of the supply of labor to the economy, when the labor supply curve is upward sloping, is leisure a normal or inferior good? Incomes increase. Which of the following statements about the factors that influence demand is true? Consider the following economic event that could potentially help trigger a recession. Explain the reasons behind the shift and how that has influenced the equilibrium price. A group of people buying and selling goods or services. List three factors that could affect the demand for tablet computers. the growth rate of potential GDP A. iii only B. i and ii C. ii o... For each of the following situations, use demand and supply to show how the specified market will be affected. Supply and Demand3,4,20,21\Supply and Demand\Supply,demand, equilibrium test questions.docx ____ 12. Why or why not? How does it affect a market? B. Label the new curve S'. Practice your understanding of supply and demand with the help of our fun quiz. In recent years, the cost of producing wines in the U.S. has increased largely due to rising rents for vineyards. B. The supply of the product will decrease. d. MZ. How is this different from a movement along a curve? For example, suppose th... You normally produce 100 products per day, and you currently have the workers and supplies for this amount. B. An improvement in the technology used to produce apples. a. The upward slope of supply and downward slope of demand indicate: Select ALL that apply. "If firms adjusted their prices every day, then the short-run aggregate-supply curve would be horizontal.". What happens to the d... You are a producer of tortillas. Summertime vacation is approaching and the price of a barrel of oil increases. If restaurant inspectors begin enforcing begin enforcing health and safety statutes, this would led to _____. The tsunami in Japan that caused the nuclear meltdown at Fukushima also caused major disruptions in the automotive supply chain. Refer to the graph below. Last month, a tornado at your factory eliminated 50% of your firm's production capability. This is the major market driver and hence necessary to know about. Question: Market Equilibrium And Disequilibrium Part 1 - Check Your Understanding- T he Demand And Supply Schedules For Backpacks Are Given Below. A reduced desire for take-out and fast-food dining B. A. Sciences, Culinary Arts and Personal There are two common ways to discourage tobacco use: taxes on tobacco and information campaigns on the hazards of tobacco use. Assume a drought in the Great Plains reduces the supply of wheat. Explain whether the event will increase or decrease the demand for movie tickets. Tastes and preferences C. Number of buyers D. Technological improvements in production. Directions: Use your notes and your Supply and Demand Graph to answer the following questions. C. There is not enoug... 1. Show the equilibrium by clicking on the graph to place a dot at the place where the equilibrium point is. Which of the following statements is correct? Suppose the following events occur in a particular market. Suppose the supply and demand for a certain textbook are by supply: p = 1/5q^2, demand: p = -1/5q^2 + 20, where p is the price and q is the quantity. True or false? Which of the following would cause an increase in the supply of cheese? There are 1,000 owners of lemons and 1,000 owners of good used cars.... For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. How is the equilibrium price level affected? The figure above illustrates a set of supply and demand curves for a product. In a recession, demand for cars falls, and the demand curve in the market for cars: a) shifts to the right b) remains unchanged c) shifts to the left d) slows down. Consider a perfectly competitive, profit-maximizing firm facing the following marginal product of labor function and prices. If the manufacturer of the computer games lowers the price of the games what would happen to a supply and demand graph? Suppose a new Dunkin' Donuts shop has opened in town. Last month, a flood at her factory eliminated 50% of her firm's production capability. supply and demand test 2 sg 2020.pdf determinant of supply and demand practice.pdf (determinant of supply and demand practice.pdf) Price elasticity Presentation.pdf This year an improvement in technology has resulted in an increase in the supply of computers. If the price of salmon increases relative to the price of cod, the demand for: a) salmon will decrease. a. Which of the following scenarios would result in a decrease in the wage rate of solar panel installers and an increase in the quantity of solar panel installers employed in Billy's town? b. increase in the price of a substitute of the good. Between two unionized firms, one would expect workers to receive higher wages in a firm a. that has workers with low education levels. Explain the equilibrium change from an event that decreases both demand and supply. Describe what happens to the equilibrium price and quantity traded of soft drinks in response to each of the following. C. what would happen t... Give an example of a good or service for which there has been a shift in demand or supply. The real income effect helps to explain why the supply curve slopes up. Which 2 of the following would be considered to be normal goods? what will happen to the price and quantity of battery powered flashlights? The idea that economic downturns result from an inadequate aggregate demand for goods and services is derived from the work of which economist? (Note: Students want to think in terms of the future, not the present--now.) Think about a good or service for which you believe there has been a shift in demand or supply. B. A. The table below shows different quantities of labor supplied and labor demanded at different wage rates. True or false: If buyers expect the price of a good to fall in the near future, we would expect that to cause the current price and the quantity traded to increase as a result. What would happen to the supply of oil if speculators expect the price to rise in the future? Draw a graph to explain the following situation: A pest attack on the tomato crop increases the cost of producing ketchup. Transportation companies charged too little fo... After increasing for a long time, wages for low-skill workers suddenly started to decrease in the U.S. in the 1850s. Prompt Graph Eco Analysis 1 2 It becomes known that an electronics store is going to have a sale on their computer games 3 months from now. The table below gives changes that occur in the market for peanut butter. Source: Eduardo Porter, "The... What are the laws of supply and demand telling us about the variables they are measuring in a no control economy? Another 11 of your friends have already taken the course and are willing to consider selli... What would happen to equilibrium price and equilibrium quantity of khaki pants if the price of jeans (which are a perfect substitute for khakis) increases? a. How would you expect the following events to affect the market equilibrium price you receive for a bottle of orange juice? b. more is consumed, the marginal benefit of consumption decreases. A change in the costs of resources needed to produce the good B. Which of the following changes would not shift the supply curve for a good or service? d. Resource prices fall. Answer true or false and explain: Expected future wages increase will reduce the profit of the firms and thus shift the current labor demand curve to the left direction. Firms lower the price of chocolate. Suppose sales of a product depend directly on economic growth. The demand and supply curves can shift for a number of factors. The price of chocolate is raised in order to increase sales. Which of the following would cause the supply curve to shift to the right? A new literacy program that encourages reading, b. Supply and demand affects the amount of a commodity, product, or service available and the desire of buyers for it, considered as factors regulating its price. Assume that whole milk is a normal good. What are the factors that cause a supply curve to shift? a. Its main competitor, Doughnut Delite, will likely: a. Was the price increase you observed caused by an... A record corn crop means that farmers will sell more corn, but at a lower price. a. A.) What happens in the market for oranges? c. a change in expectations a... A decrease in the price of a good will a. decrease quantity supplied. The problem set is comprised of challenging questions that test your understanding of the material covered in the course. Which of the following could cause a change in demand in the market for antibacterial soap? 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The Black-Scholes partial differential equation is the partial differentiation equation: More generally, the term Black-Scholes partial differential equation can refer to other partial differential equations similar in form to equation (1), derived in similar ways under more general modelling assumptions than the most basic Black-Scholes model. 0.1 Derivation from martingale form where is a given pay-off function or terminal condition, assumed to be a random variable; is the filtration generated by the the standard Wiener process under the probability measure (the “risk-neutral measure”). From the martingale form of the result, there are two ways to derive the PDE form (1). 0.1.1 Using the Feynman-Kac formula One way is to appeal to the Feynman-Kac formula, which states in this case, that under certain regularity conditions11 The standard assumption is that is twice continuously-differentiable and . However, for the particular PDE we consider, these assumptions can be relaxed. on the function , the function defined by Now, equations (6) and (2) look quite similar, but with an important difference: the conditional expectation on equation (6) is with respect to the event , while equation (2) is conditioning on the filtration . We claim that these are actually the same thing — that is, for fixed , the random variable can be written as a function of — provided that the terminal condition is itself a function of : We will demonstrate this claim later. Assuming that the claim holds, we may thus set , and the Black-Scholes partial differential equation follows from the Feynman-Kac formula as explained earlier. 0.1.2 Proof of Markov property Formally, any -measurable stochastic process that ensures is always a measurable function of , for any Borel-measurable , is called a Markov process, or is said to have the Markov property. Thus we want to show that the stock process has the Markov property. For this, we will need the following explicit solution for it at time in terms of an initial condition at time : Since is -measurable, it may be treated as a constant while taking conditional expectations with respect to . And is a random variable independent of (and hence of ). Therefore the expression that appears on the right-hand side above is a function of , and would be unchanged if the conditioning is changed from to . 0.1.3 Using Itō’s formula By theorems on the uniqueness of solutions to stochastic differential equations, the coefficients of the correponding and terms in equations (4) and (7) must be equal almost surely. Equating the coefficients, we find: Equation (9) is essentially equation (1), except that has been replaced by in some places. However, because is a random variable that takes on all values on , and and its derivatives are assumed to be continuous, equation (9) must hold for arbitrary values substituted for . Hence we obtain equation (1). To verify the initial assumption that is twice continuously differentiable, we write formula (2) in a more explicit form (using the Markov property for as before): where is the transition density of the stock process from to over a time interval of length . In fact, from the solution for , the density is the density for the log-normal random variable whose logarithm has mean and variance . By differentiation under the integral sign, we see that must be twice continously differentiable for and . 0.2 Comparison of the martingale and PDE forms The martingale formulation of the result, in equation (2) is more general than the PDE formulation (1): the latter imposes extra regularity conditions, and more importantly, it can only describe prices that are functions only of the time and the current stock price . In the case of the call option ( for some ), this assumption is true; but there are other sorts of contingent claims whose pay-off depend on the history of the stock process — for example, a common kind of contingent claim has a final pay-off that depends on an average of the stock price over time . However, the PDE form of the solution is often more amenable to a numerical solution. While the expectation in equation (2) can be approximated numerically with Monte-Carlo simulation, for low-dimensional problems, solutions based on finite-difference approximations of PDEs are often quicker to compute than those based on Monte-Carlo. 0.3 Analytic solution Though the solution for the Black-Scholes PDE is already known using the martingale representation (2), we can also solve the PDE directly using classical analytical methods (http://planetmath.org/AnalyticSolutionOfBlackScholesPDE). Actually, the classical solution can be quite instructive in that it shows more or less the physical meaning behind the PDE, and also the behavior of solutions as the terminal condition is varied. (For example, the PDE is a transformation of a diffusion equation, and consequently its solutions are always infinitely smooth, as long as the terminal condition satisfies some local-integrability properties.) 0.4 Quantity of stock needed to replicate option Another important result that must not be left unmentioned is equation (8), derived during the course of showing the Black-Scholes PDE. In the article on the Black-Scholes pricing formula, a solution for was produced that satisfied (4), but the -adapted process that appears in that stochastic differential equation was obtained by appealing to an existence theorem. However, under the assumption that , equation (8) gives an actual computable formula: which is crucial in practice, for the theoretical “price” of a stock option would be useless if one cannot produce in reality that option for the stated price. (To be written.) |Date of creation||2013-03-22 16:31:08| |Last modified on||2013-03-22 16:31:08| |Last modified by||stevecheng (10074)| |Synonym||Black-Scholes partial differential equation|

Example- Magnetic field of a toroid As we have seen earlier, from the solenoid example, by sending the same amount of current through a helical system, we can generate a very strong magnetic field, which is directly proportional to the number of turns that we have. We can generate another magnetic field geometry. In this case, if we take the solenoid and connect its both ends together, therefore generating a system something like this. If we look at this configuration from the horizontal cross sectional point of view, in other words if we just slice it down, we’re going to end up with two branches. An inner branch, something like this, and the outer branch which is going to be something like this. As the current flows through one branch into the plane, then it is going to be coming out of plane through the other branch. In other words, let’s say if the current is coming out of plane through the inner branch, then it is going to be flowing into the plane through the outer branch like this. If we look at the magnetic field geometry generated by such a current flow, considering each one of these turns one by one, for the inner branch, current is coming out of plane and if we hold our right hand thumb in the direction of flow of current which is coming out of plane, then curling the right hand fingers about the thumb, we will see that for this current, the magnetic field lines are going to be in the form of concentric circles and circling in counterclockwise direction. Similarly the next one and then the next one, next one, and so on and so forth. They will be all circling in counterclockwise direction and so on and so forth. If we consider the outer branch, in this case holding the right hand thumb pointing into the plane, and curling the right hand fingers about the thumb, we will see that the associated field lines are going to be circling in clockwise direction. We can easily see that these turns are going to be very near to one another and as a result of this, the magnetic field generated by one of these turns will overlap with the next one. In other words, they will add to one another, therefore it will generate a magnetic field line going along this circular direction in counterclockwise direction. They’re known as toroidal field lines. A similar type of phenomenon will take place for the outer ones. The magnetic field lines will overlap, therefore they will generate a net magnetic field line something like this again circling in counterclockwise direction. If we look at over here, we see that the current is coming out through the inner branch and going into the plane over here and flowing through the back of the plane and coming out and then going into the plane again and again coming out of plane here, going into the plane there, and so on and so forth. This is how the current is flowing through this system. Well we call these current systems as toroids. Now we can easily see that if the current is flowing in, let’s say clockwise direction through the turns of this toroid, then it generates magnetic field lines along this region, what we call in toroidal directions, and the direction of the magnetic field lines for such a current which is going in clockwise direction is counterclockwise direction. In other words, if the current is flowing in clockwise direction through this toroid, the magnetic field lines that it will generate through the system will be in counterclockwise direction. Of course the magnetic field will be tangent to the field line passing through the point of interest. Now we will try to determine the magnetic field of a toroid. Okay. First, let’s give some dimensions to this toroid. Let’s say the inner radius is a and the outer radius is b. In other words, it is such that this radius is a and the outer radius is b. Let’s say that we’re trying to figure out the magnetic field at a specific point inside of this toroid. Let’s say somewhere over here at point p. If we consider the magnetic field line passing through point p will be a magnetic field line in the form of a circular field line and it is going to be in toroidal direction. It will be in counterclockwise direction. Magnetic field vector at point p will be tangent to this field line, therefore it is going to be something like this. We would like to calculate the magnitude of this field line. In order to do this, we’re going to apply Ampere’s law which is integral of b dot d l along a closed contour c is equal to Mu zero times i enclosed. We will choose a closed loop which will satisfy the conditions to apply Ampere’s law. In order to that, we will choose a loop in the form of a circle, which is coinciding with the field line passing through the point of interest. Now if we choose a closed hypothetical loop that coincides with the field line passing through the point of interest, then the magnitude of the magnetic field at every point along this loop will have the same magnitude. In other words, if we consider for example the magnetic field at this point, it will be tangent to that field line and here it will be tangent to the field line like this. All these magnetic field vectors will have the same magnitude because they are tangent to the same field line. Therefore the first condition is satisfied in order to apply Ampere’s law. In other words, the d magnitude along loop c, and this is the loop c, is constant. Furthermore, if we look at the angle between b and incremental displacement vector along this loop, we can easily see that that angle will always be zero degrees because d l is an incremental displacement vector along this loop c. Wherever we go the angle between b and d l will be zero degrees. Therefore we can say that the angle Theta is going to be zero degrees all of the time along c, therefore the second condition is also satisfied. Then if we write down the left hand side in explicit form, we will have b magnitude d l magnitude times cosine of the angle between them which is zero degrees integrated along this loop c will be equal to Mu zero times i enclosed. Well cosine of zero is just 1 and b is constant along this loop, so we can take it outside of the integral. Therefore we end up with b times integral over loop c of d l is equal to Mu zero times i enclosed. Integral of d l over loop c means all the incremental displacement vector magnitude d l’s, these distances, are added to one another along this loop c. If we do that, we’re going to end up with the length of that loop. In this case, it is the circumference of this loop c. Let’s say that our point of interest is r distance away relative to the center of the toroid, so the radius of this loop is little r. Then from here, this integral is going to give us the circumference of that circle which is 2 Pi r. So the left hand side will be b times 2 Pi r and that will be equal to Mu zero times i enclosed. Well the left hand side of the Ampere’s law is done. Now we will look at the right hand side. i enclosed is the net current passing through the region or surface surrounded by Amperian loop or the closed loop c. That region is this shaded region. So we will look at the net current passing through this surface. When we look at that region, we see that it encloses all the turns of this toroid. In other words, all the turns of this toroid are coming out of this surface. We know that each turn is carrying current i, therefore the total current passing through that surface will be equal to total number of turns of the toroid times current i. If we say that n represents the total number of turns of toroid, then i enclosed is going to be equal to, since all these turns are passing through the surface of interest, and each one of them carries current i, therefore i enclosed will be equal to n times i. From here, b becomes equal to Mu zero, for i enclosed we will have n times i divided by 2 Pi r. That is the magnitude of the magnetic field that a toroid will generate. If we look at over here, we see that the magnetic field will be proportional with 1 over r and r is the distance from the center of the toroid. It means that since b is proportional to 1 over r, the distance, then as the distance increases, the strength of magnetic field decreases. If we look at our diagram over here, it means that we’re gonna end up with a stronger magnetic field nearer to the inner wall relative to the magnetic field nearer to the outer wall. That can easily be understood directly from the geometry. Once we put the solenoid in this form by connecting both its ends, you can visualize this as a slinky connected from both ends, something like this. The number of turns in the inner region will be nearer to one another in comparing to the region of the outer part. In other words, we’re gonna end up with better overlapping of fields generated due to each turn over here along this inner wall in comparing to the outer wall. That will result with a stronger magnetic field nearer to the inner wall of the toroid in comparing to the regions which are nearer to the outer wall. When we go from inner wall to outer wall, we will see that the strength of the magnetic field generated by the current flowing through the toroid will decrease. If we look at the magnetic field outside of the toroid, which we can easily do that by placing our, let’s say, loop passing through a point which is located outside of the toroid, let’s say at this p prime. For such a region, once we place our Amperian loop, such that passing through that point and without even considering the left hand side of Ampere’s law, if we look at the right hand side, to be able to get the net current passing through the area surrounded now by this loop c 2, we can easily see that n i of current is coming out of the surface or the area surrounded by this loop, which is basically this surface here, and also the n times i of current is going into the plane. In that case that net current flowing through this surface will be equal to n i mines n i and then they will cancel and we will end up with zero. If we look at here, let’s say magnetic field b outside of the toroid is going to be such that from Ampere’s law, b dot d l, in this case we will use loop c 2, which is a circular loop passing through the point located outside of the toroid, will be equal to Mu zero times i enclosed and in this case, i enclosed will be equal to, if n i is coming out of plane, n i is going into the plane and that will give us the zero. As a result of this, b outside of a toroid will always be equal to zero. By the same token, we can look at the magnetic field in this region, in other words over here. So we choose our Amperian loop passing through a point located in this region, and then when we look at the i enclosed through that region, we will see that that will be equal to zero because none of the current flowing through the toroid will go through that region. Therefore here also magnetic field is zero. Therefore here again, the strength of the magnetic field is directly proportional with the number of turns. The greater the number of turns will result with the greater the magnetic field generated from the same current i, but as a major difference from the solenoid magnitude was is that that the magnetic field is not constant inside of the toroid. It will change from point to point whereas compared to the solenoid, the magnetic field strength was the same everywhere inside of the solenoid.

