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Nijenhuis and compatible tensors on Lie and Courant algebroids We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures. Pairs of tensor fields on manifolds, which are compatible in a certain sense, were studied by Magri and Morosi , in view of their application to integrable hamiltonian systems. Besides Poisson-Nijenhuis manifolds – manifolds equipped with a Poisson bivector and a Nijenhuis -tensor which are compatible in such a way that it is possible to define a hierarchy of Poisson-Nijenhuis structures on these manifolds, the work of Magri and Morosi also covers the study of and structures. These are pairs of tensors formed respectively, by a closed -form and a Nijenhuis tensor () and a Poisson bivector and a closed -form () satisfying suitable compatibility conditions. Another type of structure that can be considered on a manifold is a Hitchin pair. It is a pair formed by a symplectic form and a -tensor that was introduced by Crainic in relation with generalized complex geometry. All these structures, defined by pairs of tensors, were studied in the Lie algebroid setting by Kosmann-Schwarzbach and Rubtsov and by one of the authors . Finally, we mention complementary forms on Lie algebroids, which were defined by Vaisman and also considered in and , and that can be viewed as Poisson structures on the dual Lie algebroid. The aim of the present paper is to show that all the structures referred to above, although they have different nature on Lie algebroids, once carried over to Courant algebroids, are all of the same type: they are Nijenhuis tensors. In this way, we obtain a unified theory of Nijenhuis structures on Courant algebroids. In order to include Poisson quasi-Nijenhuis structures with background in this unified theory, we consider a stronger version of this notion, which we call exact Poisson quasi-Nijenhuis structure with background. This seems to be the natural definition, at least in this context. We show that the structures defined by pairs of tensors on a Lie algebroid can also be characterized using the notion of compatible pair of tensors on a Courant algebroid, introduced in . An important tool in this work is the Nijenhuis concomitant of two -tensors on a Courant algebroid. It was originally defined for manifolds by Nijenhuis in and then extended to the Courant algebroid framework in and in . We use the Nijenhuis concomitant to study the compatibility of structures from the usual point of view, i.e., we say that two structures of the same type are compatible if their sum is still a structure of the same type. Thus, we can talk about compatible Poisson-Nijenhuis, and structures, as well as compatible complementary forms and compatible Hitchin pairs. The extension to Lie algebroids of the Magri-Morosi hierarchies of Poisson-Nijenhuis structures on manifolds, was done in . As it happens in the case of manifolds, the hierarchies on Lie algebroids are constructed through deformations by Nijenhuis tensors. In this paper we construct similar hierarchies of and structures on Lie algebroids, and their deformations, and also hierarchies of complementary forms. Elements of these hierarchies provide examples of compatible structures in the sense described above. Our computations widely use the big bracket – the Poisson bracket induced by the symplectic structure on the cotangent bundle of a supermanifold. The Courant algebroids that we shall consider in this paper are doubles of protobialgebroids structures on , in the simpler cases where is a function that determines a Lie algebroid structure on , or on , sometimes in the presence of a background (a closed -form on ). The paper is organized as follows. Section 1 contains a short review of Courant and Lie algebroids in the supergeometric framework while, in section 2, we recall the notion of Nijenhuis tensor on a Courant algebroid and of Nijenhuis concomitant of two tensors. In section 3, we characterize Poisson bivectors and closed -forms on a Lie algebroid as Nijenhuis tensors on the Courant algebroid . In section 4, we show how Poisson-Nijenhuis, and structures and also Hitchin pairs on a Lie algebroid can be seen either as Nijenhuis tensors or compatible pairs of tensors on the Courant algebroid . Considering, in section 5, the Courant algebroid with background , we see exact Poisson quasi-Nijenhuis structures with background as Nijenhuis tensors on this Courant algebroid, recovering a result in . For Poisson quasi-Nijenhuis structures (without background) a special case where two -forms involved are exact is also considered. The case of complementary forms is treated in section 6. Section 7 is devoted to the compatibility of structures on a Lie algebroid, defined by pairs of tensors. Sections 8, 9 and 10 treat the problem of defining hierarchies of structures on Lie algebroids. We start by showing, in section 8, that when a pair of tensors defines a certain structure on a Lie algebroid, the same pair of tensors defines a structure of the same kind for a whole hierarchy of deformed Lie algebroids. Then, in section 9, we construct hierarchies of structures defined by pairs of tensors and lastly, in section 10, we show that within one hierarchy, all the elements are pairwise compatible. We recall that if one relaxes the Jacobi identity in the definition of a Lie (respectively, Courant) algebroid we obtain what is called a pre-Lie (respectively, pre-Courant) algebroid. The proof of our results does not use the Jacobi identity of the bracket, whether if it is a Lie or a Courant algebroid bracket. Therefore, they also hold in the more general settings of pre-Lie and pre-Courant algebroids, respectively. 1. Courant and Lie algebroids in supergeometric terms We begin this section by introducing the supergeometric setting, following the same approach as in [19, 15]. Given a vector bundle , we denote by the graded manifold obtained by shifting the fibre degree by . The graded manifold is equipped with a canonical symplectic structure which induces a Poisson bracket on its algebra of functions . This Poisson bracket is sometimes called the big bracket (see ). Let us describe locally the Poisson bracket of the algebra . Fix local coordinates , , in , where are local coordinates on and are their associated moment coordinates. In these local coordinates, the Poisson bracket is given by while all the remaining brackets vanish. The Poisson algebra of functions is endowed with a -valued bidegree. We define this bidegree locally but it is well defined globally (see [19, 15] for more details). The bidegrees are locally set as follows: the coordinates on the base manifold , , , have bidegree , while the coordinates on the fibres, , , have bidegree and their associated moment coordinates, and , have bidegrees and , respectively. The algebra of functions inherits this bidegree and we set where is the -module of functions of bidegree . The total degree of a function is equal to and the subset of functions of total degree is noted . We can verify that the big bracket has bidegree , i.e., and consequently, its total degree is . Thus, the big bracket on functions of lowest degrees, and , vanish. For , is an element of and is given by where is the canonical fiberwise symmetric bilinear form on . Let us recall that a Courant structure on a vector bundle equipped with a fibrewise non-degenerate symmetric bilinear form is a pair , where the anchor is a bundle map from to and the Dorfman bracket is a -bilinear (not necessarily skew-symmetric) map on satisfying for all and . In this paper we are only interested in exact Courant algebroids. Although many of the properties and results we recall next hold in the general case, we shall consider the case where the vector bundle is the Whitney sum of a vector bundle and its dual, i.e., , and is the canonical fiberwise symmetric bilinear form. So, from now on, all the Courant structures will be defined on . From we know that there is a one-to-one correspondence between Courant structures on and functions such that . The anchor and Dorfman bracket associated to a given are defined, for all and , by the derived bracket expressions For simplicity, we shall denote a Courant algebroid by the pair instead of the triple . A Courant structure can be decomposed using the bidegrees: with and . We recall from that, when , is a Courant structure on if and only if is a Lie algebroid. The anchor and the bracket of the Lie algebroid are defined, respectively by for all and , while the Lie algebroid differential is given by 2. Nijenhuis concomitant of two tensors Let be a Courant algebroid and a vector bundle endomorphism of , . If for all , is said to be skew-symmetric. Vector bundle endomorphisms of will be seen as -tensors on . The deformation of the Dorfman bracket by a -tensor is the bracket defined, for all , by When is skew-symmetric, the deformed structure is given, in supergeometric terms, by . The deformation of by the skew-symmetric -tensor is denoted by , i.e., while the deformed Dorfman bracket associated to is denoted by . Recall that a vector bundle endomorphism is a Nijenhuis tensor on the Courant algebroid if its torsion vanishes. The torsion is defined, for all , by or, equivalently, by where . When , for some , (5) is given, in supergeometric terms, by The notion of Nijenhuis concomitant of two tensor fields of type on a manifold was introduced in . In the case of -tensors and on a Courant algebroid , the Nijenhuis concomitant of and is the map (in general not a tensor) defined, for all sections and of , as follows: where is the Dorfman bracket corresponding to . Equivalently, while if and anti-commute, i.e., , then For any -tensors and on , we have The concomitant of two skew-symmetric -tensors and on a Courant algebroid is given by : In other words, for all . The notion of Nijenhuis concomitant of two -tensors on a Lie algebroid can also be considered. If is a Lie algebroid and are -tensors on , is given by (7), adapted in the obvious way. Equations (8), (9), (10) and (14) also hold in the Lie algebroid case. As in the case of Courant algebroids, for a Lie algebroid , we use the following notation: , if is either a bivector, a -form or a -tensor on . 3. Tensors on Lie algebroids Let be a Lie algebroid and consider a -tensor , a bivector and a -form on . Associated with , , , and , we consider the skew-symmetric -tensors on , , , and given, in matrix form, respectively by In all the computations using the big bracket, instead of writing , , and , we simply write , , and . We use the -tensors on above to express the properties of being Nijenhuis, Poisson and closed on the Lie algebroid . Proposition 3.1 (). Let be a -tensor on such that , for some . Then, is a Nijenhuis tensor on the Lie algebroid if and only if is a Nijenhuis tensor on the Courant algebroid . The assumption is equivalent to . In this case, the torsion of on is given by (6), with , and coincides with the torsion of on . ∎ Let be the -tensor on , defined by The 2-form is closed on if and only if is a Nijenhuis tensor on the Courant algebroid . Recall that a bivector field on is a Poisson tensor on if or, equivalently, . Proposition 3.3 (). The bivector is a Poisson tensor on if and only if is a Nijenhuis tensor on the Courant algebroid . We have and, from (6), we get . ∎ Notice that the -tensors and anti-commute. Thus, from (14), we have Denoting by the -tensor on defined by and taking into account the fact that Proposition 3.3 admits the following equivalent formulation: The bivector is a Poisson tensor on if and only if is a Nijenhuis tensor on the Courant algebroid . 4. Pairs of tensors on Lie algebroids In we introduced a notion of compatibility for a pair of anti-commuting skew-symmetric -tensors on a Courant algebroid. Definition 4.1 (). A pair of skew-symmetric -tensors on a Courant algebroid with Courant structure is said to be a compatible pair, if and anti-commute and . In this section we show that well known structures defined by pairs of tensors on a Lie algebroid , can be seen either as compatible pairs, or as Nijenhuis tensors on the Courant algebroid . Let be a Lie algebroid. Recall that a pair , where is a bivector and is a -tensor on is a Poisson-Nijenhuis structure ( structure, for short) on if A pair formed by a -form and a -tensor on is an structure on if where or, equivalently, . A pair formed by a -form and a -tensor on is a Hitchin pair on if A pair formed by a bivector and a -form on is a structure on if where is the -tensor on defined by . Let us denote by the anti-commutator of two skew-symmetric tensors and , i.e., Let be a Lie algebroid, a Nijenhuis -tensor and a closed -form on . Then, the pair is an structure on if and only if is a compatible pair on . We start by noticing that , so that and anti-commute if and only if . Taking into account the fact that is closed, we have where in the last equality we used . Thus, the -form is closed if and only if . ∎ In the case where , for some , we have the following characterization of an structure. Let be a Lie algebroid, a closed -form on and a -tensor on such that , for some . Then, the pair is an structure on if and only if is a Nijenhuis tensor on and . and, by counting the bi-degrees, we have that is equivalent to For Hitchin pairs we obtain the following result: Let be a Lie algebroid, a -tensor on and a symplectic form on . Then, the pair is a Hitchin pair on if and only if is a compatible pair on . In the case of PN structures, we have: Let be a Lie algebroid, a Nijenhuis -tensor on and a Poisson bivector on . Then, the pair is a Poisson-Nijenhuis structure on if and only if is a compatible pair on . Notice that , so that and anti-commute if and only if . Also, we have . ∎ When , for some , we recover a result from , which is a characterization of Poisson-Nijenhuis structures. Let be a Lie algebroid, a bivector on and a -tensor on such that , for some . Then, the pair is a Poisson-Nijenhuis structure on if and only if is a Nijenhuis tensor on and . In we showed that, given a bivector and a -tensor on such that , for some , then is a PN structure on if and only if is a Poisson-Nijenhuis pair on the Courant algebroid . For structures, we have the following: Let be a Lie algebroid, a Poisson bivector on and a closed -form on . Consider the -tensor on defined by , and the corresponding -tensor on , . Then, the pair is a structure on if and only if is a compatible pair on . It is easy to see that and anti-commute. The -form being closed we have, taking into account the fact that , So, the -form is closed if and only if . ∎ 5. Exact Poisson quasi-Nijenhuis structures (with background) Let be a Lie algebroid, a closed -form on and consider the Courant algebroid with background . Poisson quasi-Nijenhuis structures with background on Lie algebroids were introduced in . We recall that a Poisson quasi-Nijenhuis structure with background on is a quadruple , where is a bivector, is a -tensor and and are closed -forms such that and , for all , , for all , with , for all , where means sum after circular permutation on , and . A Poisson quasi-Nijenhuis structure with background is called exact if , and condition (iv) is replaced by is proportional to , where , for all . In it is proved 222The quadruple considered in is and should be . that if is a Nijenhuis tensor on and satisfies , with , then the quadruple is a Poisson quasi-Nijenhuis structure with background on . It is easy to see that the same result holds for any . It is worth noticing that , , is equivalent to the three conditions: , and . Using the notion of exact Poisson quasi-Nijenhuis structure with background, we deduce the following (see the proof of Theorem 2.5 in ): Let be a Lie algebroid, a bivector, a -form, a closed -form and a -tensor on such that , and is proportional to . Then, is a Nijenhuis tensor on the Courant algebroid if and only if the quadruple is an exact Poisson quasi-Nijenhuis structure with background on . Notice that in Theorem 5.1, if , for some , then the constant of proportionality that should be considered in (iv’) is , i.e., . A Poisson quasi-Nijenhuis structure on a Lie algebroid is a Poisson quasi-Nijenhuis structure with background, with . This notion was introduced, on manifolds, in and then extended to Lie algebroids in . An exact Poisson quasi-Nijenhuis structure with background is called an exact Poisson quasi-Nijenhuis structure. In this case, the -form is also exact. Next, we consider and a special case where the assumption , , in Theorem 5.1 is replaced by , . Let be a Lie algebroid, a bivector, a -form and a -tensor on such that , and , for some . If is a Nijenhuis tensor on the Courant algebroid , then the triple is an exact Poisson quasi-Nijenhuis structure on . and, by counting the bi-degrees, we obtain that if and only if Applying both members of iii) to any , we get which gives, using (7),

Note: Please read the disclaimer. The author is not providing professional investing advice or recommendations. But one thing that bothered me when reading Greenblatt’s book was my memory of the Foolish Four. It was a similar market-trouncing “magic formula” that gained popularity in the late 90’s, only to later be discredited and deemed an artifact of data-mining. My understanding of how the wind was taken out of the Foolish Four’s sails was by investigating its performance over a longer time period. For example, it’s yearly excess return over the Dow goes from 10% to something closer to 2% when back-tested over 50 years instead of the original 20 years. So of course that becomes an undeniable hintergedanke when seeing the Magic Formula’s measly 17-year sample size. 🙁 And for those hung up on the fact that even a 2% alpha is respectable for the Foolish Four, it shrinks even further when you factor in the capital gains incurred by having to shuffle the deck, as it were, each year. Learning quantitative techniques to analyze problems such as these (i.e. is 17 years worth of data enough to conclude that the Magic Formula outperforms the S&P 500?) is exactly why I enrolled in the CFA program. And after getting a couple hundred pages into the first study guide at Level One, they cover precisely this sort of conundrum. So here is how it works. We do what is called a hypothesis test in statistics, whereby we hypothesize that the average return of the Magic Formula and the S&P 500 are actually identical! And then we do a few computations based on confidence intervals to see whether we can reject that hypothesis or not, and with what degree of confidence. As a big fan of the Magic Formula, I have to say I’m secretly hoping that it does have a statistically significant higher average yearly return than the S&P 500… First, Some Assumptions… Now in order to proceed, we have to make two assumptions. The first is that the yearly returns of these two investing techniques follow a mostly normal distribution. We’re supposed to feel comfortable making this assumption due to the central limit theorem. Briefly, it says that the distribution of any variable that is a function of a bunch of other random variables always tends to end up looking like a bell curve. The second assumption we’ll make is that the returns of the Magic Formula and S&P 500 are not independent. The CFA study guide advises that when comparing two investing strategies covering the same period of time, the returns of both depend on the same underlying market and economic forces present at that time, and therefore have some things in common. Step #1: Sample Mean Difference The first step is to compute the average difference of the yearly returns between the two strategies. This turns out to be 19.04%. Step #2: Standard Error of the Mean Difference Next we compute the sample standard deviation of the difference (in Excel, use stdev). This comes out to be 22.24%. We transform this into standard error of the mean difference (SEMD) by dividing by the square root of the sample size. The years 1988-2004 comprise 17 years. Step #3: Compute Test Statistic For normal, or mostly normal distributions and small sample sizes we use the t-test to check for statistical significance. Basically this just gives us a number that we can compare to a t-test table in order to determine whether the difference we’re seeing between the Magic Formula and S&P 500 appears to be important given the sample size. The smaller the sample size, the greater the t-test hurdle our data will have to clear in order to be able to conclude that the two don’t have the same mean. The test statistic is simply the sample mean difference minus the hypothesized mean difference, divided by the SEMD. Step #4: Pick a Significance Level and Compare Finally we need to decide on a level of significance and do our table compare. Most of these sorts of tests in the CFA curriculum seem to use 5%. This means that in our comparison, there will only be a 5% chance that there is in reality no difference between the Magic Formula and S&P 500, but we fail to detect this. In addition to level of significance, the only other parameter we require to do our table look-up is the degrees of freedom. But that’s easy as it’s simply the sample size minus 1. Given these two parameters we could now find a t-test table to do a critical value look-up, but this can be a little tedious, not to mention the additional step of converting one-sided to two-sided. I prefer to just use Excel’s tinv function. And since our t-value from Step #3 was 3.526, which is much > 2.120 we can easily reject the hypothesis that the means are equal at the 5% level of significance. Not only that, we can assume the means are different by as much as 7.59% before we start running up against our critical value of 2.120. So the Magic Formula’s outperformance appears not to just be an artifact of small sample size, and also appears to be of significant magnitude. But What About Risk? Things are truly looking rosy for the Magic Formula. But a seasoned finance student will also compare the standard deviations of yearly returns in our first table up top and notice that the Magic Formula’s is higher (24.26% versus 17.87%). Standard deviation is a common (though debatable) quantifier for risk so it wouldn’t be uncommon to argue that the Magic Formula should have higher returns to compensate the investor for taking more risk. Well just as we tested the hypothesis that the means were equal, we can do the same with variance (square of standard deviation)… Step #5: Test Equality of Two Variances I’ll cut to the chase here just to say that there’s a simple equivalent of the t-test when testing for the equality of two variances, and it’s called the F-test. We come up with our F parameter by just computing the ratio of the two variances. And again we could do a manual look-up using an F-table, but why not just let Excel compute it for us with finv…. Therefore at the 5% significance level, because 1.843 < 2.333 we cannot reject the hypothesis that the variances are the same. In conclusion, past performance may be no indication of future results… blah blah blah… The important thing is that the Magic Formula points toward having the best of both worlds, a statistically significant higher annual rate of return versus the S&P 500 without a statistically significant higher level of risk. Win-win! It is interesting to see how confidence intervals allow us to tread into gray areas. A newbie might stop at Step #1 and claim that the Magic Formula beats the S&P 500 by an average of 19.04% per year. His antagonist might point to the small sample size and say that it makes that 19.04% estimate of mean… meaningless! But a statistician can state that he’s 95% sure that the Magic Formula outperforms the S&P 500 by at least 7.59% per year…. assuming normality. And much is indeed hinging upon our assumption of normality, which can and should be tested for. But that’s for another day…

Diagnostic odds ratio In medical testing with binary classification, the diagnostic odds ratio (DOR) is a measure of the effectiveness of a diagnostic test. It is defined as the ratio of the odds of the test being positive if the subject has a disease relative to the odds of the test being positive if the subject does not have the disease. The rationale for the diagnostic odds ratio is that it is a single indicator of test performance (like accuracy and Youden's J statistic) but which is independent of prevalence (unlike accuracy) and is presented as an odds ratio, which is familiar to medical practitioners. The diagnostic odds ratio is defined mathematically as: where , , and are the number of true positives, false negatives, false positives and true negatives respectively. As with the odds ratio, the logarithm of the diagnostic odds ratio is approximately normally distributed.[clarification needed] The standard error of the log diagnostic odds ratio is approximately: From this an approximate 95% confidence interval can be calculated for the log diagnostic odds ratio: Exponentiation of the approximate confidence interval for the log diagnostic odds ratio gives the approximate confidence interval for the diagnostic odds ratio. The diagnostic odds ratio ranges from zero to infinity, although for useful tests it is greater than one, and higher diagnostic odds ratios are indicative of better test performance. Diagnostic odds ratios less than one indicate that the test can be improved by simply inverting the outcome of the test – the test is in the wrong direction, while a diagnostic odds ratio of exactly one means that the test is equally likely to predict a positive outcome whatever the true condition – the test gives no information. Relation to other measures of diagnostic test accuracy The diagnostic odds ratio may be expressed in terms of the sensitivity and specificity of the test: It may also be expressed in terms of the Positive predictive value (PPV) and Negative predictive value (NPV): It is also related to the likelihood ratios, and : The log diagnostic odds ratio is sometimes used in meta-analyses of diagnostic test accuracy studies due to its simplicity (being approximately normally distributed). Traditional meta-analytic techniques such as inverse-variance weighting can be used to combine log diagnostic odds ratios computed from a number of data sources to produce an overall diagnostic odds ratio for the test in question. The log diagnostic odds ratio can also be used to study the trade-off between sensitivity and specificity by expressing the log diagnostic odds ratio in terms of the logit of the true positive rate (sensitivity) and false positive rate (1 − specificity), and by additionally constructing a measure, : It is then possible to fit a straight line, . If b ≠ 0 then there is a trend in diagnostic performance with threshold beyond the simple trade-off of sensitivity and specificity. The value a can be used to plot a summary ROC (SROC) curve. Consider a test with the following 2×2 confusion matrix: by “Gold standard”) We calculate the diagnostic odds ratio as: This diagnostic odds ratio is greater than one, so we know that the test is discriminating correctly. We compute the confidence interval for the diagnostic odds ratio of this test as [9, 134]. The diagnostic odds ratio is undefined when the number of false negatives or false positives is zero – if both false negatives and false positives are zero, then the test is perfect, but if only one is, this ratio does not give a usable measure. The typical response to such a scenario is to add 0.5 to all cells in the contingency table, although this should not be seen as a correction as it introduces a bias to results. It is suggested that the adjustment is made to all contingency tables, even if there are no cells with zero entries. - Sensitivity and specificity - Binary classification - Positive predictive value and negative predictive value - Odds ratio - ^ a b c d e f g h Glas, Afina S.; Lijmer, Jeroen G.; Prins, Martin H.; Bonsel, Gouke J.; Bossuyt, Patrick M.M. (2003). "The diagnostic odds ratio: a single indicator of test performance". Journal of Clinical Epidemiology. 56 (11): 1129–1135. doi:10.1016/S0895-4356(03)00177-X. PMID 14615004. - ^ Macaskill, Petra; Gatsonis, Constantine; Deeks, Jonathan; Harbord, Roger; Takwoingi, Yemisi (23 December 2010). "Chapter 10: Analysing and presenting results". In Deeks, J.J.; Bossuyt, P.M.; Gatsonis, C. (eds.). Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy (PDF) (1.0 ed.). The Cochrane Collaboration. - ^ Glas, Afina S.; Lijmer, Jeroen G.; Prins, Martin H.; Bonsel, Gouke J.; Bossuyt, Patrick M.M. (November 2003). "The diagnostic odds ratio: a single indicator of test performance". Journal of Clinical Epidemiology. 56 (11): 1129–1135. doi:10.1016/S0895-4356(03)00177-X. PMID 14615004. - ^ Gatsonis, C; Paliwal, P (2006). "Meta-analysis of diagnostic and screening test accuracy evaluations: Methodologic primer". AJR. American Journal of Roentgenology. 187 (2): 271–81. doi:10.2214/AJR.06.0226. PMID 16861527. - ^ a b c d Moses, L. E.; Shapiro, D; Littenberg, B (1993). "Combining independent studies of a diagnostic test into a summary ROC curve: Data-analytic approaches and some additional considerations". Statistics in Medicine. 12 (14): 1293–316. doi:10.1002/sim.4780121403. PMID 8210827. - ^ a b Dinnes, J; Deeks, J; Kunst, H; Gibson, A; Cummins, E; Waugh, N; Drobniewski, F; Lalvani, A (2007). "A systematic review of rapid diagnostic tests for the detection of tuberculosis infection". Health Technology Assessment. 11 (3): 1–196. doi:10.3310/hta11030. PMID 17266837. - ^ Cox, D.R. (1970). The analysis of binary data. London: Methuen. ISBN 9780416104004. - Glas, Afina S.; Lijmer, Jeroen G.; Prins, Martin H.; Bonsel, Gouke J.; Bossuyt, Patrick M.M. (2003). "The diagnostic odds ratio: a single indicator of test performance". Journal of Clinical Epidemiology. 56 (11): 1129–1135. doi:10.1016/S0895-4356(03)00177-X. PMID 14615004. - Böhning, Dankmar; Holling, Heinz; Patilea, Valentin (2010). "A limitation of the diagnostic-odds ratio in determining an optimal cut-off value for a continuous diagnostic test". Statistical Methods in Medical Research. 20 (5): 541–550. doi:10.1177/0962280210374532. PMID 20639268. S2CID 21221535. - Chicco, Davide; Starovoitov, Valery; Jurman, Giuseppe (2021). "The benefits of the Matthews correlation coefficient (MCC) over the diagnostic odds ratio (DOR) in binary classification assessment". IEEE Access. 9: 47112–47124. doi:10.1109/ACCESS.2021.3068614.