Ebscohost serves thousands of libraries with premium essays, articles and other content including statistics anxiety and mathematics anxiety: some interesting differences i get access to over 12 million other articles. Do you have math anxiety a self test rate your answers from 1 to 5 dents struggle with math anxiety, and that this issue has con- in this essay we will take a constructive look at math anxiety. Overcoming math anxiety [sheila tobias] on amazoncom free shipping on qualifying offers sheila tobias said it first: mathematics avoidance is not a failure of intellect, but a failure of nerve when this book was first published in 1978. Tobias, sheila papers of sheila tobias, 1947-2005: a finding aid in 1978 she wrote overcoming math anxiety, in which she addressed gendered attitudes towards math, such as the belief that boys are more skilled at math than girls, and. Anxiety disorder research paper by lauren thesis or dissertation on anxiety disorder topics at our professional custom essay writing service which science papers computer science papers technology papers biology papers geography papers physics papers chemistry papers mathematics papers. Math can cause a lot of anxiety to people however, the larc can help and offers 10 tips to help overcome math anxiety. Add keywords: mathematics anxiety rating scale (mars keywords: research, curriculum, and professional development in mathematics and mathematics education the mathematics teacher articles, teachers and computers, essay-writing in mathematics, conception of numbers. Subject: is it ld or just anxiety/phobia about writing please help hi everyone i need to make some behavioral changes in how i approach writing papers compared to math or science. Fear of mathematics - causes of fear of maths and how to overcome maths phobia. We bring together the mathematics anxiety and stereotype threat literatures by suggesting that these two phenomena share a common underlying mechanism research papers mathematics anxiety and stereotype threat: research in mathematics education published online: 22 feb 2012. Let's scrutinize the psychology of math phobia in our today's blog post what are the causes and what to do. Math anxiety dissertation writing service to assist in writing a doctorate math anxiety dissertation for a master's dissertation defense. Nature and structure of mathematics print reference this published: 23rd march 1998:344) causes of mathematics anxiety may be unrelated to events inside classrooms and can be triggered by a lack of support or understanding and in mathematics essay writing service essays more. How to deal effectively with math anxiety and math phobia. My philosophy of mathematics education jonathan lewin contents prologue 1 nothing wrong with a feeling of anxiety when the fear that generates it is well were preparing to distribute those question papers i promised myself that i would never inÀict this sort of thing upon my own. Math anxiety 1 running head: math anxiety math anxiety: causes, effects, and preventative measures megan r smith a senior thesis submitted in partial fulfillment. How to cope with math phobia mathematics has been the foundation of many of our scientific and technological discoveries however, to many children and adults, math is a 4-letter word that evokes emotions ranging from dislike and. Unlike the commonly recognized anxiety, math anxiety is not psychological, it is emotional anyone that has math anxiety experiences stress or discomfort with math what is anxiety essay 1019 words | 5 pages. The purpose is to provide students with math anxiety a set of guidelines and skills for dealing with stress related to math. Thursday, january 31, 2013 math anxiety. Math anxiety causes children to fear math this article offers parents and children tips to overcome the frustration and symptoms that are related to. Mathematics anxiety has been the subject of several books and numerous research papers this paper presents data taken from over 2500 mathematics test papers in order to compare the levels of accuracy and the frequency of the use of the no child development research is a peer. Free essay: anxiety is commonly defined as a fear that causes immense amounts of stress when thinking of anxiety one does not consider it associated with. How can parents and students understand and alleviate math anxiety it turns out tutoring and parental guidance can significantly impact a child's fear of math. This report consists of five professional papers: 1 math anxiety: real and complex 2 math anxiety and middle school students 3 math anxiety and college freshmen 4 math anxiety and elementary teachers 5 math anxiety: conclusions, discussions, and remedies. 100 weird phobias that really exist arithmophobia means to the fear of numbers generally, but can also refer to the fear of math, numerals, certain numbers this anxiety disorder means several things: the fear of magic and a magic wand. Effective teaching strategies for alleviating math anxiety and increasing self-efficacy in secondary students by alaina hellum-alexander a project submitted to the faculty of. So how can teachers help students overcome their fear of maths how can they instil a love of a subject that so many students (and adults) find intimidating the first step is to build confidence it's no surprise that confidence is a huge factor in students' anxiety towards mathematics. Get access to math anxiety essays only from anti essays listed results 1 - 30 get studying today and get the grades you want only at antiessayscom. Math anxiety is the feeling of nervousness and apprehension toward math problems, classes, or exams it generally begins when.

Title, Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. Booktitle, Advances in Cryptology – CRYPTO ’99, 19th Annual International. Download Citation on ResearchGate | Cryptanalysis of the HFE Public Key Finally, we develop a new relinearization method for solving such systems for any. Finally, we develop a new relinearization method for solving such systems for any constant ffl? Cryptanalysis of the HFE Public Key Cryptosystem (). |Country:||Saint Kitts and Nevis| |Published (Last):||9 December 2014| |PDF File Size:||15.94 Mb| |ePub File Size:||3.71 Mb| |Price:||Free* [*Free Regsitration Required]| So and satisfy the following equations derived from the bilinear equations, namely, where and all the coefficients in. So the adversary cannot derive from the publicly known map a low-rank matrix. Patarin developed other schemes. However, some simple variants of HFE, such as the minus variant and the vinegar variant allow one to strengthen the basic HFE against all known attacks. Performance and Comparisons To make a comparison between the proposed HFE modification and the original HFE schemes in a uniform platform, we hfs the HFE scheme defined over and its extension field. Then two invertible affine transformations are applied to hide the special structure of the central map [ 25 ]. The encryption scheme consists of three subalgorithms: Suggested Parameters Considering the aforementioned discussions, we suggest choosing and. Notations Let be a -order finite field with being a prime power. It is based on a ground and cryptosysrem extension field. It is shown that the proposed public key encryption scheme is secure against known attacks including the MinRank attack, the algebraic attacks, and the linearization equations attacks. This section does not cite any sources. Retrieved from ” https: Thus we have some additional equations that associate with the plaintext ; namely, forwe have. However, the original Cryptozystem scheme was insecure, and the follow-up modifications were shown to be still vulnerable to attacks. The HFE scheme firstly defines a univariate map over an extension field: We observe that the equation can be used to further destroy the special structure of the underlying central map of the HFE scheme. In the modified scheme, the public key isand hence we need not to store the coefficients of the square terms of the public key. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To illustrate why the proposed modification of the Cryotanalysis scheme is secure against the MinRank attack [ 78 ], we just need to show that when lifted to the extension fieldthe quadratic part of the public key is not connected with a low-rank matrix. Security and Communication Networks We represent the published system of multivariate polynomials by a single univariate polynomial of a special form over an extension field, and use it to reduce the cryptanalytic problem to a system of fflm 2 quadratic equations in m variables over the extension field. Signatures are generated using the private key and are verified using the public key as follows. In this paper, we proposed a novel modified HFE encryption scheme. Given the ciphertextwe want to solve the plaintext from the quadratic equations: Conflicts of Interest The authors declare that they have no conflicts of interest. El Din, and P. Without loss of generality, we assume that the two invertible affine transformations and are linear [ 21 ] and define the terms of in in 1. History of cryptography Cryptanalysis Outline of cryptography. Description The encryption scheme consists of three subalgorithms: Security and Communication Networks. Loosely speaking, when we apply two linear transformations on the input and output of the mapthe rank of the corresponding matrix remains at most. Firstly, we define an HFE map in 1 and randomly choose two invertible affine transformations and. As a new multivariate public key encryption, the security of the proposal needs to be furthered. In addition to HFE, J. In certain cases those polynomials could be defined over both a ground and an extension field. That is to say Or equivalently, The above equation says that we can lift the quadratic part of the public key to the extension field under some unknown linear transformations to derive and hence. Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography. If we fail to derive a vector in form all the preimageswe output the symbol designating an invalid ciphertext. Thus by solving the MinRank problem we can determine the matrix and the coefficients of the linear transformation. For a plaintextwe just compute as the ciphertext. Multivariate cryptography – Wikipedia These equations are called linearization equations and can be efficiently computed from the public polynomials. In the proposed modification HFE encryption scheme, we impose some restrictions on the plaintext space. Views Read Edit View history. Then we merge the coefficients of the square and linear terms ofthat is, forand get the public key of the modified HFE scheme, namely, quadratic polynomialswhere, forThe secret key consists of, cryotanalysis. Abstract The RSA public key cryptosystem is based on a single modular equation in one variable. Advanced Search Include Citations. We analyze the security of the proposed HFE modified encryption scheme. J-GLOBAL – Japan Science and Technology Agency In this paper we consider Patarin’s Hidden Field Equations HFE scheme, which is believed to be one of the strongest schemes of this type. The construction admits a standard isomorphism between the extension field and the vector space ; namely, for an elementwe have. The proposed method is a universal padding scheme and hence can be used to other multivariate cryptographic constructions. To make a comparison between the proposed HFE modification and the original HFE schemes in a uniform platform, we consider the HFE scheme defined over and relinearizztion extension field. However, all known modification methods only can impose partial nonlinear transformation on the special structure of the HFE central map, and hence they are still vulnerable to some attacks [ 15 — 17 ]. So both cryptanalyss have the same secret key sizes and decryption costs. Solving cryptosysfem of multivariate polynomial equations is proven to be NP-hard or NP-complete. Table of Contents Alerts.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest? Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice? Imagine we have four bags containing numbers from a sequence. What numbers can we make now? Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice? Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48. Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13. List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it? Can you explain the strategy for winning this game with any target? Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number? Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why? A collection of resources to support work on Factors and Multiples at Secondary level. Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice. Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . . Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn. 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? You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku. 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. The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse? Is there an efficient way to work out how many factors a large number has? A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6. Can you find any perfect numbers? Read this article to find out more... Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th Given the products of adjacent cells, can you complete this Sudoku? Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why? A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target. The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B? Given the products of diagonally opposite cells - can you complete this Sudoku? Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? This article for teachers describes how number arrays can be a useful reprentation for many number concepts. Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off? Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters. 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. 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? 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? Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings? Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears. Follow this recipe for sieving numbers and see what interesting patterns emerge. 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? 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? Got It game for an adult and child. How can you play so that you know you will always win? What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2? A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?" In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time? Have a go at balancing this equation. Can you find different ways of doing it? Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why? Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see? Can you work out some different ways to balance this equation?

Department of Applied Mathematics and Theoretical Physics, Xiamen University, Fujian, China Received: 09-Feb-2022, Manuscript No. JSMS-22-55238; Editor assigned: 11- Feb-2022, Pre QC No. JSMS-22-55238 (PQ); Reviewed: 25- Feb-2022, QC No. JSMS-22-55238; Accepted: 28-Feb-2022, Manuscript No. JSMS-22-55238 (A); Published: 07-Mar-2022, DOI: 10.4172/ J Stats Math Sci.8.2.e001. Visit for more related articles at Research & Reviews: Journal of Statistics and Mathematical Sciences Probability is a branch of mathematics concerned with numerical explanations of the possibility of an event occurring or the accuracy of a thesis. The probability of an occurrence is a number between 0 and 1, with 0 indicating impossibility and 1 indicating surety, general terms. The higher the probability of an occurrence, the more likely it is that the event will occur. The tossing of a balanced coin is a basic illustration. The two possibilities ("heads" and "tails") are equally likely since the coin is fair; the probability of "heads" equals the probability of "tails"; and because no other outcomes are possible, the probability of either "heads" or "tails" is half. These theories have been given an axiomatic mathematical formulation in probability theory, which is widely used in fields like statistics, mathematics, science, finance, betting, artificial intelligence, machine learning, computer science, game theory, and philosophy to draw assumptions about the expected distribution of data, for example. Probability theory is also used to understand the underlying mechanics and regularities of complex systems . Probabilities may be mathematically characterized as the number of desired outcomes divided by the total number of all possibilities when dealing with random and well-defined experiments in a purely theoretical environment (like flipping a coin). Tossing a coin twice, for example, will result in "head-head," "head-tail," "tail-head," and "tail-tail" results. The probability of having a "head-head" result is 1 out of every 4 outcomes, or 1/4, 0.25, or 25% in numerical terms. When it comes to practical application, however, there are two primary rival groups of probability interpretations, each of which adheres to a differing opinion of probability's fundamental nature [2-4]. Objectivists utilize numbers to express a physical or objective condition of events. The most widely accepted version of objective probability is frequents probability, which states that the probability of a random event represents the relative frequency of occurrence of an experiment's outcome when repeated indefinitely [5,6]. In this understanding, probability is defined as the relative frequency of outcomes "in the long run”. Propensity probability is a variation of this, in which probability is defined as the likelihood of an experiment yielding a specific result, even if it is only conducted once . Subjectivists give numerical values based on subjective likelihood, or a degree of belief. Although not commonly agreed upon, the degree of belief has been defined as "the price at which you would purchase or sell a bet that pays 1 unit of utility if E, 0 if not E". Bayesian probability is the most prevalent type of subjective probability, which uses expert knowledge as well as experimental evidence to generate probabilities . Some (subjective) prior probability distribution is used to represent expert knowledge. A probability function is used to include these data. When the prior and probability are added together and standardized the outcome is a posterior probability distribution that integrates all of the information available to date. The theory of probability, like other theories, is a formal representation of it8s concepts-that means, words that may be regarded independently of their significance. The laws of mathematics and logic are used to handle these formal concepts, and any outcomes are interpreted or translated back into the problem area. The Kolmogorov formulation and the Cox formulation are two examples of successful attempts to formalize probability. Sets are understood as events in Kolmogorov's formulation and probability is a measure on a class of sets. Probability is treated as a primitive and it is not further analysed in Cox's theorem, and the focus is on creating a consistent assignment of probability values to propositions. Except for technical differences, the rules of probability are the same in both circumstances . Other approaches for measuring uncertainty exist, such as the Dempster-Shafer theory or possibility theory, but they are fundamentally different and incompatible with generally accepted probability principles. Risk evaluation and modelling are examples of how probability theory is used in everyday life. Actuarial science is used by the insurance sector and markets to establish pricing and make trading choices. Probability techniques are used in environmental control, entitlement analysis, and financial regulation . The influence of the estimated probability of a broad Middle East conflict on oil prices, which has rippling effects across the economy, is an example of how probability theory is used in market trading. A commodities trader's estimate that a conflict is more likely can cause the price of that commodity to rise or fall, as well as alert other traders of that position. As a result, the probabilities are neither objectively analysed nor always logically calculated. Probability may be used to study patterns in biology (e.g., disease transmission) and ecology, in addition to financial evaluation (e.g., biological Punnett squares). Risk assessment, like economics, may be used as a statistical tool to calculate the possibility of adverse occurrences occurring and can help with the implementation of procedures to avoid them. Probability is used to develop games of chances . Some other important use of probability theory in daily life is reliability. Reliability theory is used in the design of many consumer items, such as vehicles and consumer electronics, to minimize the probability of failure. Failure probabilities may have an impact on a manufacturer's warranty decisions. The database language model, as well as other statistical language models used in natural language processing, are instances of probability theory applications .

Future value (FV) is the value of a current asset at a future date based on an assumed rate of growth. The future value is important to investors and financial planners, as they use it to estimate how much an investment made today will be worth in the future. The future value of an annuity refers to how much money you’ll get in the future based on the rate of return, or discount rate. An annuity’s value is the sum of money you’ll need to invest in the present to provide income payments down the road. Financial calculators (you can find them online) also have the ability to calculate these for you with the correct inputs. So, let’s assume that you invest $1,000 every year for the next five years, at 5% interest. These recurring or ongoing payments are technically referred to as «annuities» (not to be confused with the financial product called an annuity, though the two are related). When calculating future values, one component of the calculation is called the future value factor. What Is The Present Value Of An Annuity? Will your new balance be exactly double, more than double, or less than double? The formula for the future value of an ordinary annuity is indeed easier and faster than performing a series of future value calculations for each of the payments. At first glance, though, the formula is pretty complex, so the various parts of the formula are first explored in some detail before we put them all together. With simple interest, it is assumed that the interest rate is earned only on the initial investment. With compounded interest, the rate is applied to each period’s cumulative account balance. The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate. If your annuity promises you a $50,000 lump sum payment in the future, then the present value would be that $50,000 minus the proposed rate of return on your money. An annuity is a contract between https://intuit-payroll.org/how-to-attract-startups-for-accounting/ you and an insurance company that’s typically designed to provide retirement income. You buy an annuity either with a single payment or a series of payments, and you receive a lump-sum payout shortly after purchasing the annuity or a series of payouts over time. The graph below shows the timelines of the two types of annuity with their future values. Get 5 FREE Video Lessons With Uncommon Insights To Accelerate Your Financial Growth The fact that a renter or car owner makes payment on December 1 before enjoying the use of their apartment or vehicle during the rest of the month is what makes it annuity due. Annuity accounts grow without being taxed and annuity funds can be taken out without a penalty after age 59.5 years. As you may recall, the disbursements you’ll get later will be taxed as ordinary income. It’s okay if you are a little put off by all the annuity terms and rules. The one thing to remember is that money saved in an annuity now can be a steady stream of retirement income Accounting for In-Kind Donations to Nonprofits later. Once you do set up an annuity, you may be curious about its future worth. You’ll also have to take into consideration whether you have an ordinary annuity or an annuity due. What Is Future Value? You purchase the contract through either a lump sum payment or a series of payments and then receive monthly payments in retirement. There are both fixed and variable annuities, with different levels of risk and reward. If you wish to factor in the impact of inflation on annuity payments, you can use a separate inflation rate in your calculations. Keep in mind that doing so will likely increase the initial investment required to achieve the desired future income. An annuity payment is a fixed amount of money received periodically from an insurance company or investment firm in return for a lump-sum payment or series of contributions. The future value formula could be reversed to determine how much something in the future is worth today. In other words, assuming the same investment assumptions, $1,050 has the present value of $1,000 today. The concept of future value is often closely tied to the concept of present value. Why is it important to know the future value of annuity? By the same logic, a lump sum of $5,000 today is worth more than a series of five $1,000 annuity payments spread out over five years. The future value of annuity is used to measure the financial outcome of an investment over a specific time. The future value calculation considers the time value of money.The future value is the total cost of a series of cash installments and does not consider the time value of money. An example of future value of annuity would be if someone invested $1,000 today and received an annual payment of $100 for the next 10 years. The future value of this annuity would be $2,614.87 at the end of 10 years. This is calculated by multiplying the cash value ($100) by the number of payments (10) and then multiplying that result by the interest rate (10%). To understand the core concept, however, simple and compound interest rates are the most straightforward examples of the future value calculation. Since an annuity’s present value depends on how much money you expect to receive in the future, you should keep the time value of money in mind when calculating the present value of your annuity. When you plug the numbers into the above formula, you can calculate the future value of an annuity. Here’s an example that should hopefully make it clearer how the formula works and what you should plug in where. Sustainable Investing Topics They have multiple options which range from long-term investments to immediate payouts. However, the appeal of immediate or consistent payouts can blind individuals to the financial reality of their investment options. Thankfully, the future value of annuity formula provides a much simpler solution to finding this cash value. - However, we believe that understanding it is quite simple, even for a beginning in finance. - The payments in a typical annuity are distributed at the end of a pay period. - Therefore, Lewis is expected to have $69,770 in case of payment at month-end or $70,119 in case of payment at month start. - For the issuer, the total cost of making the annuity payments is the sum of the cash payments made to you plus the total reduction of income the issuer incurs as the payments are made. Using the same example of five $1,000 payments made over a period of five years, here is how a present value calculation would look. It shows that $4,329.58, invested at 5% interest, would be sufficient to produce those five $1,000 payments. You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas.

Necessary & Conditional Truth Given “x,y,z are natural numbers and x>y and y>z” the proposition “x>z” is true (I am assuming logical knowledge here, which I don’t discuss until Chapter 2). But it would be false in general to claim, “It is true that ‘x>z‘.” After all, it might be that “x = 17 and z = 32“; if so, “x>z” is false. Or it might be that “x = 17 and z = 17“, then again “x>z” is false. Or maybe “x = a boatload and z = a humongous amount”, then “x>z” is undefined or unknown unless there is tacit and complete knowledge of precisely how much is a boatload and how much is a humongous amount (which is doubtful). We cannot dismiss this last example, because a great portion of human discussions of uncertainty are pitched in this way. Included in the premise “x,y,z are natural numbers and x>y and y>z” are not just the raw information of the proposition about numbers, but the tacit knowledge we have of the symbol >, of what “natural numbers” are, and even what “and” and “are” mean. This is so for any argument which we wish to make. Language, in whatever form, must be used. There must therefore be an understanding of and about definitions, language and grammar, in any argument if any progress is to be made. These understandings may be more or less obvious depending on the argument. It is well to point out that many fallacies (and the best jokes) are founded on equivocation, which is the intentional or not misunderstanding double- or multiple-meanings of words or phrases. This must be kept in mind because we often talk about how the mathematical symbols of our formulae translate to real objects, how they matter to real-life decisions. A caution not heard frequently enough: just because a statement is mathematically true does not mean that the statement has any bearing on reality. Later we talk about how the deadly sin of reification occurs when this warning is ignored. We have an idea what it means to say of a proposition that it is true or false. This needs to be firmed up considerably. Take the proposition “a proposition cannot be both true and false simultaneously”. This proposition, as I said above, is true. That means, to our state of mind, there exists evidence which allows us to conclude this proposition is true. This evidence is in the form of thought, which is to say, other propositions, all of which include our understanding of the words and English grammar, and of phrases like “we cannot believe its contrary.” There are also present tacit (not formal) rules of logic about how we must treat and manipulate propositions. Each of these conditioning propositions or premises can in turn be true or false (i.e. known to be true or false) conditional on still other propositions, or on inductions drawn upon sense impressions and intellections. That is, we eventually must reach a point at which a proposition in front of us just is true. There is no other evidence for this kind of truth other than intellection. Observations and sense impressions will give partial support to most propositions, but they are never enough by themselves except for the direct impressions. I explore this later in the Chapter on Induction. In mathematics, logic, and philosophy popular kinds of propositions which are known to be true because induction tells us so are called axioms. Axioms are indubitable—when considered. Arguments for an axiom’s truth are made like this: given these specific instances, thus this general principle or axiom. I do not claim, and it is not true, that everybody knows every axiom. The arguments for axioms must first be considered before they are believed. A good example is the principal of non-contradiction, a proposition which we cannot know is false (though, given we are human, we can always claim it is false). As said, for every argument we need an understanding of its words and grammar, and, for non-contradiction specifically, maybe the plain observation of a necessarily finite number of instance of propositions that are only true or only false, observations which are consonant with the axiom, but which are none of them the full proof of the proposition: there comes a point at which we just believe and, indeed, cannot do other than know the truth. Another example is one of Peano’s axioms. For every natural number, if x = y then y = x. We check this through specific examples, and then move via induction to the knowledge that it is true for every number, even those we have not and, given our finiteness, cannot consider. Axioms are known to be true based on the evidence and faith that our intellects are correctly guiding us. This leads to the concept of the truly true, really true, just-plain true, universally, absolutely, or the necessarily true. These are propositions, like those in mathematics, that are known to be true given a valid and sound chain of argument which leads back to indubitable axioms. It is not possible to doubt axioms or necessary truths, unless there be a misunderstanding of the words or terms or chain of proof or argument involved (and this is, of course, possible, as any teacher will affirm). Necessary truths are true even if you don’t want them to be, even if they provoke discomfort, which (again of course) they sometimes do. Peter Kreeft said: “As Aristotle showed, [all] ‘backward doubt’ terminates in two places: psychologically indubitable immediate sense experience and logically indubitable first principles such as ‘X is not non-X’ in theoretical thinking and ‘Good is to be done and evil to be avoided’ in practical thinking”. A man in the street might look at the scratchings of a mathematical truth and doubt the theorem, but this is only because he doesn’t comprehend what all those strange symbols mean. He may even say that he “knows” the theorem is false—think of the brave soul who claims to have squared the circle. It must be stressed that this man’s error arises from his not comprehending the whole of the argument. Which of the premises of the theorem he is rejecting, and this includes tacit premises of logic and other mathematical results, is not known to us (unless the man makes this clear). The point is that if it were made plain to him what every step in the argument was, he must consent. If he does not, he has not comprehended at least one thing or he has rejected at least one premise, or perhaps substituted his own unaware. This is no small point, and the failure to appreciate it has given rise to the mistaken subjective theory of probability. Understanding the whole of an argument is a requirement to our admitting a necessary truth (our understanding is obviously not required of the necessary truth itself!). From this it follows that when a mathematician or physicist says something akin to, “We now know Flippenberger’s theorem is true”, his “we” does not, it most certainly does not, encompass all of humanity; it applies only to those who can and have followed the line of reason which appears in the proof. That another mathematician or physicist (or man in the street) who hears this statement, but whose specialty is not Flippenbergerology, conditional on trusting the first mathematician’s word, also believes Flippenberger’s theorem is true, is not making (to himself) the same argument as the theory’s proponent. He instead makes a conditional truth statement: to him, Flippenberger’s theorem is conditionally true, given the premise of accepting the word of the first mathematician or physicist. Of course, necessary truths are also conditional as I have just described, so the phrase “conditional truth” is imperfect, but I have not been able to discover one better to my satisfaction. Local or relative truth have their merits, but their use could encourage relativists to believe they have a point, which they do not. Besides mathematical propositions, there are plenty other of necessary truths that we know. “I exist” is popular, and only claimed to be doubted by the insane or (paradoxically) by attention seekers. “God exists” is another: those who doubt it are like circle-squarers who have misunderstood or have not (yet) comprehended the arguments which lead to this proposition. “There are true propositions” always delights and which also has its doubters who claim it is true that it is false. In Chapter 2 we meet more. There are an infinite number and an enormous variety of conditional truths that we do and can know. I don’t mean to say that there are not an infinite number of necessary truths, because I have no idea, though I believe it; I mean only that conditional truths form a vaster class of truths in everyday and scientific discourse. We met one conditional truth above in “x>z“. Another is, given “All Martians wear hats and George is a Martian” then it is conditionally true that “George wears a hat.” The difference in how we express this “truth is conditional” is plain enough in cases like hat-wearing Martians. Nobody would say, in a general setting, “It’s true that Martians wear hats.” Or if he did, nobody would believe him. This disbelief would be deduced conditional on the observationally true proposition, “There are no Martians”. We sometimes hear people claim conditional truths are necessary truths, especially in moral or political contexts. A man might say, “College professors are intolerant of dissent” and believe he is stating a necessary truth. Yet this cannot be a necessary truth, because no sound valid chain of argument anchored to axioms can support it. But it may be an extrapolation from “All the many college professors I have observed have been intolerant of dissent”, in which case the proposition is still not a necessary truth, because (as we’ll see) observational statements like this are fallible. Hint: The man’s audience, if it be typical, might not believe the “All” in the argument means all, but only “many”. But that substitution does not make the proposition “Many college professors are intolerant of dissent” necessarily true, either. Another interesting possibility is in the proposition “Some college professors are intolerant of dissent,” where some is defined as at least one and potentially all. Now if a man hears that and recalls, “I have met X, who is a college professor, and she was intolerant of dissent”, then conditional on that evidence the proposition of interest is conditionally true. Why isn’t it necessarily true? Understand first that the proposition is true for you, too, dear reader, if we take as evidence “I have met X, etc.” Just as “George wears a hat” was conditionally true on the other explicit evidence. It may be that you yourself have not met X, nor any other intolerant-of-dissent professor, but that means nothing for the epistemological status of these two propositions. But it now becomes obvious why the proposition of interest is not necessarily true: because the supporting evidence “I have met X, etc.” cannot be held up as necessarily true itself: there is no chain of sound argument leading to indubitable axioms which guarantees it is a logically necessity that college professors must be intolerant of dissent. (Even if it sometimes seems that way.) We only have to be careful because when people speak or write of truths they are usually not careful to tell us whether they have in mind a necessary or only a conditional truth. Much grief is caused because of this. One point which may not be obvious. A necessary truth is just true. It is not true because we have a proof of it’s truth. Any necessary truth is true because of something, but it makes no sense to ask why this is so for any necessary truth. Why is the principle of non-contradiction true? What is it that makes it true? Answer: we do not know. It is just is true. How do we know it is true? Via a proof, by strings of deductions from accepted premises and using induction, the same way we know if any proposition is true. We must ever keep separate the epistemological from the ontological. There is a constant danger of mistaking the two. Logic and probability are epistemological, and only sometimes speak or aim at the ontological. Probability is always a state of the mind and not a state of the universe.

I love all kinds of music. However, while I work I prefer a quiet music. People have different names for it: chill, chillout, lounge, smooth jazz etc. I am not good at those names, I just want something that allows me to concentrate when I want and also is a pleasure to listen when I make it louder. There is plenty of such music on Youtube. So I tune in one of online radios, for example this is good: When I like a melody, I select its name, right-click and then click "Search Google". Most of the time it lands me on the corresponding Youtube video. Here comes the magic: there is a nice FREE program called 4K Video Downloader. You can download the whole video or you can extract just the audio. This is how my chillout collection reached 1000. They have other good products too. Last semester I tried to explain theory through numerical examples. The results were terrible. Even the best students didn't stand up to my expectations. The midterm grades were so low that I did something I had never done before: I allowed my students to write an analysis of the midterm at home. Those who were able to verbally articulate the answers to me received a bonus that allowed them to pass the semester. This semester I made a U-turn. I announced that in the first half of the semester we will concentrate on theory and we followed this methodology. Out of 35 students, 20 significantly improved their performance and 15 remained where they were. a. Define the density of a random variable Draw the density of heights of adults, making simplifying assumptions if necessary. Don't forget to label the axes. b. According to your plot, how much is the integral Explain. c. Why the density cannot be negative? d. Why the total area under the density curve should be 1? e. Where are basketball players on your graph? Write down the corresponding expression for probability. f. Where are dwarfs on your graph? Write down the corresponding expression for probability. This question is about the interval formula. In each case students have to write the equation for the probability and the corresponding integral of the density. At this level, I don't talk about the distribution function and introduce the density by the interval formula. Recently I enjoyed reading Jack Weatherford's "Genghis Khan and the Making of the Modern World" (2004). I was reading the book with a specific question in mind: what were the main reasons of the success of the Mongols? Here you can see the list of their innovations, some of which were in fact adapted from the nations they subjugated. But what was the main driving force behind those innovations? The conclusion I came to is that Genghis Khan was a genial psychologist. He used what he knew about individual and social psychology to constantly improve the government of his empire. I am no Genghis Khan but I try to base my teaching methods on my knowledge of student psychology. Problem 1. Students mechanically write down what the teacher says and writes. Solution. I don't allow my students to write while I am explaining the material. When I explain, their task is to listen and try to understand. I invite them to ask questions and prompt me to write more explanations and comments. After they all say "We understand", I clean the board and then they write down whatever they understood and remembered. Problem 2. Students are not used to analyze what they read or write. Solution. After students finish their writing, I ask them to exchange notebooks and check each other's writings. It's easier for them to do this while everything is fresh in their memory. I bought and distributed red pens. When they see that something is missing or wrong, they have to write in red. Errors or omissions must stand out. Thus, right there in the class students repeat the material twice. Problem 3. Students don't study at home. Solution. I let my students know in advance what the next quiz will be about. Even with this knowledge, most of them don't prepare at home. Before the quiz I give them about half an hour to repeat and discuss the material (this is at least the third repetition). We start the quiz when they say they are ready. Problem 4. Students don't understand that active repetition (writing without looking at one's notes) is much more productive than passive repetition (just reading the notes). Solution. Each time before discussion sessions I distribute scratch paper and urge students to write, not just read or talk. About half of them follow my recommendation. Their desire to keep their notebooks neat is not their last consideration. The solution to Problem 1 also hinges upon active repetition. Problem 5. If students work and are evaluated individually, usually there is no or little interaction between them. Problem 6. Some students don't want to work in teams. They are usually either good students, who don't want to suffer because of weak team members, or weak students, who don't want their low grades to harm other team members. Solution. The good students usually argue that it's not fair if their grade becomes lower because of somebody else's fault. My answer to them is that the meaning of fairness depends on the definition. In my grading scheme, 30 points out of 100 is allocated for team work and the rest for individual achievements. Therefore I never allow good students to work individually. I want them to be my teaching assistants and help other students. While doing so, I tell them that I may reward good students with a bonus in the end of the semester. In some cases I allow weak students to write quizzes individually but only if the team so requests. The request of the weak student doesn't matter. The weak student still has to participate in team discussions. Problem 7. There is no accumulation of theoretical knowledge (flat learning curve). Solution. a) Most students come from high school with little experience in algebra. I raise the level gradually and emphasize understanding. Students never see multiple choice questions in my classes. They also know that right answers without explanations will be discarded. b) Normally, during my explanations I fill out the board. The amount of the information the students have to remember is substantial and increases over time. If you know a better way to develop one's internal vision, let me know. c) I don't believe in learning the theory by doing applied exercises. After explaining the theory I formulate it as a series of theoretical exercises. I give the theory in large, logically consistent blocks for students to see the system. Half of exam questions are theoretical (students have to provide proofs and derivations) and the other half - applied. d) The right motivation can be of two types: theoretical or applied, and I never substitute one for another. Problem 8. In low-level courses you need to conduct frequent evaluations to keep your students in working shape. Multiply that by the number of students, and you get a serious teaching overload. Solution. Once at a teaching conference in Prague my colleague from New York boasted that he grades 160 papers per week. Evaluating one paper per team saves you from that hell. In the beginning of the academic year I had 47 students. In the second semester 12 students dropped the course entirely or enrolled in Stats classes taught by other teachers. Based on current grades, I expect 15 more students to fail. Thus, after the first year I'll have about 20 students in my course (if they don't fail other courses). These students will master statistics at the level of my book. This year I am teaching AP Statistics. If the things continue the way they are, about half of the class will fail. Here is my diagnosis and how I am handling the problem. On the surface, the students lack algebra training but I think the problem is deeper: many of them have underdeveloped cognitive abilities. Their perception is slow, memory is limited, analytical abilities are rudimentary and they are not used to work at home. Limited resources require careful allocation. Short and intuitive names are better than two-word professional names. Instead of "sample space" or "probability space" say "universe". The universe is the widest possible event, and nothing exists outside it. Instead of "elementary event" say "atom". Simplest possible events are called atoms. This corresponds to the theoretical notion of an atom in measure theory (an atom is a measurable set which has positive measure and contains no set of smaller positive measure). Then the formulation of classical probability becomes short. Let denote the number of atoms in the universe and let be the number of atoms in event If all atoms are equally likely (have equal probabilities), then The clumsy "mutually exclusive events" are better replaced by more visual "disjoint sets". Likewise, instead of "collectively exhaustive events" say "events that cover the universe". The combination "mutually exclusive" and "collectively exhaustive" events is beyond comprehension for many. I say: if events are disjoint and cover the universe, we call them tiles. To support this definition, play onscreen one of jigsaw puzzles (Video 1) and produce the picture from Figure 1. Figure 1. Tiles (disjoint events that cover the universe) The philosophy of team work We are in the same boat. I mean the big boat. Not the class. Not the university. It's the whole country. We depend on each other. Failure of one may jeopardize the well-being of everybody else. You work in teams. You help each other to learn. My lectures and your presentations are just the beginning of the journey of knowledge into your heads. I cannot control how it settles there. Be my teaching assistants, share your big and little discoveries with your classmates. I don't just preach about you helping each other. I force you to work in teams. 30% of the final grade is allocated to team work. Team work means joint responsibility. You work on assignments together. I randomly select a team member for reporting. His or her grade is what each team member gets. This kind of team work is incompatible with the Western obsession with grades privacy. If I say my grade is nobody's business, by extension I consider the level of my knowledge a private issue. This will prevent me from asking for help and admitting my errors. The situation when students hide their errors and weaknesses from others also goes against the ethics of many workplaces. In my class all grades are public knowledge. In some situations, keeping the grade private is technically impossible. Conducting a competition without announcing the points won is impossible. If I catch a student cheating, I announce the failing grade immediately, as a warning to others. To those of you who think team-based learning is unfair to better students I repeat: 30% of the final grade is given for team work, not for personal achievements. The other 70% is where you can shine personally. Breaking the wall of silence Team work serves several purposes. Firstly, joint responsibility helps breaking communication barriers. See in Video 2 my students working in teams on classroom assignments. The situation when a weaker student is too proud to ask for help and a stronger student doesn't want to offend by offering help is not acceptable. One can ask for help or offer help without losing respect for each other. Video 2. Teams working on assignments Secondly, it turns on resources that are otherwise idle. Explaining something to somebody is the best way to improve your own understanding. The better students master a kind of leadership that is especially valuable in a modern society. For the weaker students, feeling responsible for a team improves motivation. Thirdly, I save time by having to grade less student papers. On exams and quizzes I mercilessly punish the students for Yes/No answers without explanations. There are no half-points for half-understanding. This, in combination with the team work and open grades policy allows me to achieve my main objective: students are eager to talk to me about their problems. Set operations and probability After studying the basics of set operations and probabilities we had a midterm exam. It revealed that about one-third of students didn't understand this material and some of that misunderstanding came from high school. During the review session I wanted to see if they were ready for a frank discussion and told them: "Those who don't understand probabilities, please raise your hands", and about one-third raised their hands. I invited two of them to work at the board. Video 3. Translating verbal statements to sets, with accompanying probabilities Many teachers think that the Venn diagrams explain everything about sets because they are visual. No, for some students they are not visual enough. That's why I prepared a simple teaching aid (see Video 3) and explained the task to the two students as follows: I am shooting at the target. The target is a square with two circles on it, one red and the other blue. The target is the universe (the bullet cannot hit points outside it). The probability of a set is its area. I am going to tell you one statement after another. You write that statement in the first column of the table. In the second column write the mathematical expression for the set. In the third column write the probability of that set, together with any accompanying formulas that you can come up with. The formulas should reflect the relationships between relevant areas. Table 1. Set operations and probabilities 1. The bullet hit the universe 2. The bullet didn't hit the universe 3. The bullet hit the red circle 4. The bullet didn't hit the red circle 5. The bullet hit both the red and blue circles (in general, this is not equal to ) 6. The bullet hit or (or both) 7. The bullet hit but not 8. The bullet hit but not 9. The bullet hit either or (but not both) During the process, I was illustrating everything on my teaching aid. This exercise allows the students to relate verbal statements to sets and further to their areas. The main point is that people need to see the logic, and that logic should be repeated several times through similar exercises. Law of total probability - you could have invented this A knight wants to kill (event ) a dragon. There are two ways to do this: by fighting (event ) the dragon or by outwitting () it. The choice of the way ( or ) is random, and in each case the outcome ( or not ) is also random. For the probability of killing there is a simple, intuitive formula: This is easy to generalize to the case of many conditioning events. Suppose are mutually exclusive (that is, disjoint) and collectively exhaustive (that is, cover the whole sample space). Then for any event one has This equation is call the law of total probability. Application to a sum of continuous and discrete random variables Let be independent random variables. Suppose that is continuous, with a distribution function , and suppose is discrete, with values . Then for the distribution function of the sum we have The relativity theory says that what initially looks absolutely difficult, on closer examination turns out to be relatively simple. Here is one such topic. We start with a motivating example. Large cloud service providers have huge data centers. A data center, being a large group of computer servers, typically requires extensive air conditioning. The intensity and cost of air conditioning depend on the temperature of the surrounding environment. If, as in our motivating example, we denote by the temperature outside and by a cut-off value, then a cloud service provider is interested in knowing the probability for different values of . This is exactly the distribution function of temperature: . So how do you estimate it? It comes down to usual sampling. Fix some cut-off, for example, and see for how many days in a year the temperature does not exceed 20. If the number of such days is, say, 200, then 200/365 will be the estimate of the probability . It remains to dress this idea in mathematical clothes. Empirical distribution function If an observation belongs to the event , we count it as 1, otherwise we count it as zero. That is, we are dealing with a dummy variable The total count is and this is divided by the total number of observations, which is 365, to get 200/365. It is important to realize that the variable in (1) is a coin (Bernoulli variable). For an unfair coin with probability of 1 equal to and probability of zero equal to the mean is and the variance is For the variable in (1) , so the mean and variance are Generalizing, the probability is estimated by where is the number of observations. (3) is called an empirical distribution function because it is a direct empirical analog of . Applying expectation to (3) and using an equation similar to (2), we prove unbiasedness of our estimator: Further, assuming independent observations we can find variance of (3): Intro to option greeks: delta and its determinants I started trading stocks in 2010. I didn't expect to make big profits and wasn't actively trading. That's until 2015, when I met a guy who turned $10,000 into $140,000 in four years. And then I thought: why am I fooling around when it's possible to make good money? Experienced traders say: trading is a journey. That's how my journey started. Stocks move too slowly, to my taste, so I had to look for other avenues. Two things were clear to me. I didn't want to be glued to the monitor the whole day and didn't want to study a lot of theory. Therefore I decided to concentrate on the futures market. To trade futures, you don't even need to know the definition of a futures contract. The price moves very quickly, and if you know what you are doing, you can make a couple of hundreds in a matter of minutes. It turned out that the futures markets are the best approximation to the efficient market hypothesis. Trend is your friend (until the end), as they say. In the futures markets, trends are rare and short-lived. Trading futures is like driving a race car. The psychological stress is enormous and it may excite your worst instincts. After trying for almost two years and losing $8,000 I gave up. Don't trade futures unless you can predict a big move. Many people start their trading careers in the forex market because the volumes there are large and transaction fees are low. I never traded forex and think that it is as risky as the futures market. If you want to try it, I would suggest to trade not the exchange rates themselves but indexes or ETF's (exchange traded funds) that trace them. Again, look for large movements. One more market I don't want to trade is bonds. Actions of central banks and macroeconomic events are among strong movers of this market. Otherwise, it's the same as futures. Futures, forex and bonds have one feature in common. In all of them institutional (large) traders dominate. My impression is that in absence of market-moving events they select a range within which to trade. Having deep pockets, they can buy at the top of the range and sell at the bottom without worrying about the associated loss. Trading in a range like that will kill a retail (small) investor. Changes in fundamentals force the big guys to shift the range, and that's when small investors have a chance to profit. I tried to avoid options because they require learning some theory. After a prolonged resistance, I started trading options and immediately fell in love with them. I think that anybody with $25,000 in savings can and should be trading options. Definition. In Math, the Greek letter (delta) is usually used to denote change or rate of change. In case of options, it's the rate of change of the option price when the stock price changes. Mathematically, it's a derivative where is the call price and is the stock price. In layman's terms, when the stock price changes by $1, the call price changes (moves in the same direction) by dollars. The basic features of delta can be understood by looking at how it depends on the strike price, when time is fixed, and how it changes with time, when the strike price is fixed. As before, we concentrate on probabilistic intuition. How delta depends on strike price Figure 1. AAPL option chain with 26 days to expiration Look at the option chain in Figure 1. For the strikes that are deep in the money, delta is close to one. This is because if a call option is deep in the money, the probability that it will end up in the money by expiration is high (see how the call price depends on the strike price). Hence, stock price changes are followed by call price changes almost one to one. On the other hand, if a strike is far out of the money, it is likely to remain out of the money by expiration. The stock price changes have little effect on the call price. That's why delta is close to zero. How delta depends on time to expiration Figure 2. AAPL option chain with 5 days to expiration Now let us compare that option chain to the one with a shorter time to expiration (see Figure 2). If an option is to expire soon, the probability of a drastic stock movement before expiration is low, see the comparison of areas of influence with different times to expiration. Only a few options with strikes lower than at the money strike have deltas different from one. The deeper in the money calls have deltas equal to one: their prices exactly repeat the stock price. Similarly, only a few out of the money options have deltas different from zero. If the strike is very far out of the money, the call delta is 0 because the call is very likely to expire worthless and its dependence on the stock price is negligible. Interest rate - the puppetmaster behind option prices Figure 1. Call as a function of interest rate The interest rate is the last variable we need to discuss. The dependence of the call price on the interest rate that emerges from the Black-Scholes formula is depicted in Figure 1. The dependence is positive, right? Not so fast. This is the case when common sense should be used instead of mathematical models. One economic factor can influence another through many channels, often leading to contradicting results. As interest rates in the economy increase, the expected return required by investors from the stock tends to increase. This suggests a positive dependence of the call price on the interest rate. On the other hand, when the interest rate rises, the present value of any future cash flow received by the long call holder decreases. In particular, this reduces the payoff if at expiration the option is in the money. The combined impact of these two effects embedded in the Black-Scholes formula is to increase the value of the call options. However, experience tells the opposite The Fed changes the interest rate at discrete times, not continually. Two moments matter: when the rumor about the upcoming interest rate change hits the market and when the actual change takes place. The market reaction to the rumor depends on the investors' mood - bullish or bearish. A bullish market tends to shrug off most bad news. In a bearish market, even a slight threat may have drastic consequences. By the time the actual change occurs, it is usually priced in. In 2017, the Fed raised the rate three times: on March 15 (no reaction, judging by S&P 500 SPDR SPY), June 14 (again no reaction) and December 13 (a slight fall). The huge fall in the beginning of February 2018 was not caused by any actual change. It was an accumulated result of cautiousness ("This market has been bullish for too long!") and fears that the Fed would increase the rates in 2018 by more than had been anticipated. Many investors started selling stocks and buying bonds and other less risky assets. The total value of US bonds is about $20 trillion, while the total market capitalization of US stocks is around $30 trillion. The two markets are comparable in size, which means there is enough room to move from one to another and the total portfolio reshuffling can be considerable. Thus far, the mere expectation that the interest rate will increase has been able to substantially reduce stock prices and, consequently, call prices. All this I summarized to my students as follows. When interest rates rise, bonds become more attractive. This is a substitution effect: investors switch from one asset to another all the time. Therefore stock prices and call prices fall. Thus the dependence of call prices on interest rates is negative. The first explanation suggested by Hull neglects the substitution effect. The second explanation is not credible either, for the following reason. As I explained, stock volatility has a very strong influence on options. Options themselves have an even higher volatility. A change in interest rates by a couple percent is nothing in comparison with this volatility. Most investors would not care about the resulting reduction in the present value of future cash flows. Main idea: look at geometry in the two-dimensional case Here is an example. The norm of a vector is defined by The combination of squaring and extracting a square root often makes it difficult to understand how this construction works. Here is a simple inequality that allows one to do without this norm (or, put it differently, replace it with another norm). Take We have proved that This easily generalizes to : Application. The set is called bounded if there is a constant such that for all (1) implies an equivalent definition: the set is called bounded if there is a constant such that for all See p.35 of Baltovic's guide, where the inequality is sufficient for proving boundedness of the set Theorem 2.2 (The Cauchy-Schwarz Inequality). This inequality does not have serious applications in the guide. For a nontrivial application of the Cauchy-Schwarz inequality see my post. 2.1.8. Avoid using the definition of continuity in terms of (Definition 2.18). Use Definition 2.19 in terms of sequences instead. 2.6.2. Definition 2.21 for many students is indigestible. Just say that the matrix consists of partial derivatives of components of Theorem 2.11. The proof is really simple in the one-dimensional case. By the definition of the derivative, for any sequence Multiplying this equation by we get , which proves continuity of at 3.3.1. There is Math that happens on paper (formulas) and the one that happens in the head (logic). Many students see the formulas and miss the logic. Carefully read this section and see if the logic happens in your head. 3.4. The solution to Example 3.2 is overblown. A professional mathematician never thinks like that. A pro would explain the idea as follows: because of Condition 2, the function is close to zero in some neighborhood of infinity . Therefore, a maximum should be looked for in the set . Since this is a compact, the Weierstrass theorem applies. With a proper graphical illustration, the students don't need anything else. 4.2 First-order conditions for optima. See the proof. 4.4 Second-order conditions for optima. See explanation using the Taylor decomposition. Example 6.4. In solutions that rely on the Kuhn-Tucker theorem, the author suggests to check the constraint qualification condition for all possible combinations of constraints. Not only is this time consuming, but this is also misleading, given the fact that often it is possible to determine the binding constraints and use the Lagrange method instead of the Kuhn-Tucker theorem or, alternatively, to use the Kuhn-Tucker theorem for eliminating simple cases. The same problem can be solved using the convexity theory. Example 6.5. In this case Baltovic makes a controversial experiment: what happens if we go the wrong way (expectedly, bad things happen), without providing the correct solution. Solution to Exercise 6.1. In this exercise, the revenue is homogeneous of degree 2 and the cost is homogeneous of degree 1, which indicates that the profit is infinite. No need to do a three-page analysis! 7.6 The Bellman equations. There are many optimization methods not covered in Sundaram's book. One of them, Pontryagin's maximum principle, is more general that the Bellman approach. p. 172. The bound is obvious and does not require the Cauchy-Schwarz inequality.