Add your own favourite book of extension material for teachers and lecturers of mathematics students aged 16 to 19, with a brief description. Follow the style of the entries below. Felix Klein, Elementary Mathematics from an Advanced Standpoint (Dover 2004), 2 Vols These are the first two volumes of the three-volume German edition, Elementarmathematik vom höheren Standpunkte aus (J. Springer, Berlin, 1924-1928). László Lovász, Trends in Mathematics: How they could Change Education? (2006) H. Behnke, F. Bachmann, K. Fladt (Eds.), Fundamentals of Mathematics (MIT Press, 1974) Translation from the German of Grundzüge der Mathematik, Vandenhoeck & Ruprecht, Göttingen, 1962. 3 Vols. The book is the translation of a work commissioned by the ICMI in order to support the scientific foundations of instruction in mathematics, which was one of the topics chosen by the Commission at a meeting in Paris in 1954, in preparation for the International Congress of Mathematicians in Edinburgh in 1958. The English book has three volumes with 42 chapters: the first volume concerns the foundations of Mathematics, the Real Number System and Algebra; the second volume is devoted to geometry and the third to Analysis. There are, in general, two authors for each chapter: one a university researcher, the other a teacher of long experience in the German educational system. And the whole book has been coordinated in repeated conferences, involving altogether about 150 authors and coordinators. Martin Gardner, Mathematical Puzzles and Diversions (Penguin, 1956) The first of his series of books derived from his columns in Scientific American from 1956 to 1981 . Categorised as “Recreational Mathematics”, these are accessible to those with secondary school mathematics and a willingness to explore mathematical ideas. Albert Cuoco, Mathematical Connections: A Companion for Teachers (2005) This is a resource book for high school teachers. See the review by Steve Maurer. Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, 1979) GEB takes the form of an interweaving of various narratives. The main chapters alternate with dialogues between imaginary characters, inspired by Lewis Carroll's "What the Tortoise Said to Achilles" , in which Achilles and the Tortoise discuss a paradox related to modus ponens. Hofstadter bases the other dialogues on this one introducing characters such as a Crab, a Genie, and others. These narratives frequently dip into self-reference and metafiction. Word play also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's Magnificat in D; SHRDLU, Toy of Man's Designing" with Bach's Jesu, Joy of Man's Desiring; and "Typographical Number Theory", which inevitably reacts explosively when it attempts to make statements about itself. One Dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid and musical varieties), which is titled "Djinn and Tonic". One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines which, upon close inspection, double as an answer to a question in the next line. Zalman Usiskin, Dick Stanley, Anthony Peressini, Elena Anne Marchisotto, Mathematics for High School Teachers: An Advanced Perspective (Prentice Hall, 2003) Philip Davis & Reuben Hersh, The Mathematical Experience (Birkhauser, 1981) [This book] discusses the practice of modern mathematics from a historical and philosophical perspective. It won the 1983 National Book Award in the Science category. It is frequently cited by mathematicians as a book that was influential in their decision to continue their studies in graduate school and has been hailed as a classic of mathematical literature. The book drew a critical review from Martin Gardner, who disagreed with some of the authors' philosophical opinions, but was well-received otherwise. A study edition and also a study guide for use with the book have been released, both co-authored with Elena A. Marchisotto. The authors wrote a follow-up book, Descartes' Dream: The World According to Mathematics, and both have separately written other books on related subjects, such as Davis' Mathematics And Common Sense: A Case of Creative Tension and Hersh's What is Mathematics, Really? Philip Davis & Reuben Hersh, Descartes' Dream: The World According to Mathematics (Houghton Mifflen, 1986) Martin Aigner & Günter Ziegler, Proofs from THE BOOK (Springer-Verlag, 1998) Lynn Arthur Steen (Ed), Mathematics Today: Twelve Informal Essays. (Springer-Verlag, 1978) L.A. Steen has written many books and articles on [http:// mathematics], from this very early one to a follow-up Mathematics Tomorrow (Springer-Verlag, 1981)

It isn't, but if any reference other than neutral is chosen, you have said, the 2-phase no longer exists. So why must the neutral point be the reference in order to determine the number of phases? Because of the way the system is defined. Don't get the idea that I am saying that voltages 'disappear'. What I am saying is that the voltage will not fit in the particular system phase-counting bucket. You have to pick a reference in order to define a voltage. There is no universal designation for picking a reference point. Neither is there a universal designation to say which voltage has the angle from which all other angles will be measured. There are several systems we can define using the transformer terminals. The transformer can supply any of these different systems. If the transformer is capable of supplying more than one order of system, I am saying the transformer is a supply of the highest order. There are several type loads we can serve. Single-phase loads use only one pressure wave. Two-phase loads use two different pressure waves. Three-phase loads... you get the idea. Whether the voltages to these loads will all fit into a single phase-counting system bucket has to be determined. Let's use the 240/480 volt transformer with the X1, X23, X4 terminals for example so we can put in some real numbers for some example systems: System 1: A 240 volt single-phase system System 2: A 480 volt single-phase system System 3: A 240 volt two-phase system Now pick a voltage reference and let's see what possible systems we can have: X1 (or X4) Reference: We can have two single-phase systems, each with one voltage (one at 240 volts and one at 480 volts). Both voltages are present, but they are classified into two different systems. X23 Reference: We can have two single-phase systems with one voltage each. One single-phase system is 240@0 and the other single-phase system is 240@180. This system can serve two independent loads from separate sides of the winding. We can also have one two-phase system with two voltages. This type system serves loads that use both voltages and one voltage is 240@0 and the other voltage is 240@180. Whether we have two systems with one voltage each or one system with two voltages, both voltages are used. I was discussing the single current on the primary side of the transformer creating a corresponding current 'direction' in a single center tapped secondary winding. Un-equal loads can cause the "neutral" to be different from the natural neutral point. With these loads, the neutral is just a grounded conductor. It is forcing the common connection point away from a point of natural equilibrium. Regardless of the direction you pick to be positive, there are times when the current in one side of the coil is positive and the current in the other side of the coil is negative. They are actually flowing in two different directions. Even when in the same direction, the flux can be different because of a magnitude difference. Looking at the primary current is not representative of what is going on in the secondary because the net current in the primary is going to be based on the net flux of two different secondary currents. By looking at the primary current, the real secondary currents are hidden from you because you can't see the original individual fluxes. Without the neutral, the currents would stabilize such that the voltage balance point was located inside the load away from the common connection point. In that case, there is no unbalanced current and the current in both sides of the secondary coil would be exactly the same. At that point, you would just have one single-phase load, one flux, etc. I did not say that a grounded conductor must be counted, only that it may be. It is you who is putting requirements on which conductors must be used. I am looking for a definition that does not change based on which reference is used. I have given you a method but you haven't been able to see it yet. It is a general definition that can apply to more than one type system. The formula to determine the available system types and which voltages you can put into each system bucket for counting phases does not change. How you configure the source can restrict the number of available systems. Saying you don't want the neutral to make a change is like saying you are okay with saying we have a 480 volt source, but don't want to recognize that you can also get 240 volts if you use the center-tap. The neutral gives you an option you did not have without it. There are two pairs of two L-L voltages. L1-L2 and L1'-L2'. L1-L1' and L2-L2' were never considered as valid connections (similar to the high leg in a 240/120 connection). This is why I refer to this as 2-phase 5-wire. The high-leg is a valid connection but since its voltage is higher than the others, it is put in a system phase-counting bucket by itself. The only unique voltage in that bucket is one 208 volt high-leg so it becomes a single-phase 208 volt source. The four L-L voltages in the 5-wire are just as legitimate as any other voltage. They are real voltages. Whether or not we can find a practical use for them is immaterial. With your logic, the presence or absence of a center tapped neutral changes the number of phases, but for some reason it doesn't affect a wye connection. With my logic it makes no difference. Then I think you are misunderstanding something I have said. A 120/208 wye source with no possible connection to the neutral can only supply a single-phase 208 system. If we can use the neutral, the transformer can supply the following: 1) one single-phase system with one 208 voltage 2) two single-phase systems with one voltage each (one at 0 deg and one at 120 deg) 3) one two-phase system with two voltages (one at 0 deg and one at 120 deg) FWIW, the utility industry considers the open-wye distribution system to be two-phase. The load of the system as a whole appears as a two-phase load. If an open-wye is labeled "single-phase" it would be because of the nature of the loads. As I have said before, the load types have been used as part of the labeling. If it is serving single-phase loads, it can get labeled as a single-phase supply. That does not mean it is no longer a valid source for two-phase loads. If you consider two voltages and how you are going to classify systems for them to be counted in, there is a difference between two single-phase systems and one two-phase system. Not all grounded connections use a neutral, not all 'non-end point' taps are neutrals. For example, there is a standard control power connection of 24/120V (x1-X23= 24V, and X1-X4=120V). I know that. And not all "neutrals" are true neutral points. I believe I have said there is a single current direction, created by a single magnetic field direction, such as X1->X23->X4. But through the magic of math, X1->X23 and X23->X1 can be interchanged, by following proper 'signing' rules, giving the appearance of two currents. Pick any direction you want, but the currents will not always flow in the same direction or have the same magnitude. It is a physical reality, not math magic. Do you say a high-leg 4-wire delta (one winding is center tapped) has these 3-phases: 2@180?, and 1@90? while ignoring the 3@120?? If you mention them all is this a 6-phase transformer? No. To be included in a system phase count, the voltage must have the same magnitude and a different angle from another counted voltage. All of those phases are not classified in the same system as each system phase-counting bucket has its own voltage level. Here are examples of some of the systems a 120/240 high-leg transformer could supply: 1) Three 240 volt single-phase systems with one voltage each. 2) One 208 volt single-phase system with one voltage. 3) One 120 volt single-phase system with one voltage. 4) One 120 volt two-phase system with two voltages. 5) One 240 single-phase system with one voltage. Each system has one voltage magnitude to use for counting voltages..

FINELG is a general non linear finite element program which has been first written by F. Frey . Major contributions have been made by V. de Ville de Goyet who developed efficient schemes for 2D and 3D steel beams. The concrete orientation of the beam elements and the time resolution algorithm have been developed by P.Boeraeve . The program is able to simulate the behaviour of structures undergoing large displacements and moderate deformations. All beam elements are developed using a total co-rotational description. For convenience, a short description of the beam is now given. We consider a 2D Bernoulli fibre beam element with 3 nodes and 7 degrees of freedom. The total number of DOF corresponds to two rotational DOF at end nodes and 5 translational DOF (see Figure ). The intermediate longitudinal DOF is necessary to allow to represent strong variations of the centroïd position when the behaviour of the cross-section is not symmetric. Such behaviour is observed for example in concrete sections as soon as cracking occurs. Figure . 3 nodes plane beam element - DOFs As usual for fibre element, internal forces at the element nodes are computed on the basis of a longitudinal and transversal integration scheme. The integration along the beam length is performed using 2, 3 or 4 integration points (see Figure ,a). For each longitudinal integration point LIPi, a transversal integration is performed using the trapezoidal scheme. The section is divided into layers (see Figure ,b) each of which being assumed in uni-axial stress state. The state of strain and stress is computed at each integration point TIPj. Figure . Integration scheme : (a) longitudinal integration with 4-point Gauss scheme; (b) transversal integration with trapezoidal scheme. The software can be used for both static and dynamic non linear analyses. FINELG has been extensively validated for static non linear analyses. Development and validation of non linear dynamic analysis has been realized in the context of the joint research program DYNAMIX between University of Liège and Greisch . Dynamic computations have been re-assessed at the beginning of the OPUS project . INLDA analysis and definition of the structural capacity A first assessment of the seismic behavior of each frame is performed by carrying out incremental non linear dynamic analyses considering nominal values of the material properties. It was observed that all buildings exhibit a similar seismic behavior. Indeed, for these highly redundant moment resisting frames, the only active failure criterion is the rotation capacity of the plastic hinges. No global instability, no local instability nor storey mechanism was observed, even for seismic action levels equal to 3 times the design level. In the OPUS project, rotation demand and capacity were computed according to FEMA356 recommendations. The rotation demand is defined in Figure for both beams and columns. The rotation capacity is estimated to be equal to 27 mrad for steel columns. However, since no indication is given in FEMA 356 regarding composite beam capacities, a detailed study has been undertaken to better estimate the rotation capacity of composite beams. This study relies on the plastic collapse mechanism model developed by Gioncu . Figure shows that the sagging zone is large with a significant part of it having a quasi-constant moment distribution near the joint. As a consequence, the plastic strains are low in steel as well as in concrete, and no crushing of the concrete is observed. On the contrary, the hogging zone is shorter but with high moment gradient. This results in a concentration of plastic deformations and in a more limited rotation capacity. The ductility demand in plastic hinges is computed according to the actual position of contra-flexure point. Rotations of plastic hinges (θb1 and θb2) in beams are calculated as follows, where v1, v2 and v3 are defined in Figure . The resisting moment rotation curve in the hogging zone is determined by using an equivalent standard beam (see Figure .a) as commonly suggested in many references (Spangemacher and Sedlacek and Gioncu and Petcu ,). A simply supported beam is subjected to a concentrated load at mid span. The post-buckling behavior is determined based on plastic collapse mechanisms (see Figure ). Two different plastic mechanisms are considered (in-plane and out-of-plane buckling, see Figure .c and d respectively). The behavior is finally governed by the less dissipative mechanism (see Figure .b). Elastic and hardening branches of the M- curve have been determined using a multi-fiber beam model. When the hardening branch intersects the M- curve representative of the most critical buckling mechanism (softening branch), the global behavior switches from the hardening branch to the corresponding softening branch. The equation of the softening branches can be found in . The method has been implemented in MATLAB and validated against experimental results and F.E. results. For OPUS buildings, it has been found from the analysis of the results that the length of the hogging zone was approximately equal to 2 m. As a consequence, an equivalent simply supported beam with L = 4 m is considered. The resulting M- curves are depicted for the composite beams of cases 1 and 2 in Figure a and b. Figure . Typical bending moment diagram in a beam showing rotation in plastic hinges considering the exact position of the contraflexure point. Figure . Model of Gioncu : (a) equivalent beam, (b) Moment rotation curve, (c) in plane buckling mechanism, (d) out of plane buckling mechanism Figure . Moment-rotation curve of the composite beam IPE330 (a) (Case studies 1 and 2) and IPE 360 (b) (case studies 3 and 4) The moment-rotation curve obtained from the model of Gioncu describes the static behavior. According to Gioncu, when a plastic hinge is subjected to a cyclic loading, its behavior remains stable as long as no buckling appears. When buckling is initiated, damage accumulates from cycle to cycle. M- curves of the composite beam of OPUS exhibit a steep softening branch that does not allow for a long stable behavior. Consequently, in the following developments, the rotation max corresponding to the maximum moment is considered as the maximum rotation capacity under cyclic loading. The ultimate rotation max, the theoretical plastic rotation p, and the ratio max/p are reported on table 12. Since the ratio max/p is larger for case studies 3 and 4, the relative ductility is larger, and this leads, as it will be shown in the following, to a better seismic behavior of these buildings even if they were designed for the low seismicity. While this seems to be a paradox, it is nevertheless logical. The rotation capacity is defined by the local buckling limit. Lower steel grade used for low seismicity cases is favorable for this phenomenon, as it reduces the maximum stresses attained in the steel. Table : characteristic rotations of the composite beams 1 and 2 3 and 4 The evolution of the maximum rotation demand of the hinges in the hogging zones of the beams and at the bottom of the columns for increasing seismic acceleration agR is represented in Figure for all case studies. The failure level fixed by the beam rotation criterion is considerably lower than the one fixed by the column ultimate rotation. As a consequence, the statistical analysis will focus on the ductility criterion of the beams in the hogging zone. Figure . Results of the incremental dynamic analysis – case study 1(a), 2 (b), 3 (c), 4 (d)