Some general properties of a fractional Sumudu transform in the class of Boehmians Keywords:Boehmian, fractional Sumudu transform, fractional derivative, Mittag- Leffler function, Sumudu transform. AbstractIn literature, there are several works on the theory and applications of integraltransforms of Boehmian spaces, but fractional integral transforms of Boehmians havenot yet been reported. In this paper, we investigate a fractional Sumudu transform ofan arbitrary order on some space of integrable Boehmians. The fractional Sumudutransform of an integrable Boehmian is well-defined, linear and sequentially completein the space of continuous functions. Two types of convergence are also discussed indetails. Alieva, T., Bastiaans, M.J. & Calvo, M.L. (2005) Fractional transforms in optical information processing, EURASIP. Journal on Applied Signal Processing, 10:1498-1519. Asiru, M.A. (2002) Further properties of the Sumudu transform and its applications. 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(2001) The fractional Fourier transform with applications in optics and signal processing. John Wiley and Sons. Ltd. Pei, S.C. & Ding, J.J. (2002) Fractional cosine, sine, and Hartley transforms, IEEE Trans. Signal Rajendran, S.M. & Roopkumar, R. (2014) Curvelet transform for Boehmians. Arab Journal of Mathematical Sciences, 20(2):264-279. Torre, A. (2003) Linear and radial canonical transforms of fractional order. Journal of Computational and Applied Mathematics, 153(1-2):477-486. Weerakoon, S. (1998) Complex inversion formula for Sumudu transforms. International Journal of Mathematical Education in Science and Technology, 29(4):618-621. Watugala, G.K. (1993) Sumudu transform: a New integral transform to solve differential equations and control engineering problems. International Journal of Mathematical Education in Science and Weerakoon, S. (1994) Application of Sumudu transform to partial differential equations. International Journal of Mathematical Education in Science and Technology, 25:277-283. Watugala, G.K. (1998) Sumudu transform new integral transform to solve differential equations and control engineering problems. Mathematical Engineering in Industry, 6(4):319-329. Watugala, G.K. (2002) The Sumudu transform for functions of two variables. Mathematical Engineering in Industry, 8(4):293-302. Yu, L., Lu, Y., Zeng, X., Huang, M. & Chen, M. et al. (1998) Deriving the integral representation of a fractional Hankel transform froma fractional Fourier transform. Optics Letters, 23(15):1158-1160. Zhang, J. (2007) A Sumudu based algorithm for solving di_erential equations. Computer Science Journal of Moldova, 15(45):303-313.

Postgraduate CourseSample vs populationWe want to know about these(population: N)We have to work with these(sample: n)population mean: μselectionsample mean: X_fit? Postgraduate CourseLaw of large numbersThe larger the sample size (or the number ofobservations), the more accurate the predictions of thecharacteristics of the whole population, and smallerthe expected deviation in comparisons of outcomes.As a general principle it means that, in the long run,the average (mean) of a large number of observationswill be close to (or: may be taken as the best estimateof) the true mean’ of the population.Sample vs population Postgraduate CourseSample size: why does it matter? Law of the large numbers: a reliable and accuraterepresentation of the population Statistical power: to prevent a type 2 error / falsenegativeSample vs population Don‟t confuse: representativeness and reliabilityThe sample size has no direct relationship withrepresentativeness; even a large random sample can beinsufficiently representative.Postgraduate CourseSample vs population Postgraduate CourseVariablesPostgraduate CourseVariable: anything that can be measured and candiffer across entities or timeIndependent variable: predictor variable (value doesnot depend on any other variables)Dependent variable: outcome variable (valuedepends on other variables) Postgraduate Course3. Level of measurementPostgraduate Course Postgraduate CourseLevel of measurementPostgraduate CourseRelationship between what is being measured andthe numbers that represent what is being measured. Postgraduate CourseCategoricalContinuousNominalOrdinalIntervalRatioLevel of measurement Postgraduate CourseNominal scaleClassification of categorical data. There is no order to thevalues, they are just given a name („nomen‟) or a number.The numbers can‟t be used to calculate … (you can‟tcalculate the mean of fruit) .. only frequencies1 = Apples2 = Oranges3 = Pineapples4 = Banana’s5 = Pears6 = Mango’s Postgraduate CourseOrdinal scaleClassification of categorical data. Values can berank-ordered, but the distance between thevalues have no meaning. The numbers canonly be used to calculate a modus or a median1. Full Professor2. Associate professor3. Assistant professor4. PhD5. Master6. Bachelor Postgraduate CourseInterval scaleClassification of continuous data. Values canbe rank-ordered, and the distance betweenthe values have meaning. However, there isno natural zero point1. John (1932)2. Denise (1945)3. Mary (19524. Marc (1964)5. Jeffrey (1978)6. Sarah (1982) Postgraduate CourseRatio scaleClassification of continuous data. Values canbe rank-ordered, the distance between thevalues have meaning and there is a naturalzero point.1. Jeffrey (192 cm)2. John (187 cm)3. Sarah (180 cm4. Marc (179 cm)5. Mary (171 cm)6. Denise (165 cm) Postgraduate CourseNominal Ordinal Interval RatioClassification Yes Yes Yes YesRank-order No Yes Yes YesFixed and equal intervals No No Yes YesNatural 0 point No No No YesNominal Ordinal Interval RatioMode Yes Yes Yes YesMedian No Yes Yes YesMean No No Yes YesLevels of measurementCategorical Continuous Postgraduate CourseLevels of measurementOrdinal or interval? Can I calculate a mean?Q3: Every organization is unique, hence the findings from scientificresearch are not applicable.☐ Strongly agree☐ Somewhat agree☐ Neither agree or disagree☐ Somewhat disagree☐ Strongly disagree Postgraduate Course4. Central tendencyThe aim is to find a single number that characterises the typical value ofthe variable in the sample. Which one you use depends in part on thelevel of measurement of the variable. Postgraduate CourseCentral tendencyCentral tendency of a set of data / numbers(what number is most representative of the dataset / population?)7, 9, 9, 9, 10, 11,11, 13, 13 Mean = 10,2 Median = 10 Mode = 9 Postgraduate CourseCentral tendencyCentral tendency of a set of data / numbers(what number is most representative of the dataset / population?)3, 3, 3, 3, 3, 3, 100 Mean = 16,9 Median = 3 Mode = 3 Postgraduate Course“It is easy to obtain evidence in favor of virtually any theory,but such „corroboration‟ should count scientifically only if itis the positive result of a genuinely „risky‟ prediction, whichmight conceivably have been false.… A theory is scientific only if it is refutableby a conceivable event. Every genuine testof a scientific theory, then, is logically anattempt to refute or to falsify it.”Hypothesis: falsifiabilityCarl Popper Postgraduate CourseHypothesis Null hypothesis (H0): Big Brother contestants andmembers of the public will not differ in their scores onpersonality disorder questionnaires Alternative hypothesis (H1): Big Brother contestants willscore higher on personality disorder questionnairesthan members of the public. Postgraduate CourseHypothesis: type I vs type II errornull hypothesisis true& was rejected(type I error)αnull hypothesisis false& was rejected(correct conclusion)null hypothesisis true& was accepted(correct conclusion)null hypothesisis false& was accepted(type II error)βH0 is true H0 is falsereject H0accept H0 Postgraduate CourseA confidence interval gives an estimated rangeof values which is likely to include an unknownpopulation parameter (e.g. the mean).Confidence intervals are usually calculated sothat this percentage is 95% (95% CI)Confidence intervals Postgraduate CourseWhen you see a 95% confidence interval for amean, think of it like this: if we‟d collected 100samples and calculated the mean for eachsample, than for 95 of these samples the meanwould fall within the confidence interval.Confidence intervals Postgraduate Course2008 20094,54,03,55,03,0“According to the federalgovernment, theunemployment rate hasdropped from 4.3% to 3.8%.”95% CI= 4,1 - 3,5.This means theunemployment rate couldhave increased from 4.0 to4,1 !Confidence intervals Postgraduate CourseWhen a point estimate (e.g. mean,percentage) is given, always check: standard deviationor confidence intervalConfidence intervals 110 130Postgraduate Course1. Is there a difference / an effect?2. How certain is it that the difference / effect found is not achance finding?X_0 X_1Statistical significance Testing multiple hypothesisWhen you test 20 different hypotheses (or independentvariables), there is a high chance that at least one will bestatistically significant.example:Does apples, bacon, cheese, eggs, fish, garlic, hazelnuts, icecream, ketchup, lamb, melons, nuts, oranges, peanut butter,roasted food, salt, tofu, vinegar, wine or yoghurt causecancer?Postgraduate CourseStatistical significance Significance testing:always prospective, never retrospectivePostgraduate CourseStatistical significance Sample size Effect size(Significant increase in IQ)4 1025 4100 210.000 0,2Postgraduate CourseStatistical powerThe statistical power: the power to detect a meaningfuleffect, given sample size, significance level, and effect size. Postgraduate CourseOverpowered: sample size too large, highprobability of making a Type I errorUnderpowered: sample size too small, highprobability of making a Type II error.Statistical power Postgraduate CourseEffect sizeEffect size: a standardized measure of themagnitude of effect, independent ofsample sizestandardized > makes it possible to compare effect sizesacross different studies that have measured differentvariables, or have used different scales of measurement Postgraduate CourseEffect sizes Cohen‟s d Pearson‟s r other - Hedges‟ g- Glass‟ Δ- odds ratio OR- relative risk RR Postgraduate CourseEffect sizes Cohen‟s dEffect size based on means or distancesbetween/among meansInterpretation< .10 = small.30 = moderate> .50 = large Postgraduate CourseEffect sizes Pearson‟s rEffect size based on ‘variance explained’Interpretation< .10 = small (explains 1% of the total variance).30 = moderate (explains 9% of the total variance)> .50 = large (explains 25% of the total variance) Postgraduate Course12. Critical appraisalWhen you critically appraise a study, what characteristicsof the findings will you consider to determine its statisticalsignificance and magnitude? Postgraduate CourseCritical appraisalWhen you critically appraise a study, what characteristicsof the findings will you consider to determine its statisticalsignificance and magnitude? p-value confidence interval sample size / power effect size practical relevance

Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve We study first-order quantum phase transitions in models where the mean-field traitment is exact, and the exponentially fast closure of the energy gap with the system size at the transition. We consider exactly solvable ferromagnetic models, and show that they reduce to the Grover problem in a particular limit. We compute the coefficient in the exponential closure of the gap using an instantonic approach, and discuss the (dire) consequences for quantum annealing. Quantum statistical mechanics Phase transitions: general studies Many important practical problems involve the minimization of a function of discrete variables. Solving such combinatorial problems by temperature annealing is a classical strategy in computer science : the idea is to use thermal fluctuations to avoid trapping the system in local minima, and thereby efficiently visit the whole configuration space. It has been proposed to extend this approach to quantum fluctuations; it is thus of interest to ask whether annealing by tuning down the amplitude of a quantum mechanical kinetic operator such as a transverse magnetic field can outperform the classical approach. In particular, can problems that normally take exponential time be solved in only polynomial time? Some considerable effort has been devoted to this question in the context of difficult combinatorial problems (see for instance ) which have a counterpart in statistical physics where they corresponds to mean-field spin-glass models [4, 5]. However, most of the studies were purely numerical and thus restricted to very small sizes due to the difficulty of simulating quantum mechanics without a quantum computer. In a recent Letter (see also ), we argued that with the usual implementation of the quantum annealing it is likely that the most difficult systems undergo a quantum transition of the first order as the transverse field is tuned; this is a generic feature for many quantum spin glasses. More recently, a first order transition has indeed been indentified in the phase diagram of one of the most studied random optimization problems, called XORSAT . As we shall see, this implies the failure of quantum annealing for the hardest optimization problems. The reason why quantum annealing is not an efficient strategy for finding the ground state across a first-order transition can be understood from a simple argument. Quantum annealing could in principle be more efficient than thermal annealing for certain classes of problems: From the WKB approximation it is well known that a quantum particle tunnels rapidly through very high (in energy) but thin (in distance) energy barriers. Thermal annealing is much better at low, but deep barrier crossing. However, in a first-order transition the two states whose free energies cross are generally far from each other in the phase space; quantum tunneling must be inefficient. To make this argument more precise, and to compute how slow an annealing should be such that the tunelling do happens, one can consider the Landeau-Zener theory of level crossings . During an avoided crossing, the time needed in order to actually reach the ground state is bounded by the inverse of the energy gap between these states. If the total annealing is longer , then the adiabtic theorem imply that at each time step, the systems remains in the ground state. Otherwise, the system misses the crossing and is not in the ground state at the end of the computation. A good estimates of the running time of the algoritmh is thus obtain by the minimal energy gap during the annealing process . We will see that mean-field first-order transitions have generically an exponentially small gap where is the system size. This implies , that is to say: quantum annealing is an exponentially slow algorithm for a mean-field system with a first-order transition111In finite dimensions one expects that nucleation will help. However, optimization problems are not finite dimensional generically.. The goal of this Letter is to illustrate these features via a complete analytical and detailed numerical analysis for a family of models. We consider the ferromagnetic -spin model, which reduces to a mean-field ferromagnet for the case and to the Grover problem when . We show how to solve the thermodynamics of these models using standard tools of statistical physics. We perform extensive numerical studies of the gap for the case of finite and odd. By introducing an ansatz for the dominant instantonic pathways, we show how to compute the coefficient in the exponential decay of the gap. 1 The simplest quantum ferromagnet We consider a Hamiltonian with Pauli spins of the form where is a function of the longitudinal values of the spins. is thus diagonal in the representation. We focus on the ferromagnetic -spin model: where we have defined the longitudinal magnetization and the transverse one and their magnetization by site and . That sort of models were introduced initially in a spin-glass context in [10, 4]. The ground state of the classical problem, when , corresponds to all spins aligned in the same direction. Whereas both the up and down states are valid ground states for even , the up state is the unique ground state for odd , and we will concentrate on this case for simplicity. The case is the usual Curie-Weiss model, where the transition is continuous [11, 12]. For however, both quantum and thermal transitions are discontinuous. Of special interest is the limit where for odd if , and zero otherwise. It leads to: where the function is if is true and zero otherwise. We now specialize to this limit. 2 The limit 2.1 The classical case: The model is trivial in the limit where there are only two levels with nonzero energies and . The partition sum is thus so that One recognizes a first-order transition at between two phases that are always locally stable (no spinodal): a ferromagnetic phase that consists of the classical configuration where all spins are up for and a trivial paramagnetic phase at larger temperature. 2.2 The extreme quantum case: When is large the classical part of can be neglected; we then find, in the basis, independent classical spins in a field : The entropy density is given by the logarithm of a binomial in : this is a perfect quantum paramagnet. 2.3 The general case For and inverse temperature we saw that the classical model is just a model where (almost) all configurations have zero energy. In this case, we thus can ignore the two nonzero levels and we expect the quantum paramagnetic free energy to be valid for all . A simple perturbation computation – given in the next section– shows that this is true in the low-temperature phase as well, when . The system thus has two distinct phases, the first a quantum paramagnetic and the second a ferromagnetic phase. A first-order transition occurs when the free energies cross so that . The phase diagram of the model is very simple: For low and , the free-energy density is that of the classical model in the ferromagnetic phase, while for larger it jumps to the quantum paramagnetic free energy; a first-order transition separates the two different behaviors at the value such that ; this happens on the line defined by where the magnetization jumps from to (see Fig. 1). The zero-temperature behavior can be understood from standard Rayleigh-Schrödinger perturbation theory . Consider the set of eigenvalues and eigenvectors of the unperturbed model, when . The series for the lowest perturbed eigenvalue reads Since if and only if the two configuration and differ by a single spin flip, odd orders do not contribute in Eq. (5). Noting that reduces to a sum over the levels connected to by a single spin flip, and using the fact that all (except ), succesives terms are easyly computed and one finds, to all (finite) orders (see for a similar computation): The expansion can also be performed using now as a starting point and with as perturbation. Consider the eigenvalue . In the base corresponding to the eigenvalues of 222Note that in the basis the ground-state vector has elements ., we obtain Denoting the elements of the vector in the basis, the first-order term in this expansion reads . Since the are of order the first-order shift is tiny. The next term involves a sum over the levels The last sum is entropically dominated by the states with and therefore gives a negligible contribution (as one can check term by term). Subsequent terms are treated similarly. This yields the ground-state energy: 2.4 Exponential closure of the gap Near the transition the treatment must be refined: There is an (avoided) level crossing at in the large limit between the paramagnetic and the ferromagnetic ground state. We now compute the behavior of the quantum gap around . We write the Hamiltonian in the basis: where is the state corresponding to all spins aligned in the direction expressed in the basis. and s are the (binomially distributed) energies due to the quantum interaction. With an appropriate convention for the eigenvectors we can take for the vector . In this basis, on multiplying with an eigenvector of eigenvalue , we find Multiplying again by , we find The qualitative behavior of the eigenvalues can now be understood graphically: Between each pole in the denominator of Eq. (14) the function passes from to passing through unity. All interior roots to the function are thus bracketed by a comb of poles separated by . In the small phase this rigorously brackets almost all the eigenvalues near . The exception is the lowest eigenvalue which can split off from the comb, a sign of the phase transition in the large limit. In the paramagnetic phase, the lowest eigenvalue is very close to . In this case is very small so that we can write . In addition the overwelming majority of eigenvalues are close to zero 333Systematic corrections to this approximation do not change the result.; Eq. (14) then implies, at the transition when so that finally at the critical point and The gap closes exponentially fast at the transition. We have an extremely simple model with a first-order mean-field transition and most of the physics discussed in this Letter is already present in this model: difficult problems, such as this one where only one in configurations has a low energy, manifest themselves by a first-order transition in the quantum annealing path, and consequently by an exponentially small gap. The reader could at this point argue that we have not shown that all choices of the quantum interaction lead to this result; perhaps a more intelligent choice would turn the transition to second order, and make the gap polynomial? We know that for this precise model, this is just impossible. In fact, this model is nothing else than the Grover problem , that is: searching for a minimum value in an unsorted database. The best algorithm is known, and it is an exponential one . It is obtained by adjusting the evolution rate of the Hamiltonian in the quantum annealing process so as to keep the evolution adiabatic on each infinitesimal time interval. In doing so, the total running time can be , which is still exponential. There is thus no way to avoid the exponential gap in this situation. 3 Behavior for general We now consider finite value of and begin by calculating the phase diagram in the static approximation. We then consider closure of the gap using numerical diagonalization and an instantonic calculation which we then compare. 3.1 Phase diagram We shall first use the Suzuki-Trotter formula in order to map onto a classical model with an additional “time” dimension: We then introduce closure relations : with the convention that . Applying times the integral representation of the delta function , one finds: The saddle point condition imposes that . Writing and performing the limit we obtain: We now use the “static” approximation, which we also check numerically [11, 12], and remove all “time” indices for to finally obtain: All thermodynamic quantities can now be computed. For instance, the self-consistent equation for the magnetization reads (for ) It is easy to check that the former expression leads to first-order (quantum and classical) transitions when its minima cross. In particular, the free energy for is simply for and otherwise, as we obtained in the first section. The phase diagram of the model is plotted in Fig. 1. The energy is given by , and thus at low : In the low-temperature , the energy of a system with excited states with an energy gap is , and this computation thus shows that there are levels with an energy gap where is discontinuous at the transition. This is, however, only a crude description of the phenomenology of the low-lying states. If indeed only one level is closing at the transition, then we expect the energy to behave as , and therefore one needs to compute the correction to the energy in order to take this into account. The former computation thus misses this behavior and indeed, numerical results show that the first excited state is unique. Worse, we expect the energy gap between the ground state and the excited state to close exponentially fast at the transition, and therefore, in order to be able to investigate this behavior, we should be looking for an exponentially small gap: in that case we thus need to look for exponentially small correction to the free energy! Fortunatly, there is a way to deal with this problem: we now turn to a numerical study of the gap and to the instantonic approach. 4 Closure of the gap 4.1 Numerical methods We use two complementary methods to study the spectrum of the -spin model for . The full matrix representation of the Hamiltonian is a sparse operator of dimension . For such sparse matrices Laczos methods are particularly useful and can be used to extract extremal eigenvalues from the spectrum for . We note in particular that that for the transition occurs between two states with the maximum possible angular momentum . Considerable improvements in efficiency are obtained by realazing that the total angular momentum commutes with . Thus the transition occurs in a subspace of dimension . In this subspace the Hamiltonian has diagonal elements corresponding to different values of . Standard methods from the theory of angular momentum show that the off-diagonal elements of the matrix in this subspace are only those labeled by . The matrix is symmetric with off-diagonal elements The resulting tri-diagonal matrix an be treated with very high efficiency allowing one to study systems of in just a few seconds. The limiting factor in the study of even larger systems is the reduction of the gap to double precision machine accuracy so that floating point round-off errors dominate the results. Fig. 3 shows the dependence of the minimum gap for some values of . We see that for all the gap closes exponentially in . Fig. 2 shows the dependence of the gap as a function function of for and different . indeed closes fast at the transition that arises exactly at the critical value predicted analytically. The region where the gap closes is getting narrow as increases, and one has to be very careful in scanning in order not to miss it: this is an important message for future numerical simulations. Fig. 3 shows the dependence of the minimum value of the gap as a function of for some values of . For all , the gap decays exponentially as . The different values of are given in Table 1. As we expected, the gap closes exponentially fast at the first-order transition point. We want now to show how the coefficient in the exponent can be computed analytically. 4.2 The Instantonic approach It is well known that the tunneling between quantum states can be computed using an instantonic approach . Let us briefly explain how this can be understood via corrections to the saddle-point computation. At the transition, two solutions (the ferromagnetic one and the paramagnetic one ) have the same free energies . Let us assume now that we are able to find another time-dependent path —which we shall call instantonic— that spends some time in the ferromagnetic state and then jumps to the paramagnetic state where it spends a time , and that exactly at the transition, one has with in the zero-temperature limit. Since we are summing over all periodic paths, one should now take into account all such instantonic paths that jump an even number of times to compute the correction to Eq. (3.1). Each of these jumps can occur at any time and the saddle-point computation thus reads, at the transition: where the factor comes from the counting of all possible paths with jumps. One then recognizes the series expansion of an effective two level system: Diagonalizing the effective Hamiltonian at one sees immediately that the gap goes as : the energy cost of the instanton thus provides the exponent of the gap at the transition. 4.3 Computing the Instanton We can consider various ansätze to compute the optimal instanton, all of them giving lower bounds on the coefficient. The simplest one is just a sharp wall when jumps abruptly from the value to . The gap thus reads in this approximation:444This can be seen in the discrete Suzuki-Trotter formalism where where and are the eigenvectors of the matrix Exactly at the transition, this gives an estimates on the gap . In particular for , we find that , as was previously found in the first section. For finite , however this yields only a crude lower bound on the value of the exponent (see Table 1). We thus use a shape for and compute numerically the cost, by integrating Eq.(16). We use the width of the function as a variational parameter which we vary in order to minimize the estimate of the instanton free energy from which we deduce the gap. The results of this procedure are given in Table 1. When we now compare the numerical data from exact diagonalization with the prediction from the instantonic computation, we observe that there is no detectable difference within our numerical precision between the instantonic prediction from the shape and the numerical estimation of the coefficient. We have thus obtained from first-principle computations. Quantum annealing has been presented as a new way of solving hard optimization problems with complicated and rough configuration spaces. In this paper we have shown that even in systems with trivial energy landscapes quantum annealing can fail (and there is thus no need for more complex phenomena to explain this failure, as for instance in ). Already the ferromagnet exhibits a first-order phase transition with an exponentially closing gap: A scenario which is very pessimistic for the success of the quantum annealing algorithm. We have also shown that the limit of the ferromagnetic model is related to the Grover problem. This is a clear indication that these first-order transition carry the signature of the most difficult problems. Models presented in this Letter allow a complete analytical and numerical treatment. Their disordered counterpart can be studied using the generalized instanton introduced in [6, 7]. It would be interesting to extend this approach to dilute mean-field system and random optimization problems, using the quantum cavity of [12, 9]. Acknowledgements.We thank P. Boniface, S. Franz, A. Rosso, G. Semerjian, L. Zdeborová and F. Zamponi for discussions. - S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Science, 220 (1983) 671. - A. B. Finnila et al., Chem. Phys. Lett., 219 (1994) 343. T. Kadowaki and H. Nishimori, Phys. Rev. E, 58 (1998) 5355. E. Farhi et al., Science, 292 (2001) 472. G. E. Santoro et al., Science, 295 (2002) 2427. - A. P. Young, S. Knysh and V. N. Smelyanskiy, Phys. Rev. Lett., 101 (2008) 170503 and Phys. Rev. Lett. 104, 020502 (2010). E. Farhi et al., arXiv:0909.4766. - M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond, (World Scientific, Singapore) 1987. - M. Mézard and A. Montanari, Physics, Information, Computation, (Oxford University Press, Oxford) 2009. - T. Jörg et al., Phys. Rev. Lett., 101 (2008) 147204. - T. Jörg et al., arXiv:0910.5644, 2009. - G. Biroli and L. F. Cugliandolo, Phys. Rev. B, 64 (2001) 014206. L. F. Cugliandolo et al., Phys. Rev. Lett., 85 (2000) 2589. - T. Jörg et al., arXiv:0911.3438, 2009. - B. Derrida, Phys. Rev. Lett., 45 (1980) 79. - L. Chayes et al., J. Stat. Phys., 133 (2008) 131. - F. Krzakala et al., Phys. Rev. B, 78 (2008) 134428. - N. March, W. Young and S. Sampanthan, The many-body problem in quantum theory, (Cambridge University) 1967. - L. K. Grover, Proceedings, 28th Annual ACM Symposium on the Theory of Computing, May 1996. - J. Roland and N. J. Cerf, Phys. Rev. A, 65 (2002) 042308. - J. Zinn-Justin, Path Integrals in Quantum Mechanics, (Oxford University, Oxford) 2004. - B. Altshuler, H. Krovi and J. Roland, arXiv:0908.2782 and arXiv:0912.0746.

Plot Multiple Error Bars Here's Why Members Love Eng-Tips Forums: Talk To Other Members Notification Of Responses To Questions Favorite Forums One Click Access Keyword Search Of All Posts, And More... One Account Your MATLAB Central account is tied to your MathWorks Account for easy access. I will test it with my own data and see if this answer is more clear always or only with simpler data. Watch lists Setting up watch lists allows you to be notified of updates made to postings selected by author, thread, or any search variable. http://exactcomputerrepair.com/error-bars/plot-error-bars-idl.html Tags make it easier for you to find threads of interest. I got those std from the Matlab example! (aah this Matlab examples) –Ander Biguri Feb 17 '15 at 10:36 1 Oh i think using the std is fine; in my Learn more You're viewing YouTube in German. I'm supposed to be incognito How are beats formed when frequencies combine? https://www.mathworks.com/matlabcentral/answers/171168-plotting-multiple-lines-on-errorbar Matlab Error Bars On Bar Graph Share charts, dashboards, Jupyter notebooks, and presentations through our enterprise-grade cloud. Play games and win prizes! For anyone having the same issue:errorbar(label1, value1, sdevi1); hold on; errorbar(label2, value2, sdevi2, 'r'); 0 Comments Show all comments Log In to answer or comment on this question. Did millions of illegal immigrants vote in the 2016 USA election? This method creates error bar values by calculating either: A user-specified percentage of each data point. You can change this preference below. Spam Control Most newsgroup spam is filtered out by the MATLAB Central Newsreader. Matlab Errorbar No Line The MATLAB Central Newsreader posts and displays messages in the comp.soft-sys.matlab newsgroup. When error bars are added to a data plot, the error data is output to a new column on the source worksheet. There are several advantages to using MATLAB Central. The newsgroups are a worldwide forum that is open to everyone. https://www.mathworks.com/matlabcentral/newsreader/view_thread/244739 Newsgroup content is distributed by servers hosted by various organizations on the Internet. Matlab Errorbar Horizontal Draw error bars in polar graphs as arcs. look at this web-site I fear that, if the data is too messy this may be misleading (and data tend to be messy). Matlab Error Bars On Bar Graph The orange parts is hard to recognize due to the red and the sharp contrast of the other data. Matlab Add Error Bars To Existing Plot News & Events Careers Distributors Contact Us All Books Origin Help Graphing Adding Data Labels and Error Bars User Guide Tutorials Quick Help Origin Help X-Function Origin C LabTalk Programming Reload the page to see its updated state. this content This function will allow you to plot as many functions as you want in one figure. Related Content 1 Answer the cyclist (view profile) 33 questions 2,632 answers 1,090 accepted answers Reputation: 6,067 Vote1 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/267402-how-to-draw-errorbars-for-multiple-data-lines-in-a-single-plot#answer_209222 Cancel Copy to Clipboard Answer by In this tab, you can: Specify plus and/or minus directions. Error Bars Matlab Scatter Close Tags for this Thread multiple error bar...ploterrplt What are tags? I want this plots for a journal article (hopefully), so they need to be clear. –Ander Biguri Feb 17 '15 at 14:23 1 @AnderBiguri Glad it worked! Based on your location, we recommend that you select: . weblink Note that plt uses the helper functions "ebar" and "vbar" to create error bars and vertical bars respectively. (Type "help ebar" for more information). Example: clear all close all clc x = 0:pi/10:pi; y = sin(x); y2=cos(x); y3=atan(x); e = std(y)*ones(size(x)); e2 = std(y2)*ones(size(x)); e3 = std(y3)*ones(size(x)); figure hold on errorbar(x,y,e) errorbar(x,y2,e2) errorbar(x,y3,e3) My idea Matlab Errorbar Width Set up your worksheet so that the columns are designated Y1, yEr1, Y2, yEr2, Y3, yEr3... (the error bar column must be to the right of the Y data column). How do I add an item to my watch list? Can my address as well as final 4 digits and name on credit card be stored without PCI compliance? MATLAB Central is hosted by MathWorks. MAC where key is provided afterwards Would Earth's extraterrestrial colonies have a higher average intelligence? Learn more MATLAB and Simulink resources for Arduino, LEGO, and Raspberry Pi Learn more Discover what MATLAB® can do for your career. Standard Error Matlab Melde dich bei YouTube an, damit dein Feedback gezählt wird. What kind of supernatural powers don't break the masquerade? Is there a rule for how to handle creative use of spells? Related Content 1 Answer M Carvalho (view profile) 1 question 1 answer 1 accepted answer Reputation: 0 Vote0 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/171168-plotting-multiple-lines-on-errorbar#answer_165617 Cancel Copy to Clipboard Answer by check over here Apply Today MATLAB Academy New to MATLAB? Wird geladen... Browse other questions tagged matlab or ask your own question. Which answer best completes the sequence? Baker SDSU Marketing 45.936 Aufrufe 10:05 Weitere Vorschläge werden geladen… Mehr anzeigen Wird geladen... share|improve this answer answered Oct 29 '10 at 7:22 Rogach 9,0101051119 also: take a look at hold in the documentation. –rubenvb Oct 31 '10 at 16:46 add a comment| Play games and win prizes!