Light-front Hamiltonian field theory towards a relativistic description of bound states Lichtfront Hamiltoniaanse veldentheorie Naar een relativistische beschrijving van gebonden Leescommissie: prof.dr. J.F.J. van den Brand dr. A.E.L. Dieperink prof.dr. G. McCartor prof.dr. P.J.G. Mulders prof.dr. H.-C. Pauli |Grafisch ontwerp:||Alex Henneman| Dit werk maakt deel uit van het onderzoekprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM) die financieel wordt gesteund door de Nederlandse Organisatie voor Wetenschappelijk ©1998, Nico Schoonderwoerd Light-front Hamiltonian field theory [.5] towards a relativistic description of bound states ter verkrijging van de graad van doctor aan [.4] de Vrije Universiteit te Amsterdam, [.4] op gezag van de rector magnificus [.4] prof.dr. T. Sminia, [.4] in het openbaar te verdedigen [.4] ten overstaan van de promotiecommissie [.4] natuurkunde en sterrenkunde [.4] van de faculteit der exacte wetenschappen [.4] op donderdag 14 januari 1999 om 13.45 uur [.4] in het hoofdgebouw van de universiteit, [.4] De Boelelaan 1105 Nicolaas Cornelis Johannes Schoonderwoerd geboren te Breukelen |Promotor:||prof.dr. P.J.G. Mulders| |Copromotor:||dr. B.L.G. Bakker| Aan Joop, Bea en Cosander - § 1 Forms of relativistic dynamics - § 2 Light-front quantization - § 3 Feynman rules - § 4 Divergences in the Yukawa model - § 5 Instantaneous terms and blinks - § 6 Pair contributions in the Breit-frame - § 7 Introduction - § 8 Example: the one-boson exchange correction - § 9 Equivalence of the fermion self-energy - § 10 Equivalence of the boson self-energy - § 11 Conclusions - § 12 Formulation of the problem - § 13 Minus regularization - § 14 Equivalence for the fermion triangle § 15 Equivalence for the one-boson exchange diagram - § 15.1 Covariant calculation - § 15.2 BPHZ regularization - § 15.3 Light-front calculation - § 15.4 Equivalence - § 16 Conclusions - § 17 Formulation of the problem - § 18 The Lippmann-Schwinger formalism - § 19 The box diagram - § 20 A numerical experiment - § 21 Light-front versus instant-form dynamics - § 22 Numerical results above threshold - § 23 Numerical results off energy-shell - § 24 Analysis of the on energy-shell results - § 25 Analysis of the off energy-shell results - § 26 Conclusions - § 26.0.1 Equivalence of light-front and covariant perturbation theory - § 26.0.2 Entanglement of Fock-space expansion and covariance - .0.3 Lichtfront Hamiltoniaanse veldentheorie Naar een relativistische beschrijving van gebonden toestanden - .0.4 Hoofdstuk Introductie van lichtfront Hamiltoniaanse dynamica - .0.5 Hoofdstuk Het Yukawa-model - .0.6 Hoofdstuk Longitudinale divergenties in het Yukawa-model - .0.7 Hoofdstuk Transversale divergenties in het Yukawa-model - .0.8 Hoofdstuk Verstrengeling van de Fock-ruimteontwikkeling en covariantie - .0.9 Hoofdstuk Samenvatting en conclusies I Introduction to light-front Hamiltonian dynamics Einstein’s great achievements, the principle of relativity, imposes conditions which all physical laws have to satisfy. It profoundly influences the whole of physical science, from cosmology, which deals with the very large, to the study of the atom, which deals with the very small. This quote reflects the work that I have done in physics so far, which began with an investigation of new black hole solutions to the Einstein equation when I was a student in Groningen, and which now ends with the research presented in this Ph.D.-thesis on models to describe bound states of elementary particles [2, 3, 4, 5, 6, 7]. The above quote contains the first two lines of an important article that Dirac wrote in 1949, in the middle of a century that has produced enormous progress in the understanding of the properties of matter. Not only the development of relativity, but also the rise of Quantum Mechanics was instrumental for this progress. At the end of this century, these and many other advances have resulted in a model that ambitiously is called the Standard Model. It describes all elementary particles that have been discovered until now; the leptons and the quarks, and their interactions. However, this does not imply we have to call it the end of the day for high-energy physics. A number of problems of a fundamental nature remain in the Standard Model. As an example we mention the question of the neutrino mass, that may or may not point to physics not included in the Standard Model. There are many practical problems when one wants to calculate a physical amplitude. If the interactions are sufficiently weak, perturbation theory is usually applied and gives in many cases extremely accurate results. However, in the case of strong interactions, or when bound states are considered, nonperturbative methods must be developed. We have to ensure that in such methods covariance is maintained. We mean by this that measured quantities, like cross sections and masses, are relativistic invariants. When the equations are written down in covariant form it is clear that the outcome will satisfy relativistic invariance and we refer to such methods as manifestly covariant. For example, the Bethe-Salpeter equation is manifestly covariant, but suffers from numerical intractability beyond the ladder approximation. Great progress is made by Lattice Field Theory, which is now able to give quantitative predictions. However, it depends on the choice of a specific frame of reference and the advances in its application rely strongly on a continued increase of the speed of computers. A very intuitive picture of a bound state is provided by Hamiltonian methods. However, the “classical” method of setting up a relativistic Hamiltonian theory by quantization on the equal-time plane, so-called instant-form (IF) quantization, suffers from problems such as a square root in the energy operator, which results in the existence of both positive and negative energy eigenstates, and the complexity of the boost operators, which keeps us away from determining the wavefunctions in an arbitrary reference frame. Weinberg proposed to use the Infinite Momentum Frame (IMF), because in this limit time-ordered diagrams containing vacuum creation or annihilation vertices vanish, and therefore the total number of contributing diagrams is significantly reduced. It is found that it provides a picture which connects to the one of the constituent quark model. However, its big disadvantage is that the IMF is connected to the rest frame by a boost for which one takes the limit of the boost parameter to infinity. It is dubitable whether this limit commutes with others that are taken in field theory. It was only in the seventies that one began to realize that a theory with the same advantages as the IMF, but without the disadvantages, had already been suggested by Dirac some decades before: light-front (LF) quantization, i.e., quantization on a plane tangent to the light-cone. Of the ten Poincaré generators, seven are kinematic, i.e., can be formulated in a simple way and correspond to conserved quantities in perturbation theory. Most important is that these seven operators include one of the boost operators, allowing us to determine the wavefunction in a boosted frame if it is known in the rest frame. This property is not found in IF quantization. As a drawback one finds that not all rotations are kinematic, and therefore rotational invariance is not manifest in LF quantization, a problem which is discussed frequently in the literature. In particular, our interest was triggered by an article by Burkardt and Langnau who claimed that rotational invariance is broken for -matrix elements in the Yukawa model. Instead of a lack of manifest rotational invariance, we prefer to talk about lack of manifest covariance, as this is a property that all Hamiltonian theories share. Because in each form of quantization dynamical operators that involve creation or annihilation of particles are present, in any relativistic Hamiltonian theory particle number is not conserved, implying that each eigenstate has to be represented as a sum over Fock states of arbitrary particle number. However, light-front dynamics (LFD) is the only Hamiltonian dynamical theory which has the property that the perturbative vacuum is an eigenstate of the (light-front) Hamiltonian, provided that zero-modes are neglected (in this thesis zero-modes will not explicitly be discussed). Bound states are also eigenstates and are distinct from the LF vacuum, which simplifies their analysis. In this thesis we shall not solve the eigenvalue problem, an interacting Hamiltonian will not even be written down! The goal of this thesis will not be to calculate a spectrum, but to illuminate two important properties of LF Hamiltonian dynamics. The first is: 1. Light-front dynamics provides a covariant framework for the treatment of bound states. Although the calculation of bound states requires nonperturbative methods, these usually involve ingredients encountered in perturbation theory, e.g., the driving term in a Lippmann-Schwinger or Bethe-Salpeter approach. We prove that LF perturbation theory is equivalent to covariant perturbation theory. By equivalent we mean that physical observables in LF perturbation theory are the same as those obtained in covariant perturbation theory. This can be done by showing that the rules for constructing LF time-ordered diagrams can be obtained algebraically from covariant diagrams by integration over the LF energy . Two technical difficulties, namely that the integration over can be ill-defined, and that divergences in the transverse directions may remain, are solved in Chapters Light-front Hamiltonian field theory towards a relativistic description of bound states and Light-front Hamiltonian field theory towards a relativistic description of bound states respectively, for the Yukawa model, which is introduced in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. In Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we discuss the entanglement of covariance and the Fock-space expansion, and show another important property of LF Hamiltonian dynamics: 2. Higher Fock state contributions in LF Hamiltonian field theory are typically small, in particular much smaller than in IF Hamiltonian field theory, and therefore the ladder approximation gives accurate results for the spectrum. It has been known for a long time that on the light-front one has to take into account fewer diagrams than in the instant-form of Hamiltonian dynamics. On top of this, diagrams involving higher Fock states are numerically smaller, as we will show. We look at two nucleons interacting via boson exchange, and we compare the contributions of the diagrams with one boson in the air, to diagrams where two bosons are simultaneously exchanged. The latter are ignored if we use the ladder approximation. We show in numerical calculations involving scalar particles that this approximation is viable for both scattering amplitudes and off energy-shell states, if masses and momenta are chosen in such a way that they are relevant for the deuteron. § 1 Forms of relativistic dynamics An important first step on the path to a Hamiltonian description of a dynamical system was taken by Dirac in 1949, in his famous article ’Forms of Relativistic Dynamics’ . One foot of this work is in special relativity, when Dirac writes: …physical laws shall be invariant under transformations from one such coordinate system to another. The other foot of his method is in Quantum Mechanics because Dirac writes: …the equations of motion shall be expressible in the Hamiltonian form. In more technical terms, this condition tells us that any two dynamical variables have a Poisson bracket, later to be associated with (anti-)commutation relations. We restrict the transformations further to continuous ones, therefore excluding space inversion and time reversal. In the forthcoming subsections we are going to work out these two principles and construct the generators of the Poincaré group. § 1.1 The Poincaré group The transformations mentioned in the first quote are the four translations , the three rotations , and the three boosts , where is an anti-symmetric tensor. These transformations should satisfy Setting up a dynamical system is equivalent to finding a solution to these equations. The solution of the ten generators is generally such that some of them are simple, and correspond to conserved quantities. These are labeled as kinematical, indicating that they do not contain any interaction. Others are more complicated and describe the dynamical evolution of the system as the Hamiltonian does in nonrelativistic dynamics. Therefore these are called dynamical, which means that they do contain interaction. It seems obvious that one should want to setup the framework in such a way that the number of dynamical operators is small. A simple solution of the Eqs. (I-1)-(I-3) can be found if we define a point in space-time to be given by the dynamical variable and its conjugate momentum by . Using a solution is now given by As already mentioned by Dirac, this solution may not be of practical importance, however, it can serve as a building block for future solutions. Another important ingredient for the dynamical theory is that we have to specify boundary conditions. We do this by taking a three-dimensional surface in space-time not containing time-like directions, at which we specify the initial conditions of the dynamical system. The ten generators then split into two groups, namely those that leave invariant and those that do not. The first group is called the stability group. The larger the stability group of , the smaller the dynamical part of the problem. We can ensure a large stability group by demanding that it acts transitively on : every point on may be mapped on any other point of by applying a suitable element of the stability group. This ensures that all points on the initial surface are equivalent. |(a) the instant-form||(b) the light-front| The restriction of relativistic causality reduces the number of world lines, and therefore increases the number of surfaces that one can choose for . Dirac found three independent choices for the initial surface that fulfill these conditions. In total there are five, as was pointed out by Leutwyler and Stern . We, however, only discuss the two most important ones. They are listed in Fig. I-1. In IF Hamiltonian dynamics one quantizes on the equal-time plane, given by This is the form of dynamics closest to nonrelativistic Quantum Mechanics. Another important possibility for quantization is offered by a plane tangent to the light-cone. The light-front is given by the equation Notice that this plane contains light-like directions. It is common to use the -direction to define the light-front. The different status of the other space-like directions and leads to the fact that the symmetry of rotational invariance becomes nonmanifest on the light-front. In explicitly covariant LFD , one defines the light-front by its normal vector , which is not fixed. This method will be encountered in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. § 1.2 Light-front coordinates In the remainder of this chapter, we show some of the advantages of LF quantization. We define so-called longitudinal coordinates and transverse coordinates such that the spatial coordinates and define a coordinate system on the light-front, and plays the role of time. From now on we will put the velocity of light equal to unity: . The indices of the four-vectors can be lowered and raised using the following LF metric : where the first and second row/column refer to the longitudinal components, and the third and fourth to the transverse components. Unfortunately, a number of conventions are frequently used for LF coordinates. We will stick to the one given above, commonly referred to as the Kogut-Soper convention . The stability group on the light-front has seven elements, as can be verified by writing out the commutation relations between these operators and : The other three operators are dynamical, as can be seen by the fact that they do not commute with , If we look at Fig. I-1b, we see that we can describe the operation of as a translation perpendicular to the light-front. The operators correspond to rotations of the light-front about the light-cone. Using these two words in one sentence clearly indicates why the common expression “light-cone quantization” is badly chosen. We prefer to use the phrase “light-front quantization”. § 1.3 The initial surface In LF quantization, we first solve the Poincaré algebra on the surface . The stability group is the group generated by transformations of this surface into itself. We already met these operators in Eq. (§ 1.2). As is fixed, the dynamical variable has lost its meaning and, according to Dirac, it should be eliminated. We can add to the generators in Eq. (I-5) multiples of : We then construct the and in such a way that on the light-front the -dependence drops from these equations. For the elements of the stability group we find: and for the three dynamical operators we find: When one quantizes in the instant-form, one finds four operators to be dynamical, which is one more than in LF quantization. However, more important is the form of the energy operator. In the instant-form, it is The presence of the square root causes the degeneracy of positive and negative energy solutions in IF dynamics, whereas on the light-front they are kinematically separated, as can be seen from Eq. (I-16): positive longitudinal momentum corresponds to positive LF energy , and vice versa. This effect leads to the spectrum condition, which is explained in the next section. The dynamical operators (I-16) and (I-17) reveal a little of the problems encountered on the light-front: the infrared problem for , which can be associated with the so-called zero-modes. As the path of quantization on the light-front is beset by problems, such as the nonuniqueness of the solution of the Cauchy problem, attempts have been made to find another path. Inspiration may be found in Quantum Field Theory, which leads to expressions for propagators of particles, and finally for -matrix elements. They may serve as a starting point to derive rules for time-ordered diagrams. § 2 Light-front quantization The first to set foot on the new path towards a LF perturbation theory were Chang and Ma , and Kogut and Soper . Their work relies on the Feynman rules that are constructed in Quantum Field Theory. To determine the LF time-ordered propagator we take the Feynman propagator and integrate out the energy component. For the types of theories that are discussed in this thesis, two are of importance: the scalar propagator and the fermion propagator. § 2.1 The scalar propagator The Klein-Gordon propagator for a particle of mass is well-known: where the subscript “Min” denotes that the integral is over Minkowski space. The inner products of the Lorentz vectors can be written in LF coordinates: Following Kogut and Soper , we separate the energy integral from the integral over the kinematical components of , indicated by : We then find for the propagator of Eq. (I-19): in which we use the definition and where is the on mass-shell value, or, in other words, the pole in the complex -plane: Forward propagation in LF time requires . Then, we can only evaluate the integral over by closing the contour in the lower complex half-plane, because of the presence of the factor in the integrand. For the pole is below the real axis. Therefore application of Cauchy’s theorem gives a nonvanishing result only in this region: where the on mass-shell four-vector is given by § 2.2 The fermion propagator The well-known propagator for a spin- particle is related to the Klein-Gordon propagator by the following relation: where is short for . We interchange differentiation and integration. Differentiation of the integrand in Eq. (I-19) gives: where the Feynman slash for an arbitrary four-vector is defined by An important difference with Eq. (I-23) is that the numerator contains the LF energy . We can remove it by rewriting the numerator, Upon substitution of this expansion into Eq. (I-29) we see that the first term of Eq. (I-31) cancels against a similar factor in the denominator. Integration over the LF energy gives the LF time-ordered fermion propagator: The first term is the same as the result for the scalar propagator (I-26), except for the factor . The second term on the right-hand side of (I-32) has lost its propagating part, resulting in the appearance of a -function in . This explains why it is called the instantaneous part. The decomposition of the covariant propagator into the propagating and the instantaneous fermion will occur frequently in this thesis and is an important ingredient in establishing equivalence between LF and covariant perturbation theory. § 2.3 The spectrum condition An important result that we infer from the previous subsections is that the time-ordered propagators (I-26) and (I-32) contain -functions restricting the longitudinal momentum. This will severely reduce the size of phase-space. Moreover, the longitudinal momentum is a conserved quantity and therefore all LF time-ordered diagrams containing either vacuum creation or annihilation contributions will vanish, as can be explained by looking at Fig. I-2. According to Eq. (I-16), every massive particle in Fig. I-2 should have positive -momentum. As the longitudinal momentum is a kinematical quantity, it should be conserved at each vertex. However, the vacuum has . Therefore diagrams containing vacuum creation or annihilation vertices are not allowed in a series of LF time-ordered diagrams. In IF dynamics there is no such reduction of the number of diagrams, because there is no restriction on the IF momentum . § 2.4 The energy denominator From now on, we shall write the Feynman diagrams in the momentum representation. In this subsection we show where the energy denominators originate from. Let us choose as a simple example Compton-like scattering in theory: where is the total momentum, and is the momentum of the intermediate particle. Because of momentum conservation, they are the same. However, we make this distinction, to be able to write it in the following form: where is the on mass-shell value of , conform Eq. (I-25). To stress that the diagram on the right-hand side is a time-ordered diagram, we draw a vertical thin line, indicating an energy denominator. The first denominator in Eq. (I-34) is the phase-space factor, constructed by taking for each intermediate particle the plus-momentum and a factor 2 because of the Kogut-Soper convention [13, 15]. The direction of the momenta should be chosen forward in time, such that the plus-component, satisfying the spectrum condition, is positive. The energy denominator is constructed by taking the total energy and subtracting from it the on mass-shell values of the minus-momentum of the particles in the corresponding intermediate state. Thus, it is proportional to the energy that is “borrowed” from the vacuum. This explains why highly off energy-shell intermediate states are suppressed. In the next subsection we present examples where the energy denominators are more complicated because different time-orderings of the vertices are involved. § 2.5 Light-front time-ordering The most trivial example of time-ordering of vertices was already discussed in the previous subsection. In Compton-like scattering there are two time-orderings, however, one, the so-called Z-graph, is excluded because of the spectrum condition. § 2.5.1 The one-boson exchange If we look at a similar amplitude as Eq. (I-33), now with the exchanged particle in the -channel, both time-orderings can contribute. where the momentum of the intermediate particle . The sign of determines the time-ordering of the vertices: where , which, if the external states are on energy-shell, coincides with the energy of the system. Again we see that the energy denominators are constructed by subtracting from the total energy the on mass-shell values of the minus-momentum of the particles in the intermediate state. Because of our choice of momenta, for the diagram (I-38) the momentum flow of the intermediate particle is backward in time. If we substitute , then the plus-momentum becomes positive, and the particle can be reinterpreted as going forward in LF time. In Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we will again encounter these two time-orderings when we describe the interaction of two nucleons by the exchange of bosons. § 2.5.2 The scalar shower The next example is used to illustrate that some algebraic manipulations are needed to construct all LF time-ordered diagrams. We look at the decay of a particle into four scalars, again in theory. where the two intermediate scalars have momentum and respectively. We now use the algebraic identity This splitting can be used for the covariant amplitude in Eq. (I-39). We find: § 2.6 Loop diagrams In the case of a loop diagram, covariant Feynman rules require an integration over the internal four-momentum . Time-ordered diagrams have an integration over the three kinematical components. As was found by Kogut and Soper , a relation between these types of diagrams can be established if we integrate out the energy component from the covariant diagram. Upon doing this integration one finds all LF time-ordered diagrams with the vertices time-ordered in all possible ways, however, respecting the spectrum condition. For an arbitrary number of particles in a loop, the proof was only recently given by Ligterink and Bakker . As an example, which also shows some of the problems encountered in LF Hamiltonian dynamics, we discuss the electromagnetic form factor in theory, given earlier by Sawicki : An essential difference between the instant-form and the light-front occurs if we write the Feynman propagator in terms of the poles in the energy plane. In terms of IF coordinates we find: and on the light-front we have: We see that the Feynman propagator is quadratic in the IF energy but only linear in the LF energy . In the former case it leads to the presence of both positive and negative energy eigenstates, whereas on the light-front only positive energy states occur. In the instant-form, half of the poles occur above the real axis, and the other half below. Therefore contour integration will always give a nonvanishing result. In contrast to this, on the light-front the poles can cross the real axis. If all poles are on the same side of the real axis, the contour can be closed in the other half of the complex plane, and contour integration gives a vanishing result. Because of this effect, four of the six time-ordering in Fig. I-3 disappear. Only the first two remain. This is another manifestation of the spectrum condition. If we then turn to the Breit-frame (), also the second diagram of Fig. I-3 vanishes, as will follow from the analysis we present below. Most important in our analysis is the sign of the imaginary part of the poles. Because of our choice of the Breit-frame, these are identical for the first and the second Feynman propagator in Eq. (I-44), namely . The imaginary part of the third Feynman propagator is . In Fig. I-4 we show the location of these poles for different intervals. (a) (b) (c) We see in (a) and (c) of Fig. I-4 that the contour can be closed in such a way that no poles are inside the contour, and therefore contour integration leads to a vanishing result. In case that we calculate the component of the current, application of Cauchy’s theorem is not valid because there is a contribution to the integral from a pole at infinity, i.e., for large absolute values of the integrand goes as . Therefore we restrict ourselves in this example to the components and . Only one LF time-ordered diagram contributes to the current: where we have drawn vertical lines in the LF time-ordered diagram to indicate the energy denominators and to avoid confusion with the covariant diagram. The kinematics are given in Fig. I-3a. The photon line is vertical to indicate that we are in the Breit-frame. The imaginary parts have been omitted. For the result above to be correct three assumptions are essential: Interchange of the limit and -integration is valid, There is no contribution of poles at infinity upon doing the -integration, The amplitude is well-defined and finite. All three assumptions can be justified in this case. De Melo et al. have shown that the interchange mentioned under assumption 1 may cause pair creation or annihilation contributions to become nonvanishing. In § 6 of Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we show that this effect may also occur in the Yukawa model. However, it is not a violation of the spectrum condition. The second assumption can be justified by looking at Eq. (I-44). As is linear in , we see that the integration over the minus component is well-defined for each component of the current. In a theory with fermions, this integration can be ill-defined, leading to longitudinal divergences and the occurrence of so-called forced instantaneous loops (FILs). Divergences for the Yukawa model are classified in § 4 of Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states and the longitudinal ones are dealt with in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states, where it is shown that the FILs vanish upon using an appropriate regularization method: “minus regularization” . The third assumption is also satisfied, since the superficial degree of divergence for integration over the perpendicular components is smaller than zero for all components of the current. If any transverse divergences occur, they can be attacked with the method of extended minus regularization presented in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. The phase space factor contains endpoint singularities in . However, these are canceled by identical factors in the energy denominators. Ii The Yukawa model In particle physics several models are used to describe existing elementary particles, interactions and bound states. Many of these models are just used to highlight certain properties, or to make exact or numerical calculation possible. Although the latter are referred to as toy models, they are helpful because they are stripped from those properties that are of no concern to the investigation that is done. In this thesis we are going to “play around” with two models. One of them we already met in the introductory chapter: theory. In this model one can very nicely demonstrate that higher Fock states are much more suppressed on the light-front than in instant-form Hamiltonian dynamics, as will be done in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. This model only contains scalar particles. The simplest model including fermions is the Yukawa model, which has the following Lagrangian: The field describes the fermions and the field describes the scalar particles, from now on referred to as bosons. The last term is the interaction between the fermion–anti-fermion field and the boson field. Yukawa introduced this model to describe the interaction of nucleons (fermions) via pions (bosons). The strength of the interaction is given by . In our calculations we limit ourselves to a scalar coupling. § 3 Feynman rules Using perturbation theory one can deduce from the Lagrangian the well-known rules for Feynman diagrams. Summing over these diagrams one then finds the -matrix. The first term of the Lagrangian (II-48) leads to the following propagator: for a fermion with momentum and mass . For a (scalar) boson with momentum and mass we have the following Feynman rule: The full set of Feynman rules to compute the scattering amplitude in the Yukawa model can be found in many text books such as Itzykson and Zuber . Our goal is to translate these rules to rules for diagrams that one uses in LFD. In Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we introduced the -integration to obtain the rules for the LF time-ordered diagrams. A complication in this procedure was already mentioned there: the integration over may be ill-defined, and the resulting integral may be divergent. Before solving these problems, we first classify the divergences. § 4 Divergences in the Yukawa model In the previous subsection we described how to construct each covariant Feynman diagram. Covariant diagrams may contain infrared and ultraviolet divergences. Therefore we are not surprised that both in the process of constructing the LF time-ordered diagrams as in the diagrams themselves divergences can be encountered. The first type can be classified as longitudinal divergences, and the second as transverse divergences. § 4.1 Longitudinal divergences We can deduce what (superficial) divergences we are going to encounter upon integration over . We denote the longitudinal degree of divergence by . Suppose we have a truncated one-loop diagram containing bosons and fermions. In the fermion propagator Eq. (II-49) the factor occurs both in the numerator and in the denominator, and therefore it does not contribute to . Each boson will, according to Eq. (II-50) contribute to the degree of divergence, and the measure of the loop contributes , resulting in Longitudinally divergent diagrams, i.e., , contain one boson in the loop, or none. Since every loop contains at least two lines, a longitudinally divergent diagram contains at least one fermion. For the model we discuss, the Yukawa model with a scalar coupling, the degree of divergence is reduced. For scalar coupling it turns out that and therefore two instantaneous parts cannot be neighbors. The longitudinal degree of divergence for the Yukawa model with scalar coupling is where the subscript “entier” denotes that we take the largest integer not greater than the value between square brackets. § 4.2 Transverse divergences The transverse degree of divergence of a LF time-ordered diagram is the divergence one encounters upon integrating over the perpendicular components. In most cases this degree of divergence is the same as what is known in covariant perturbation theory as the superficial degree of divergence of a diagram. In that case it is the divergence one finds if in the covariant amplitude odd terms are removed and Wick rotation is applied. For a one-loop Feynman diagram in four space-time dimensions with internal fermion lines and internal boson lines the transverse degree of divergence is In case of space-time dimensions we have to replace the term by . In Table II-1 all one-loop diagrams up to order that are candidates to be divergent have been listed with their longitudinal and transverse degree of divergence. § 5 Instantaneous terms and blinks As was already illustrated in the introduction, in the case of fermions we have to differentiate between propagating and instantaneous parts. Therefore this distinction plays an important role in the Yukawa model. The covariant propagator in momentum representation for an off-shell spin-1/2 particle can be written analogously to Eq. (I-32): The first term on the right-hand side is the propagating part. The second one is the instantaneous part. The splitting of the covariant propagator corresponds to a similar splitting of LF time-ordered diagrams. For any fermion line in a covariant diagram two LF time-ordered diagrams occur, one containing the propagating part of the covariant propagator, the other containing the instantaneous part. For obvious reasons we call the corresponding lines in the LF time-ordered diagrams propagating and instantaneous respectively. For a general covariant diagram the -singularity in the propagating part cancels a similar singularity in the instantaneous part. Therefore the LF time-ordered diagrams with instantaneous lines are necessary; they are usually well-defined. If the -singularities are inside the area of integration we may find it necessary to combine the propagating and the instantaneous contribution again into the so-called blink, introduced by Ligterink and Bakker , such that there is a cancellation of the singularities: The thick straight line between fat dots is a blink. The bar in the internal line of the third diagram is the common way to denote an instantaneous fermion. When a LF time-ordered diagram resembles a covariant diagram, we draw a vertical line as in the second diagram of Eq. (II-55). If no confusion is possible, we omit it in the remainder of this thesis. The difference between Eqs. (II-54) and (II-55) lies in the fact that the former uses covariant propagators, and the latter has energy denominators. In this case the difference is only formal. However, in more complicated diagrams there is a big difference, as we will see later. Examples of blinks are discussed in the next section, and in § 8 of Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states where we discuss the one-boson exchange correction to the vertex. § 6 Pair contributions in the Breit-frame In Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we found that for massive particles the spectrum condition applies: there can be no creation from or annihilation into the vacuum. This gives a significant reduction of the number of diagrams that one has to incorporate in a light-front calculation. In any frame where the particles have positive plus-momentum this is valid. In Leutwyler and Stern it was already noted that on the light-front the regions , and are kinematically separated, another manifestation of the spectrum condition. This fact should already make us aware that the Breit-frame, where one takes the limit of the plus-momentum of the incoming virtual photon going to zero, is dangerous. Indeed one finds that pair creation or annihilation contributions play a role in this limit. This was first found by De Melo et al. and later by Choi and Ji . They discuss as an example the electro-magnetic current in theory, and find a pair creation contribution for the component of the current. We have shown in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states that for the other components and pair creation contributions vanish. Because De Melo et al. discuss a scalar theory, we infer that this effect is not related to the presence of fermions in the theory. In a theory with fermions, such as the Yukawa model, we now show that the pair creation/annihilation term is also nonvanishing, and that its omission leads to a breaking of covariance and rotational invariance. In the presence of fermions, the individual time-ordered diagrams may contain singularities that cancel in the full sum. Because this cancellation has nothing to do with the time-ordering of the diagrams, we combine the LF time-ordered diagrams into blink diagrams. After that, we have a clear view on the point we want to discuss. Again, we use kinematics as in Fig. I-3a. Two blinks contribute to the current, provided we have chosen the plus-component of the momentum of the outgoing boson . where the diagrams containing blinks are given by: The diagram (§ 6) is an example of a “double” blink. The total blink is the thick line between the two fat dots. For both blinks we see that the energy denominators are the same as for usual LF time-ordered diagrams. However, the numerators are different. In the next subsection we show how the numerator of the blink is constructed. § 6.1 Construction of the blink As an example, we show how we can construct the blink (§ 6). For the fat line we have to substitute the propagating and instantaneous part. The propagating contribution is and the instantaneous contribution, denoted by the perpendicular tag, is We see that both have a singularity at the upper boundary of the integration interval over . These cancel in the sum: the blink Eq. (§ 6). It is obtained by making the denominators common for the two diagrams. We can verify, using the relation , that the lower boundary at does not cause any problems, neither for the LF time-ordered diagrams, nor for the blink. In an analogous way the double blink is constructed. It consists out of the following LF time-ordered diagrams: We see that one diagram is missing: the diagram with two instantaneous fermions. Because it contains two neighboring matrices, it vanishes. It is an example of a forced instantaneous loop (FIL), which is related to longitudinal divergences and will be discussed in the next chapter. The LF time-ordered diagrams have the same integration interval as the double blink (§ 6). The last diagram on the right-hand side is the same as the diagram in Eq. (§ 6.1), the only difference being the integration range. The instantaneous fermions have been ’tilted’ a little in these two diagrams, to indicate that the integration interval is such that the instantaneous fermions carry positive plus-momentum. The second diagram on the right-hand side of Eq. (II-62) has no tilted instantaneous fermion, because the integration range has not been split as in the previous case. We will not give the formulas for the LF time-ordered diagrams in Eq. (II-62), but we have verified that their end-point singularities are removed when the diagrams are combined into the double blink. § 6.2 The Breit-frame What happens to the current in Eq. (II-56) in the limit of ? Relying on the spectrum condition, one may expect that diagrams like (§ 6) disappear, and that only the double blink (§ 6) contributes. This is confirmed by the fact that the integration area of the single blink (§ 6) goes to zero. However, it could be that the integrand obtains singularities in the limit that cause a nonzero result. We denote this limit in the diagrams by drawing the line of the outgoing boson vertically. For the numerator of the blink (§ 6) we use the relation The integral is dominated by factors and . Identifying these factors in (§ 6) we find: We write Eq. (II-64) in internal coordinates , and find that the dependence on the integration range drops. Moreover, the integration contains no singularities in the internal variable . If we disregard for a moment the transverse integration, we see that in the Breit-frame there is a finite contribution of pair-creation/annihilation to the current. This agrees with the result of De Melo et al. . Furthermore, we see that it is not covariant, and therefore its omission will not only lead to the wrong amplitude, but also to breaking of Lorentz covariance and rotational invariance. Iii Longitudinal divergences in the Yukawa model If the doors of perception were cleansed everything would appear as it is, infinite. William Blake, The marriage of heaven and hell For a number of reasons mentioned in the previous chapters, quantization on the light-front is nontrivial. Subtleties arise that have no counterpart in ordinary time-ordered theories. We will encounter some of them in this chapter and show how to deal with them in such a way that covariance of the perturbation series is maintained. In LFD, or any other Hamiltonian theory, covariance is not manifest. Burkardt and Langnau claimed that, even for scattering amplitudes, rotational invariance is broken in naive light-cone quantization (NLCQ). In the case they studied, two types of infinities occur: longitudinal and transverse divergences. They regulate the longitudinal divergences by introducing noncovariant counterterms. In doing so, they restore at the same time rotational invariance. The transverse divergences are dealt with by dimensional regularization. We would like to maintain the covariant structure of the Lagrangian and take the path of Ligterink and Bakker . Following Kogut and Soper they derive rules for LFD by integrating covariant Feynman diagrams over the LF energy . For covariant diagrams where the -integration is well-defined this procedure is straightforward and the rules constructed are, in essence, equal to the ones of NLCQ. However, when the -integration diverges the integral over must be regulated first. We stress that it is important to do this in such a way that covariance is maintained. In this chapter, we will show that the occurrence of longitudinal divergences is related to the so-called forced instantaneous loops (FILs). If these diagrams are included and renormalized in a proper way we can give an analytic proof of covariance. FILs were discussed before by Mustaki et al. , in the context of QED. They refer to them as seagulls. There are, however, some subtle differences between their treatment of longitudinal divergences and ours, which are explained in § 9. Transverse divergences have a different origin. However, they can be treated with the same renormalization method as longitudinal divergences. We shall present an analytic proof of the equivalence of the renormalized covariant amplitude and the sum of renormalized LF time-ordered amplitudes in two cases, the fermion and the boson self-energy. In the other cases we have to use numerical techniques. They will be dealt with in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. § 7 Introduction In the previous chapters we already introduced instantaneous fermions. For a discussion on longitudinal divergences they play an important role. Without fermions there are no longitudinal divergences! The longitudinal divergences can be both seen from a “pictorial” and a mathematical point of view. The pictorial view is the following. When a diagram contains a loop where all particles but one are instantaneous, a conceptual problem occurs. Should the remaining boson or fermion be interpreted as propagating or as instantaneous? Loops with this property are referred to as forced instantaneous loops (FILs). Loops where all fermions are instantaneous are also considered as FILs. However, they do not occur in the Yukawa model with (pseudo-)scalar coupling. Examples of these three types of FILs are given in Fig. III-5. Mathematically this problem also shows up. The FILs correspond to the part of the covariant amplitude where the -integration is ill-defined. The problem is solved in the following way. First we do not count FILs as LF time-ordered diagrams. Second we find that this special type of diagram disappears upon regularization if we use the method of Ligterink and Bakker : minus regularization. § 7.1 Minus regularization The minus-regularization scheme was developed for the purpose of maintaining the symmetries of the theory such that the amplitude is covariant order by order. It can be applied to Feynman diagrams as well as to ordinary time-ordered or to LF time-ordered diagrams. Owing to the fact that minus regularization is a linear operation, it commutes with the splitting of Feynman diagrams into LF time-ordered diagrams. We explain very briefly how the method works. Consider a diagram defined by a divergent integral. Then the integrand is differentiated with respect to the external energy, say , until the integral is well defined. Next the integration over the internal momenta is performed. Finally the result is integrated over as many times as it was differentiated before. This operation is the same as removing the lowest orders in the Taylor expansion in . For example, if the two lowest orders of the Taylor expansion with respect to the external momentum of a LF time-ordered diagram are divergent, minus regularization is the following operation: The point is chosen in this example as the renormalization point. This regularization method of subtracting the lowest order terms in the Taylor expansion is similar to what is known in covariant perturbation theory as BPHZ (Bogoliubov-Parasiuk-Hepp-Zimmermann) . Some advantages of the minus-regularization scheme are preservation of covariance and local counterterms. Another advantage is that longitudinal as well as transverse divergences are treated in the same way. A more thorough discussion on minus regularization can be found in the next chapter. § 7.2 Proof of equivalence for the Yukawa model The proof of equivalence will not only hold order by order in the perturbation series, but also for every covariant diagram separately. In order to allow for a meaningful comparison with the method of Burkardt and Langnau we apply our method to the same model as they discuss, the Yukawa model, as introduced in Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. In this model we have to distinguish four types of diagrams, according to their longitudinal () and transverse degrees () of divergence. These divergences were classified also in Table II-1 on page II-1. The proof of equivalence is illustrated in Fig. III-6. We integrate an arbitrary covariant diagram over LF energy. For longitudinally divergent diagrams this integration is ill-defined and results in FILs. A regulator is introduced which formally restores equivalence. Upon minus regularization the -dependence is lost and the transverse divergences are removed. We can distinguish Longitudinally and transversely convergent diagrams (). No FILs will be generated. No regularization is needed. The LF time-ordered diagrams may contain -poles, but these can be removed using blinks. A rigorous proof of equivalence for this class of diagrams is given by Ligterink and Bakker . Longitudinally convergent diagrams () with a transverse divergence (). In the Yukawa model there are three such diagrams: the four fermion box, the fermion triangle and the one-boson exchange correction. Again, no FILs occur. Their transverse divergences and therefore the proof of equivalence will be postponed until Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states. However, because the one-boson exchange correction illustrates the concept of -integration, the occurrence of instantaneous fermions and the construction of blinks, it will be discussed as an example in § 8. In Chapter Light-front Hamiltonian field theory towards a relativistic description of bound states we gave an example for a longitudinally convergent diagram in theory: the electro-magnetic current. Longitudinally divergent diagrams () with a logarithmic transverse divergence (). In the Yukawa model with a scalar coupling there is one such diagram: the fermion self-energy. Upon splitting the fermion propagator two diagrams are found. The troublesome one is the diagram containing the instantaneous part of the fermion propagator. According to our definition it is a FIL and needs a regulator. In § 9 we show how to determine the regulator that restores covariance formally. Since can be chosen such that it does not depend on the LF energy, the FIL will vanish upon minus regularization. Longitudinally divergent diagrams with a quadratic transverse divergence (). In the Yukawa model only the boson self-energy is in this class. We are not able to give an explicit expression for . However, in § 10 it is shown that the renormalized boson self-energy is equal to the corresponding series of renormalized LF time-ordered diagrams. This implies that the contribution of FILs has again disappeared after minus regularization. § 8 Example: the one-boson exchange correction We will give an example of the construction of the LF time-ordered diagrams, the occurrence of instantaneous fermions and the construction of blinks. It concerns the correction to the boson–fermion–anti-fermion vertex due to the exchange of a boson by the two outgoing fermions. Here, and in the sequel, we drop the dependence on the coupling constant and numerical factors related to the symmetry of the Feynman diagrams. A boson of mass with momentum decays into a fermion anti-fermion pair with momenta and respectively. The covariant amplitude for the boson exchange correction can be written as

If you are connected to any kind of financial market or watch the financial news even for 5 minutes every day, it is likely that you have heard the word, financial derivatives many times. The media is flush with articles wherein derivatives are criticized or appreciated. Most of the times, commentators are in awe of the mind-numbingly large amounts behind these contracts. It is often said that the total amount of derivatives contracts in the worlds, is actually greater than the total amount of money available in the world! Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Derivatives In this chapter we will start looking at the next major topic in a calculus class, derivatives. This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter. Here is a listing of the topics covered in this chapter. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function. Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function. Differentiation Formulas — In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule — In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions — In this section we will discuss differentiating trig functions. Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. Logarithmic Differentiation — In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule. More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.1. Take derivative 2. Plug given x value into derivative (for slope) 3. Negative reciprocal of slope 4. Plug into point-slope form. Study Guide Resource Home Textbook Instructor's Manual Study Guide Computing in Calculus (PDF - MB) 2: Derivatives. The Derivative of a Function Powers and Polynomials The Slope and the Tangent Line. Calculus Here is a list of skills students learn in Calculus! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. Beginner’s Guide to NCFM Certification Exam:If you are considering undertaking NCFM modules, there are a lot many of them to caninariojana.com of course cannot consider giving all of them at the same time. Here through this article, I wish to provide you with a brief summary on their modules which might help you decide which one you should choose from. Aug 07, · Derivatives Essentials: An Introduction to Forwards, Futures, Options, and Swaps (Wiley, ) by Aron Gottesman is an excellent textbook/self-study guide. It . will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y ′ (from the chain rule). View Test Prep - DERIVATIVES study guide from ACCT I S at University of Wisconsin. DERIVATIVES Derivative financial instruments have become the key tools of RISK MANAGEMENT. Derivatives financial. The project that you can use to engage the students is to use word art as a way of creating a study guide of some sort. There are several terms associated with derivatives, so good way to remember the key terms would be to write the most important terms from . Calculus 1 Class Notes, Thomas' Calculus, Early Transcendentals, 12th Edition Copies of the classnotes are on the internet in PDF format as given below. Introduction to .