GEOSTATS: adaptive interpolation - A few thoughts 1. Adaptive can mean several different things. The kriging estimator/predictor is already "adaptive" in several senses. a. It "adapts" to the spatial pattern of the data locations by incorporating the spatial correlation for pairs of data locations. (The data locations need not be on a grid). b. It "adapts" to the spatial relationship of the location to be estimated to the data locations used in the estimation c. By using a search neighborhood it makes several adaptations. Note however that using search neighborhoods may introduce discontinuities in the interpolated surface. One of the supposed justifications for using a search neighborhood is that for most variogram models the data locations outside the search neighborhood will have small or zero weights even if included (this obviously depends on the choice of the search neighborhood and the variogram model and its parameters) hence there is no substantial change by not including them. One of the presumed advantages of using a search neighborhood is that the constant mean assumption can be slightly weakened, i.e., the mean is only assumed constant (or nearly constant) within each search neighborhood (separately). See Journel et al (Math Geology) on "Why do we need a trend d. By allowing the variogram model to be "adapted" to the data, the kriging estimator incorporates several aspects of the data including a directional effect in the spatial correlation. Most of the above are what might be called pre-adaptations, i.e, they do not occur during the actual kriging. 2. In order that the kriging variance be non-negative (remember the kriging variance is the minimized estimation and mimimizing non-negative valued functions is very different from mimimizing functions that can take on both postive and negative values) and that the kriging system have a unique solution it is (essentially) necessary that the variogram be conditionally negative definite (or that the covariance be positive definite). If the variogram is allowed to change from one search neighborhood to another there are two potential problems, (i) do the multiple forms collectively determine a valid model, (ii) if the neighborhoods overlap then which one does one use in the intersection (for consistency they should in fact match on the intersection). One of the practical reasons that we commonly use only a small number of standard models for variograms is that it is difficult to check whether a given function is a valid variogram. 3. It is well-known that the kriging estimator can be re-written in the so-called "dual" form, which shows the relationship to radial basis function interpolation. However there is a significant practical difference. In the usual form of the kriging estimator the coefficients change with the location to be estimated whereas in the dual form the coefficients do not depend on the location to be estimated. That is, the radial basis form produces an interpolating function whereas the kriging estimator only implicitly determines the interpolating function (by giving its values one at a time for each location). Hence it is not so simple to use a search neighborhood since in that case one must actually confront the piecing together functions defined on separate (but usually overlapping) regions and hence the discontinuities will become very apparent. (See Myers, "Interpolation with Positive Definite Functions", Sciences de la 4. If the function to be interpolated is really so variable that one would have to consider a locally varying variogram then maybe it is too complicated to be represented by second order properties. 5. Finally one must remember that geostatistics is essentially based on the idea that the data is a (non-random) sample from one realization of a random function. The variogram or covariance function is a characteristic of the random function, not of the data. The sample variogram (or other variogram estimator) is a statistic based on the one realization, some form of ergodicity and stationarity is needed to use this spatial statistic to estimate/model the variogram which is an ensemble statistic. Ideally one should have data from multiple realizations at each data location, i.e., replications. In some instances where one has spatial temporal data, it might be assumed that there is no temporal dependence and hence data collected at multiple time points for a given location might be treated as replications. This would allow local fitting/estimation of variograms and without invoking (spatial) stationarity. This doesn't mean that it might not be possible to have a locally varying variogram but it does mean that one should be careful both in the modeling step and in using it for kriging because there are hidden implications. Department of Mathematics University of Arizona *To post a message to the list, send it to ai-geostats@.... *As a general service to list users, please remember to post a summary of any useful responses to your questions. *To unsubscribe, send email to majordomo@... with no subject and "unsubscribe ai-geostats" in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list!