NPV and Alternative Investment Rules November 27, 2001 Objectives of lecture on captital budgeting rules In previous lectures we have already argued that the appropriate goal from shareholders point of view is to maximize NPV. But in practice other method are used as well. ⇒ Comparison of different capital budgeting techniques. • Payback period rule • Discounted payback period rule • Average accounting return (AAR) • Internal rate of return (IRR) • Profitability index Then will discuss in more detail problems using NPV in capital budgeting. Why should investment rule be based on NPV? When we discussed the separation theorem, we have shown that management which invests in projects with positive NPV ⇔ increases shareholder value. NPV is appropriate rule since: • NPV uses cash flows not earnings: cash flows really what is available to pursue cor- porate investment plans. • NPV uses all cash flows: Cash flows exhaustively describe an investment opportunity from the point of view of valuation. Other methods ignore cash flows after a certain point in time. • NPV discounts the cash flows properly: Captures the time value of money as revealed by the markets. Why are other methods used in practice? • Historical reasons and convenience. • Alternative methods less complicated. • Lack of appropriate estimates for cash flows and risk adjusted discount rates. Pay Back Period Rule Idea: Many investment projects start with a cash outflow followed by cash inflows in the future. ⇒ How many years does it take until initial cash outflow has been paid back ? Answer: Pay back period . Application: Management decides about what it considers as acceptable (upper limit) time until initial investment is paid back. ⇒ Rule: Undertake all investment projects with a pay back period less than say 5 years. ⇒ Ranking Criteria: Undertake investment project with shortest pay back period. Pros of Payback Period • Upper limit on cut off time limits risk exposure of investment projects. • Favors liquidity of firm. • Respects short term investment horizon. Cons of Payback Perido Projects considered as identical with respect to payback period can be extremely different: • Different cash flows after cut off period are ignored. • Timing of cash flows before cut off period is not taken into account: Two projects with same initial cash outflow of $100 followed by cash inflows of $10 over 10 year for one project and no cash inflows for 9 year but $ 100 in the 10th year have the same pay back period! (Should prefer the first project since intermediate cash inflows can be reinvested.) ⇒ Payback period method ignores time value of money. • Biased against long-term projects and therefore against shareholder interests. • Criteria may not exist and/or acceptance criteria is arbitrary. Discounted Pay Back Period Idea: Correct payback period method such that time value of money is taken into account. Pros: Same as pay back period. But in addition reflect opportunity costs of money as valued by the financial markets. Cons: As pay back period. But in addition get more complicated since have to discount as for NPV rule. Averge Accounting Return (AAR) Idea: It is often easier to determine projects earning rather than cash flows. Calculate rate of return of projects earning by dividing its earnings by the project’s book value. Since both vary offer time take time averages ⇒ Average Accounting Return. Average net income Application:Proceed in two steps 1. Determine average net income: Average Net Income = (Revenuen − Expensesn −Depreciationn − Taxesn ) 2. Determine average investment: Average Investment = (Value of investmentn − Depreciationn ) Decison Rule for ARR ⇒ Rule: Management defines hurdle rate. Undertake investment when AAR is higher than this hurdle rate. ⇒ Ranking criteria: Invest in project with highest hurdle rate. Pros of ARR • Accounting information available. • Easy to calculate. • Does always exist. Cons of ARR • Ignores the time value of money: average net income can be same but income per period can be very different. • Benchmark (hurdle rate) arbitrary. • Based on book values: Do not reflect means available for investment project as cash flows to and book values do not reflect valuation by the market. Internal Rate of Return (IRR) Idea: Given the cash flow structure of a given investment project at which interest rate is its NPV equal to zero ? Application: Management fixes target rate of return. IRR solves the following equation: (1 + IRR)n IRR corresponds to the “yield” of the investment project. Investing and Financing 1. Investing: Undertake investment if IRR is higher than a specified hurdle rate which may correspond to discount rate from financial markets. (Opportunity cost of capital). 2. Financing: Accept financing project if IRR is lower than a specified hurdle rate which may correspond to the discount rate from financial markets (Opportunity cost Ranking criteria: Undertake investment project with highest IRR and accept financing project with lowest IRR. NPV and IRR Remark: IRR is very close to NPV: • Both are based on all cash flows and capture time value of money. • Both evaluate investment project by a simple number But IRR does not always exist or is not necessarily unique. Pros of IRR • Uses all cash flows. • Not based on accounting information. • Easy to understand. • Easy to communicate. Cons of IRR • Does not capture different scale of project. • Does not always exist (“is not always real”) and may not be unique: Example: Consider cash flows C0 , C1 , C2 . Then IRR is given by solution C0 + + =0 1 + IRR (1 + IRR)2 then define X = 1+IRR and try to find X. The equation defining IRR is quadratic and you may remember that the general solution for X is then −C1 + C1 − 4C0 C2 −C1 − 2 C1 − 4C0 C2 and consequently we will have two solutions for IRR if C1 − 4C0 C2 is not equal to 2 − 4C C < 0 we have one IRR which is not a real number ! zero. Furthermore if C1 0 2 Remark : As you can see this does not occur if the initial investment is negative and all other cash flows are positive (Investment) or vice versa all but the last cash flows are positive (Financing). Cons IRR continued • Does not distinguish between investing and financing. Example: If we multiply all cash flows by the same number, the IRR does not change ! Consequently if we compare an investment project with a financing project we obtain the same IRR!. • Re-investment rate assumption: IRR rule does not discount at opportunity costs of capital, it implicitly assumes that the time value of money is the IRR. Implicit assumption: ⇒ Re-investment rate assumption. Therefore the IRR assumes that investor can reinvest their money at the IRR which is different from the market-determined opportunity costs. It is logically not coherent that the re-investment rate should be different if they are in the same risk class. • Value Additivity Principle: The value of the firm should be equal to the sum of the values of the individual projects of the firm. Illustration: Violation of Value Additivity Principle Therefore if we have choice between two mutually exclusive projects and an indepen- dent project, it should not occur that project one is preferred to project two but if mixed with the independent project, project two is preferred. But this can happen – Project 1: -100,0,550 ⇒ IRR=134.4% – Project 2: -100,225,0 ⇒ IRR=125.0% – Project 3: -100,450,0 ⇒ IRR=350 % – Project 1+3: -200,450,550 ⇒ IRR=212.8% – Project 2+3: -200,675,0 ⇒ IRR=237.5% Idea: IRR is attractive since it is not dependent on scale of investment. It is a “per dollar” quantity. The profitability index does the same thing for NPV: Amount of NPV generated for each investment dollar. PV of cash inflows initial cash outflow PV of cash outflows PIF in = initial cash inflow Application: Management defines benchmark for index ⇒ Undertake investment if profitability index is higher than benchmark. ⇒ Undertake investment with highest profitability index. Pros and Cons for Profitability Index • respects limited availability of investment funds. • easy to understand and communicate • same decision as NPV for independent projects Cons: As IRR non-additive ⇒ Problems when evaluate mutually exclusive projects. Conclusion: Valuation principles Only NPV satisfies all of the following requirements: • Considers all cash flows. • Cash flows are discounted at the true opportunity cost. • Selects from mutually exclusive projects the one that maximizes shareholder value. • Respects value-additivity principle. Conflict among Criteria Exercise: Consider the following projects: • A: -1000,100,900,100,-100,-400 • B: -1000,0,0,300,700,1300 • C: -1000,100,200,300,400,1250 • D: -1000,200,300,500,500,600 Show that payback rule chooses A, ARR rule B, NPV with discount rate 10% C and IRR Difficulties Using NPV Rule In practice several issues make the use of NPV rule difficult: • Measurement of cash flows • Incremental cash flows not accounting income • Measurement of opportunity costs of capital • Investments with unequal lives Measurement of cash flows • Incremental cash flow: What part of the cash flow is exclusive consequence of invest- ment project ? ⇒ Ignore Sunk costs ⇒ Include Opportunity costs ⇒ Correct for Side • Projection of future cash flows. • Real versus nominal cash flows. • Capital Cost Allowance (CCA). . If we assume straight-line depreciation then the net cash flow (NCF) can be obtained in two equivalent ways: 1. Total project cash flow approach 2. Tax Shield Approach Remark: In practice we have to use Declining Balance CCA Total project cash flow approach N CF = (N 0I − CCA) × (1 − tc ) + CCA • tc : Corporate tax rate • NOI: Net operating income • NOI-CCA: Taxable income • (N OI − CCA) × (1 − tc ): Net income Tax shield approach N CF = N OI × (1 − tc ) + tc × CCA • N OI × (1 − tc ): After tax net operating income. • tc × CCA: Tax shield (tax savings). Measurement of opportunity costs of capital • Real versus nominal interest rate (The Fisher relation). ⇒ Right discount rate de- pends on whether cash flows are real or nominal. • Correction of discount factor for risk. Investments with unequal lives • Matching cycles: ⇒ repeat projects until timing matches. Example Cost project A −500, 120, 120, 120 project B −600, 100, 100, 100, 100. To compare consider smallest common multiple 12 and compare PV of 4 repetitions of projection A with 3 repetitions of project B. • Equivalent annual costs: Consider example as before. 1 1 1 EAC A = P V A [ + + ]−1 1 + r (1 + r)2 (1 + r)3 1 1 1 1 EAC B = P V B [ + 2 1 + r (1 + r) (1 + r) (1 + r)4

Bạn đang xem: Angular frequency What does 2pif mean? 97 1. frequency-number of rotations in a period of time, usually one s. w-angular velocity-angle in an amount of time. 2pi-one full circle. hence w=2pif. What is 2pi frequency? Frequency is in cycles per second. Multiplying by 2π gives the frequency in radians per second, where a radian is a measure of angle such that 2π radians = 360°. Why is omega equal to 2pif? The angular frequency ω is another way of expressing the number of turns, in terms of radians. One full circle consists of 2π radians of arc, so we multiply the “number of circles per second” by 2π to get the “number of radians per second”—which we call the angular frequency, ω. What is V 2πr T? v=T2πr. In Physics, Uniform Circular Motion is used to describe the motion of an object traveling at a constant speed in a circle. … Here, r represents the radius of the circle, T the time it takes for the object to make one complete revolution, called a period. Is time a frequency? Frequency is a rate quantity. Period is a time quantity. Frequency is the cycles/second. Period is the seconds/cycle. What is the relationship between Omega and frequency? Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π. This figure uses the symbol ν, rather than f to denote frequency. A sphere rotating around an axis. Points farther from the axis move faster, satisfying ω = v / r. What is W Omega? Angular frequency (ω), also known as radial or circular frequency, measures angular displacement per unit time. Its units are therefore degrees (or radians) per second. Angular frequency (in radians) is larger than regular frequency (in Hz) by a factor of 2π: ω = 2πf. How do you get Omega? ω = v/r, where ω is the Greek letter omega. Angular velocity units are radians per second; you can also treat this unit as “reciprocal seconds,” because v/r yields m/s divided by m, or s-1, meaning that radians are technically a unitless quantity. What is 2pi by Omega? It’s an invention. Originally Answered: What is the proof of the theory “omega”= (2*pi)/T in circular motion? T is the period of the motion, the time for full cycle or 360 degrees – which is 2π radians. w (omega) is the angular velocity in radians per second. What is difference between Omega and F? In general, ω is the angular speed – the rate change of angle (as in a circular motion). Frequency (f) is 1/T or the number of periodic oscillations or revolutions during a given time period. Angular speed or angular frequency, relates the same idea to angles – how much angle is covered over a time period. What does Theta mean in physics? Angular Position, Theta. The angle of rotation is a measurement of the amount (the angle) that a figure is rotated about a fixed point— often the center of a circle. What is the angular frequency of the oscillations? The angular frequency ω is given by ω = 2π/T. The angular frequency is measured in radians per second. The inverse of the period is the frequency f = 1/T. The frequency f = 1/T = ω/2π of the motion gives the number of complete oscillations per unit time. What is V 2 in circular motion? Centripetal acceleration The centripetal acceleration is the special form the acceleration has when an object is experiencing uniform circular motion. It is: ac = v2 / r. and is directed toward the center of the circle. What does capital T stand for in physics? A capital T is often used to mean: Time period () Temperature. Kinetic energy. What is the difference between V and Omega? The symbol for the linear velocity is v. The symbol for angular velocity is a Greek letter, i.e., ω (pronounced as omega). … We measure the linear velocity in m/s. We measure the angular velocity in both degrees and radians. What is temporal frequency? The term temporal frequency is used to emphasise that the frequency is characterised by the number of occurrences of a repeating event per unit time, and not unit distance. How many types of frequency are there? Frequency Band NameAcronymFrequency RangeMedium FrequencyMF300 to 3000 kHzHigh FrequencyHF3 to 30 MHzVery High FrequencyVHF30 to 300 MHzUltra High FrequencyUHF300 to 3000 MHz How do you tell the difference between frequency and period? Period refers to the amount of time it takes a wave to complete one full cycle of oscillation or vibration. Frequency, on the contrary, refers to the number of complete cycles or oscillations occur per second. Period is a quantity related to time, whereas frequency is related to rate. How do you get rid of time period? each complete oscillation, called the period, is constant. The formula for the period T of a pendulum is T = 2π Square root of√L/g, where L is the length of the pendulum and g is the acceleration due to gravity. What is difference between angular frequency and frequency? Either one can be used to describe periodic or rotational motion. Frequency (usually with the symbol or ) is cycles (or revolutions) per second. Angular frequency (usually with the symbol ) is radians per second. 1 cycle (or revolution) equals radians, so . What is angular frequency and frequency? Frequency definition states that it is the number of complete cycles of waves passing a point in unit time. The time period is the time taken by a complete cycle of the wave to pass a point. Angular frequency is angular displacement of any element of the wave per unit time. What is R Omega? R-Omega is a phospholipid rich DHA and EPA omega-3 supplement from herring roe. … R-Omega contains 340mg of DHA and 100mg of EPA per two capsule dose. What is the weird W in physics? (mathematics, set theory) The first (countably) infinite ordinal number, its corresponding cardinal number ℵ0 or the set of natural numbers (the latter of which are often defined to equal the former). What does ω stand for? Greek Letter Omega The 24th and last letter of the Greek alphabet, Omega (Ω), essentially means the end of something, the last, the ultimate limit of a set, or the “Great End.” Without getting into a lesson in Greek, Omega signifies a grand closure, like the conclusion of a large-scale event. Where can I find Omega in GTA 5? Travel to the green “?” in the eastern section of Sandy Shores to meet Omega. Approach him and the Far Out mission will begin. How do you find angular frequency from period? It is the reciprocal of the period and can be calculated with the equation f=1/T. Some motion is best characterized by the angular frequency (ω). The angular frequency refers to the angular displacement per unit time and is calculated from the frequency with the equation ω=2πf. What is natural angular frequency? When calculating the natural frequency, we use the following formula: f = ω ÷ 2π Here, the ω is the angular frequency of the oscillation that we measure in radians or seconds. We define the angular frequency using the following formula: ω = √(k ÷ m) How do you calculate omega t? At a particular moment, it’s at angle theta, and if it took time t to get there, its angular velocity is omega = theta/t. So if the line completes a full circle in 1.0 s, its angular velocity is 2π/1.0 s = 2π radians/s (because there are 2π radians in a complete circle). Is Omega an Radian? Angular frequency ω(Ordinary) frequency1 radian per secondapproximately 0.159155 Hz1 radian per secondapproximately 57.29578 degrees per second What is meant by theta in maths? The Greek letter θ (theta) is used in math as a variable to represent a measured angle. For example, the symbol theta appears in the three main trigonometric functions: sine, cosine, and tangent as the input variable. How do you write theta? To Type Theta θ in MS Word, another method is to use it from symbols. In symbols, goto subset Greek and Coptic and select θ from it. Third Method: One of the method to type theta θ in MS Word is press and hold alt key + 952 from numpad. What is difference between angular velocity and angular frequency? Angular velocity is that velocity which act on revolution or rotation of a body with a certain angle with axis with valid radius. … Angular frequency also known as radial or circular frequency, measures angular displacement per unit time. Its units are therefore degrees (or radians) per second. Does mass affect angular frequency? With other variables held constant, as mass increases, angular momentum increases. Thus, mass is directly proportional to angular momentum. What is oscillation frequency? The frequency of oscillation is the number of full oscillations in one time unit, say in a second. A pendulum that takes 0.5 seconds to make one full oscillation has a frequency of 1 oscillation per 0.5 second, or 2 oscillations per second. What is M Squared Omega? Or. Or. V= velocity of the body. w=angular velocity(omega) R=Radius of the circular path in which the body is moving. What"s the formula of tension? Tension Formula. The tension on an object is equal to the mass of the object x gravitational force plus/minus the mass x acceleration. T = mg + ma. T = tension, N, kg-m/s2. Xem thêm: Depth Of Field Là Gì Tại Sao Lại Có Câu Depth Of Field Trong Game Là Gì What is MG force? The weight of an object is the force of gravity on the object and may be defined as the mass times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is the newton. Density is mass/volume.

3 2010 #1 Agricultural experts are trying to develop a bird deterrent to reduce costly damage to crops in the United States. An experiment is to be conducted using garlic oil to study its effectiveness as a nontoxic, environmentally safe bird repellant. The experiment will use European starlings, a bird that causes considerable damage annually to the corn crop in the United States. Food granules made from corn are to be infused with garlic oil in each of five concentrations of garlic – 0 percent, 2 percent, 10 percent, 25 percent, and 50 percent. The researchers will determine the adverse reaction of the birds to the repellent by measuring the number of food granules consumed during a two-hour period following overnight food deprivation. There are forty birds available for the experiment, and the researchers will use eight birds for each concentration of garlic. Each bird will be kept in a separate cage and provided with the same number of food granules. a) For the experiment, identify i. the treatments ii. the experimental units iii. the response that will be measured 4 i. The treatments are the different concentrations of garlic in the food granules. Specifically, there are five treatments: 0 percent, 2 percent, 10 percent, 25 percent and 50 percent. ii. The experimental units are the birds (starlings), each placed in an individual cage. iii. The response is the number of food granules consumed by the bird. 5 After performing the experiment, the researchers recorded the data shown in the table below. i. Construct a graph of the data that could be used to investigate the appropriateness of a linear regression model for analyzing the results of the experiment. 7 ii. Based on your graph, do you think a linear regression model is appropriate? Explain. The curved pattern in this scatterplot reveals that a linear regression model would not be appropriate for modeling the relationship between these variables. 9 2003B #2 A simple random sample of adults living in a suburb of a large city was selected. The age and annual income of each adult in the sample were recorded. The resulting data are summarized in the table below.Annual IncomeAge Category$25,000-$35,000$35,001-$50,000Over $50,000Total21-30815275031-452232358946-60121453Over 60537476496207 10 a) What is the probability that a person chosen at random from those in this sample will be in the age category?b) What is the probability that a person chosen at random from those in this sample whose incomes are over $50,000 will be in the age category? Show your work.Annual IncomeAge Category$25,000-$35,000$35,001-$50,000Over $50,000Total21-30815275031-452232358946-60121453Over 60537476496207 11 c) Based on your answers to parts (a) and (b), is annual income independent of age category for those in this sample? Explain. If annual income and age were independent, the probabilities in (a) and (b) would be equal. Since these probabilities are not equal, annual income and age category are not independent for adults in this sample. 12 2009B #2 The ELISA tests whether a patient has contracted HIV 2009B #2 The ELISA tests whether a patient has contracted HIV. The ELISA is said to be positive if it indicates that HIV is present in a blood sample, and the ELISA is said to be negative if it does not indicate that HIV is present in a blood sample. Instead of directly measuring the presence of HIV, the ELISA measures levels of antibodies in the blood that should be elevated if HIV is present. Because of variability in antibody levels among human patients, the ELISA does not always indicate the correct result. As part of a training program, staff at a testing lab applied the ELISA to 50 blood samples known to contain HIV. The ELISA was positive for 489 of those blood samples and negative for the other 11 samples. As a part of the same training program, the staff also applied the ELISA to 500 other blood samples known to not contain HIV. The ELISA was positive for 37 of those blood samples and negative for the other 463 samples. 13 a) When a new blood sample arrives at the lab, it will be tested to determine whether HIV is present. Using the data from the training program, estimate the probability that the ELISA would be positive when it is applied to a blood sample that does not contain HIV.The estimated probability of a positive ELISA if the blood sample does not have HIV present is 14 b) Among the blood samples examined in the training program that provided positive ELISA results for HIV, what proportion actually contained HIV? A total of = 526 blood samples resulted in a positive ELISA. Of these, 489 samples actually contained HIV. Therefore the proportion of samples that resulted in a positive ELISA that actually contained HIV is 15 c) When a blood sample yields a positive ELISA result, two more ELISAs are performed on the same blood sample. If at least one of the two additional ELISAs is positive, the blood sample is subjected to a more expensive and more accurate test to make a definitive determination of whether HIV is present in the sample. Repeated ELISAs on the same sample are generally assumed to be independent. Under the assumption of independence, what is the probability that a new blood sample that comes into the lab will be subjected to the more expensive test if that sample does not contain HIV? 16 From part (a), the probability that the ELISA will be positive, given that the blood sample does not actually have HIV present, is Thus, the probability of a negative ELISA, given that the blood sample does not actually have HIV present, is 1 – = P(new blood sample that does not contain HIV will be subjected to the more expensive test) = P(1st ELISA positive and 2nd ELISA positive OR 1st ELISA positive and 2nd ELISA negative and 3rd ELISA positive | HIV not present in blood) = P(1st ELISA positive and 2nd ELISA positive | HIV not present in blood) + P(1st ELISA positive and 2nd ELISA negative and 3rd ELISA positive | HIV not present in blood) = (0.074)(0.074) + (0.074)(0.926)(0.074) = = ≈ 17 P(new blood sample that does not contain HIV will be subjected to the more expensive test) = P(1st ELISA positive and not both the 2nd and 3rd are negative)= (0.074)( )= (0.074)( )=≈ 19 2002 #3 There are 4 runners on the New High School team 2002 #3 There are 4 runners on the New High School team. The team is planning to participate in a race in which each runner runs a mile. The team time is the sum of the individual times for the 4 runners. Assume that individual times of the 4 runners are all independent of each other. The individual times, in minutes, of the runners in similar races are approximately normally distributed with the following means and standard deviations.MeanStandard DeviationRunner 14.90.15Runner 24.70.16Runner 34.50.14Runner 44.8 20 a) Runner 3 thinks that he can run a mile in less than 4 a) Runner 3 thinks that he can run a mile in less than 4.2 minutes in the next race. Is this likely to happen? Explain.It is possible but unlikely that runner 3 will run a mile in less than 4.2 minutes on the next race. Based on his running time distribution, we would expect that he would have times less than 4.2 minutes less than 2 times in 100 races in the long run. OR It is possible but unlikely that runner 3 will run a mile in less than 4.2 minutes on the next race because 4.2 is more than 2 standard deviations below the mean. Since the running time has a normal distribution, it is unlikely to be more than 2 standard deviations below the mean. 21 b) The distribution of possible team times is approximately normal b) The distribution of possible team times is approximately normal. What are the mean and standard deviation of this distribution? The runners times are independently distributed, therefore 22 c) Suppose the team’s best time to date is 18. 4 minutes c) Suppose the team’s best time to date is 18.4 minutes. What is the probability that the team will beat its own best time in the next race? 23 2010 #4 An automobile company wants to learn about customer satisfaction among the owners of five specific car models. Large sales volumes have been recorded for three of the models, but the other two models were recently introduced so their sales volumes are smaller. The number of new cars sold in the last six months for each of the models is shown in the table below. The company can obtain a list of all individuals who purchased new cars in the last six months for each of the five models shown in the table. The company wants to sample 2,000 of these owners.Car ModelABCDETotalNumber of new cars sold in the last six months112,33896,17483,2413,2782,323297,354 24 a) For simple random samples of 2,000 new car owners, what is the expected number of owners of model E and the standard deviation of the number of owners of model E? Because the population size is so large compared with the sample size (≈ 149 times the sample size), far greater than the usual standard of 10 or 20 times larger, we can use the binomial probability distribution even though this is technically sampling without replacement. The parameters of this binomial distribution are the sample size, n, which has a value of n = 2,000, and the proportion of new car buyers who bought model E, p, which has a value of p = The expected value of the number of model E buyers in a simple random sample of 2,000 is therefore n× p = 2,000× ≈ The variance is n× p×(1− p) = 2,000×0.0078×(1− ) ≈15.50, so the standard deviation is the square root of ≈ 3.94. 25 b) When selecting a simple random sample of 2,000 new car owners, how likely is it that fewer than 12 owners of model E would be included in the sample? Justify your answer. For the reason given in part (a), the binomial distribution with n = 2,000 and p ≈ can be used here. The probability that the sample would contain fewer than 12 owners of model E is calculated from the binomial distribution to be This probability is small enough that the result (fewer than 12 owners of model E in the sample) is not likely, but this probability is also not small enough to consider the result very unlikely. 26 c) The company is concerned that a simple random sample of 2,000 owners would include fewer than 12 owners of Model D or fewer than 12 owners of Model E. Briefly describe a sampling method for randomly selecting 2,000 owners that will ensure at least 12 owners will be selected for each of the 5 car models.Stratified random sampling addresses the concern about the number of owners for models D and E. By stratifying on car model and then taking a simple random sample of at least 12 owners from thepopulation of owners for each model, the company can ensure that at least 12 owners are included in the sample for each model while maintaining a total sample size of 2,000. For example, the companycould select simple random samples of sizes 755, 647, 560, 22 and 16 for models A, B, C, D and E, respectively, to make the sample size approximately proportional to the size of the owner population foreach model. 28 2006 #3 The depth from the surface of Earth to a refracting layer beneath the surface can be estimated using methods developed by seismologists. One method is based on the time required for vibrations to travel from a distant explosion to a receiving point. The depth measurement (M) is the sum of the true depth (D) and the random measurement error (E). That is, M = D + E. The measurement error (E) is assumed to be normally distributed with mean 0 feet and standard deviation 1.5 feet. a) If the true depth at a certain point is 2 feet, what is the probability that the depth measurement will be negative? Since M = D + E (a normal random variable plus a constant is a normal random variable), we know that M is normally distributed with a mean of 2 feet and a standard deviation of 1.5 feet. Thus, 29 b) Suppose three independent depth measurements are taken at the point where the true depth is 2 feet. What is the probability that at least one of these measurements will be negative? P(at least one measurement < 0) = 1 – P(all three measurements 0) = 1 – (1 – )3 = 1 – (0.9082)3 = 1 – = 30 c) What is the probability that the mean of the three independent depth measurements taken at the point where the true depth is 2 feet will be negative? Let denote the mean of three independent depth measurement taken at a point where the true depth is 2 feet. Since each measurement comes from a normal distribution, the distribution of is normal with a mean of 2 feet and a standard deviation of feet. Thus, 31 2009 #2 2. A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and the stopping distances are approximately normally distributed. a) What is the 70th percentile of the distribution of stopping distances? Let X denote the stopping distance of a car with new tread tires where X is normally distributed with a mean of 125 feet and a standard deviation of 6.5 feet. The z-score corresponding to a cumulative probability of 70 percent is z = Thus, the 70th percentile value can be computed as: 32 b) What is the probability that at least 2 cars out of 5 randomly selected cars in the study will stop in a distance that is greater than the distance calculated in part (a)? From part (a), it was found that a stopping distance of feet has a cumulative probability of Thus the probability of a stopping distance greater than is 1– 0.70 = Let Y denote the number of cars with the new tread pattern out of five cars that stop in a distance greater than feet. Y is a binomial random variable with n = 5 and p = 0.30. 33 c) What is the probability that a randomly selected sample of 5 cars in the study will have a mean stopping distance of at least 130 feet? Let denote the mean of the stopping distances of five randomly selected cars. All tires have the new tread pattern. Because the stopping distance for each of the five cars has a normal distribution, the distribution of is normal with a mean of 125 feet and a standard deviation of feet. Thus, 35 2010 #3A humane society wanted to estimate with 95 percent confidence the proportion of households in its county that own at least one dog.a) Interpret the 95 percent confidence level in this context.The 95 percent confidence level means that if one were to repeatedly take random samples of the same size from the population and construct a 95 percent confidence interval from each sample, then in the long run 95 percent of those intervals would succeed in capturing the actual value of the population proportion of households in the county that own at least one dog. 36 The humane society selected a random sample of households in its county and used the sample to estimate the proportion of households that own at least one dog. The conditions for calculating a 95 percent confidence interval for the proportion of households in this county that own at least one dog were checked and verified, and the resulting confidence interval was ± b) A national pet products association claimed that 39 percent of all American households owned at least one dog. Does the humane society’s interval estimate provide evidence that the proportion of dog owners in its county is different from the claimed national proportion? Explain. No. The 95 percent confidence interval ± is the interval (0.298, 0.536). This interval includes the value 0.39 as a plausible value for the population proportion of households in the county that own at least one dog. Therefore, the confidence interval does not provide evidence that the proportion of dog owners in this county is different from the claimed national proportion. 37 c) How many households were selected in the humane society’s sample c) How many households were selected in the humane society’s sample? Show how you obtained your answer. The sample proportion is 0.417, and the margin of error is Determining the sample size requires solving the equation Thus, so the humane society must have selected 66 households for its sample. 39 2006B #6 Sunshine Farms wants to know whether there is a difference in consumer preference for two new juice products – Citrus Fresh and Tropical Taste. In an initial blind taste test, 8 randomly selected consumers were given unmarked samples of the two juices. The product that each consumer tasted first was randomly decided by the flip of a coin. After tasting the two juices, each consumer was asked to choose which juice he or she preferred, and the results were recorded. a) Let p represent the population proportion of consumers who prefer Citrus Fresh. In terms of p, state the hypotheses that Sunshine Farms is interested in testing. H0 : p = 0.5 versus Ha : p ≠ 0.5 40 b) One might consider using a one-proportion z-test to test the hypotheses in part (a). Explain why this would not be a reasonable procedure for this sample. The conditions for the large sample one-proportion z-test are not satisfied. np = n(1 – p) = 8 x 0.5 = 4 < 5. 41 c) Let X represent the number of consumers in the sample who prefer Citrus Fresh. Assuming there is no difference in consumer preference, find the probability for each possible value of X. Record the x-values and the corresponding probabilities in the table below. X will follow a binomial distribution with n = 8 and p = 8. The possible values of X and their corresponding probabilities are given in the table below: 42 d) When testing the hypotheses in part (a), Sunshine Farms will conclude that there is a consumer preference if too many or too few individuals prefer Citrus Fresh. Based on your probabilities in part (c), is it possible for the significance level (probability of rejecting the null hypothesis when it is true) for this test to be exactly 0.05? Justify your answer. No, there is no possible test with a p-value of exactly 0.5. The probability that none of the individuals (X = 0) or all of the individuals (X = 8) prefer Citrus Fresh is 2 x = , which is less than The probability that one or fewer of the individuals (X 1) or seven or more of the individuals (X 7) prefer Citrus Fresh is 2 x ( ) = , which is greater than 0.05. 43 e) The preference data for the 8 randomly selected consumers are given in the table below. Based on these preferences and your previous work, test the hypotheses in part (a).IndividualJuice Preference1Tropical Taste2Citrus Fresh345678 44 For the preference data provided, X = 2 For the preference data provided, X = 2. From the table of binomial probabilities computed in part (c), the probability that two or fewer of the individuals (X 2) or six or more of the individuals (X 6) prefer Citrus Fresh when p = 0.05 is 2 x ( ) = Because the p-value of is greater than any reasonable significance level, say , we would not reject the null hypothesis that p = 0.5. That is, we do not have statistically significant evidence for a consumer preference between Citrus Fresh and Tropical Taste. 45 f) Sunshine Farms plans to add one of these two new juices – Citrus Fresh or Tropical Taste – to its production schedule. A follow-up study will be conducted to decide which of the two juices to produce. Make one recommendation for the follow-up study that would make it better than the initial study. Provide a statistical justification for your recommendation in the context of the problem. 46 Increase the number of consumers involved in the preference test Increase the number of consumers involved in the preference test. More consumers will give you more data, and you will be better able to detect a difference between the population proportion of consumers who prefer Citrus Fresh and 0.5. The sample proportion in the initial study was only 0.25 (2/8), but we were not able to reject the null hypothesis that p = ½. By increasing the number of consumers, a difference of that magnitude would allow the null hypothesis to be rejected. For example, with n = 80 and X = 20 the large sample z- statistic would be and the p-value would be approximately zero.