Shamit Kachru,††, , , Renata Kallosh, Andrei Linde, Juan Maldacena, ††, , Liam McAllister, and Sandip P. Trivedi Department of Physics, Stanford University, Stanford, CA 94305, USA SLAC, Stanford University, Stanford, CA 94309, USA Institute for Advanced Study, Princeton, NJ 08540, USA Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, INDIA We investigate the embedding of brane inflation into stable compactifications of string theory. At first sight a warped compactification geometry seems to produce a naturally flat inflaton potential, evading one well-known difficulty of brane-antibrane scenarios. Careful consideration of the closed string moduli reveals a further obstacle: superpotential stabilization of the compactification volume typically modifies the inflaton potential and renders it too steep for inflation. We discuss the non-generic conditions under which this problem does not arise. We conclude that brane inflation models can only work if restrictive assumptions about the method of volume stabilization, the warping of the internal space, and the source of inflationary energy are satisfied. We argue that this may not be a real problem, given the large range of available fluxes and background geometries in string theory. Inflation provides a compelling explanation for the homogeneity and isotropy of the universe and for the observed spectrum of density perturbations [1,2]. For this reason, we would hope for inflation to emerge naturally from any fundamental theory of microphysics. String theory is a promising candidate for a fundamental theory, but there are significant obstacles to deriving convincing models of inflation from string theory. One problem is that string compactifications come with moduli fields which control the shape and size of the compactification manifold as well as the string coupling. Inflation is possible only if these fields are either stable or else have relatively flat potentials which do not cause fast, non-inflationary rolling in field space. Controlling the moduli in this way is a difficult problem. In particular, the potential for the dilaton and for the compactification volume tends to be a rather steep function . A second problem is that the inflaton potential itself must be exceptionally flat to ensure prolonged slow-roll inflation. A successful microphysical theory would naturally produce such a flat potential. Since the flatness condition for the potential involves the Planck scale one should ensure that quantum gravity corrections do not spoil it. Hence, the problem should be analyzed in a theory of quantum gravity, such as string theory. The hope of brane-antibrane inflation scenarios is that the brane-antibrane interaction potential can play the role of the inflaton potential (see for a nice review), but it is well known that this potential is not naturally flat. Since in string theory one cannot fine-tune by hand, but only by varying background data (like the compactification manifold or the choice of flux), one concludes that in generic compactifications, brane inflation will not work. However, the many choices of flux and compactification make possible a considerable degree of discrete fine-tuning, so for very special choices of the background one would expect to find potentials which are sufficiently flat for inflation. In this note we discuss these problems in the concrete context of the warped type IIB compactifications described in e.g. [5,6]. One reason for working in this setting is that one can sometimes stabilize all the moduli in a geometry of this type, avoiding the first problem mentioned above. In addition, the constructions of naturally admit D3-branes and anti-D3-branes transverse to the six compact dimensions. Furthermore, one could wish for a model which accommodates both inflation and the present-day cosmic acceleration. This might be possible if one could construct inflationary models which asymptote at late times to the de Sitter vacua of (or variants on that construction, as described in e.g. [8,9]; earlier constructions in non-critical string theory appeared in ). As these vacua included one or more anti-D3-branes in a warped type IIB background, it is quite natural to consider brane-antibrane inflation in this context. Our idea, then, is to begin with the de Sitter vacua constructed in , add a mobile D3-brane, and determine whether the resulting potential is suitable for inflation. For the impatient reader, we summarize our findings here. We find that modest warping of the compactification geometry produces an extremely flat brane-antibrane interaction potential, provided that we neglect moduli stabilization. This solves the second problem listed above. However, a new problem appears when we incorporate those terms in the potential which led, in the construction of , to the stabilization of the volume modulus. We show that generic volume-stabilizing superpotentials also impart an unacceptably large mass to the inflaton, halting inflation. While these conclusions are “generic,” it is very important to emphasize that the problem of the inflaton mass might be circumvented in at least two different ways. First, the stabilization mechanism for the moduli might be different from that in . For example, the volume modulus could be stabilized by corrections to the Kähler potential, which, as we will see, can naturally circumvent this problem. Second, the mobile brane might be located not at a generic point in the compact manifold but close to some preferred point. If the location of the D3-brane is appropriately chosen then there could be significant corrections to the superpotential. In general models, the superpotential may be a rather complicated function of both the brane positions and the volume modulus. Little is known about the form of these nonperturbative superpotentials in string compactifications. Our arguments show that if the functional form of the superpotential is generic then inflation does not occur. Nevertheless, it seems quite likely, given the range of available fluxes and background geometries, that cases exist which are sufficiently non-generic to permit inflation, although with predictions which are altered from those of naive brane inflation.†† This point is made more quantitative in Appendix F, where we explain that the degree of non-genericity required corresponds roughly to a fine-tune of one part in 100. Our conclusions should be viewed as a first pass through the class of brane inflation models, in the context of the moduli stabilization mechanism which has recently been developed in [5,7]. Once the non-perturbative superpotentials involved in such constructions are better understood, and/or as soon as other mechanisms for moduli stabilization become available, one could re-examine brane inflation in light of this further concrete knowledge. This may well lead to a precise determination of the non-generic cases where working models of brane inflation in string theory can be realized. Our analysis clearly indicates that any viable inflation scenario in string theory has to address the moduli stabilization problem. Since essentially all papers on the subject, to the best of our knowledge, have ignored the problem of moduli stabilization, their conclusions are questionable in view of our results. In particular, should a more detailed analysis reveal the possibility of inflation in various non-generic situations, as suggested above, we expect that the resulting inflationary parameters will typically be quite different from those calculated in the existing literature by neglecting moduli stabilization. This paper is organized as follows. In §2 we review basic facts about brane-antibrane inflation [11,12], with special attention to the case of D3-branes, and discuss some generic problems for such models. In §3 we show that warping of the geometry can help with some of these problems. In §4 we explain one method of embedding the warped inflation scenario into string theory, using the warped compactifications of . In §5 we describe further problems that arise in the string theory constructions when one tries to stabilize the overall volume modulus. Generic methods of stabilization (e.g. via a nonperturbative superpotential) modify the inflaton potential and make inflation difficult to achieve. We discuss several ways to overcome this problem. We conclude with some general remarks in §6. Appendix A contains a general discussion of the gravitational interaction of an (unwarped) brane-antibrane pair, and demonstrates that the potentials which arise are typically not flat enough to lead to prolonged inflation. In Appendix B we specialize to a warped background and derive the interaction potential. In Appendix C we explore the detailed properties of inflation in warped brane-antibrane models, assuming that a solution to the challenges of §5 has been found. In Appendix D we explain that eternal inflation may be possible in this scenario. In Appendix E we discuss the exit from inflation and point out that the production of undesirable metric perturbations due to cosmic strings, which are typically created during brane-antibrane annihilation, is highly suppressed in warped models. Finally, in Appendix F we discuss the possibility of fine-tuning of the inflaton potential in order to achieve an inflationary regime. After completing this work, we became aware of the papers , in which related issues are addressed. 2. Brief Review of Inflation In brane-antibrane inflation one studies the relative motion of a brane and an antibrane which are initially separated by a distance on the compactification manifold . One should assume , so that the force is well approximated by the Coulomb attraction due to gravity and RR fields. Then the potential takes the form where is the ten-dimensional Planck scale, defined by , and is the tension of a D3-brane. In terms of a canonically normalized scalar field , one can rewrite this as It was suggested in that for large fields (large ), one may obtain inflation from this potential. A basic (and well known ) problem with this scenario is the following. The standard inflationary slow-roll parameters and are defined via One generally wants to get slow-roll inflation with sufficient e-foldings. Is this possible in the model (2.2)? The four-dimensional Planck mass appearing in (2.4) is where is the volume of M. This implies that is Hence, is possible only for – but two branes cannot be separated by a distance greater than in a manifold of size ! One can try to evade this constraint by considering anisotropic extra dimensions or non-generic initial conditions which yield flatter potentials than (2.2). We argue in Appendix A that this is not possible. There are always some tachyonic directions in the potential with . This implies that the slow-roll approximation cannot be maintained for a large number of e-foldings. In §3 we will explore another possibility that successfully evades this problem – we will modify the potential (2.2) by considering branes and antibranes in a warped geometry. We should mention that there are other proposals which might evade the above problem, such as branes at angles or branes with fluxes, see [14,15,16,17,18]. However, all of these models have an unsolved problem: moduli stabilization. For an internal manifold of size , the correct four-dimensional Einstein-frame potential is not quite (2.2). If one assumes that the main contribution to the inflationary energy comes from the D3-brane tension then one finds, for , that The energy in the brane tensions sources a steep potential for the radial modulus of the internal manifold. Therefore, in the absence of a stabilization mechanism which fixes with sufficient mass so that the variation of in (2.6) is negligible, one will find fast-roll in the direction of large rather than slow-roll in the direction of decreasing . This means that it is important to study concrete scenarios where the volume modulus has already been stabilized. However, we will show that not every means of volume stabilization is compatible with inflation, even when the naive inter-brane potential is flat enough to inflate. We will return to the issue of volume stabilization in §5 , where we will discuss a new and generic problem which appears when one considers the issue in detail. 3. Inflation in a Warped Background: Essential Features Our modified brane-antibrane proposal is that inflation might arise from the interaction potential between a D3-brane and an anti-D3-brane which are parallel and widely separated in five-dimensional anti de Sitter space ().†† This is a slight simplification; in §4 we will construct compact models which deviate from both in the infrared and in the ultraviolet. It is nevertheless convenient to work out the essential features of the model in this simpler geometry. The anti-D3-brane is held fixed at one location in the infrared end of the geometry (this is naturally enforced by the dynamics, as we shall explain). The D3-brane is mobile; it experiences a small attractive force towards the anti-D3-brane. The distance between the branes plays the role of the inflaton field. The forces on the brane and antibrane arise as follows. A single D3-brane experiences no force in an AdS background: electrostatic repulsion from the five-form background exactly cancels gravitational attraction. The addition of a distant anti-D3-brane results in a relatively weak interaction potential arising from the attraction between the brane and the antibrane. We interpret this as a slowly varying potential for the inflaton. We will demonstrate in §3.2 and in Appendix B that this potential is much flatter than the interaction potential for a brane-antibrane pair in flat space. In the remainder of this section we explain this key idea in more detail. §3.1 is a review of gravity in a warped background. §3.2 deals with the motion of a brane probe in such a background. It is important to point out that throughout this discussion, we will ignore the possibility that other moduli (or the effects which stabilize them) interfere with inflation. In the context of the string constructions of §4, the relevant other modulus is the compactification volume, and the generic problems associated with its stabilization are the subject of §5. In fact we will see that this modulus problem will generically stop inflation. 3.1. Gravity in an AdS Background We first consider a compactification of string theory on where is a five-dimensional Einstein manifold.†† The detailed form of will not matter for the moment. For concreteness the reader may imagine that . This arises in string theory as a solution of ten-dimensional supergravity coupled to the five-form field strength . The solution is given in Poincaré coordinates by the metric There is, in addition, a five-form flux: if the geometry (3.1) arises as the near-horizon limit of a stack of D3-branes, then the five-form charge (in units of the charge of a single D3-brane) is . , the characteristic length scale of the geometry, is related to the five-form charge by where the constant depends on . It will be useful to recall that is a maximally symmetric, constant curvature spacetime. Its curvature scales like and is independent of the radial location . As long as this curvature is small and supergravity analysis is reliable. We will choose to truncate to the region . The reader will notice that, apart from the additional manifold , this background is identical to that considered by Randall and Sundrum in . Two physical insights from will be crucial for our model. First, one can see from the warped metric (3.1) that the region of small is the bottom of a gravitational well. Energies along the coordinates therefore get increasingly redshifted as decreases. (The region of significant redshift is consequently referred to as the infrared end of the geometry.) Second, as a result of truncating the AdS region, the four-dimensional effective theory which governs low-energy dynamics will have a finite gravitational constant, and will include four-dimensional gravity described by the Einstein-Hilbert action: †† The graviton zero modes have polarizations parallel to , are constant on , and have a profile identical to the warped background. Recall also that in , the truncation of AdS space was achieved in a brute force manner by placing two branes, conventionally called the Planck brane and the Standard Model brane, at and , respectively. In the string theory constructions of , the truncation arises because the compactification geometry departs significantly from that of away from the region . In the ultraviolet, in the vicinity of , the AdS geometry smoothly glues into a warped Calabi-Yau compactification. In the infrared, near , the AdS region often terminates smoothly (as in the example of ). The infrared smoothing prevents the redshift factor from decreasing beyond a certain minimum whose value will be very important for our model. 3.2. Brane Dynamics We mentioned above that the warped nature of the geometry gives rise to a redshift dependent on the radial location. It will be important in the discussion below that the redshift results in a very significant suppression of energies at the location of the antibrane; that is, the ratio is very small. Also note that within the truncated AdS geometry, , we have chosen to place the anti-D3-brane at the infrared cutoff , where it has minimum energy due to the redshift effect. The five-form background is given by In a suitable gauge the corresponding four-form gauge potential takes the form The D3-brane stretches along the directions . Its location in the radial direction of AdS space will be denoted by . In the discussion below we will assume (self-consistently) that the D3-brane has a fixed location along the angular coordinates of the space. The motion of the D3-brane is then described by the Born-Infeld plus Chern-Simons action The indices denote directions parallel to the D3-brane along the coordinates, and is the metric along these directions. The D3-brane tension, , is For future purposes we note here that since an anti-D3-brane has the same tension as a D3-brane but opposite five-form charge, it is described by a similar action where the sign of the second term is reversed. Now consider a D3-brane slowly moving in the background given by (3.1) and (3.4), with no antibranes present. It is easy to see that because of a cancellation between the Born-Infeld and Chern-Simons terms, the D3-brane action at low energies is just that of a free field, This in accord with our comment above that the net force for a D3-brane in the background (3.1),(3.4) vanishes due to gravitational and five-form cancellations. We are now ready to consider the effect of an antibrane on the D3-brane. Physically this arises as follows. The anti-D3-brane has a tension and a five-form charge and perturbs both the metric and the five-form field. This in turn results in a potential energy dependent on the location of the D3-brane. The potential between a brane located at and an antibrane located at , in the limit when , is given by: For a derivation see Appendix B. The first term in the potential is independent of the location of the D3-brane and can be thought of as a constant potential energy associated with the anti-D3-brane. It is proportional to the tension . For the antibrane the force exerted by gravity and the five-form field are of the same sign and add, so we have a factor of 2. In addition, the warped geometry gives rise to a redshift, which reduces the effective tension of the antibrane by a factor . The second term in (3.9) depends on the location of the D3-brane; its negative sign indicates mutual attraction between the pair. Two features of this term will be important in the subsequent discussion. First, the term varies slowly, as the inverse fourth power of the radial location of the D3-brane. Second, due to the warping of the background, the coefficient of this second term is highly suppressed, by a redshift factor . Two more comments are in order at this stage. We have assumed that the antibrane is fixed at . From (3.9), we see that this is in fact a good approximation to make. In the limit the first term in (3.9) is much bigger than the second, and most of the energy of the anti-D3-brane arises due to interaction with the background. This is minimized when the anti-D3-brane is located at in the truncated AdS spacetime. Second, in our analysis above, we are working in the approximation . We will see below that the D3-brane is far away from the anti-D3-brane while the approximately sixty e-foldings of inflation occur, so this condition is met during the inflationary epoch. Eventually the D3-brane approaches the antibrane, , and this approximation breaks down. The potential then becomes quite complicated and more model dependent (e.g. it depends on the separation between the brane and antibrane along ). The resulting dynamics is important for reheating. A summary of the discussion so far is as follows. We have considered a D3-brane moving in an background with five-form flux, in the presence of a fixed anti-D3-brane. This system is described by an action: The reader will notice in particular that , the location of the D3-brane, is a scalar field in the effective four-dimensional theory. Once we cut off the space as in the Randall-Sundrum models we will find that we can add to (3.10) the four-dimensional Einstein action. However, we should also add an extra coupling of the form coming from the fact that the scalar field describing the position of the D3-brane is a conformally coupled scalar . This unfortunately leads to a large contribution to . We will discuss this phenomenon in more generality (from the perspective of the effective low-energy four-dimensional supergravity) in §5. The model described above has several appealing features in addition to the flatness of the potential. We study these properties in Appendices C,D, and E, with the assumption that one can somehow overcome the problems of §5 (which must be tantamount to cancelling the conformal coupling). In Appendix C we compute the inflationary parameters and show that observational constraints are easily met. In Appendix D we argue that eternal inflation can be embedded into this model, and in Appendix E we point out that the warped geometry suppresses the production of metric perturbations due to cosmic strings (which naturally form during the brane/anti-brane annihilation). 4. A Concrete Example in String Theory We now show how to realize our proposal in a specific class of string compactifications. In §4.1 we present the compactifications and explain why they contain warped throat regions. As the warped throat is well-described by the Klebanov-Strassler (KS) solution , we dedicate §4.2 to a very brief review of the KS geometry. In §4.3 we show that a brane moving in the KS background might give rise to inflation, realizing the general idea presented in §3. Throughout this discussion, we ignore the problem of stabilizing the overall volume modulus, which is unfixed in the constructions of . We consider the problem of volume stabilization in §5, where we will find that generic methods of volume stabilization can perturb the inflaton enough to stop inflation. 4.1. The Compactification Our starting point is type IIB string theory compactified on a six-dimensional Calabi-Yau orientifold. More generally one could use F-theory on an elliptically-fibered Calabi-Yau fourfold. We choose to turn on background fluxes: the three-form fluxes present in the theory are placed along cycles in the internal space (and is fixed as in ). These fluxes induce warping of the background. One can show that the resulting space is a warped product of Minkowski space and the Calabi-Yau: where are coordinates on the compactification manifold and is the Calabi-Yau metric. As was discussed in , one expects that with a generic choice of flux, all the complex structure moduli of the Calabi-Yau, as well as the dilaton-axion, will be fixed. We will assume that the compactification has only one Kähler modulus, the overall volume of the internal space. As described in , one can use the above construction to compactify the warped deformed conifold solution of Klebanov and Strassler (KS). We spend the next section reviewing a few facts about this geometry, as certain details will be important for inflation. 4.2. The Klebanov-Strassler Geometry The Klebanov-Strassler geometry is a noncompact ten-dimensional solution to type IIB supergravity in the presence of background fluxes. The spacetime naturally decomposes into a warped product of a Minkowski factor and a six-dimensional internal space. The six-dimensional space has a tip which is smoothed into an of finite size. Far from this tip the geometry can be approximated by a cone over the Einstein manifold , which is topologically . Our coordinates will be five angles on , which we can consistently neglect in the following, and a radial coordinate which measures distance from the tip. The background fluxes are given by where A is the at the tip and B is its Poincaré-dual three-cycle. We will require that and ; these conditions are important in deriving the solution. The exact metric is known, but for our purposes a simpler form, valid far from the tip, will be more useful. For large we may express the complete ten-dimensional solution as Neglecting the logarithmic corrections and the second term on the right, this takes the form †† The second term on the r.h.s. of (4.4) can easily be included. For the numerical values discussed in Appendix C, this gives a three percent correction. When the KS geometry is embedded in a compactification then at some location the warped throat geometry is smoothly joined to the remainder of the warped Calabi-Yau orientifold. Near this gluing region, departures from the geometry are noticeable; eventually the must end. In terms of redshift this location corresponds to the deep ultraviolet, and so the gluing region plays the role of the ultraviolet cutoff (Planck brane) in the of §3. The exact solution likewise shows departures from (4.3) in the far infrared, near the tip, although the geometry remains smooth.†† The radius of curvature is , so the tip is smooth provided . The details of the deviation from (4.3), although known, are unimportant here; it will suffice to know the redshift at the tip. This can be modeled by cutting off the radial coordinate at some minimum value , which is the location of the tip. It was shown in that the minimal redshift satisfies This can be extremely small given a suitable choice of fluxes. 4.3. Inflation from Motion in the KS Region In additional anti-D3-branes were introduced in the KS region. These anti-D3-branes minimize their energy by sitting at the location where the redshift suppression is maximum, i.e. at the very tip of the deformed conifold, where (the dynamics of anti-D3-branes in the KS geometry was studied in ). Thus we see that the string construction outlined above has all the features of the general model of §3: a truncated geometry, an associated five-form flux of the correct strength, and anti-D3-branes fixed at the location of maximum redshift. In addition most of the moduli associated with the compactification, including the dilaton, are stabilized. The one exception is the volume modulus; we will discuss the complications its stabilization introduces separately, in §5. No mobile D3-branes were included in the construction of , but it is easy to incorporate them. One needs to turn on somewhat different values of three-form flux, which allow the four-form tadpole to cancel in the presence of the additional D3-branes. This is straightforward to do and does not change any of the features discussed above. We will take one such D3-brane to be present in the KS region of the compactification. The general discussion of §3 applies to this brane. Since the D3-brane is described by the action (3.6), with now given by (4.5), the calculation of the brane-antibrane potential follows the discussion in Appendix C, which we outline here. In the KS model the warp factor (4.1) is given in terms of a function which obeys a Laplace equation, with the fluxes and branes acting as sources. In particular, a single D3-brane located at will correct the background according to Here is the background given in (4.4) and is the correction due to the D3-brane. In a region where the original warp factor is very small we see that , so that the total warp factor can be expanded as This warp factor yields the contribution to the energy due to the presence of an antibrane. If this typically gives a very flat potential. The small warp factor and the consequent exponential flatness are the heart of our proposal, so an alternative explanation of the origin of these small numbers may be helpful. Recall that there is a holographic dual gauge theory which describes the geometry of the KS model. This gauge theory is approximately scale invariant in the deep ultraviolet, with slowly running gauge couplings. It undergoes duality cascades before leading in the infrared to a confining gauge theory with a mass gap. Then the smallness of the redshift factor, can be ascribed to the exponential smallness of the confinement scale in such a gauge theory. In summary, we have seen that one can construct concrete examples of string compactifications which lead to the general behavior described in §3. One of their virtues is that they automatically lead to very flat inflaton potentials, without the need for large brane separation or excessive fine-tuning of initial conditions. The primary source of this flatness is the redshift suppression (4.7) which is exponentially sensitive to the (integer) choice of fluxes and . However, all of these virtues must be re-examined in the light of concrete ideas about how to stabilize the closed string moduli. In this general class of flux compactifications, the fluxes stabilize many moduli but not e.g. the overall volume. We now turn to the discussion of volume stabilization. 5. Volume Stabilization: New Difficulties for D-brane Inflation The results of §3,4 indicate that warped geometries provide a promising setting for making models of inflation with naturally small and . However, as emphasized in §2, one must ensure that the compactification volume is stabilized in order to avoid rapid decompactification instead of inflation. We will now demonstrate that in the concrete models of this is far from a trivial constraint. In these models the four-dimensional supergravity at low energies is of the no-scale type. The Kähler potential for the volume modulus and the D-brane fields takes the form †† The variable is called in . Let us pause for a moment to explain how this is obtained. In the tree level compactification the massless fields are the volume, the axion and the position of the branes. The axion comes from a four-form potential proportional to a harmonic four-form in the internal manifold . At first sight one would think that the moduli space is simply a product of the moduli space for , which is just the internal Calabi-Yau manifold, and the space spanned by the volume and the axion. This is not correct; the axion describes a circle which is non-trivially fibered over the moduli space. This structure arises from the coupling of the four-form potential to the worldvolume of the moving D3-brane. The moduli space has a metric of the form where is proportional to the volume of the Calabi Yau (in the notation of , ). If we tried to work with a complex variable then (5.2) would not follow from a Kähler potential. It turns out that the good complex variable is , which is defined as follows. The imaginary part of is the axion, while the real part of is defined by It is then possible to see that (5.1) gives rise to (5.2). This type of definition of arises when we Kaluza-Klein compactify supergravity theories; see for example . The superpotential is of the form where is a constant (we assume the D-branes are on their moduli space, so we do not write down the standard commutator term). This arises from the (0,3) part of the three-form flux in the full theory including the complex structure moduli and the dilaton. We have not yet included the anti-D3-branes used in §3,4; these will be incorporated at the end of the discussion. It is important that with the Kähler potential (5.1), one obtains the no-scale cancellation in the potential where run over and . †† The easiest way to check (5.6) is to note that in expression (5.6) we can switch back to the variables in (5.2). In these variables is only a function of . Using (5.5), it is clear that a generic will yield a potential for the D-brane fields , but that the potential for the modulus will vanish if the solution for the fields has . It is also clear that a superpotential, as in (5.4), gives no potential to the fields. This is consistent with the analysis in , where the pseudo-BPS nature of the flux background leaves the D3-brane moduli unfixed. We are interested in finding a situation where the D-branes can move freely in the Calabi-Yau (so the fields are ), but the volume is stabilized. Before we discuss various scenarios for such a stabilization, it is important to distinguish carefully between the chiral superfield, and the actual volume modulus, , which controls the expansion. The Kähler potential (5.1) has the following peculiar feature. Let us imagine that there is one D-brane, and hence a triplet of fields describing its position on the Calabi-Yau space. Then should be the Kähler potential for the Calabi-Yau metric itself, at least at large volume. However, under Kähler transformations of , the expression (5.1) is not well behaved. This can be fixed by assigning the transformation laws This is a manifestation of the fact that the circle described by the axion is non-trivially fibered over the moduli space. Note that the physical volume of the internal dimensions, which is given by , (5.3), is invariant under (5.7). Armed with this knowledge, and given (5.1) and (5.4) as our starting point, we can now explore various scenarios for volume stabilization. 5.1. Scenario I: Superpotential Stabilization Perhaps the most straightforward method of stabilizing the volume involves a nonperturbative contribution to the superpotential. Various sources of nonperturbative superpotentials for the modulus are known; one instructive example described in involves a superpotential where and are constants and is the contribution (5.4) of the three-form flux. For the remainder of this section we will consider to be a general holomorphic function of . In the presence of D3-branes the superpotential must in addition develop some dependence on , as it should be invariant under (5.7). For instance, as argued in , the superpotential due to Euclidean brane instantons or gauge dynamics on D7-branes has to vanish when a D3-brane hits the relevant cycle. This can be understood directly by examining and integrating out the massive D3-D7 strings in the latter case. This subtlety must be accounted for to get a globally well-defined , and we will see in a moment that this actually changes the inflaton mass term. Nevertheless, we will first study the simpler case , both because it reflects the essential features of the problem and because the full dependence of on is not known. Let us start by presenting a general argument which highlights a problem faced by any inflationary model involving a moving D3-brane in the models of . The main point is that one will choose some configuration with a positive energy . When the compact manifold is large then this energy will go to zero rather quickly, as a power of the volume modulus : where is a number of order one and the form of depends on the source of energy. This follows because in existing proposals the inflationary energy arises either from brane tensions or from fluxes, and all known brane and flux energies vanish as some power of . On the other hand the stabilization mechanism would fix (or else some combination of and rather than . This implies that as the brane moves and changes there will be a change in the potential, This will lead to a contribution to of order one, unless there is a compensating contribution to the mass term from some other source. One possible source of such a cancellation is a dependence of the superpotential on , not just . If then we would get an additional contribution to the mass term, so that at the minimum we find In principle the second contribution to the mass term might substantially cancel the first, alleviating the problem of the inflaton mass. This would certainly require fine-tuning at the level of one percent (in order to make sufficiently small to allow sixty e-foldings). More importantly, the dependence of on is not known, so the question of which models admit such fine-tuning cannot be answered at present. We should emphasize that the problem we are discussing is quite general, but one might well be able to find non-generic configurations in which the problem is absent. Let us discuss these issues more concretely for the case of a brane-antibrane pair transverse to a stabilized Calabi-Yau. In principle one should be able to compute the inflaton potential directly, by substituting the complete superpotential into the supergravity F-term potential and possibly including the effects of D-term contributions. This turns out to be a rather subtle problem, essentially because of the breaking of supersymmetry in the brane-antibrane system. We will begin instead by understanding the (supersymmetric) system of a single D3-brane transverse to a Calabi-Yau. We will find that superpotential stabilization of the volume necessarily generates mass terms for the scalars which describe the motion of the D3-brane. An implicit assumption in brane-antibrane inflation scenarios is that the brane and antibrane are free, in the absence of interactions, to move around the Calabi-Yau; the gentle force from their Coulomb interaction is then expected to lead to a relatively flat inflaton potential. Significant mass terms for the D3-brane (or any external forces on the D3-brane) invalidate this assumption and make inflation impossible. Let us therefore consider the effective potential governing a D3-brane transverse to a Calabi-Yau manifold. We substitute the superpotential and the Kähler potential (5.1) into (5.11), where the physical volume modulus is given by (5.3). The resulting four-dimensional effective potential is In the vicinity of a point in moduli space where , this can be simplified to We must now incorporate the effects of an anti-D3-brane. In the scenario of the superpotential (5.8) stabilized the compactification volume and generated a negative cosmological term . The positive, warped tension of an anti-D3-brane was added to this to produce a small positive cosmological constant. In our notation, the anti-D3-brane induces an additional term in the effective potential (5.12), where is a positive constant. Notice that this induced term differs from the one in by a factor of . This arises because the anti-D3 tension in the warped compactifications of scales like , and in the highly warped regime, . This does not alter the conclusions of , though it changes the numerology. Suppose that the potential (5.14) has a de Sitter minimum at . We will now compute the mass of the D3-brane moduli in an expansion about this minimum. To simplify the analysis we assume that at the minimum is real, and also that for real , is real. The canonically normalized scalar which governs the motion of the D3-brane is not but is instead a rescaled field ; it is the mass of which we will compute. First, we rewrite (5.14) as where primes denote derivatives with respect to , and define by This means that the field acquires the mass This is in fact precisely the result one would obtain for a conformally coupled scalar in a spacetime with cosmological constant . This is most easily understood by setting and studying the resulting . The four-dimensional AdS curvature is , so that (5.18) corresponds to a coupling If the D3-brane is in a highly warped region this result could have been anticipated, since this highly warped region is dual to an almost conformal four-dimensional field theory and the scalar field describing the motion of the brane is conformally coupled (see )†† Note that the kinetic term for is of the form .. The derivation of (5.19) is also valid even when the D3-brane is far from the near horizon region. We now see that the D3-brane moduli masses are necessarily of the same scale as the inflationary energy density , since during inflation, the extra antibrane(s) simply sit at the end of the throat and provide an energy density well-modeled by (5.14). It is straightforward to verify that such masses lead to a slow-roll parameter , incompatible with sustained slow-roll inflation. It is instructive to compare this result with the well-known -problem, which bedevils most models of F-term inflation in supergravity. One begins by asking whether slow-roll inflation is possible in a model of a single field with any type of Kähler potential and any superpotential . For a minimal Kähler potential and a generic superpotential one typically has a inflaton mass , and hence no inflation, just as in the generic case considered in the present paper. But this does not mean that inflation in supergravity is impossible. Various superpotentials with non-generic dependence on have been found, some of which permit inflation. For example, in supergravity with the canonical Kähler potential and a linear superpotential for the inflaton, the mass term contribution to the potential cancels: A similar effect occurs for the superpotential , which leads to a simple realization of F-term hybrid inflation . Moreover, the dangerous mass terms for the inflaton do not appear at all in D-term inflation . It is quite possible, therefore, that one could find a consistent inflation scenario in string theory by studying superpotentials which depend on the inflaton field. As mentioned above, this would undoubtedly require a fine-tuned configuration in which two contributions to the mass cancel to high precision. We treat this question in detail in Appendix F, where we show that the introduction of a superpotential depending on the inflaton field leads to a modification of the mass-squared of the inflaton field which could make it much smaller (or much greater) than . This issue merits further investigation, which should become possible as we learn more about the detailed dependence of on the background geometry and on the fluxes in string compactifications. 5.2. Scenario II: Kähler Stabilization One model of stabilization that would be compatible with the inflationary scenario of §3,4 is the following. We have seen that the true Kähler-invariant expansion parameter which controls the expansion in these models, is . Furthermore, and have independent kinetic terms. A method of directly stabilizing could freeze the volume directly, without stopping inflation. Since is not a chiral superfield itself, stabilization via effects in the superpotential cannot accomplish this. However, given that , one imagine that corrections to the Kähler potential could directly stabilize . In fact, Kähler stabilization has been proposed earlier for rather different reasons (see e.g. , which discusses Kähler stabilization of the heterotic string dilaton). Here we would need the corrections to (5.1) to break the no-scale structure and fix . Some of these corrections have been calculated (see e.g. ). The subset of terms presented in does not lead to this kind of stabilization, though there are likely to be other terms at the same orders which could change this conclusion. However, Kähler stabilization would be very difficult to find in a controlled calculation, so one might simply have to state it as a model-building assumption. If one does assume that is stabilized by corrections to the Kähler potential, then the models of §3,4 could be realized in the framework of . In Appendix C we show that in these models one can easily satisfy observational constraints such as the number of e-foldings and the size of the density perturbations. One of the most promising ideas for obtaining inflation in string theory is based on brane cosmology. However, brane-antibrane inflation suffers from various difficulties when one tries to embed it in full string compactifications with moduli stabilization, such as the (metastable) de Sitter vacua of . We have argued here that some of these difficulties can be resolved by introducing highly warped compactifications. The warped brane-antibrane models introduced in general form in §3 and in a compact string theory example in §4 give rise to slow-roll inflation with an exponentially flat potential. In the compact example, the slow-roll parameters and the density perturbations can be fixed at suitable values by an appropriate choice of discrete fluxes in the warped region. The above discussion assumes a suitable stabilization mechanism for the volume modulus of the compactification manifold. As described in §5, this is a highly nontrivial issue. Indeed, we have found that if one stabilizes the moduli as in then this field acquires an effective mass-squared , making inflation impossible. As discussed in §5.1, fine-tuned dependence of the superpotential on could reduce this mass. With generic dependence on the problem persists. The arguments leading to our conclusion that generic methods of stabilization stop inflation are rather general, and should apply to any system where the energy density depends on the volume modulus as with . There are general arguments that this should always be the case, for the sources of energy we know about in string theory . Thus, it appears very difficult to achieve slow-roll brane inflation in a manner compatible with stabilization of the compactified space in string theory. At the very least, it is challenging to find a model which works for generic forms of the stabilizing superpotential, which itself varies in a way that depends on all of the microscopic details of the compactification. In those non-generic cases where inflation is possible, the inflationary predictions will depend on the details of the moduli stabilization. One should note that the degree of fine-tuning required for slow-roll inflation in these models is not extraordinary (see Appendix F), and may well be attainable within the large class of known models. Moreover, even though fine-tuning is certainly undesirable, it may not be a grave problem. Indeed, if there exist many realizations of string theory, then one might argue that all realizations not leading to inflation can be discarded, because they do not describe a universe in which we could live. Meanwhile, those non-generic realizations which lead to eternal inflation (see Appendix D) describe inflationary universes with an indefinitely large and ever-growing volume of inflationary domains. This makes the issue of fine-tuning less problematic. It will not escape the reader’s notice that this argument is anthropic in nature [31,32,33]. It is worth pointing out that it is an independent, presumably well-defined mathematical question, whether or not string theory has solutions which are consistent with present experiments (e.g. which contain the standard model of particle physics, have sufficiently small cosmological term, and allow early inflation). This question can of course be studied directly (see e.g. for recent work in this direction), and is an important one for string theorists to answer. Only if string theory does admit such solutions, does anthropic reasoning in this context become tenable. The large diversity of string vacua makes it reasonable to be optimistic on this score. We have primarily focused on the implications of superpotential stabilization of the moduli for D3-brane/anti-D3-brane inflation. Our analysis has implications for other models of brane inflation as well. These include systems and -branes at angles with . In these cases, Chern-Simons couplings will generically induce a -brane charge on the branes due to the presence of a non-trivial field. Such a charge will also be generated due to the curvature couplings for generic topologies of the cycles the branes wrap. If the induced charge is of order unity or more, the discussion of the previous section will apply. The volume modulus and the inflaton field will mix non-trivially in the Kähler potential and as a result a superpotential of the kind considered in §5.1, or in fact any source of energy which scales like , will generically impart an unacceptably big mass to the inflaton. It would be interesting to explore the special cases where such a charge is not induced, to see if one can make simple working models of brane inflation. Other existing proposals for brane inflation depend on Fayet-Iliopoulos terms in the low-energy field theory . The status of these FI terms in the effective supergravity arising from compactified string theory therefore merits careful investigation. String theory models with D-terms were realized in brane constructions [14,17] without consideration of volume stabilization. A consistent embedding of this model into compactified string theory is under investigation . We would like to thank C. Burgess, M. Dine, M. Fabinger, A. Guth, L. Kofman, S. Prokushkin, F. Quevedo, N. Seiberg, A. Sen, M. M. Sheikh-Jabbari, S. Shenker, E. Silverstein, L. Susskind, S. Thomas, H. Tye and E. Witten for helpful discussions. This research was supported by NSF grant PHY-9870115. The work of S.K. was also supported in part by a David and Lucile Packard Foundation Fellowship for Science and Engineering and the DOE under contract DE-AC03-76SF00515. The research of JM is supported in part by DOE grant DE-FG02-90ER40542. The work of L.M. was supported in part by a National Science Foundation Graduate Research Fellowship. S.P.T. acknowledges support from the Swarnajayanti Fellowship, DST, Govt. of India, and most of all from the people of India. Appendix A. General Discussion of Brane-Antibrane Potentials Here we compute the gravitational force between a D3-brane and an anti-D3-brane which are transverse to a general compact six-dimensional space. We assume that there is no warping before we add the D-branes. Our objective is to compute the expression for the slow-roll parameter (2.4) in this setup. For this purpose we note that the brane tension as well as the ten-dimensional Planck mass drop out from the expression for if we express it in terms of the physical distance. We can therefore set , , to avoid clutter in the equations. The action for the system has the form where is a constant we will determine below. Here is the gravitational potential on the internal space. The equation of motion from (A.1) is Treating one brane as the source and the other as a probe and comparing with (2.1) we see that .†† Note that (2.1) contains a contribution both from gravity and from the Ramond fields, so the gravity contribution is half of that in (2.1). The expression for the energy of branes is thus where the subscript in indicates that we compute the potential due to the other branes, with , and evaluate it at . There is also a self-energy correction. We assume that the latter is independent of position. This is true in homogeneous spaces, such as tori. The equation of motion (A.2) is not consistent since all the charges on the left hand side of (A.2) have the same sign. A minimal modification that makes the equation consistent is to write it as where is the volume of the compact manifold. This term comes naturally from the curvature of the four-dimensional spacetime, which, in the approximation that we neglect the potential, is de Sitter space. This positive curvature gives rise to a negative contribution to the effective potential in the six internal dimensions. It is reasonable to assume that the negative term is smeared over the compact space as in this minimal modification, as long as the transverse space is approximately homogeneous.†† In compactifications with orientifold planes, there would also be localized negative terms. However, these would be cancelled by the tensions of the branes which are present even after brane/antibrane annihilation. The extra energy of the inflationary brane/antibrane pair can be expected to induce a smeared negative contribution over and above the orientifold plane contribution. Note that this term does not arise for the Ramond fields since the total charge is zero. Let us now consider, for simplicity, the case of a single brane and a single antibrane. In order to compute we compute the Laplacian of the potential with respect to . We get The subscript in indicates that this is the potential due to the brane at . For a pair of branes the potential is . The Laplacian has a constant negative value (A.5). We see that this implies that there exists at least one direction in which the second derivative has a value , since there are six transverse dimensions. When we compute the contribution to the factor cancels out. When there are many fields, one should consider as a matrix. In order to have slow roll inflation we need to demand that the matrix has no negative eigenvalue that is too large. If we have a large negative eigenvalue, then even if the scalar field is not initially rolling in that direction, it will typically start moving in this direction after a few e-foldings. The discussion above implies that , viewed as a matrix, has an eigenvalue more negative than This implies that at least one of the moduli acquires a tachyonic mass , which typically prevents a prolonged stage of inflation. A similar analysis can be carried out for the general case of a Dp-brane/anti-Dp-brane system. It is easy to see that the only change is that the coefficient in (A.6) is replaced by . More interestingly, the above analysis can also be applied to the case of Dp-branes at angles. By this we mean a system of slightly misaligned branes and orientifold planes, . The supersymmetry breaking scale in such a system is controlled parametrically by an angle which measures the relative orientation of the branes. For small values of this angle, the vacuum energy, , obtained by summing over all the branes and planes, can be much smaller than the tension of any individual brane or plane. The force on a brane in such a system arises due to graviton-dilaton and RR exchange. In these systems there can be a cancellation between the graviton-dilaton and the RR force in such a way that the resulting force, computed with non-compact “internal” dimensions, is parametrically smaller than the value of the cosmological constant. Once the internal dimensions are compact, we have to make some modification of the gravitational equation in order to make it consistent. The simplest modification is to add a constant term on the right hand side of the corresponding Laplace equation. In this case the constant term will be proportional to the effective four dimensional cosmological constant. Then, repeating the analysis above, one finds that the resulting potential satisfies the inequality As a result, once again one obtains a value of , (A.6), with the coefficient replaced by . In other words, both the potential and its second derivative scale in the same way with the small angle which supresses supersymmetry breaking, making independent of this angle. Appendix B. Computation of the Potential in Warped Geometries To calculate the potential it is actually easier to turn things around and view the D3-brane as perturbing the background and then calculate the resulting energy of the anti-D3-brane in this perturbed geometry. This of course gives the same answer for the potential energy of the brane-antibrane pair. The coupling of the metric and the five-form to the D3-brane is given by (3.6). On general grounds one expects that the changes in the metric and caused by the D3-brane will vary in the directions transverse to the brane. These directions are spanned by the radial coordinate and the directions along . It is useful to observe that the background can be written as follows: where is the line element on , and is given by It is easy to check that is a harmonic function in a six-dimensional space spanned by and the directions along , with metric Adding one additional D3-brane at a radial location results in a perturbed background which is of the form (B.1), but with a harmonic function now given by solves the equation in the six-dimensional space (B.4).†† The constant is determined by the tension of the D3-brane. For a simple calculation shows that independent of and the detailed metric on . In (B.6) the coefficient arises because the ambient background is supported by units of charge, whereas the perturbation we are interested in arises due to a single D3-brane. From (B.5) the resulting harmonic function is To determine the potential we now couple this new background to the anti-D3-brane. The anti-D3-brane is described by an action of the form (3.6), except that, as was mentioned before, the sign of the Chern-Simons term is reversed relative to the case of a D3-brane. We also remind the reader that the antibrane is located at ; we will assume that . Combining all these results, after a simple calculation one recovers the desired potential (3.9). This calculation of the potential is valid for one brane-antibrane pair. For one brane and antibranes, to leading order, (3.9) is simply multiplied by . Corrections to this leading-order potential are suppressed for small . Appendix C. Warped Inflation In this appendix we discuss how inflation would look if one managed to fix the overall volume modulus without giving a mass to the brane motion. We argued above that the low energy dynamics of the system is described by the action (3.10). The radial position of the D3-brane, , will play the role of the inflaton below. We define a canonically normalized field and . The effective action is then given by We have assumed that there are no significant additional terms in the effective action (C.2). This inflaton potential is extremely flat: the first term in the potential, which is independent of the inflaton, is larger than the second term by a factor proportional to . This factor can be interpreted as the relative redshift between the brane location and the antibrane location ; as we explained in §4, this redshift is exponentially sensitive to the parameters of the model: where is the string coupling and are integers that specify fluxes turned on in the compactification. The slow-roll parameters can now be calculated in standard fashion. We will use conventions where . One finds that Slow-roll requires that . Of these the condition on is more restrictive. It can be met by taking The number of e-foldings is given by Requiring can be achieved by taking to be sufficiently large and is compatible with the bound (C.5). Finally, the adiabatic density perturbations are given by (, page 186) This quantity should be equal to at , when the perturbations responsible for the large scale structure of the observable part of the universe are produced. After some algebra, can be expressed in terms of as follows: is a constant which is somewhat model dependent; using (C.7) and (C.2), one has and after using (4.5), (3.7) one finds that for the model of §4.†† increases by a factor when there are antibranes. While making numerical estimates we set . The four-dimensional Planck scale () is given by where is the volume of the Calabi-Yau. This formula is strictly applicable only to a Kaluza-Klein compactification, not a warped compactification of the kind considered here. However, the approximation is a good one since the graviton zero mode has most of its support away from the regions with large warping (where its wave function is exponentially damped.) We may express the brane tension as This dimensionless ratio evidently depends on the string coupling constant and the volume of the six compact dimensions. The value is quite reasonable: it corresponds to and a Calabi-Yau volume of a characteristic size . Larger values of lead to smaller values for , which make it easier to meet the density perturbation constraints. More important, for present purposes, is the factor , which has its origins in the redshift suppression of the potential that was emphasized in the discussion above. By taking this factor to be small enough we see that the constraint on , (C.7), can be met. As an example, taking and , we find that for . This condition on can easily be met for reasonable values of the flux integers . Taking , we get , with . The latter condition can be achieved using moderate values of flux, e.g. . Now that we have ensured that the various constraints can be met in our model, it is worth exploring the resulting inflationary scenario a little more. The energy scale during inflation can be expressed in terms of . One finds from (C.8), and using the fact that the potential is well approximated by the first term in (3.9), that Taking , , and GeV one finds that the energy scale is This is considerably lower than the GUT scale GeV. This low scale of inflation is a generic feature of the scenario. Next, it is straightforward to see that is given in terms of and by Solving for from (C.12) gives Taking , , gives a very small number. The ratio of the anisotropy in the microwave background generated by gravitational waves to that generated by adiabatic density perturbations is given (at large ) by In our model this is very small, so the anisotropy is almost entirely due to density perturbations. Finally, can be related to , and is given by Clearly, as we mentioned above, . The tilt parameter is given by