Gimlet Rule - a simplified visual demonstration using one hand to correctly multiply two vectors. The geometry of the school course implies pupils' awareness of the scalar product. In physics, the vector is often found. The Concept of a Vector We believe that there is no sense in interpreting the driller rule with no knowledge of the definition of a vector. It is required to open a bottle - the knowledge of the correct actions will help. A vector is a mathematical abstraction that does not really exist, showing these signs: - A directional segment, indicated by an arrow. - The starting point is the point of action of the force described by the vector. - The length of the vector is equal to the modulus of the force, the field, the other described quantities. Not always affect force. The vectors describe the field. The simplest example is shown to schoolchildren by teachers of physics. We imply lines of magnetic field intensity. Vectors are usually drawn along a tangent along. In the illustrations of the action on the conductor with a current you will see straight lines. Vector values are often devoid of application space, action centers are selected by agreement. The moment of force emanates from the axis of the shoulder. Required to simplify addition. Suppose that levers of different lengths are affected by different forces applied to the shoulders with a common axis. By simple addition, subtraction of moments, we find the result. Vectors help to solve many everyday problems and, although they act as mathematical abstractions, they really work. On the basis of a number of regularities, it is possible to predict the future behavior of an object along with scalar values: the population of the population, the ambient temperature. Environmentalists are interested in directions, the speed of flight of birds. Displacement is a vector quantity. The Gimlet Rule helps to find the vector product of vectors. This is not a tautology. Just the result of the action will also be a vector. The gimlet rule describes the direction to which the arrow will point. As for the module, you need to apply the formula. The gimlet rule is a simplified purely qualitative abstraction of a complex mathematical operation. Analytical geometry in space Everyone knows the problem: standing on one side of the river, determine the width of the channel. It seems to the mind incomprehensible, solved in two ways by the methods of simplest geometry, which students learn. Let's do a number of simple actions: - Detect a prominent landmark on the opposite bank, an imaginary point: a tree trunk, the mouth of a stream that flows into a stream. - At the right angle of the opposite bank line, make a notch on this side of the channel. - Find a place from which the landmark is visible at an angle of 45 degrees to the shore. - The width of the river is equal to the distance of the end point from the notch. We use the tangent of an angle. Not necessarily equal to 45 degrees. Need more accuracy - the angle is better to take sharp. Just the tangent of 45 degrees is one, the solution of the problem is simplified. Similarly, it is possible to find answers to burning questions. Even in the electron-controlled microcosm. We can definitely say one thing: to the uninitiated the rule of a gimlet, the vector product of vectors seems to be boring, boring. A handy tool that helps in understanding many processes. Most will be interested in the principle of operation of the electric motor( regardless of the design).Can easily be explained using the left-hand rule. In many branches of science, two rules follow side by side: the left, the right hand. Vector product can sometimes be described in one way or another. Sounds vague, we suggest to immediately consider an example: - Suppose an electron moves. A negatively charged particle plows a constant magnetic field. Obviously, the trajectory will be bent due to the Lorentz force.skeptics will argue, according to some scientists, the electron is not a particle, but rather a superposition of fields. But the principle of uncertainty Heisenberg consider another time. So, the electron moves: By placing the right hand so that the vector of the magnetic field perpendicularly enters the palm, the extended fingers indicate the direction of the particle's flight, bent 90 degrees to the side, the thumb will stretch in the direction of the force. The right-hand rule is another expression of the gimlet rule. Synonyms. It sounds different, in fact - one. - We give the phrase Wikipedia, giving a weirdness. When reflected in a mirror, the right three of vectors becomes left, then you need to apply the rule of the left hand instead of the right. The electron flew in one direction, according to the methods adopted in physics, the current moves in the opposite direction. As if reflected in the mirror, therefore the Lorentz force is already determined by the left hand rule: If you place your left hand so that the magnetic field vector perpendicularly enters the palm, the extended fingers indicate the direction of the current flow, bent 90 degrees to the side of the thumb, stretching, indicating the action vectorstrength You see, situations are similar, the rules are simple. How to remember which one to apply? The main principle of uncertainty of physics. The vector product is calculated in many cases, with one rule being applied. What is the rule to apply The words are synonyms: arm, screw, gimlet First we analyze the word-synonyms, many began to ask themselves: if the story here should affect the gimlet, why does the text constantly touch the hands. We introduce the concept of the right three, the right coordinate system. Total, 5 words-synonyms. It was necessary to find out the vector product of vectors, it turned out: this does not work in school. Clarify the situation inquisitive schoolchildren. School graphics on the board are drawn in the Cartesian coordinate system X-Y.The horizontal axis( positive part) is directed to the right - we hope, the vertical axis points up. We take one step, getting the right three. Imagine: from the beginning of the counting, the Z axis looks to the class. Now the schoolchildren know the definition of the right three vectors. In Wikipedia it is written: it is permissible to take left triples, right, when calculating a vector product, they disagree. Usmanov is categorical in this respect. With the permission of Alexander Evgenievich, we give an exact definition: a vector product is a vector that satisfies three conditions: - The product module is equal to the product of the modules of the original vectors and the sine of the angle between them. - The result vector is perpendicular to the original( together they form a plane). - The triple vectors( in order of context) the right. The right three know. So, if the X axis is the first vector, Y is the second, Z will be the result. Why was called the right three? Apparently, it is connected with screws, gimlets. If the imaginary gimlet is twisted along the shortest path, the first vector is the second vector, the translational axis of the cutting tool will begin to move in the direction of the resulting vector: - The gimlet rule applies to the product of two vectors. - The driller rule qualitatively indicates the direction of the resultant vector of this action. Quantitatively, the length is the expression mentioned( the product of the modules of the vectors and the sine of the angle between them). Now everyone understands: the Lorentz force is found according to the rule of a left-handed thread. The vectors are collected by the left triple, if mutually orthogonal( perpendicular to one another), the left coordinate system is formed. On the board, the Z-axis would look in the direction of the view( from the audience behind the wall). Simple techniques for memorizing the rules of the gimlet People forget that it is easier to determine the Lorentz force by the rule of the gimlet with left-handed thread. One who wants to understand the principle of operation of an electric motor should double-click like nuts. Depending on the design, the number of rotor coils is significant, or the circuit degenerates, becoming a squirrel cage. Knowledge seekers are helped by the Lorentz rule, which describes the magnetic field where copper conductors move. To memorize, let's present the physics of the process. Suppose an electron moves in a field. The rule of the right hand is applied to find the direction of the force. It is proved: the particle carries a negative charge. The direction of the force on the conductor is the left-hand rule, remember: physicists completely from the left resources took that electric current flows in the opposite direction to where the electrons went. And this is wrong. Therefore it is necessary to apply the rule of the left hand. Do not always go such a wilds. It would seem that the rules are more confusing, not quite. The right-hand rule is often used to calculate the angular velocity, which is the geometric product of the acceleration radius: V = ω x r. Many people will be helped by visual memory: - The vector of the radius of the circular path is directed from the center to the circle. - If the acceleration vector is directed upwards, the body moves counterclockwise. Look, the right-hand rule is again here: if you position the palm so that the acceleration vector enters perpendicularly into the palm, extend your fingers in the direction of the radius, bent by 90 degrees, the thumb indicates the direction of movement of the object. It is enough to draw once on paper, remembering at least half a life. The picture is really simple. More on the physics lesson will not have to wrestle with a simple question - the direction of the vector of angular acceleration. Similarly, the moment of force is determined. It comes perpendicularly from the axis of the shoulder, coincides with the direction with angular acceleration in the figure described above. Many will ask: what is needed? Why is the moment of force not a scalar quantity? Why the direction? In complex systems is not easy to trace the interaction. If there are a lot of axes, forces, vector addition of moments helps. You can greatly simplify the calculations.

Below is a list of the best Ice cream scoop size chart voted by readers and compiled and edited by our team, let’s find out Update: Also see my post on dishers for left-handed use. To ensure that cookies, cupcakes, and muffins bake evenly, one of the steps should be to divide the dough or batter into uniform quantities. For the best way to do this, the common advice is to use portion-control tools. These are known as dishers in the food-service industry, although to the rest of us they look like ice cream scoops. In addition to their use in baking, dishers can also ensure consistent portioning of meatballs or hamburger patties. You can buy them at your local restaurant-supply store. In the US, commercial-grade dishers are denominated in sizes (numbers) that represent quart fractions—for example, a No. 12 disher should hold 1/12 of a quart (in other words, it takes 12 scoops to fill a quart), a No. 16 holds 1/16 quart, etc. Using this standard, we can create a chart of disher sizes and their equivalent nominal volumes, in both US customary and metric units: * Color codes are only available on dishers with plastic handles. Size Color* fl oz tbsp cup (fraction) mL 6 White 5.33 10.7 0.667 (2/3) 158 8 Gray 4.00 8.00 0.500 (1/2) 118 10 Ivory 3.20 6.40 0.400 94.6 12 Green 2.67 5.33 0.333 (1/3) 78.9 16 Blue 2.00 4.00 0.250 (1/4) 59.1 20 Yellow 1.60 3.20 0.200 47.3 24 Red 1.33 2.67 0.167 39.4 30 Black 1.07 2.13 0.133 31.5 40 Orchid 0.800 1.60 0.100 23.7 50 Rust 0.640 1.28 0.0800 18.9 60 Pink 0.533 1.07 0.0667 15.8 70 Plum 0.457 0.914 0.0571 13.5 100 Orange 0.320 0.640 0.0400 9.46 You’ll notice there are gaps between sizes. As far as I know, these are the only ones available for the US food-service market, and not all manufacturers make this entire size range. But that’s not the problem. The real problem is that the quart-fraction standard is only followed loosely, and actual disher capacities vary both from the nominal size and among different manufacturers. How far off are these scoop sizes, you ask? Let’s look at the numbers. The following table contains a semi-random sampling of product lines and shows how much the scoops’ specified capacities deviate from nominal sizes. These calculations are based on the manufacturers’ specifications, which I’ve collected into a spreadsheet you can either view online (HTML) or download (Excel, with formulas). The file is also available in Google Spreadsheets format if you’re logged in to a Google account. In addition to the manufacturers’ specifications and my calculations, the spreadsheet also contains details such as the dishers’ scoop diameters shown in both inches and centimeters. Deviation of specified capacity from nominal size. Bold indicates sizes accurate to within 2%. Details. Size Adcraft (PDF) Fox Run Hamilton Beach Johnson Rose Norpro OXO Vollrath (metal) Vollrath (plastic) Zeroll 6 12.5% 12.6% 12.5% 0.0% 12.6% 8 0.0% 0.0% 9.0% 8.3% 0.0% 0.0% 9.0% 10 17.2% 0.3% 14.6% 2.3% 1.6% 0.3% 12 21.9% 4.3% 25.0% 3.1% 0.0% 4.3% 16 37.5% 0.0% 3.5% 0.0% 0.0% 0.0% 3.5% 20 56.3% 0.0% 10.6% 4.2% 6.3% 6.3% 6.3% 1.6% 10.6% 24 31.3% 11.8% 0.0% 3.1% 0.0% 11.8% 30 17.2% 3.4% 6.3% 6.3% 17.2% 6.3% 3.4% 40 9.4% 6.3% 15.0% 16.7% 6.3% 9.4% 6.3% 11.3% 50 2.3% 0.0% 2.3% 1.6% 60 5.5% 6.3% 37.5% 5.5% 0.6% 70 9.4% 0.6% 6.0% 0.6% 100 17.2% 0.0% 4.2% 17.2% 0.0% As you can see, disher size accuracy can be way off the mark. In this table, scoops accurate to within 2% of the nominal size are highlighted in bold. If this seems to be too tight of an allowance for size variations, think about it this way: a 12-inch ruler that’s off by 2% will be either too long or too short by roughly a quarter of an inch, making for a potential variation of nearly half an inch from one ruler to the next. While it’s possible that dishers are made to match nominal sizes but their true capacities are rounded off for publication formatting (thus calculations based on published specifications will show more deviation than actually exists), seeing that there is no uniformity even within one manufacturer’s own two product lines leads me to believe this is not the case, since there is no reason to think different rounding standards would be used here. Moreover, by comparing the ratios between sizes, one is likely to find that manufacturers’ guidance for real-world applications (e.g., hamburger patties) further deviate from both the nominal and specified volumes. Makes one wonder which numbers are really right. Please note, however, that in spite of these discrepancies, inaccurate dishers aren’t necessarily defective or inferior. This is because dishers are primarily portioning tools instead of measuring tools, so if you find some that fit your recipes, there’s not much point to worrying about whether they match some fixed standard or not. You just need to be aware that size designations are not reliable indicators of actual capacity, and dishers of identical (nominal) size from different manufacturers may hold different amounts of material. Now, how do you know which dishers will fit your recipes without buying every size and trying them? That’s a good question. From what I’ve seen, recipes generally don’t tell you that. At least for muffins and cupcakes, since standard muffin tins have a capacity of 1/2 cup (4 fl. oz., or 8 tbsp) per pocket, we should figure to place only about 1/4 to 1/3 cup of batter in each to avoid overflowing during baking—in other words, #12 and #16 (nominal size) should work in most cases. However, I’ve also seen recipes that exceed this quantity to deliberately create overflowing muffin tops, so exceptions do exist. Disher (ice cream scoop) sizes can be inconsistent, so it’s better to know their actual (or at least specified) capacities than to rely on nominal sizes. In terms of nominal size, 12 and 16 should, in principle, fit most muffin and cupcake recipes.

Magnetised Neutron Stars : An Overview\abstracts In the presence of strong magnetic field reported to have been observed on the surface of some neutron stars and on what are called Magnetars, a host of physical phenomenon from the birth of a neutron star to free streaming neutrino cooling phase will be modified. In this review I will discuss the effect of magnetic field on the equation of state of high density nuclear matter by including the anomalous magnetic moment of the nucleons into consideration. I would then go over to discuss the neutrino interaction processes in strong as well as in weak magnetic fields. The neutrino processes are important in studying the propagation of neutrinos and in studying the energy loss, Their study is a prerequisite for the understanding of actual dynamics of supernova explosion and on the stabilization of radial pulsation modes through the effect on bulk viscosity. The anisotropy introduced in the neutrino emission and through the modification of the shape of the neutrino sphere may explain the observed pulsar kicks. Large magnetic fields Gauss have been reported to exist on the surface of pulsars. Recent observations of -ray repeaters and spinning X-ray pulsars (Magnetars) hints to the existence of fields in excess of Gauss. It then follows from the Scalar Virial theorem that the fields in the core could even reach a value as large as Gauss. There is however, an upper limit on the magnetic field discussed by Chandershekhar beyond which the magnetic energy exceeds the gravitational energy and the star is no longer stable.In the presence of such magnetic fields, neutron star properties in all its phases from its evolution from proto-neutron star to cold neutrino emitting phase would be modified. This arises because in the presence of magnetic field motion of the charged particles is quantised in a plane perpendicular to the magnetic field and the charged particles occupy discrete Landau levels. This has the effect of not only modifying the energy eigenvalues but also the particle wave functions.The quantum state of a particle in magnetic field is specified by momentum components , spin s and Landau quantum number . The anomalous magnetic moments of protons and neutrons further modify the energy eigenvalues. The time scales involved during all phases of neutron stars from birth to neutrino burst through thermal neutrino emission from trapped neutrino sphere to freely streaming neutrino cooling phase are large compared to the interaction time scales of strong, electromagnetic and weak interactions and the matter is in equilibrium. The magnetic field would modify the equilibrium and all neutrino interaction processes including scattering, absorption and production. The stategy then is to first solve the Dirac equation for all particles in magnetic field including their anamolous magnetic moments, obtain the energy eigenvalues, construct the Grand Partition Function taking into account strong interactions in some model dependent way and obtain the equation of state (EOS). The next step is to calculate the scattering cross-sections for all neutrino processes by using the exact wave functions and by modifying the phase space integrals for arbitrary values of degenracy, density, temperature and magnetic field. The various phenomenon that I will address are : Composition of matter in neutron stars, proton fraction, effective nucleon mass etc. Cooling of neutron stars in the free streaming regime. Neutrino transport in neutron and collapsing stars which is an essential prerequisite for an understanding of supernova explosion, structure of proto-neutron star and observed pulsar kicks. Damping of radial oscillations and secular instability through the calculation of bulk viscosity. 2 Nuclear Matter Composition For determining the composition of dense, hot, magnetised matter, we employ a relativistic mean field theoretical approach in which the baryons (protons and neutrons) interact via the exchange of mesons in a constant uniform magnetic field. Following reference in a uniform magnetic field B along z axis corresponding to the choice of the gauge field ,the relativistic mean field Lagrangian can be written as in the usual notation with and as the anomalous magnetic moments given by where and are the Lande’s g-factor for protons and neutrons respectively. Replacing the meson fields in the relativistic mean field approximation by their density dependent average values , and , the equations of motion satisfied by the nucleons in the magnetic field become The equations are first solved for the case when momentum along the magnetic field direction is zero and then boosting along that direction till the momentum is . For the neutrons and protons we thus get where , where and n being integer known as Landau quantum number and principal quantum number respectively. indicates whether the spin is along or opposite to the direction of the magnetic field and is the effective baryon mass. The energy spectrum for electrons is given by The mean field values , and are determined by minimizing the energy at fixed baryon density or by maximizing the pressure at fixed baryon chemical potential . We thus get where and (i=e,p) are the number and scalar number densities for proton and neutron. In the presence of the magnetic field phase space volume is replaced as The expressions for number densities, scalar densities for neutron and proton are given by and the net electron and neutrino number densities are given by The thermodynamic potential for the neutron, proton, electron and neutrino are given by and the thermodynamic potential for the system is given by where . The various chemical potential are determined by the conditions of charge neutrality and chemical equilibrium. In later stages of core collapse and during the early stages of protoneutron star, neutrinos are trapped and the chemical potentials satisfy the relation These situations are characterized by a trapped lepton fraction where is the net electron fraction and is the net neutrino fraction. The evolution of a protoneutron star begins from a neutrino-trapped situation with to one in which the net neutrino fraction vanishes and chemical equilibrium without neutrinos is established. In this case the chemical equilibrium is modified by setting . In all cases, the conditions of charge neutrality requires In the nucleon sector, the constants , , , b and c are determined by nuclear matter equilibrium density and the binding energy per nucleon , the symmetry energy , the compression modulous and the nucleon Dirac effective mass at . Numerical values of the coupling constants so chosen are : 3 Weak Rates and Neutrino Emissivity The dominant mode of energy loss in neutron stars is through neutrino emission.The important neutrino emission processes leading to neutron star cooling are the so called URCA processes At low temperatures for degenerate nuclear matter, the direct URCA process can take place only near the fermi energies of participating particles and simultaneous conservation of energy and momentum require the inequality to be satisfied in the absence of the magnetic field. This leads to the well known threshold for the proton fraction thus leading to strong suppression in nuclear matter. This condition is satisfied for ( where ,is the nuclear saturation density), in a relativistic mean field model of interacting n-p-e gas for . The standard model of the long term cooling is the modified URCA: which differ from the direct URCA reactions by the presence in the initial and final states of a bystander particle whose sole purpose is to make possible conservation of momentum for particles close to the Fermi surfaces.For weak magnetic field the matrix element for the process remains essentially unaffected and the modification comes mainly from the phase space factor. Treating the nucleons non-relativistically and electrons ultra-relativistically, the matrix element squared and summed over spins is given by where is the axial-vector coupling constant. The emissivity expression is given by where the phase space integrals are to be evaluate over all particle states. The statistical distribution function ,where are the Fermi-Dirac distributions. We can now evaluate the emissivity in the limit of extreme degeneracy, a situation appropriate in neutron star cores by using the standard techniques to perform the phase space integrals where . In the limit of vanishing magnetic field,the sum can be replaced by an integral and we recover usual expression i.e. the case for B=0. Modified URCA process are considered to be the dominant process for neutron star cooling. Similarly the energy loss expression with appropriate electron phase space, for modified URCA process is calculated to be where is the coupling constant and has been estimated to be . The above equation in the limit goes over to the standard result In the case of super strong magnetic fields such that ( Gauss) all electrons occupy the Landau ground state at T=0 which corresponds to state with electron spins pointing in the direction opposite to the magnetic field. Charge neutrality now forces the degenerate non-relativistic protons also to occupy the lowest Landau level with proton spins pointing in the direction of the field. In this situation we can no longer consider the matrix elements to be unchanged and they should be evaluated using the exact solutions of Dirac equation. Further because nucleons have anomalous magnetic moment, matrix elements need to be evaluated for specific spin states separately. The electron in state has energy and the positive energy spinor in state is given by Protons are treated non-relativistically and the energy in state is and the non-relativistic spin up operator given by where . For neutrons we have and in the non-relativistic limit.The neutrino wave function is given by Here is the usual free particle spinor, is the spin spinor and the wave function has been normalised in a volume . Using the explicit spinors given above we can now calculate the matrix element squared and summed over neutrino states to get The neutrino emissivity is calculated by using the standard techniques for degenerate matter and we get where and are the neutron Fermi momenta for spins along and opposite to the magnetic field direction respectively and are given by We thus see as advertised that in the presence of quantising magnetic field the inequality is no longer required to be satisfied for the process to proceed, regardless of the value of proton fraction and we get non zero energy loss rate. 4 Bulk Viscosity of Magnetised Neutron Star Matter The source of bulk viscosity of neutron star matter is the deviation from equilibrium, and the ensuing nonequilibrium reactions, implied by the compression and rarefaction of the matter in the pulsating neutron star. These important reactions are the URCA and the modified URCA processes. Since the source of bulk viscosity is the deviation from equilibrium, these reactions are driven by non zero value of . We calculate the bulk viscosity of neutron star matter in the presence of magnetic field for direct URCA processes in the linear regime i.e. . The bulk viscosity is defined by Here is the specific volume of the star in equilibrium configuration, is the amplitude of the periodic perturbation with period and The quantity is the mean dissipation rate of energy per unit mass and is given by the equation The pressure can be expressed near its equilibrium value , as The change in the number of neutrons, protons and electrons per unit mass over a time interval due to URCA reactions (23) is given by The net rate of production of protons, is given by the difference between the rates and of the URCA reactions. At equilibrium the two rates are obviously equal and the chemical potentials satisfy the equality A small volume perturbation brings about a small change in the chemical potentials and the above inequality is no longer satisfied, now is not zero and consequently the reaction rates are no longer equal. The net rate of production of protons will thus depend upon the value of . In the linear approximation, , the net rate can be written as Using the thermodynamic relation and employing the above relation we obtain The change in the chemical potential arises due to a change in the specific volume and changes in the concentrations of various species, viz, neutrons, protons and electrons. Thus and we arrive at the following equations for : Since for small perturbations, , A and C are constants, equation (46) can be solved analytically to give and we obtain the following expressions for Given the number densities of the these particle species in terms of their respective chemical potentials, one can determine the coefficients A and C; given the rates and for the two URCA processes one can determine and hence for any given baryon density and temperature. For weak magnetic field several Landau levels are populated and the matrix elements remain essentially unchanged and one needs to account for the correct phase space factor. For non-relativistic degenerate nucleons the decay rate constant is given by For strong magnetic field, the electrons are forced into the lowest Landau level. Using the exact wave functions for protons and electrons in the lowest Landau level and carrying out the energy integrals for degenerate matter, the decay constant is given by It is clear from above that in the case of completely polarised electrons and protons the direct URCA decay rate always gets a non-zero contribution from the second term in the last square bracket irrespective of whether the triangular inequality is satisfied or not. 5 Neutrino Opacity in Magnetised Hot and dense Nuclear Matter We calculate neutrino opacity for magnetised, interacting dense nuclear matter for the following limiting cases: a) nucleons and electrons, highly degenerate with or without trapped neutrinos, b) non-degenerate nucleons, degenerate electrons and no trapped neutrinos and finally, c) when all particles are non-degenerate. The important neutrino interaction processes which contribute to opacity are the neutrino absorption process and the scattering processes both of which get contributions from charged as well as neutral current weak interactions. For the general process The cross-section per unit volume of matter or the inverse mean free path is given by where is the density of states of particles and the transition rate . Weak Magnetic Field : For weak magnetic fields, several Landau levels are populated and the matrix element remain essentially unchanged and one needs to only account for the correct phase space factor. We first consider the neutrino-nucleon processes. In the presence of weak magnetic fields, the matrix element squared and summed over initial and final spins in the approximation of treating nucleons non-relativistically and leptons relativistically is given by where 1, 1.23 for the absorption process; 1, 1.23 for neutrino scattering on neutrons and 0.08, 1.23 for neutrino We now obtain neutrino cross-sections in the limits of extreme degeneracy or for non-degenerate matter. Degenerate Matter : The absorption cross-section for highly degenerate matter can be calculated by using (79) in (78) by the usual techniques and we get for small B The case of freely streaming, untrapped neutrinos is obtained from the above equation by putting 0 and replacing by . When the magnetic field is much weaker than the critical field for protons, only electrons are affected and the neutrino-nucleon scattering cross-section expression remain unchanged by the magnetic field. The numerical values however, are modified due to changed chemical composition. The cross-sections are given by If neutrinos are not trapped, we get in the elastic limit The neutrino-electron scattering cross section is which in the untrapped regime goes over to Non-Degenerate Matter : We now treat the nucleons to be non-relativistic non-degenerate such that and thus the Pauli-blocking factor can be replaced by 1, the electrons are still considered degenerete and relativistic. The various cross-sections are given by where is the nuclear density and . If the electrons too are considered non-degenerate we get Quantising Magnetic Field : For quantizing magnetic field the square of the matrix elements can be evaluated in a straight forward way and we get

Evercore Restructuring Interview Questions You Need to KnowUpdated: Ranking restructuring investment banks is hard. However, there is no question that among the Tier 1 restructuring investment banks you'll find Evercore. Evercore's restructuring practice - founded by the legendary David Ying - has grown quickly and has picked up some of the largest out-of-court deals in recent years. Evercore's restructuring practice is also entirely siloed from their other business lines. Meaning that you'll be applying directly for a position within restructuring and will only be interviewed by members of the RX team. Below are five of the most frequent RX questions that crop up in Evercore restructuring interviews. In every restructuring interview - but in particular for Evercore - you should expect some of the following questions: - Bond math questions (on YTM at various durations) - Waterfall questions (where you'll be given a cap table and asked what recovery values are) - Restructuring specific accounting questions (around PIK, asset write downs, etc.) If you'd like even more questions (roughly 500 more, to be exact), be sure to check out the Restructuring Interviews course. A bond is worth $80 and has a 10% coupon maturing in one year. What's the YTM? What if it matures in two years, not one? Before beginning to answer any question on bond math, you always want to make sure you have all the information you need. For example, you haven't been told here whether or not the coupon is paid semi-annually or annually. This makes a difference. Note: The YTM will be lower if it's paid twice a year. For nearly all bond math questions - because they don't want you breaking out a calculator - they'll be annual calculations. However, you should have an intuition for how yields work nevertheless. So, if we have a coupon being paid annually then we can use the simple, generalized formula C + (P-FV) / P where C is coupon and FV is face value. Note: We weren't told in the question what the face value was. However, you can assume it's always $100 if the price is x < $100. So we will have $10 of coupon payments, a spread between P and FV of $20 on maturity, and a current price of $80. Therefore, we have 30/80 = 37.5%, which is your answer (as can be verified with a YTM calculator). Now what happens if the bond matures in two years, not one? Well we're getting one more coupon payment ($10), which is good, but we're now delaying getting the spread between FV and P for two years. So YTM is going to invariably be lower. Remember that YTM hinges on reinvesting proceeds at the same rate so cash flow today is always heavily preferred. When dealing with maturities beyond just one year, we need to use the more formal estimated YTM formula. This won't get us the exact YTM - as that would involve a much more complicated set up - instead it'll get us a reasonably close approximation. It's important to point that out in an interview. The estimated YTM formula, for when you have maturities equal or greater to two years, is simply the following where n is the number of years until maturity: Using this formula we'll get down to (10+10)/90, which is 22.22%. You can verify that this is the correct estimated YTM with the YTM calculator linked above (which provides both the exact YTM and the traditional estimated YTM as well). Enterprise value (EV) is $200 and we have a TL of $100, Senior Secured Notes of $50, and Unsecured Notes of $100 and another tranche of Unsecured Notes (maturing two years after the first) of $50. What are the recovery values throughout? Here we have a slight spin on a typical waterfall question. First, we know that the TL and the Senior Secured Notes are going to be fully covered leaving $50 behind. Now we have two groups of Notes (bonds) within the same class. As an aside, which may be worth keeping in mind, in a Chapter 11 you'll have a Plan of Reorganization (POR) that will need to be submitted by the debtor. One of the requirements of the POR is that it treats all claims within the same class equally, unless otherwise consented to by one of the claims in the class. In other words, if you're in the same class you have to actively agree to take less than your proportional share (which, as you can imagine, is rare to find!). So in this question we have two separate Notes (differing in maturity and size, but not in their seniority) and they must be treated equally. So this class of claims should be thought of as being $150 ($100+$50) and the amount they can lay claim to is $50. So the recovery rate for both is 33.33%. It's also important to have the right verbiage down in an interview. These Unsecured Notes represent an impaired class that in the event of a Chapter 11 would be the ones that would vote on a POR. So the recovery rate of this impaired class is 33.33%. Let's say we have $100 in debt with 15% in PIK. How does this flow through the three statements? Let's assume a 20% tax rate. PIK accounting questions are very common in RX interviews because so many of the out-of-court restructurings done will involve some PIK. Note: Why is this the case? The obvious answer is because the company likely doesn't have much cash on hand (negative FCF, limited liquidity) so PIK allows them to avoid imminent cash crunches. The less obvious answer, perhaps, is that PIK allows the company to offer a much higher interest rate (in this case 15%), which current holders who may be exchanging bonds into will find enticing. So let's go through it. On the income statement (IS) you will have $15 in new interest expense in the form of newly issued debt. This creates a tax shield (another reason why you can have higher interest rate) of $3 ($15*20%). Therefore, net income is down by $12. Moving to the top of the cash flow statement (CFS) you have $12, you then add back the $15 as it's a non-cash expense (that's the primary reason to do PIK!), so you have cash flow from operations up by $3. On the balance sheet (BS) you have assets (cash) up by $3, on the liabilities side you have debt up by $15, and within shareholders equity (retained earnings) you have a $12 decrease from net income. So both sides of the equation are up by $3. If you know a certain class of debt will be heavily impaired for a public company in distress, why might equity be trading above zero? Aren't they at the bottom of the capital structure? While it's true that equity is (of course) at the bottom of the capital structure, equity also has hypothetically unlimited upside in the event the company turns things around. It's entirely common for a company to be in obvious distress - where there will be impaired class(es) in the capital structure - yet equity is trading above zero. This reflects the optionality of equity. In the event of a turnaround of some kind, equity could have incredibly large gains (whereas debt, even if trading at a heavy discount, will have more lacklustre gains as it will just creep back up to around par). So as a company enters into distress equity increasingly begins to look like a call option with a capped downside (equity going to zero in the event of a Chapter 11) or having incredibly large upside if the company can turn things around. Tupperware is a good, practical example of this where bonds were trading below fifty cents on the dollar in March of 2020 (so clearly quite distressed) yet equity was still trading above $1 per share. Tupperware successfully did an out-of-court restructuring - pushing out maturities until 2023 - and the equity is trading well over $30 per share in 2021. If we have a leverage ratio of 5 and a coverage ratio of 5, what is the yield on the debt? This is one of my favorite questions. It's a bit of a brainteaser, because when you hear the answer you'll realize just how easy it is. I've written a rather long post on it over here where I go over two different ways to solve it (one brute force, one a bit more generalizable). So for the sake of brevity I won't cover it step-by-step here. Let's start by noting our formulas. For the leverage ratio we have Debt / EBITDA and for the coverage ratio we have EBITDA / Interest Expense. The key to this problem is being able to isolate the yield on debt (interest rate or r), but it's obviously not directly in any of the formulas here. The key is to notice that all interest expense represents is Debt*(r). So we can say that the coverage ratio is 5 = EBITDA / ((r)(Debt)) and isolate r by noticing that EBITDA / Debt is just the inverse of the leverage ratio (so it's 1/5) and we therefore arrive at r = 1/25 or 4%. As a final note, one thing I should mention is that in all restructuring interviews one of the most important things to do is show that you have the contextual understanding of what RX really is in practice. This is particularly true now as so many people are trying to get into restructuring positions. This is largely the reason why the Restructuring Interviews course is not just a set of hundreds of Q&A (although those are there), but also includes a nearly 100-page guide on what restructuring is in practice, what the day-to-day job entails, and what examples of deliverables you'll need to produce look like. The single best way to stand out and give your interviewer confidence that you're worth extending an offer to is by showing you know what you're getting involved in and that it is much different than "traditional" M&A investment banking.

Algebra and Trigonometry: Graphing and Data Analysis, 1/e Michael Sullivan, Chicago State University Michael Sullivan, South Suburban College Published December, 1997 by Prentice Hall Engineering/Science/Mathematics Copyright 1998, 1137 pp. Sign up for future mailings on this subject. See other books about: Algebra/Trig with Graphing Calculators-Mathematics Designed for the Precalculus course covering Algebra and Trigonometry. This text covers right angle trigonometry first and then develops the unit circle approach. This text requires student use of graphing calculators or a computer based software program. For schools who wish to cover unit circle first, please see Precalculus: Graphing and Data Analysis. The goal of this text is to provide a solid mathematical foundation via visualization of real world data. Technology is used as a tool to solve problems, motivate concepts, explore and preview mathematical concepts and to find curves of best fit to the data. Most mathematical concepts are developed and illustrated both algebraically and graphically - with the more intuitive and appropriate method presented first. @BREAKNOLINALT = Mathematics The authors use their extensive teaching and writing experiences to guide and support students through the typical difficult areas. Each section opens with the mathematical objectives of the section. Each objective is referenced as it is encountered in the Examples are worked out step-by-step, both numerically and Many examples include the Now Work feature which suggests a similar odd-numbered problem from the section exercise set. This allows for immediate reinforcement of concepts through Historical Notes are provided in context, enhancing student interest, provide anecdotal information on how and where mathematical concepts have come from. Exercises are carefully crafted beginning with confidence builders and visualization exercises, then practice and drill, followed by the more challenging and application driven problems. Discussion, Writing and Research questions are clearly called out by the red icon in the margin. Each chapter opens by listing the concepts (and page references) that the student will need to review Before Getting Started. The chapters conclude with a detailed chapter review, including Important Formulas, Theorems and Definitions, a list of Things to Know and Do, True/False Questions, Fill-in-the-Blank items, and Review Exercises. @BREAKNOLINALT = Technology The authors approach the use of technology as an enhancement to the learning of mathematics not as a replacement for learning. Graphing utilities are used to help students analyze data and find curves of best fit. Types of curve fitting discussed include: linear, quadratic, cubic, power, exponential, logarithmic, logistic, Using the power of the grapher, students are able to approach problems and concepts that may have been beyond them without the grapher. Real TI-83 screen shots are used as the illustrations for the purpose of clear visualization of the materials. @BREAKNOLINALT = Data Sourced data connects the mathematical concepts to other disciplines and other interests of the students - adding relevancy Applications involving data analysis utilize real world sources such as the US Census Bureau, Government Agencies and the Each chapter has an Internet Exploration. These optional explorations introduce students to live data via the Internet. Multiple questions follow each exploration encouraging the use of Polya's problem solving strategies. The links to the sites are all maintained via the Prentice Hall Companion Website for Sullivan (NOTE: Chapters end with Chapter Review.) Data and its Representation. Rectangular Coordinates; Graphing Utilities; Data in Ordered Pairs. Graphs of Equations. Lines. Parallel and Perpendicular Lines; Circles. Linear Curve Fitting. Variation. 2. Functions and Their Graphs. Functions. More About Functions. Graphing Techniques. Operations on Functions; Composite Functions. Mathematical Models: Constructing 3. Equations and Inequalities. Solving Equations Using A Graphing Utility. Linear and Quadratic Equations. Setting Up Equations: Applications. Other Types of Equations. Inequalities. Equations and Inequalities Involving Absolute Value. 4. Polynomial and Rational Functions. Quadratic Functions; Curve Fitting. Power Functions; Curve Fitting. Polynomial Functions; Curve Fitting. Rational Functions. The Real Zeros of a Polynomial Function. Complex Numbers; Quadratic Equations with a Negative Discriminant. Complex Zeros; Fundamental Theorem of Algebra. Polynomial and Rational Inequalities. 5. Exponential and Logarithmic Functions. One-to-One Functions; Inverse Functions. Exponential Functions. Logarithmic Functions. Properties of Logarithms. Logarithmic and Exponential Equations. Compound Interest. Growth and Decay. Exponential, Logarithmic, and Logistic Curve Fitting. Logarithmic Scales. 6. Trigonometric Functions. Angles and Their Measure. Right Triangle Trigonometry. Computing the Values of Trigonometric Functions of Given Angles. Trigonometric Functions of a General Angle. Properties of the Trigonometric Functions. Graphs of the Trigonometric Functions. The Inverse Trigonometric Functions. 7. Analytic Trigonometry. Trigonometric Identities. Sum and Difference Formulas. Double-Angle and Half-Angle Formulas. Product-to-Sum and Sum-to-Product Formulas. 8. Applications of Trigonometric Functions. Solving Right Triangles. The Law of Sines. The Law of Cosines. The Area of a Triangle. Sinusoidal Graphs: Sinusoidal Curve Fitting. Simple Harmonic Motion: Damped Motion. 9. Polar Coordinates; Vectors. Polar Coordinates. Polar Equations and Graphs. The Complex Plane; Demoivre's Theorem. Vectors. The Dot Product. 10. Analytic Geometry. Conics. The Parabola. The Ellipse. The Hyperbola. Rotation of Axes; General Form of a Conic. Polar Equations of Conics. Plane Curves and Parametric Equations. 11. Systems of Equations and Inequalities. Systems of Linear Equations: Substitution; Elimination. Systems of Linear Equations; Matrices. Systems of Linear Equations: Determinants. Matrix Algebra. Partial Fraction Decomposition. Systems of Nonlinear Equations. Systems of Inequalities. Linear Programming. 12. Sequences; Induction; Counting; Probability. Sequences. Arithmetic Sequences. Geometric Sequences; Geometric Series. Mathematical Induction. The Binomial Theorem. Sets and Counting Permutations and Combinations. Probability. Topics from Algebra and Geometry. Polynomials and Rational Expressions. Radicals; Rational Exponents. Solving Equations. Completing the Square. Synthetic Division.

Hi Dears, My name is Ali, I'm trading in Forex market for more than 15 years, and I loss a lot of money before as you, but I believe that, there is a rule in this market, so I worked on many technical and fundamental systems, but I found that the market is not depend on the technical and fundamentals, there is not any restrict in the market, and market try to go to the special targets, this targets is calculated from past points, and I could found this targets since 18 months ago, and we should take profit from this targets. Of course there is two coordinates in this targets,The firs one is the value of targets and the second one is the time, So unfortunately I cant find the time exactly as the value, but I found that these targets should touch by market in 4 next candle. So I do not want to tell about this targets, but I want to play a game , a win 2 win game by me and you. but how? I create a PAMM account in real market 4 month ago and I can earn 2.5% each week in balance and equity, I'am trading on 15 pairs with out losing on balance, and you can see the historical trades in my real account and many analytic statisticts as bellow So you could monitor my trades and invest on my PAMM account (Damavand) to earn 2.5% in each week and 10% on each month, I hold 40% as wage (managers fee) of trading and the strategies so you could try only by 1000$ at the beginning, and withdraw the profit or all of your money at the end of the each investment period(after 4 weeks) By this strategy, I can trade in H4,Daily and Weekly time frames, but I prefer to trade in weekly as midterm and long term investment, Most of the positions take 2 or 3 days in average. This PAMM account as you see is a real account, and the balance is more than 100k right know. Let see the performance and some statistics of that: Monthly Return by Balance, Jan 2016 : 7.97% Feb 2016 : 9.37% Mar 2016 : 6.35% Apr 2016 : 15.68% Till Today (08 May) : 3.63% Monthly Return By Equity, Jan 2016 : 7.66% Feb 2016 : 1.14% Mar 2016 : 8.76% Apr 2016 : 19.43% Till Today (08 May) : 6.7% Weekly Return by Balance, Week No 1 10 Jan : 3% Week No 2 17 Jan : 1.7% Week No 3 24 Jan : 1.3% Week No 4 31 Jan : 1% Week No 5 07 Feb : 2.2% Week No 6 14 Feb : 2.4% Week No 7 21 Feb : 1.5% Week No 8 28 Feb : 4.8% Week No 9 06 Mar : 0.7% Week No 10 13 Mar : 0.9% Week No 11 20 Mar : 0% Nowrouz Holiday Week No 12 27 Mar : 4.2% Week No 13 03 Apr : 4.4% Week No 14 10 Apr : 2.6% Week No 15 17 Apr : 3.7% Week No 16 24 Apr : 1.1% Week No 17 01 May: 3% Weekly Return by Equity, Week No 1 10 Jan : 2.9% Week No 2 17 Jan : 1.7% Week No 3 24 Jan : 1.3% Week No 4 31 Jan : 1% Week No 5 07 Feb : 2.3% Week No 6 14 Feb : 1.9% Week No 7 21 Feb : -2.7% Week No 8 28 Feb : 5.0% Week No 9 06 Mar : -2.9% Week No 10 13 Mar : 9.2% Week No 11 20 Mar : 0% Nowrouz Holiday Week No 12 27 Mar : 0.9% Week No 13 03 Apr : 4.8% Week No 14 10 Apr : -1.7% Week No 15 17 Apr : -0.6% Week No 16 24 Apr : 9.7% Week No 17 01 May: 5.4% I'm interested, to inform you some other statistics of this PAMM account, Absolute Draw down = 0.00 % Relative Draw down = 12.41% Maximal Draw down = 8.02% MyFXbook Draw down = 17.25% Monthly Target Return = 10% and Average Monthly Return is 10.5% you can see in Goal part of the Report weekly Target Return = 2.5% and Average Weekly return is 2.4% you can see in Goal part of the Report Trade Accuracy between 75% and 85% Avg Loss $/Avg Won$ between 0.5 and 0.85 Avg Trade Length between 2 and 3 Days Z-Core(Probability) = -30.79(99.99%) ,It means 99.99% win trade followed by win trade and loss trade followed by loss trade Sharp Ratio =0.52 Close Trade = 1521 (in 4 or 5 Month) in average 12 trade a day Trade Instruments Number = 15 forex pair Risk Of Ruin <0.01% Max Consecutive win(Count)= 42 Max Consecutive loss(Count)= 22 Average Consecutive win(Count)=19 Average Consecutive loss(Count)=6 Profit Factor =3.14 after invest in PAMM account, The robot will trade for you, and at the end of investment period you can withdraw your profit PAMM Account contract stat are: Minimum Investment is : 300$ Investment Period is : 1 Week Performance fee is : 40% of profit Also you can introduce this PAMM account to others and take 20% of New investor's performance fee lets take an Example: Suppose that you invest 1000$ for 1 week and at the end of week, robot take 2.5 percent as profit your profit=1000*0.025=25$ you should pay 40% as performance fee =25*0.4=10$ your Net Profit=25-10=15$ Also if you could introduce 10 person how invest 1000$ too,you can earn 20% of theirs performance fee too, PAMM agent's fee for each investor : 25*0.4*0.2=2$ your agent's fee for 10 investor : 2*10=20$ So your Net Profit=15$+20$=35$ each week 35$ each week means 35/1000=3.5% each week or 14% return for each month or 168% as annual return, So invest and share the PAMM by your friends for maximum return as you can and earn unlimited profit each weekend This system has weekly signal as mid term and long term signal, So I am trying to publish this signal in the next week, So you could follow my trend and invest on my Real PAMM Account. This week I can earn only 0.3% as profit, but the target of the each week is 2.5 Percent, so I think the next week this system can earn more than 2.5 percent for profit 고위험 경고: 외환 거래는 모든 투자자에게 적합하지 않을 수 있는 높은 수준의 위험을 수반합니다. 레버리지는 추가적인 위험 및 손실 노출을 만듭니다. 외환 거래를 결정하기 전에 투자 목표, 경험 수준 및 위험 허용 오차를 신중하게 고려하십시오. 초기 투자의 일부 또는 전부를 잃을 수 있습니다. 잃을 여유가 없는 돈을 투자하지 마십시오. 외환 거래와 관련된 위험에 대해 스스로 교육하고 궁금한 점이 있으면 독립 금융 또는 세무사에게 조언을 구하십시오. 모든 데이터 및 정보는 정보 제공 목적으로만 있는 그대로 제공되며 거래 목적이나 조언을 위한 것이 아닙니다. 과거의 성과는 미래의 결과를 나타내는 것이 아닙니다.

The Scalar Curvature of a Causal Set A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D’Alembertian when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well approximated by curved spacetimes in which case they approximate where is the Ricci scalar curvature. This can used to define an approximately local action functional for causal sets. The coexistence of Lorentz symmetry and fundamental, Planck scale spacetime discreteness has its price: one must give up locality. Since, if our spacetime is granular at the Planck scale, the “atoms of spacetime” that are nearest neighbours to a given atom will be of order one Planck unit of proper time away from it. The locus of such points in the approximating continuum Minkowski spacetime is a hyperboloid of infinite spatial volume on which Lorentz transformations act transitively. The nearest neighbours will, loosely, comprise this hyperboloid and so there will be an infinite number of them. Where curvature limits Lorentz symmetry, it may render the number of nearest neighbours finite but it will still be huge so long as the radius of curvature is large compared to the Planck length. Causal set theory is a discrete approach to quantum gravity which embodies Lorentz symmetry Bombelli et al. (1987, 2006) and exhibits nonlocality of exactly this form Moore (1988); Bombelli et al. (1988). Nonlocality looks to be simultaneously a blessing and a curse in tackling the twin challenges that any fundamentally discrete approach to the problem of quantum gravity must face. These are to explain (1) how the fundamental dynamics picks out a discrete structure that is well approximated by a Lorentzian manifold and (2) why, in that case, the geometry should be a solution of the Einstein equations. This is often referred to as the problem of the continuum limit but in the context of a fundamentally discrete theory in which the discreteness scale is fixed and is not taken to zero but rather the observation scale is large, it is more accurately described as the problem of the continuum approximation. Consider first the problem of recovering a continuum from a quantum theory of discrete manifolds. (We adopt this term following Riemann Riemann (1868) and use it to refer to causal sets, simplicial complexes, graphs, or whatever discrete entities the underlying theory is based on.) Whenever a background principle or structure in a physical theory is abandoned in order to seek a dynamical explanation for that structure, the state we actually observe becomes a very special one amongst the myriad possibilities that then arise. The continuum is just such a background assumption. In giving it up, generally one introduces a space of discrete manifolds in which the vast majority have no continuum approximation. There will therefore be a competition between the entropic pull of the huge number of noncontinuum configurations – choose one uniformly at random and it will not look anything like our spacetime – and the dynamical law which must suppress the contributions of these nonphysical configurations to the path integral. The following general argument shows that a local dynamics for quantum gravity will struggle to provide the required suppression. Consider the partition function as a sum over histories in which the weight of each discrete manifold is where is the real Wick rotated action. As we increase the observation scale, the sum will be over discrete manifolds with an increasing number, , of atoms. If the action is local – which in a discrete setting translates to it being a sum over contributions from each atom – then it will grow no faster than times some constant, , and so each weight is no smaller than . If the number of discrete manifolds with atoms grows faster than exponentially with , and if the majority of these discrete manifolds are not continuumlike then they will overwhelm the partition function and the typical configuration will not have a continuum approximation. Even when the number of discrete manifolds is believed to grow exponentially, entropy can still trump dynamics as was seen in the lack of a continuum limit in the Euclidean dynamical triangulations programme Agishtein and Migdal (1992a, b); Ambjorn and Jurkiewicz (1992, 1994). Causal dynamical triangulations do better, see, e.g., Ambjorn et al. (2004, 2005); Ambjorn et al. (2008a, b), by restricting the class of triangulations allowed in the sum. In the case of causal sets, the number of discrete manifolds of size grows as Kleitman and Rothschild (1975) and a local action would give causal set theory little chance of recovering the continuum. So the nonlocality of causal sets holds out hope that the theory has a continuum regime and indeed there exist physically motivated, classically stochastic dynamical models for causal sets Rideout and Sorkin (2000) in which the entropically favoured configurations almost surely do not occur and those that do exhibit an intriguing hint of manifold-like-ness Ahmed and Rideout (2009). However, nonlocality poses a danger when it comes to the second challenge of recovering Einstein’s equations. If we assume that a discrete quantum gravity theory does have a 4 dimensional continuum regime, and if the theory is local and generally covariant, then the long distance physics will be governed by an effective Lagrangian which is a derivative expansion in which all diffeomorphism invariant terms are present but higher derivative terms are suppressed by the appropriate powers of the Planckian discreteness length scale, : where is the Ricci scalar, and are dimensionless couplings of order 1, and the dots denote further curvature squared terms as well as cubic and higher terms. The coefficient of the leading term, , is also naturally of order 1 which would make it 120 orders of magnitude larger than its observed value. However, that would also produce curvature on Planckian scales and so would not be compatible with the assumption of a continuum approximation. In a discrete theory, the question of why the cosmological constant does not take its natural value is the same question as why there is a continuum regime at all and we must look to the fundamental dynamics for its resolution. Assuming there is a resolution and a continuum regime exists, locality and general covariance then pretty much guarantee Einstein’s equations due to the natural suppression of the curvature squared and higher terms compared to the Einstein-Hilbert term. So, Lorentz symmetry and discreteness together imply nonlocality, but nonlocality blocks the recovery of general relativity, and if causal sets were incorrigibly nonlocal, this would be fatal. Suppose, however, that the nonlocality were somehow limited to length scales shorter than a certain , which could be much larger than the Planckian discreteness scale, , but yet have remained experimentally undetected to date. There is already evidence that this is possible and indeed causal sets admit constructions that are local enough to approximate the scalar D’Alembertian operator in 2 dimensional flat spacetime Henson (2006); Sorkin (2006). We add to this evidence here by exhibiting a family of discrete operators that approximate the scalar D’Alembertian in 4 dimensional flat spacetime. Further, both the 2D and 4D operators, when applied to scalar fields on causal sets which are well described by curved spacetimes approximate , where is the Ricci scalar curvature. We use this to propose an action for a causal set which is approximately local. We recall that a causal set (or causet) is a locally finite partial order, i.e., it is a pair where is a set and is a partial order relation on , which is (i) reflexive: , (ii) acyclic , and (iii) transitive , for all . Local finiteness is the condition that the cardinality of any order interval is finite, where the (inclusive) order interval between a pair of elements is defined to be . We write when and . We call a relation a link if the order interval contains only and : they are nearest neighbours. Sprinkling is a way of generating a causet from a -dimensional Lorentzian manifold . It is a Poisson process of selecting points in with density so that the expected number of points sprinkled in a region of spacetime volume is . This process generates a causet whose elements are the sprinkled points and whose order is that induced by the manifold’s causal order restricted to the sprinkled points. We say that a causet is well approximated by a manifold if it could have been generated, with relatively high probability, by sprinkling into . We propose the following definition of a discrete D’Alembertian, , on a causet that is a sprinkling, at density , into 4D Minkowski space . Let be a real scalar field, then where the sums run over 4 layers , and . So, for example, layer is the set of all elements that are linked to and as described above, they will be distributed close to a hyperboloid that asymptotes to the past light cone of and is proper time away from . This sum will not in general be uniformly convergent if it is over the elements of a sprinkling into infinite so we introduce an IR cutoff, , by embedding in and summing over the finitely many elements sprinkled in the intersection of the causal past of and a ball of radius centred on . The details of the calculation that shows why 4 layers are necessary in 4D will appear elsewhere, however see Sorkin (2006) for an explanation of why 3 layers are needed in 2D and the conjecture that 4D will require 4 layers. Now let be a real test field of compact support on . If we fix a point (which we always take to be included in ) and evaluate on a sprinkling into , its expectation value in this process is given by where , is the volume of the causal interval between and and there is an implicit cutoff , the size of the support of , on the integration range. It can be shown that this mean converges, as the discreteness scale is sent to zero, to the continuum D’Alembertian of , and that is well approximated by when the characteristic length scale, , on which varies is large compared to . is therefore effectively sampling the value of the field only in a neighbourhood of of size of order and the mean, at least, of is about as local as it can possibly be, given the discreteness. To see roughly how this can happen, notice that the integrand in (4) is negligible for where is such that . The significant part of the integration range therefore lies between the past light cone of and the hyperboloid and comprises a part within a neighbourhood of of size – whence the local contribution – and the rest which stretches off far down the light cone. It is this second part of the range which threatens to introduce nonlocality but because it can be coordinatized by itself and some coordinates on the hyperboloid the integration over it will be proportional to If is nearly constant over length scale , the integration is close to zero and the contribution is suppressed. The fluctuations in , however, are a different matter: if the physical IR cutoff is fixed and the discreteness scale sent to zero, i.e., the number of causet elements grows, simulations show the fluctuations around the mean grow rather than die away and will not be approximately equal to the continuum . To dampen the fluctuations we follow Sorkin (2006) and introduce an intermediate length scale and smear out the expressions above over this new scale, with the expectation that when the inhering averaging will suppress the fluctuations via the law of large numbers. Thus we seek a discrete operator, , whose mean is given by (4) but with replaced by : where now . Working back, one can show that the discrete operator, , with this mean is reduces to when . effectively samples over elements in 4 broad bands with a characteristic depth , the bands’ contributions being weighted with the same set of alternating sign coefficients as in . Since (7) is just (4) with replaced by , the mean of is close to when the characteristic scale over which varies is large compared to . Now, however, numerical simulations show that the fluctuations are tamed. Points were sprinkled into a fixed causal interval in between the origin and on the axis, at varying density , where volume . For each , 100 sprinklings were done and for each sprinkling, was calculated at the topmost point of the interval for and . For , the mean was and the standard deviation . For , and and for , and . These results indicate that the fluctuations do die away, as anticipated, as increases and are consistent with the dependence . Further results will appear elsewhere. The operators and derived in both 2D (in Sorkin (2006)) and 4D are defined in terms of the order relation on alone and so can be applied to a scalar field on any causet. If, therefore, is a (2D or 4D) curved spacetime and is a scalar field on , we can compute on a sprinkling into and calculate its mean. Let and be the volumes of the intervals in 2D and 4D respectively, and . Then, in the presence of curvature, in 2D and 4D respectively. These expressions can be evaluated using Riemann normal coordinates and in both cases we find The limit is a good approximation to the mean when the field varies slowly over length scales and the radius of curvature . If the damping of fluctuations found in simulations in flat space are indicative of what happens in curved space then, for a fixed large enough IR cutoff, , the nonlocality length scale can be chosen such that and the value of for a single sprinkling will be close to the mean. If is applied to the constant field , we therefore obtain an expression that is close to the scalar curvature of the approximating spacetime. In each of 2D and 4D, we can now define a one parameter family of candidate actions, , for a causal set, , by summing over the elements of , times to get the units right, times a number of order one which in 4D is the ratio of to , where is the rationalized Planck length. When the nonlocality length equals the discreteness length , and the action, takes a particularly simple form as an alternating sum of numbers of small order intervals in . Up to factors of order one, we have in 2D and 4D, respectively: where is the number of elements in and is the number of () element inclusive order intervals in . Because is the most non-nonlocal of the operators in the family, the action is a sum of contributions each of which is not close to the value of the Ricci scalar at the corresponding point of the continuum approximation. However, one might expect that if the curvature is slowly varying on some intermediate scale, which we might as well call , the averaging involved in the summation might perform the same role of suppressing the fluctuations as the smearing out of the operator itself so that the whole action is a good approximation to the continuum action when is the appropriate size. There are many new avenues to explore. Can we use these results to define a quantum dynamics for causal sets? In 2D is there a relation with the Gauss-Bonnet theorem? Can we analytically continue the action in an appropriate way Sorkin (2009) to enable Monte-Carlo simulations of the path sum? What sort of phenomenology might emerge from such actions? To answer this latter question, we need to know how big must be so that the action is a good approximation to the Einstein-Hilbert action of the continuum . In Sorkin (2006), a rough estimate is reported that in dimension 4, . Taking to be the Hubble scale, that would mean that in the continuum regime, only spacetimes whose curvature was constant over a scale would be able to have an approximately local fundamental action. One might expect therefore that the phenomenological IR theory of gravity that could emerge from such a fundamental theory would be governed by an effective Lagrangian where and are of order 1, is set to its observed value, and where varies with epoch and today is much larger than the Planck scale. The phenomenological implications of these ideas remain to be explored. We end by pointing out that these results have a relevance beyond causal set theory as they provide a “proof of concept” for the mutual compatibility of Lorentz invariance, fundamental spacetime discreteness, and approximate locality. Acknowledgements.We thank Rafael Sorkin for invaluable help and Michael Delph and Joe Henson for useful discussions. We also thank David Rideout for help with the simulations using his CausalSets toolkit in the Cactus framework (www.cactuscode.org). DMTB is supported by EPSRC. FD is supported by by EC Grant No. MRTN-CT-2004-005616 and Royal Society Grant No. IJP 2006/R2. We thank the Perimeter Institute for Theoretical Physics, Waterloo, Canada where much of this work was done. - Bombelli et al. (1987) L. Bombelli, J.-H. Lee, D. Meyer, and R. Sorkin, Phys. Rev. Lett 59, 521 (1987). - Bombelli et al. (2006) L. Bombelli, J. Henson, and R. D. Sorkin (2006), eprint gr-qc/0605006. - Moore (1988) C. Moore, Phys. Rev. Lett. 60, 655 (1988). - Bombelli et al. (1988) L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 60, 656 (1988). - Riemann (1868) B. Riemann, Über die hypothesen, welche der geometrie zu grunde liegen,1854, Riemann’s Habiliationsschrift, Göttingen, (1868). - Agishtein and Migdal (1992a) M. E. Agishtein and A. A. Migdal, Mod. Phys. Lett. A7, 1039 (1992a). - Agishtein and Migdal (1992b) M. E. Agishtein and A. A. Migdal, Nucl. Phys. B385, 395 (1992b), eprint hep-lat/9204004. - Ambjorn and Jurkiewicz (1992) J. Ambjorn and J. Jurkiewicz, Phys. Lett. B278, 42 (1992). - Ambjorn and Jurkiewicz (1994) J. Ambjorn and J. Jurkiewicz, Phys. 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Oriti (Cambridge University Press, 2006), eprint gr-qc/0601121. - Sorkin (2006) R. D. Sorkin, in Approaches to Quantum Gravity: Towards a New Understanding of Space and Time, edited by D. Oriti (Cambridge University Press, 2006), eprint gr-qc/0703099. - Sorkin (2009) R. D. Sorkin , in “Recent Research in Quantum Gravity”, edited by A. Dasgupta, (Nova Science Publishers, New York, to be published), eprint arxiv:0911.1479.