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Therefore, the correct ratio is 2:1. Question 8, 63% correct — calculating discount and sale price Most common wrong answer: B; students calculated the discount alone without subtracting from the total Question 9, 39% correct — measurement conversions and comparing unit price Most common wrong answer: C; this is the unit price of the bulk raisins. Our worksheets use a variety of high-quality images and some are aligned to Common Core Standards. See my Do Now in my Strategy folder that explains my beginning of class routines. For the given test scores, one natural way to set the intervals is 50 — 59, 60 — 69, 70 — 79, 80 — 89, and 90 — 100. Apply the properties of operations to generate equivalent expressions. The test scores in Mr. Some noise canceling headphones and dividers are provided. Next calculate the number of values that fall within each interval, i. The answer key is automatically generated and is placed on the second page of the file. 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If a currently accepted theory is considered to be modern, and its introduction. In classical, mechanics, the exact same mathphysics might be involved to model an electron orbit as planetary orbit. Classical physics definition at, a free online dictionary with pronunciation, synonyms and translation. Mathematics of classical and quantum physics dover books on physics. It is part of a fourvolume textbook, which covers electromagnetism, mechanics. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. This book provides an illustration of classical mechanics in the form of problems at. This first course in the physics curriculum introduces classical mechanics. Classical mechanics including an introduction to the theory of. This textbook provides an introduction to classical mechanics at a level. It provides teachers and students with new examples and tools based on contemporary research in physics. A course in classical physics 1mechanics alessandro. Free classical mechanics books download ebooks online. Alessandro bettini has fulfilled the ambitious goal of writing a treatise that covers all of classical physics with a depth. There is another book which uses classical mechanics to familiarize its readers with the math. This is a minimalist textbook for a first semester of university, calculusbased physics, covering classical mechanics including one chapter on mechanical waves, but excluding fluids, plus a brief. The authors do however remain concrete in their treatment, with realworld examples permeating the text. This classic book enables readers to make connections between classical and modern physics an. Rae, quantum mechanics, crc press, boca raton, ann arbor, london, tokyo, 5 th. Classical physics classical physics includes the traditional branches and topics that were recognized and welldeveloped before the beginning of the 20th century classical mechanics, acoustics, optics. This textbook provides an introduction to classical mechanics at a level intermediate between the typical undergraduate and advanced graduate level. Develop an understanding far beyond the standard popular science discussions. For 30 years, this book has been the acknowledged standard in advanced classical mechanics courses. Classical mechanics wikibooks, open books for an open world. Moreover, if a quantum system has a classical analogue, then for the limit h 0, it must yield the corresponding classical results. This new edition of classical mechanics, aimed at undergraduate physics and engineering students, presents ina. This firstyear, graduatelevel text and reference book covers the fundamental concepts and twentyfirstcentury applications of six major areas of classical physics that every masters or phdlevel physicist. In the case of quantum mechanics, this correspondence may be specified by claiming that, for large quantum numbers, quantum theory must be consistent with classical physics. The first volume is devoted to the mechanics of point particles and rigid bodies. List of textbooks on classical mechanics and quantum mechanics. This first volume covers the mechanics of point particles, gravitation, extended systems starting from the twobody system, the basic concepts of relativistic mechanics and the mechanics of rigid bodies and fluids. This book, although short, is a fairly good overview of classical mechanics, which emphasizes more recent developments in the theory, such as chaotic dynamical systems. Aside from the standard topics of mechanics in the physics curriculum, this book includes an introduction to the theory of elasticity and its use in selected modern. Are there modern 1st year university physics textbooks using oldschoool layout, i. For physicists, however, the real importance of classical mechanics lies not so much in the vast range of its applications as in its role as the base on which the whole pyramid of modern physics has been. Classical mechanics is concerned with bodies acted on by forces and bodies in motion and may be divided into statics study of the forces on a body or. Modern classical physics princeton university press. Classical mechanics wiley online books wiley online library. This text describes the background and tools for use in. This book is a formulation of the work there attempted. Motion in one dimension and vectors, motion in two and three dimensions and force and motion, work and kinetic energy. This is a minimalist textbook for a first semester of university, calculusbased physics, covering classical mechanics including one chapter on mechanical waves, but excluding fluids, plus a brief introduction to thermodynamics. Historically, a set of core conceptsspace, time, mass, force, momentum, torque, and angular momentumwere introduced in. Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories. This new edition of classical mechanics, aimed at undergraduate physics and. The wonderful quantum world in this video lecture, prof. The failure of classical physics and the advent of quantum. Kotkins collection of problems in classical mechanics. Classical mechanics illustrated by modern physics world scientific. This book provides an illustration of classical mechanics in the form of problems at the bachelor level inspired for most of them by contemporary research in physics, and resulting from the teaching and. Here are some of my favorite popular physics books. Shankar is a good modern introduction to quantum mechanics. This book complements the book solved problems in modern physics by the same author and published by springerverlag so that bulk of the courses for undergraduate curriculum are covered. List of textbooks on classical mechanics and quantum. A modern perspective mcgrawhill series in fundamentals of physics barger, vernon d. Classical physics classical physics includes the traditional branches and topics that were recognized and welldeveloped before the beginning of the 20th century classical mechanics, acoustics, optics, thermodynamics, and electromagnetism. Classical mechanics, second edition presents a complete account of the classical mechanics of particles and systems for physics students at the advanced undergraduate level. Walter lewin discusses classical mechanics, and in spite of all of its impressive predictive power, fails. This firstyear, graduatelevel text and reference book covers the fundamental concepts and twentyfirstcentury applications of six major areas of classical physics that every masters or phdlevel physicist should be exposed to, but often isnt. Classical mechanics is very useful for analyzing problems in which quantum and. In many fields of modern physics, classical mechanics plays a key role. This book provides an illustration of classical mechanics in the form of problems at undergraduate level inspired for the most part by contemporary research in. Classical mechanics books meet your next favorite book.1104 3 455 1434 1494 974 1164 412 1478 797 648 838 19 1379 1314 1070 840 1404 819 344 311 551 214 751 503 564 1465 1196 1511 835 990 1478 1024 877 498 1398 249 730 301 347 537 766

Evidence under Bayes' theorem This article needs additional citations for verification. (July 2013) This article possibly contains original research. (September 2008) The use of evidence under Bayes' theorem relates to the probability of finding evidence in relation to the accused, where Bayes' theorem concerns the probability of an event and its inverse. Specifically, it compares the probability of finding particular evidence if the accused were guilty, versus if they were not guilty. An example would be the probability of finding a person's hair at the scene, if guilty, versus if just passing through the scene. Another issue would be finding a person's DNA where they lived, regardless of committing a crime there. Among evidence scholars, the study of evidence in recent decades has become broadly interdisciplinary, incorporating insights from psychology, economics, and probability theory. One area of particular interest and controversy has been Bayes' theorem. Bayes' theorem is an elementary proposition of probability theory. It provides a way of updating, in light of new information, one’s probability that a proposition is true. Evidence scholars have been interested in its application to their field, either to study the value of rules of evidence, or to help determine facts at trial. Suppose, that the proposition to be proven is that defendant was the source of a hair found at the crime scene. Before learning that the hair was a genetic match for the defendant’s hair, the factfinder believes that the odds are 2 to 1 that the defendant was the source of the hair. If they used Bayes’ theorem, they could multiply those prior odds by a “likelihood ratio” in order to update her odds after learning that the hair matched the defendant’s hair. The likelihood ratio is a statistic derived by comparing the odds that the evidence (expert testimony of a match) would be found if the defendant was the source with the odds that it would be found if defendant was not the source. If it is ten times more likely that the testimony of a match would occur if the defendant was the source than if not, then the factfinder should multiply their prior odds by ten, giving posterior odds of 20 to one. Bayesian skeptics have objected to this use of Bayes’ theorem in litigation on a variety of grounds. These run from jury confusion and computational complexity to the assertion that standard probability theory is not a normatively satisfactory basis for adjudication of rights. Bayesian enthusiasts have replied on two fronts. First, they have said that whatever its value in litigation, Bayes' theorem is valuable in studying evidence rules. For example, it can be used to model relevance. It teaches that the relevance of evidence that a proposition is true depends on how much the evidence changes the prior odds, and that how much it changes the prior odds depends on how likely the evidence would be found (or not) if the proposition were true. These basic insights are also useful in studying individual evidence rules, such as the rule allowing witnesses to be impeached with prior convictions. Second, they have said that it is practical to use Bayes' theorem in a limited set of circumstances in litigation (such as integrating genetic match evidence with other evidence), and that assertions that probability theory is inappropriate for judicial determinations are nonsensical or inconsistent. Some observers believe that in recent years (i) the debate about probabilities has become stagnant, (ii) the protagonists in the probabilities debate have been talking past each other, (iii) not much is happening at the high-theory level, and (iv) the most interesting work is in the empirical study of the efficacy of instructions on Bayes’ theorem in improving jury accuracy. However, it is possible that this skepticism about the probabilities debate in law rests on observations of the arguments made by familiar protagonists in the legal academy. In fields outside of law, work on formal theories relating to uncertainty continues unabated. One important development has been the work on "soft computing" such as has been carried on, for example, at Berkeley under Lotfi Zadeh's BISC (Berkeley Initiative in Soft Computing). Another example is the increasing amount of work, by people both in and outside law, on "argumentation" theory. Also, work on Bayes nets continues. Some of this work is beginning to filter into legal circles. See, for example, the many papers on formal approaches to uncertainty (including Bayesian approaches) in the Oxford journal: Law, Probability and Risk . There are some famous cases where Bayes' theorem can be applied. - In the medical examples, a comparison is made between the evidence of cancer suggested by mamograms (5% show positive) versus the general risk of having cancer (1% in general): the ratio is 1:5, or 20% risk, of having breast cancer when a mammogram shows a positive result. - A court case which argued the probabilities, with DNA evidence, is R v Adams.

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