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S.1.3(a) The stress system applied to the plate is shown in Fig. x The point P therefore moves at an angle to the x axis given by tanÿ1 8:25 xy = 45N/mm. ÿ x Solutions to Chapter 10 Problems ± Stress analysis of aircraft components 121 Thus, from Eq. or ®nal length L is given by (1.12) 3p Solution-1-H6739.tex 24/1/2007 9:28 Page1 Aircraft Structures for engineering students Fourth Edition Solutions Manual T. H. G. Megson ÿd ÿ10y3 6d 2 y dy 0 Thus the stress function satis®es the boundary conditions for axial load in the x –10 Fig. # T.H.G. ; d Megson. Solutions to Chapter 1 Problems ± Basic elasticity 0 50AB cos ÿ 35BC sin 40AB sin 40BC cos Dividing through Eq. 1:29 8:14 q px 3 Megson. II ÿ20:2 N=mm2 ÿ @x4 @4 ii Hence, from Eqs (1.27) L0 Thus for the isotropic sheet, Eqs (1.47) become @x2 E @y2 75 ÿ 0 Hence Published in 1999 by Arnold, 338 Euston Road, London NW1 3BH, UK. ; from Eq. xy 2 y @y2 From Eqs (1.6) and assuming body forces X Y 0 2 Substituting for B from Eq. @x4 @4 2 q The resultant shear force on the plane x 0 is given by Unlike static PDF Aircraft Structures For Engineering Students 6th Edition solution manuals or printed answer keys, our experts show you how to solve each problem step-by-step. @f1 y Bd whence "y y ÿ x T 2θ = 23° td 3 (1.15) ( Instructor's Solutions Manual ) Applied Fluid Mechanics (6th Ed., Mott), © 2020 Created by AircraftOwner Online. (1.45), G E=2 1 E=2:5 and Eq. @y2 Substituting in Eq. τ N/mm2 2 2 @y2 2 @x2 @y2 and since x y Er2 T 0 ÿ3By2 ÿ C All rights reserved. closed, thin-walled beams Megson. @y2 ii y @2 Part I Elasticity Our solutions are written by Chegg experts so you can be assured of the highest quality! ÿ5l 2 y2 ÿ d 2 ÿ 5y4 6y2 d 2 ÿ d 4 3pxy 2 ÿ458 or 1358 2 x f1 y 4d 3 Thus, the stress function satis®es the boundary conditions for load in the y direction. then ÿ3:5x 10y Eq. 752 4 752 ÿ Indian Institute of Technology, Kharagpur, solution manual, Megson.pdf - Aircraft Structures for engineering students Solutions Manual T H G Megson A member of the Hodder Headline Group LONDON \u000f, 11 out of 11 people found this document helpful. Megson. S.1.4(h) S.1.5 Solutions to Chapter 1 Problems S.1.1 The principal stresses are given directly by Eqs (1.11) and (1.12) in which σ . ÿ 2 @x S.1.4(f ) I (iv) q q Solutions Manual . All rights reserved. Also, from the last of Eqs (1.47) and Eq. q @2 0 @[email protected] @x 3:5p 1 ÿ 2 1 1 x or, since G E=2 1 (see Section 1.15) The direct strains are expressed in terms of the stresses using Eqs (1.42), i.e. All rights reserved. " L ÿ L0 L0 1 T ÿ L0 E viii uÿ @x2 All rights reserved. v Solutions to Chapter 2 Problems The resultant shear force at any section of the beam is ÿP. standard texts on stress analysis, strength of materials etc. 2 ÿd ÿ Thus, using the method described in Section 1.6 and the q 2 3 @x @y Eqs (vii) and (viii) now become I 1:29 N=mm2 @y G E so that v 2:75p 10 N/mm2 10 N/mm2 (σy ) Fig. ÿ 2 x Ay B x y 0 ÿ y z II (1.12) from 0; Megson. and –60 –50 2θ = 37° C From Eq. # T.H.G. I II 15 N=mm2 and that the x and y planes are principal planes. @x2 give the state of stress shown in Fig. @x2 4p (v) 50 352 4 402 i.e. 30 @2 2, σ. y = τ0 (or vice versa) and . S.2.1 From Eqs (1.42) in which z 0 @x vii Dierentiating Eq. (iv) for "x , "y and xy from Eqs (i), (ii) and (iii) respectively gives (v) gives @y Hence Therefore, from Eq. E v 8:25p 75 N=mm2 0 @2 Aircraft Structures for Engineering Students, 3rd Edition: Solutions Manual by T.H.G. Similarly, suppose 3pxy on the plane x l. S.2.4 d=2 . 1 ÿ 2 Thus Also at any section x where y ÿd 23 S.2.2 k x2 y2 ÿ a2 i @2 @2 A # T.H.G. 2 1 @[email protected] σ Q2 (σy , –τxy ) Course Hero, Inc. ÿ 2 Solutions to Chapter 1 Problems S.1.2 Megson. corresponds to I while the second value corresponds to II . P.2.3 and taking moments about the plane x l, 4 ÿ Megson. The shear stress distribution given by Eq. (viii), x ÿd pl x = 80N/mm. 2:75p 1 3 @x2 2 d=2 y e 2G"y triangular element of unit thickness shown in Fig. Q2 (30,–5) 40 50 σ N/mm2 60 –10 Fig. Published in 1999 by Arnold, 338 Euston Road, London NW1 3BH, UK. I @ 2 xy Published in 1999 by Arnold, 338 Euston Road, London NW1 3BH, UK. 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- Why is temperature directly proportional to pressure? - Which state of matter has the lowest temperature? - Why does pressure increase when temperature decreases? - Does temperature affect pressure matter? - Does pressure decrease temperature? - Does pressure increase as temperature increases? - What is effect of change of pressure? - What is the relationship between temperature and air pressure? - How does temperature affect pressure? - Are pressure and temperature directly related? - What happens to pressure if temperature decreases? - Does water volume change with temperature? - Is temperature and pressure inversely proportional? - Does temperature affect water pressure? - What is the effect of temperature on change of state of matter? - Why does temperature increase when pressure increases? - How do you convert pressure to temperature? Why is temperature directly proportional to pressure? Gay Lussac’s Law – states that the pressure of a given amount of gas held at constant volume is directly proportional to the Kelvin temperature. If you heat a gas you give the molecules more energy so they move faster. This means more impacts on the walls of the container and an increase in the pressure.. Which state of matter has the lowest temperature? SolidSolid matter exists at the lowest temperature of the four states of matter. Why does pressure increase when temperature decreases? When this happens, the pressure (P) of the gas increases if the number of moles (n) of gas remains constant. If you keep the pressure constant, reducing the temperature (T) also causes the gas to compress. Does temperature affect pressure matter? Physical conditions like temperature and pressure affect state of matter. … When thermal energy is added to a substance, its temperature increases, which can change its state from solid to liquid (melting), liquid to gas (vaporization), or solid to gas (sublimation). Does pressure decrease temperature? For example, when the pressure increases then the temperature also increases. When the pressure decreases, then the temperature decreases. … Because there is less mass in the can with a constant volume, the pressure will decrease. This pressure decrease in the can results in a temperature decrease. Does pressure increase as temperature increases? As the temperature increases, the average kinetic energy increases as does the velocity of the gas particles hitting the walls of the container. The force exerted by the particles per unit of area on the container is the pressure, so as the temperature increases the pressure must also increase. What is effect of change of pressure? The temperature at which a solid melts to become a liquid at the atmospheric pressure is called its melting point. … Effect of change of pressure on matter: By applying pressure we can bring the particles of matter closer and closer. What is the relationship between temperature and air pressure? The relationship between the two is that air temperature changes the air pressure. For example, as the air warms up the molecules in the air become more active and they use up more individual space even though there is the same number of molecules. This causes an increase in the air pressure. How does temperature affect pressure? The temperature of the gas is proportional to the average kinetic energy of its molecules. Faster moving particles will collide with the container walls more frequently and with greater force. This causes the force on the walls of the container to increase and so the pressure increases. Are pressure and temperature directly related? The pressure of a given amount of gas is directly proportional to its absolute temperature, provided that the volume does not change (Amontons’s law). The volume of a given gas sample is directly proportional to its absolute temperature at constant pressure (Charles’s law). What happens to pressure if temperature decreases? Decreasing Pressure If temperature is held constant, the equation is reduced to Boyle’s law. Therefore, if you decrease the pressure of a fixed amount of gas, its volume will increase. … Gay-Lussac’s law states that at constant volume, the pressure and temperature of a gas are directly proportional. Does water volume change with temperature? An increase in temperature caused the water molecules to gain energy and move more rapidly, which resulted in water molecules that are farther apart and an increase in water volume. … When water is heated, it expands, or increases in volume. When water increases in volume, it becomes less dense. Is temperature and pressure inversely proportional? For a fixed mass of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional. Or Boyle’s law is a gas law, stating that the pressure and volume of a gas have an inverse relationship. If volume increases, then pressure decreases and vice versa, when the temperature is held constant. Does temperature affect water pressure? A 5% increase in absolute temperature will resultin a 5% increase in the absolute pressure. … Resultant pressure changes will vary. A useful thumb rule for water is that pressure in a water-solid system will increase about 100 psi for every 1 F increase in temperature. What is the effect of temperature on change of state of matter? As temperatures increase, additional heat energy is applied to the constituent parts of a solid, which causes additional molecular motion. Molecules begin to push against one another and the overall volume of a substance increases. At this point, the matter has entered the liquid state. Why does temperature increase when pressure increases? Pressure is created by the number of collisions that occur between the molecules and the surface of container. If the temperature in the container is increased this will cause the molecules to move faster. As molecules move faster the number of collisions that will occur will increase. How do you convert pressure to temperature? If, for instance, the gas contains 2 moles of molecules: 20 / 2 = 10. Divide the result by the gas constant, which is 0.08206 L atm/mol K: 10 / 0.08206 = 121.86. This is the gas’s temperature, in Kelvin. Subtract 273.15 to convert the temperature to degrees Celsius: 121.86 – 273.15 = -151.29.

Question. The formula for average acceleration is a = (Vf – Vi)/t, where Vf is the final velocity, Vi is the initial velocity, and t is the time in seconds. What is VF and VO in physics? vo = original velocity. vf = final velocity. Page 2. Projectile Velocity and Acceleration. A projectile does not accelerate horizontally. What does the symbol VF stand for? The “VF” in VF tires stands for “very-high flexion.” While VF isn’t the catchiest of monikers, it aptly explains the nature of a VF tire, which is essentially that they’re very, very bendy in the sidewall. How do you get a VF? - Work out which of the displacement (S), initial velocity (U), acceleration (A) and time (T) you have to solve for final velocity (V). - If you have U, A and T, use V = U + AT. - If you have S, U and T, use V = 2(S/T) – U. - If you have S, A and T, use V = (S/T) + (AT/2). What is VF in acceleration? vf = final velocity vi = initial velocity a = acceleration ∆x = displacement Use this formula when you don’t have ∆t. Dynamics. F = ma. F = force m = mass a = acceleration Newton’s Second Law. What is final velocity and initial velocity? Initial velocity describes how fast an object travels when gravity first applies force on the object. On the other hand, the final velocity is a vector quantity that measures the speed and direction of a moving body after it has reached its maximum acceleration. What is the formula for final velocity? Final velocity (v) squared equals initial velocity (u) squared plus two times acceleration (a) times displacement (s). Solving for v, final velocity (v) equals the square root of initial velocity (u) squared plus two times acceleration (a) times displacement (s). How do you find VF from VI and acceleration? What is VF VO? v.o. -or- version originale -or- VOSTF -or- version originale sous-titrée en français = original version that has NOT been dubbed in French (in original film language may it be English, German, Hindi, etc.) but will have French subtitles. v.f. -or- version française = version has been dubbed in French. Where is VF based? VF moves its corporate headquarters from Wyomissing, Pennsylvania to Greensboro, North Carolina, home to the Wrangler® brand. VF acquires the Bulwark® brand. What does VF stand for in electronics? Vf is the term used for the LEDs forward voltage. It is the voltage required to activate the LED and produce the output specified, assuming that it is drawing the recommended current. What is final speed in physics? Final velocity (v) of an object equals initial velocity (u) of that object plus acceleration (a) of the object times the elapsed time (t) from u to v. Use standard gravity, a = 9.80665 m/s2, for equations involving the Earth’s gravitational force as the acceleration rate of an object. What does V mean in physics? voltage. also called electric potential difference. volt (V) What are the 3 formulas in physics? The three equations are, v = u + at. v² = u² + 2as. s = ut + ½at² How do you find final velocity without time? What is initial velocity in physics? Initial Velocity is the velocity at time interval t = 0 and it is represented by u. It is the velocity at which the motion starts. They are four initial velocity formulas: (1) If time, acceleration and final velocity are provided, the initial velocity is articulated as. u = v – at. Is final velocity zero? If a projectile is tossed into the space, its initial velocity will be more than zero. If a car stops after applying the brake, the initial velocity will be more than zero, but the final velocity will be zero. What is the formula of initial velocity? Initial velocity is 3.5. The equation is s = ut + 1/2at^2, where s – distance, u – inititial velocity, and a – acceleration. How do you find final velocity with acceleration and distance? Solving for Final Velocity from Distance and Acceleration t = v − v 0 a . v 2 = v 0 2 + 2 a ( x − x 0 ) ( constant a ) . Is velocity a speed? Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object’s movement. Put another way, speed is a scalar value, while velocity is a vector. What is the final velocity of a projectile? Projectile (2): For the x direction, the velocity is constant, so the final velocity is equal to the initial velocity. For the y direction, there is no initial velocity. How do you calculate Vo in physics? What does XO mean in physics? •Conventions: At an inital time, which is defined to be zero the initial position is x0, and the initial. velocity is v0. At a later time t the position is x and the velocity v. The acceleration a is constant. during all times. How do you find the velocity? Equation for Velocity To figure out velocity, you divide the distance by the time it takes to travel that same distance, then you add your direction to it. For example, if you traveled 50 miles in 1 hour going west, then your velocity would be 50 miles/1 hour westwards, or 50 mph westwards. What do VF Corporation do? VF Corporation (NYSE: VFC) outfits consumers around the world with its diverse portfolio of iconic outdoor and activity-based lifestyle and workwear brands, including Vans®, The North Face®, Timberland® and Dickies®.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale. Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants." One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is about 1.262. A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.:1 Several types of fractal dimension can be measured theoretically and empirically (see Fig. 2). Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence,:97–104 river networks,:246–247 urban growth, human physiology, medicine, and market trends. The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s,:19 but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975. Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Unlike topological dimensions, the fractal index can take non-integer values, indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. For instance, a curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.:48 This general relationship can be seen in the two images of fractal curves in Fig.2 and Fig. 3 – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of 1.26. The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity. These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length or slope, as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack. The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. The length between any two points on these curves is undefined because the curves are theoretical constructs that never stop repeating themselves. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not rectifiable curves, meaning they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be characterized by finding their lengths or slopes. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard. The two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized. This sort of structure can be extended to other spaces (e.g., a fractal that extends the Koch curve into 3-d space has a theoretical D=2.5849). However, such neatly countable complexity is only one example of the self-similarity and detail that are present in fractals. The example of the coast line of Britain, for instance, exhibits self-similarity of an approximate pattern with approximate scaling.:26 Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. These include, as examples, strange attractors for which the detail has been described as in essence, smooth portions piling up,:49 the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time. Fractal complexity may not always be resolvable into easily grasped units of detail and scale without complex analytic methods but it is still quantifiable through fractal dimensions.:197; 262 The terms fractal dimension and fractal were coined by Mandelbrot in 1975, about a decade after he published his paper on self-similarity in the coastline of Britain. Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in usual linear terms. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about undifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s.:405 There was a lull in the published work on such functions for a time after that, then a renewal starting in the late 1800s with the publishing of mathematical functions and sets that are today called canonical fractals (such as the eponymous works of von Koch, Sierpiński, and Julia), but at the time of their formulation were often considered antithetical mathematical "monsters". These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after him and is frequently invoked in defining modern fractals.:44 See Fractal history for more information Role of scaling The concept of a fractal dimension rests in unconventional views of scaling and dimension. As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable stands for the number of sticks, for the scaling factor, and for the fractal dimension: This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because when as in the example above, and for squares, because when The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, when and the value of can be found by rearranging Equation 1: That is, for a fractal described by when a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in. The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, the images shown are not true fractals because the scaling described by the value of cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619. D is not a unique descriptor As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns. The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 6. Fractal surface structures The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties. Numerous surface descriptors are used to interpret the structure of nominally flat surfaces, which often exhibit self-affine features across multiple length-scales. Mean surface roughness, usually denoted RA, is the most commonly applied surface descriptor, however numerous other descriptors including mean slope, root mean square roughness (RRMS) and others are regularly applied. It is found however that many physical surface phenomena cannot readily be interpreted with reference to such descriptors, thus fractal dimension is increasingly applied to establish correlations between surface structure in terms of scaling behavior and performance. The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of contact mechanics, frictional behavior, electrical contact resistance and transparent conducting oxides. The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practice, however, fractal dimensions can be determined using techniques that approximate scaling and detail from limits estimated from regression lines over log vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent: - Information dimension: D considers how the average information needed to identify an occupied box scales with box size; is a probability. - Correlation dimension: D is based on as the number of points used to generate a representation of a fractal and gε, the number of pairs of points closer than ε to each other. - Generalized or Rényi dimensions: The box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by: - Lyapunov dimension - Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern. - Uncertainty exponent - Hausdorff dimension: For any subset of a metric space and , the d-dimensional Hausdorff content of S is defined by - The Hausdorff dimension of S is defined by Estimating from real-world data Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from sampled data using computer based fractal analysis techniques. Practically, measurements of fractal dimension are affected by various methodological issues, and are sensitive to numerical or experimental noise and limitations in the amount of data. Nonetheless, the field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena may have many practical applications in various fields including diagnostic imaging, physiology, neuroscience, medicine, physics, image analysis, ecology, acoustics, Riemann zeta zeros, and electrochemical processes. An alternative to a direct measurement, is considering a mathematical model that resembles formation of a real-world fractal object. In this case, a validation can also be done by comparing other than fractal properties implied by the model, with measured data. In colloidal physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient to speak about a distribution of fractal dimensions, and eventually, a time evolution of the latter: a process that is driven by a complex interplay between aggregation and coalescence. - Falconer, Kenneth (2003). Fractal Geometry. Wiley. p. 308. ISBN 978-0-470-84862-3. - Sagan, Hans (1994). Space-Filling Curves. Springer-Verlag. p. 156. ISBN 0-387-94265-3. - Vicsek, Tamás (1992). Fractal growth phenomena. World Scientific. p. 10. ISBN 978-981-02-0668-0. - Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–8. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. - Benoit B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. - Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer. p. 7. ISBN 978-0-387-74749-1. - Harte, David (2001). Multifractals. 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T.; Jeon, T.; Hynan, L. S.; Youn, T. S.; Kennedy, D. N.; Dickerson, B.; the Alzheimer’s Disease Neuroimaging Initiative (2009). "Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis". Brain Imaging and Behavior. 3 (2): 154–166. doi:10.1007/s11682-008-9057-9. PMC 2927230. PMID 20740072. - Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. Wiley. ISBN 0-471-13938-6. - Edgar, Gerald, ed. (2004). Classics on Fractals. Westview Press. ISBN 978-0-8133-4153-8. - Albers; Alexanderson (2008). "Benoit Mandelbrot: In his own words". Mathematical people : profiles and interviews. AK Peters. p. 214. ISBN 978-1-56881-340-0. Mandelbrot, Benoit (2004). Fractals and Chaos. Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension - Sharifi-Viand, A.; Mahjani, M. G.; Jafarian, M. (2012). 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"Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms". Investigative Ophthalmology & Visual Science. 36 (13): 2749–2755. PMID 7499097. - Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. doi:10.1023/A:1022355723781. - Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal. 85 (6): 4041–6. Bibcode:2003BpJ....85.4041L. doi:10.1016/S0006-3495(03)74817-6. PMC 1303704. PMID 14645092. - Smith, T. G.; Lange, G. D.; Marks, W. B. (1996). "Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals". Journal of Neuroscience Methods. 69 (2): 123–136. doi:10.1016/S0165-0270(96)00080-5. PMID 8946315. - Li, J.; Du, Q.; Sun, C. (2009). "An improved box-counting method for image fractal dimension estimation". Pattern Recognition. 42 (11): 2460–9. doi:10.1016/j.patcog.2009.03.001. - Dubuc, B.; Quiniou, J.; Roques-Carmes, C.; Tricot, C.; Zucker, S. (1989). "Evaluating the fractal dimension of profiles". Physical Review A. 39 (3): 1500–12. Bibcode:1989PhRvA..39.1500D. doi:10.1103/PhysRevA.39.1500. PMID 9901387. - Roberts, A.; Cronin, A. (1996). "Unbiased estimation of multi-fractal dimensions of finite data sets". Physica A: Statistical Mechanics and Its Applications. 233 (3–4): 867–878. arXiv:chao-dyn/9601019. Bibcode:1996PhyA..233..867R. doi:10.1016/S0378-4371(96)00165-3. - Al-Kadi O.S, Watson D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images" (PDF). IEEE Transactions on Biomedical Engineering. 55 (7): 1822–30. doi:10.1109/tbme.2008.919735. PMID 18595800. - Pierre Soille and Jean-F. Rivest (1996). "On the Validity of Fractal Dimension Measurements in Image Analysis" (PDF). Journal of Visual Communication and Image Representation. 7 (3): 217–229. doi:10.1006/jvci.1996.0020. ISSN 1047-3203. Archived from the original (PDF) on 2011-07-20. - Tolle, C. R.; McJunkin, T. R.; Gorsich, D. J. (2003). "Suboptimal minimum cluster volume cover-based method for measuring fractal dimension". IEEE Transactions on Pattern Analysis and Machine Intelligence. 25: 32–41. CiteSeerX 10.1.1.79.6978. doi:10.1109/TPAMI.2003.1159944. - Gorsich, D. J.; Tolle, C. R.; Karlsen, R. E.; Gerhart, G. R. (1996). "Wavelet and fractal analysis of ground-vehicle images". Wavelet Applications in Signal and Image Processing IV. 2825: 109–119. doi:10.1117/12.255224. Cite journal requires - Wildhaber, Mark L.; Lamberson, Peter J.; Galat, David L. (2003-05-01). "A Comparison of Measures of Riverbed Form for Evaluating Distributions of Benthic Fishes". North American Journal of Fisheries Management. 23 (2): 543–557. doi:10.1577/1548-8675(2003)023<0543:acomor>2.0.co;2. ISSN 1548-8675. - Maragos, P.; Potamianos, A. (1999). "Fractal dimensions of speech sounds: Computation and application to automatic speech recognition". The Journal of the Acoustical Society of America. 105 (3): 1925–32. Bibcode:1999ASAJ..105.1925M. doi:10.1121/1.426738. PMID 10089613. - Shanker, O. (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions". Journal of Physics A: Mathematical and General. 39 (45): 13983–97. Bibcode:2006JPhA...3913983S. doi:10.1088/0305-4470/39/45/008. - Eftekhari, A. (2004). "Fractal Dimension of Electrochemical Reactions". Journal of the Electrochemical Society. 151 (9): E291–6. doi:10.1149/1.1773583. - Kryven, I.; Lazzari, S.; Storti, G. (2014). "Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems". Macromolecular Theory and Simulations. 23 (3): 170–181. doi:10.1002/mats.201300140. - Mandelbrot, Benoit B.; Hudson, Richard L. (2010). The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Books. ISBN 978-1-84765-155-6. |Wikimedia Commons has media related to Fractal dimension.| - TruSoft's Benoit, fractal analysis software product calculates fractal dimensions and hurst exponents. - A Java Applet to Compute Fractal Dimensions - Introduction to Fractal Analysis - Bowley, Roger (2009). "Fractal Dimension". Sixty Symbols. Brady Haran for the University of Nottingham. - "Fractals are typically not self-similar". 3Blue1Brown.

« ΠροηγούμενηΣυνέχεια » 4 48 960 1 12 240 NOTE 1. The symbol £. stands for the Latin word libra, signifying a pound; s. for solidus, a shilling; d. for denarius, a penny; gr. for quadrans, a quarter. NOTE 2.- Farthings are sometimes expressed in a fraction of a penny; thus, 1 far. =* d.; 2 far. = d.; 3 far. = d. NOTE 3. - The term sterling is probably from Easterling, the popular name of certain early German traders in England, whose money was noted for the purity of its quality. NOTE 4. — The English coins consist of the five-sovereign piece, the double-sovereign, the sovereign, and the half-sovereign, made of gold; the crown, the half-crown, florin, the shilling, the six-pence, the four-pence, the three-pence, the two-pence, the one-and-a-half-pence, and the penny, made of silver ; the penny, the half-penny, the farthing, and the half-farthing, made of copper. The sovereign represents the pound sterling, whose legal value in United States money is $4.84 ; and the florin represents one-tenth of the pound. The value of the English guinea is 21 shillings sterling. The guinea, the five-guinea, the half-guinea, the quarter-guinea, and the seven-shilling piece, are no longer coined. The English gold coins are now made of 11 parts of pure gold, and 1 part of copper, or some other alloy; and the silver coin, of 37 parts of pure sil. ver, and three parts of copper. The present standard weight of the sovereign is 12347} grains Troy; the crown, 43611 grains; the copper penny, 2913 grains. 128. To change numbers expressed in one or more denominations to their equivalents in one or more other denominations. Ex. 1. In 48£. 12s. 7d. 2far. how many farthings ? We multiply the 48 by 20, be48£. 1 2 s. 7 d. 2 far. cause 20 shillings make 1 pound, 20 and to this product we add the 12 shillings in the question, and obtain 9 7 2 shillings. 972 shillings. We then multiply 12 by 12, because 12 pence make 1 11 6 7 1 pence. shilling, and to the product we add the 7 pence, and obtain 11671 4 pence. Again, we multiply by 4, Ans. 4 6 6 8 6 farthings. because 4 farthings make 1 penny, and to this product we add the 2 farthings, and obtain 46686 farthings, the answer sought. Ex. 2. In 46686 farthings how many pounds ? We divide by 4, because 4 far4) 4 6 6 8 6 far. things make 1 penny, and the re sult is 11671 pence, and 2 farthings 1 2 ) 11 671 d. 2 far. remaining. We then divide by 12 20 ) 9 7 2 s. 7 d. because 12 pence make 1 shilling, and the result is 972 shillings, and 48 £. 1 2 s. 7 pence remaining. Lastly, we diAns. 48£. 12s. 7d. 2 far. vide by 20, because 20 shillings make i pound, and the result is 48 pounds, and 12 shillings remaining. By annexing to the last quotient the several remainders, we obtain 48£. 12s. 7d. 2far. as the required result. From these illustrations, for the two kinds of reduction, we deduce the following RULE. — For REDUCTION DESCENDING. Multiply the highest denomination given by the number of units required of the next lower denomination to make one in the denomination multiplied. To this product add the corresponding denomination of the multiplicand, if there be any. Proceed in this way, till the reduction is brought to the denomination required. FOR REDUCTION ASCENDING. Divide the lower denomination iven by the number of units required of that denomination to make one of the next higher. The quotient thus obtained divide in like manner, and so proceed until it is brought to the denomination required. The last quotient, with the several remainders, if there be any, annexed, will be the answer. 3. In 127£. 15s. 8d. how many farthings ? 129. Avoirdupois or Commercial Weight is used in weighing almost every kind of goods, and all metals except gold and silver. NOTE 1. — The oz. stands for onza, the Spanish for ounce, and in cwt. the c stands for centum, the Latin for one hundred, and wt for weight. NOTE 2. — The laws of most of the States, and common practice at the present time, make 25 pounds a quarter, as given in thc table. But formerly, 28 pounds were allowed to make a quarter, 112 pounds a hundred, and 2240 pounds a ton, as is still the standard of the United States government in collecting duties at the custom-houses. Note 3. — The term avoirdupois is from the French avoir du poid, signifying to have weight. NOTE 4. — - The standard avoirdupois pound of the United States is the weight, taken in the air, of 2770066 cubic inches of distilled water, at its maximum density, or when at a temperature of 3918. degrees Fahrenheit, the barometer being at 30 inches. It is the same as the Imperial poand avoirdupois of Great Britain, which is the weight of 2716166 cubic inches of distilled water at the temperature of 62 degrees. 1. In 165T. 13cwt. 3qr. 191b. 14oz. how many ounces? 2. In 5302318 ounces how many tons ? 3. If a load of hay weigh 3T. 16cwt. 2qr. 181b., required the weight in ounces. 4. In 122688 ounces how many tons ? 5. Required the number of drams in 2T. 17cwt. 3qr. 161b. 15oz. 13dr. 6. In 1482749 drams how many tons ? 7. What is the value of 7T. 17cwt. at 7 cents per pound? Ans. $ 1099.00. 8. What will 19cwt. 3qr. 201b. of sugar cost at 9 cents per pound? Ans. $ 179.55. TROY OR MINT WEIGHT. 130. Troy or Mint Weight is the weight used in weighing gold, silver, jewels, and liquors; and in philosophical experiments. TABLE. 24 Grains (gr.) make 1 Pennyweight, pwt. 20 Pennyweights 1 Ounce, 12 Ounces Note 1. — Troy weight was introduced into Europe from Cairo in Egypt, in the 12th century, and was first adopted in Troyes, a city in France, where great fairs were held, whence it may have had its name. NOTE 2. A grain or corn of wheat, gathered out of the middle of the ear, was the origin of all the weights used in England. Of these grains, 32, well dried, were to make one pennyweight. But in later times it was thought sufficient to divide the same pennyweight into 24 equal parts, still called grains, being the least weight now in use, from which the rest are computed. NOTE 3. — - Diamonds and other precious stones are weighed by what is called Diamond Weight, of which 16 parts make 1 grain; 4 grains, 1 carat. 1 grain Diamond Weight is equal to 1 grains Troy, and 1 carat to 3} grains Troy. In weighing pearls, the pennyweight is divided into 30 grains instead of 24, so that 1 pearl grain is equal to grains Troy. The carat as a weight must not be confounded with the assay carat, a term whose use is to indicate a proportional part of a weight, as in expressing the fineness of gold, each carat means a twenty-fourth part of the entire mass used. Thus, pure gold is termed 24 carat gold, and gold that is not pure is termed 18 carat gold. 20 carat gold, &c., as its mass may be 18 twenty-fourths, 20 twenty-fourths, &c. pure gold. Each assay carat is subdivided into 4 assay grains, and each assay grain into 4 assay quarters. NOTE 4. — The Troy pound, the standard unit of weight adopted by the United States Mint, is the same as the Imperial Troy pound of Great Britain, and is equal to the weight, taken in the air, of 2276000 cubic inches of distilled water, at its maximum density, the barometer being at 30 inches. 1. How many grains in 28lb. 11oz. 12pwt. 15gr. Troy? 2. In 166863 grains Troy how many pounds? 3. If a silver pitcher weigh 3lb. 10oz., what is its weight in grains ? 4. How many pounds Troy in 22080 grains ? 5. What is the value of 73lb. lloz. of standard silver at $0.062 per pennyweight ? 6. How many pounds of standard silver can be purchased for $ 1099.88, at the rate of $ 0.062 per pennyweight? 7. A Californian has 571b. 7oz. of pure gold. What is its value at $ 20.593} per ounce ? Ans. $14229.9013. 8. What is the value of a mass of standard gold weighing 191b. Coz. 16pwt. at 93 cents per pennyweight ? Ans. $ 4367.28. 9. I have a lump of pure silver weighing 131b. 9oz. What is its value at $ 1.385 70 per ounce ? Ans. $ 228.6404. 131. Apothecaries' Weight is used in mixing medical prescriptions. TABLE. 20 Grains (gr.) make 1 Scruple, sc. or Ə. 3 Scruples dr. or 3. 8 Drams oz. or 3. 12 Ounces Ib. or Ib. gr. 20 Note 1. — In this weight the pound, ounce, and grain are the same as in Troy Weight. NOTE 2. — Medicines are usually bought and sold by Avoirdu pois Weight. NOTE 3. — In estimating the weight of fluids, 45 drops, or a common teaspoonful, make about 1 fluid dram; 2 common table-spoonfuls, about 1 fluid ounce; a wineglassful, about 13 fluid ounces; and a common teacupful, about 4 fluid ounces. 1. In 23lb 93 03 23 13gr. how many grains ?

Discussions concerning a new multiplicative accounting system and the advantages of using the geometric calculus in economic analysis are included in an article by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France). The geometric calculus was used by Hasan Özyapıcı (Eastern Mediterranean University in Cyprus), İlhan Dalcı (Eastern Mediterranean University in Cyprus), and Ali Özyapıcı (Cyprus International University) in their article "Integrating accounting and multiplicative calculus: an effective estimation of learning curve". From the Abstract: "The results of this study are also expected to help researchers, practitioners, economists, business managers, and cost and managerial accountants to understand how to construct a multiplicative based learning curve to improve such decisions as pricing, profit planning, capacity management, and budgeting." (The expression "multiplicative calculus" refers here to the geometric calculus.) The non-Newtonian approach to accounting [82, 121, 149, 181, 216] was advocated by Amelia Correa (St. How many L capes can Kareem make with the material? In one kind, the goal is given first, and then the mind goes from the goal to the means, that is, from the question to the solution. Great art is another matter, nothing trivial about that. Now, since we are using integer math, which has inherent round-off error, we might get some artifacts which are not strictly fractal in origin, hence the 1-map mode, but which every once in a while give something interesting. Describe how to solve the equation -1.25 + .v= 1. 25 Then solve. 2. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on [geometric] and [bigeometric] calculi is self-evident. S. at applied to the stock market, to air currents, and all of the elements (like depth, color, shape and intrigue) to create a good piece of art. Try This Use the formula you discovered to find the volume of each prism. 1. 4 cm; 3 cm 6 cm 7 cm 3. 6 cm 2 cm V Top Side 4 cm A. 2 cm 3 cm 594 Chapter 10 Measurement: Three-Dimensional Figures Activity 2 Q You can use a process similar to that in Activity 1 to develop the formula for the volume of a cylinder. In each of these two calculi, the use of multiplication/division to combine/compare numbers is crucial. We showed that the classical calculus and each non-Newtonian calculus can be 'weighted' in a manner explained in our book The First Systems of Weighted Differential and Integral Calculus (1980). Since pi is about 3.14, that means the book is correct. IVIultiple Choice Wliich expression does NOT have a value of —3? Probability 639 1^ 7.4.5 Use theoretical probability and proportions to make approximate predictions. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983. Jane Grossman, Michael Grossman, and Robert Katz. But they don't have the creativity to ask new questions. Students may be able to participate in the UC Education Abroad Program (EAP) and UCSD's Opportunities Abroad Program (OAP) while still making progress towards the major. Being fundamentally lazy, we prefer to sit down in a car while connecting directly to nodes up to tens of kilometers away -- there is no need to cross over to different modes of transport. Think: 3 out of W is how many out of 500? Multiply Divide each side by W to isolate the variable. 150 = x Caitlyn can predict that she wiU make about 150 of 500 tliree-point shots. 3 _ 10 -V 500 3 ■ 500 = 10 -X 1,500 = lOx 1,500 _ lO.v EXAMPLE 1? 5 _. Non-Newtonian calculus was recommended as a topic for the 21st-century college-mathematics-curriculum at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM) in March of 2015. (The conference was sponsored by Pearson PLC, the largest education company and the largest book publisher in the world; and the Electronic Proceedings of the conference were hosted by Math Archives (archives.math.utk.edu) with partial support provided by the National Science Foundation.) Please see item in the References section. ================================================================================================================================================================= Citations Contents Home Multiplicative Calculus Brief History Applications Citations Reviews Comments Quotations References Links/Reading Appendix 1 Appendix 2 Appendix 3 Dedication Non-Newtonian Calculus is cited in the book The Rainbow of Mathematics: A History of the Mathematical Sciences by the eminent mathematics-historian Ivor Grattan-Guinness. Non-Newtonian calculus is cited in an article on atmospheric temperature by Robert G. As we iterate powerset and union, we therefore progressively create bigger sets and more sets. That means if a device produces a magnetic field that exhibits fractal behaviour, the magnetic field would not posses dimension equal to a whole number — such as one, two, or three dimensions — but rather a fractional value such as 0.8 or 1.6 dimensions. In particular, in architectural studies of tall building skylines, the silhouette complexity significantly affected preference scores while facade complexity was of less importance (Heath et al., 2000 ). However, sometimes there are additional prior information we want to take into consideration, then the Bayesian approach is to be employed. She builds a larger tank by doubling each dimension. EXAMPLE Using the Division Property of Equality Solve the equation 240 = 42. Consumer Math Members at a swim club pay $5 per lesson plus a one-time fee of $60. Afraid of epidemics, she tried to keep him out of school. He links to a piece where he argues that this can’t be true. They provide a wide variety of mathematical tools for use in science, engineering, and mathematics. Krantz" and published in 2008 by the Journal of Mathematical Analysis and Applications. Then he turns north and rides another 15 miles before he stops to rest. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. We knew that the bigeometric calculus like the geometric calculus and maybe many other non-Newtonian calculi would be useful. Simulation games are games that try to make something as realistic as possible. Find the value of x when y = 162. y = 18. V Write the equation for the direct variation. 162 = 18. Bashirov, Mustafa Riza, and Yucel Tandogdu (all of Eastern Mediterranean University in North Cyprus); Emine Misirli Kurpinar and Yusuf Gurefe (both of Ege University in Turkey); and Ali Ozyapici (Lefke European University in Turkey). Try This Use graph paper to estimate each square root. Calling intervals the "fourth," "fifth," or "octave" (i.e. "eighth"), when they are part of a system of seven tones, is a little confusing.

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McDonald, Pearson, A study of the history and development of mathematics and its cultural impact from the. Statistics. A course will be worth a minimum of half credit and a maximum of 5 credits, depending. The degree program is structured such that in many subjects introductory courses are taught in. To equip the student with the basic mathematical ideas in the area of analysis and its. John E. Freud's Mathematical Statistics with Applications, 7 th. Edition. London; Pearson education International. 2. How Many Molecules Of Co2 Are Needed To Generate One Molecule Of Glucose? Using ATP and NADPH from the light reactions, 3-PG is reduced to glyceraldehyde-3-phosphate (G3P). Two molecules of G3P can spontaneously form a molecule of glucose (the first, energetically uphill part of glycolysis running backwards). 18/10/2009 · Since 2 atoms of oxygen are needed for each molecule of co2, then you would divide 6 by 2 to Medical Statistics & Methodology · Midwifery · Nursing Studies · Nursing · Obstetrics & Gynecology · Occupational Medicine · Ophthalmology. Price. $25 to $50 (1). $50 to $100 (5). $100 to $200 (33). More than $200 (2). Oxford Lecture Series in Mathematics and Its Applications RSS. The aim. Mathematical modelling features, as do applications to finance, engineering, and the physical and biological sciences. An Introduction to Semilinear Evolution Equations. Revised Edition. Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition LaTorre, Donald R.; Kenelly, John W.; Reed, An Introduction to Mathematical Statistics and Its Applications (6th Edition) Larsen, Richard J.; Marx, Morris L. 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Is Presentism compatible with relativity? presentism/possibilism are not only incompatible with general relativity but also with quantum theory. What is meant by simultaneity of relativity? In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer’s reference frame. What does it mean by the statement simultaneity is not absolute According to the theory of special relativity? Einstein concluded that simultaneity is not absolute, or in other words, that simultaneous events as seen by one observer could occur at different times from the perspective of another. It’s not lightspeed that changes, he realized, but time itself that is relative. Is simultaneity a consequence of special relativity? The relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer’s frame of reference. How do you explain simultaneity? So let's break it down now remember the concept of simultaneity is. The idea that any two vents that may be perceived. By one observer to be simultaneous that is they occur at the same time. How relative motion is relates to explaining two simultaneous event? Whether two events at separate locations are simultaneous depends on the motion of the observer relative to the locations of the events. (a) Two pulses of light are emitted simultaneously relative to observer B. (c) The pulses reach observer B’s position simultaneously. Do any two observers always agree on simultaneity of events Why? Two observers in relative motion will agree that two events are simultaneous: Two observers in relative motion will agree that two events are simultaneous: Never since they are both moving. Always only if they occur at the same time. What is relativity of simultaneity time dilation? Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute. Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer. Do simultaneous events exist? Only at most one will agree they happened at the same time. No, there is no absolute agreement on simultaneity. Alpha Centauri is about three light years from us. We know that it existed three years ago because we can see it in the sky as it was, for us, three years ago. What is simultaneity problem? What Causes It? Simultaneity happens when two variables on either side of a model equation influence each other at the same time. In other words, the flow of causality isn’t a hundred percent from a right hand side variable (i.e. a response variable) to a left hand side variable (i.e. an explanatory variable). Will it be possible for a moving person to observe two events happening simultaneously with respect to inertial frame given that another person? Answer and Explanation: From the special theory of relativity two events which are observed simultaneous in one inertial frame of reference may not be simultaneous in other frame of reference. Under what condition will two events at different locations be simultaneous in both frames? In order for the two events to be simultaneous in frame S′, they must be both be on its x′-axis, so, the only way it can be simultaneous for both observers is if S and S′ are not moving relative to eachother so that their worldlines are parallel. When two observers are moving at relativistic speeds relative to each other do they still agree on each other’s measurement of speeds? Explanation of Solution Thus, the speed of light measured by both the observers who are in relative motion is same. The relative speed between the two observers will be same. Therefore, they agree about their own relative speed. How is relativity used in everyday life? “Since this is the core principle behind transformers and electric generators, anyone who uses electricity is experiencing the effects of relativity,” Moore told Live Science. Electromagnets work via relativity as well. When a direct current of electric charge flows through a wire, electrons drift through the material. What is it in the general theory of relativity that does not involve the special relativity? Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to other forces of nature. Which form of relativity applies for observers who are accelerating? Special relativity deals with observers moving at constant velocity; this is a lot easier than general relativity, in which observers can accelerate with respect to each other. What is different about the reference frames that apply to special relativity and to general relativity? What is different about the reference frames that apply to special relativity and to general relativity? Special relativity applies to frames of reference moving at constant velocity, whereas general relativity includes accelerating reference frames. In special relativity, there are no forces and no acceleration. Does the principle of relativity require that every observer observe the same laws of physics explain? And, given that relativity is a study of how observers in different reference frames measure the same event, it is not necessary that every observer will observe the same laws of physics happening.

Volume 575, March 2015 |Number of page(s)||12| |Published online||09 March 2015| X atom photodesorption probabilities at Tice = 20 K, 30 K, and 60 K resulting from photoexcitation of a X2O (X = H, D) or XOY (HOD or DOH) molecule present in a specific monolayer of H2O ice. OX radical photodesorption probabilities at Tice = 20 K, 30 K, and 60 K resulting from photoexcitation of a X2O (X = H, D) or XOY (HOD or DOH) molecule present in a specific monolayer of H2O ice. Total X2O (X = H, D) or XOY (HOD or DOH) photodesorption probabilities at Tice = 20 K, 30 K, and 60 K per monolayer due to the direct and the kick-out mechanism for X2O and XOY photodissociation in H2O ice. Tables 2, 4, 5, A.1−A.3 list the total probabilities for X desorption, OX desorption, and X2O and HDO desorption (X = H, D) following a dissociation event, as a function of both monolayer and ice temperature. These tables also give the average probabilities, over the top four monolayers, for each species. For use in astrochemical models, it is useful to know the probability (per monolayer) of every potential outcome, rather than the total probability for the desorption of each species. This is because, in full gas-grain models, one is also interested in the composition of the ice mantle, as well as the gas. As discussed in the main body of this paper, there are six potential outcomes following a dissociation event which can lead to a change in composition of both the ice and gas. For example, for HDO which is dissociated into H + OD, Equation (B.6) is the process known as “kick out” whereby a neighbouring H2O is ejected from the ice via momentum transfer from an excited photofragment. The probabilities of each of these events as a function of monolayer and ice temperature have been compiled from the raw data of the molecular dynamics simulations and are available at the CDS. There is a seventh possibility in which the photofragments recombine to reform HDO which remains trapped in the ice. This process does not change the gas or ice composition and thus we have not listed the probabilities for this outcome here; however, these data are necessary if one is interested in extrapolating the probabilities to deeper monolayers, ML > 4. To determine the desorption probabilities at temperatures and in monolayers outside of those tabulated, one can simply interpolate/extrapolate using, for example, cubic spline interpolation. However, when extrapolating to determine probabilities for deeper monolayers, ML > 4, one should take care to ensure that, deep into the ice mantle, the probabilities for outcomes (B.1), (B.2), (B.3), (B.5), and (B.6) tend to 0, and the probability for outcome (B.4) tends to 1 − Precom, where Precom is the probability that the photofragments recombine to reform the molecule (which remains trapped in the ice). Deeper into the ice, desorption events become increasingly less probable and the most probable outcome becomes trapping of the photofragments (or the reformed molecule, following recombination). In addition, at very low coverage, ML < 1, the rates for all outcomes should tend to 0 as ML → 0. In Table 1 we present our fitting functions and corresponding best-fit parameters for the temperature-averaged probabilities per monolayer for each outcome. The probabities are well fitted using a Gaussian-like function with the exception of the outcomes leading to trapping of the OY radical for which an exponential-like function was found to be more appropriate for describing the asymptotic behaviour of the probabilities towards deeper monolayers (≫4). In Fig. B.1 we present the probability per monolayer at each temperature and the temperature-averaged probabilities per monolayer along with the fitted functions for the example of DOH∗. The probabilities were fitted using the non-linear least-squares (NLLS) Marquardt-Levenberg algorithm (Marquardt 1963). The probabilities are a much stronger function of monolayer than temperature; hence, our decision to fit functions with respect to monolayer only. For implementation in chemical models which adopt the rate equation method for describing the ice chemistry and gas-grain balance, the probabilities per monolayer should be multiplied by the rate of arrival of UV photons in the wavelength range 1650−1300 Å onto the grain surface times the absorption cross section of a UV photon by a grain-surface site (or molecule, in this case, HDO). The total desorption rate is then determined by integrating the desorption rate per monolayer over the total number of monolayers on the grain. The probabilities can be directly employed in stochastic chemical models in which the discrete nature of chemical reactions are taken into account (see, e.g. Cuppen & Herbst 2007). Temperature-specific probabilities, temperature-averaged probabilities, and fitted functions for each outcome as a function monolayer for HDO photodissociation into D + OH. |Open with DEXTER| This section investigates whether photodesorption ultimately also leads to fractionation of HDO/H2O in the gas. We can estimate the total photodesorption probability ratio between HDO and H2O by taking into account the direct and kicked out mechanism in both cases. The probability of HDO photodesorption through the direct mechanism is given by (C.1)In Eq. (C.1), Pdirect( HDO∗) is the probability that upon photo-excitation of HDO (the generic case) the HDO recombines and desorbs directly. It can be approximately calculated using (C.2)and rHDO is the original HDO/H2O ratio in the ice (of the order of 0.01 or less as indicated by observations). In Eq. (C.2), the probabilities on the right hand side are the probabilities for the direct mechanism for photodesorbing HDO averaged over the top four monolayers and presented in Tables 5 and A.3. The probability of H2O photodesorption through the direct mechanism is given by (C.3)In Eq. (C.3), Pdirect(H2O∗) is the probability that upon photo-excitation H2O recombines and desorbs directly. It can be obtained directly from Tables 5 and A.3. As can be seen from Table 2 and after using Eq. (C.2), Pdirect(HDO∗) and Pdirect(H2O∗) are roughly the same. As a result (C.4)meaning that there is no isotope fractionation due to the direct mechanism. Now consider the kick-out mechanism. The indirect probabilities can be written as follows: (C.5)and (C.6)In Eqs. (C.5) and (C.6), is the probability of desorption of HXO through the kick-out mechanism, where X is either H or D. Furthermore, PKO(HX1O; HX2O∗) is the probability that HX1O is kicked out after photo-excitation of HX2O, where X1 can either be H or D, and X2 can also be H or D. As for the direct mechanism, we can approximately calculate PKO(HX1O;HX2O∗) from (C.7)The two quantities on the right hand side of Eq. (C.7) have been tabulated for X1 equal to H in Tables 5 and A.3. Because we have only calculated probabilities that H2O is kicked out, we make the following approximations, The right hand values of Eqs. (C.8) and (C.9) can be directly obtained from Tables 5 and A.3. PKO( HDO; HDO∗) can be computed using the approximation in Eq. (C.10) and using Eq. (C.7) and Tables 5 and A.3. Inserting Eq. (C.12) in Eq. (C.11) yields (C.13)and inserting Eq. (C.12) in Eq. (C.6) yields (C.14)From Eqs. (C.13) and (C.14), we can derive that (C.15)meaning that there should be no isotope fractionation due to the indirect kick-out mechanism. Taken together, Eqs. (C.4) and (C.15) ensure that the ratio of desorbed HDO over desorbed H2O in the ice is given by (C.16)which means that this ratio is simply equal to the ratio of HDO and H2O in the ice. Therefore, isotope fractionation does not occur for HDO and H2O photodesorption. © ESO, 2015 Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform. Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days. Initial download of the metrics may take a while.

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- About this Journal · - Abstracting and Indexing · - Aims and Scope · - Article Processing Charges · - Articles in Press · - Author Guidelines · - Bibliographic Information · - Citations to this Journal · - Contact Information · - Editorial Board · - Editorial Workflow · - Free eTOC Alerts · - Publication Ethics · - Reviewers Acknowledgment · - Submit a Manuscript · - Subscription Information · - Table of Contents Journal of Engineering Volume 2013 (2013), Article ID 516462, 8 pages Some Variational Principles for Coupled Thermoelasticity Dipartimento di Ingegneria Strutturale, Università di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy Received 22 September 2012; Accepted 18 December 2012 Academic Editor: Oronzio Manca Copyright © 2013 Francesco Marotti de Sciarra. 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. The nonlinear thermoelasticity of type II proposed by Green and Naghdi is considered. The thermoelastic structural model is formulated in a quasistatic range, and the related thermoelastic variational formulation in the complete set of state variables is recovered. Hence a consistent framework to derive all the variational formulations with different combinations of the state variables is provided, and a family of mixed variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. A uniqueness condition is provided on the basis of a suitable variational formulation. Coupled thermomechanical problem arises in a variety of important fields of application, including casting, metal forming, machining and other manufacturing processes, structural models, and others. Green and Naghdi (GN) introduced a theory in which heat propagates as thermal waves at finite speed and does not necessarily involve energy dissipation. Another property of the GN theory of type II is the fact that the entropy flux vector is determined by means of the same potential as the mechanical stress tensor. Motivated by the procedure presented in [1, 2], this paper is concerned with the formulation of variational principles characterizing the solutions of the coupled thermomechanical problem for the GN model without dissipation, that is, type II. The variational characterization of the thermoelastic problem means the identification of a functional whose stationary points are solutions of the problem. Once this functional is known, the solutions of the problem can be identified with certain extrema of the functional. Following the pioneering work of Biot , the variational forms of the coupled thermoelastic and thermoviscoelastic problems for the classical Fourier model have been investigated in several papers; see, for example, [2, 4–9] for the case of thermomechanical coupling in dissipative materials. Moreover consistent variational principles for structural problems concerning elastic and elastoplastic models in isothermal conditions are well developed; see, for example, [10–12]. In this paper the GN thermoelastic coupled structural model without dissipation is formulated in a suitable form so that we can provide the methodology to build the complete family of all the admissible variational formulations associated with the considered GN model. It is well-known that the GN theory of type II has been developed as a dynamic theory, but an important prerequisite for its use is a thorough understanding of the corresponding problems in which the dynamical effects are disregarded; see, for example, . An advantage of such an approach consists in the fact that the boundary-value problem for the thermoelastic model is formulated in such a way that the model can be cast in terms of a multivalued structural operator defined in terms of all the state variables. This operator encompasses in a unique expression the field equation, the constitutive relations, the constraint relations and the initial conditions which describe the considered thermoelastic structural model. The related non-smooth potential can then be evaluated by a direct integration along a ray in the operator domain and depends on all the state variables involved in the model. Appraising the generalized gradient of the non-smooth potential and imposing its stationarity, the operator formulation of the problem is recovered. Hence a general procedure to derive variational formulations within the incremental framework for the considered GN coupled thermoelastic model without dissipation is formulated. It is then shown how a family of mixed variational formulations, associated with the considered GN model, can be obtained following a direct and general procedure by enforcing the fulfilment of field equations and constraint conditions. The possibility to formulate the coupled GN thermoelastic problem in a variational form has a number of consequences and some beneficial effects. For instance, the variational framework allows one to apply the tools of the calculus of variations to the analysis of the solutions of the problem. In particular, conditions for the existence (see, e.g., [16, 17]) and uniqueness of the solution are based on the variational framework. Accordingly the condition for the uniqueness of the solution of the considered model is provided by means of a minimization principle. 2. Thermoelastic Structural Problem Throughout the paper bold-face letters are associated with vectors and tensors. The scalar product between dual quantities (simple or double index saturation operation between vectors or tensors) is denoted by . A superimposed dot means differentiation with respect to time and the symbol denotes the gradient operator. In small strain analysis the theory of thermoelasticity without energy dissipation, as described in Green and Naghdi , is considered. Such a theory is based on the introduction of a scalar thermal displacement defined as where is a point pertaining to a thermoelastic body defined on a regular bounded domain of an Euclidean space, represents the temperature variation from the uniform reference temperature , and is the initial value of the thermal displacement at the time . Accordingly the time derivative of is the temperature variation, that is, . The mechanical and thermal parts of the thermoelastic model are hereafter defined. Let denote the linear space of strain tensors and denote the dual space of stress tensors . The inner product in the dual spaces has the mechanical meaning of the internal virtual work, that is The linear space of displacements is denoted by . The linear space of forces is and is placed in separating duality with by a nondegenerate bilinear form which has the physical meaning of external virtual work. For avoiding proliferation of symbols, the internal and external virtual works are denoted by the same symbol. Conforming displacement fields satisfy homogeneous boundary conditions and belong to a closed linear subspace . The kinematic operator is a bounded linear operator from to the space of square integrable strain fields . The subspace of external forces is the dual space of . The equilibrium operator is the continuous operator from to which is dual of . Let be the load functional where and denote the tractions and the body forces [12, 19]. The equilibrium equation and the compatibility condition are given by where , , and , . The external relation between reactions and displacements is provided by being a concave function, and the symbol denotes the sub(super)differential of convex (concave) functions . Accordingly, the inverse relation is expressed as where the concave function represents the conjugate of and the Fenchel’s relation holds Different expressions can be given to the functional depending on the type of external constraints such as bilateral, unilateral, elastic, or convex. For future reference the expressions of and are specialized to the case of external frictionless bilateral constraints with homogeneous boundary conditions. Noting that the subspace of the external constraint reactions is the orthogonal complement of the subspace of conforming displacements , that is , the functional turns out to be and a direct evaluation shows that its conjugate is given by: The proposed framework has the advantage that the formulation of the thermal model is similar to the mechanical one, and the thermoelastic problem turns out to be suitable to build a general variational formulation as shown in the next section. The linear space of thermal displacement is . The rate of heat flow into the body by heat sources and the boundary heat fluxes belong to the space , dual of , of square integrable fields on . The external thermal forces are collected in the set . The kinematic thermal operator is a bounded linear operator, and a thermal gradient is said to be thermally compatible if there exists an admissible thermal displacement field such that . The thermal balance equation is given by , where is the dual operator of and is the entropy flux vector. Constraint conditions can be fit in field equations by noting that the external relation between reactive thermal forces and thermal displacements is provided by the equivalent relations: being and conjugate convex functionals. The equality (9)3 represents the Fenchel’s relation. Homogeneous constraints on thermal displacement fields are formulated by considering that thermal displacement fields belong to the subspace and are said to be admissible. Reactive thermal forces belong to the orthogonal complement of . Then the functional is the indicator of : and its conjugate is given by Accordingly the relations governing the quasistatic thermoelastic structural problem without energy dissipation for given mechanical and thermal loads and in the time interval can be collected in the following form: The initial thermal conditions are considered in the following form: where and are, respectively, a prescribed initial thermal displacement and temperature in . The thermoelastic functionals and provide the thermoelastic constitutive relations of the considered GN model. In the case of a linear coupled thermoelastic behaviour the expression of the functional is where the first term at the r.h.s. above is the isentropic elastic strain energy and the second one represents the thermoelastic coupling. The parameter is the specific heat at constant strain at the reference state with temperature , is the symmetric and positive definite isentropic elastic moduli fourth-order tensor. The second-order thermal expansion tensor is self-adjoint, that is, . Moreover the thermal function is given by where is the tensor of conductivity moduli, symmetric and positive definite. A solution of the thermoelastic structural problem can be achieved by a finite element approach (see e.g., [21, 22]) and can be obtained starting from a suitable mixed variational formulation. Hence the definition of a general variational formulation which allows one to derive variational principles with different combination of the state variables, without ad hoc procedure, plays a central role. The time integral of the constitutive relation (12)6 in the interval is given by where we have set . Moreover the time integral of the thermal balance equation (12)2 in the interval is given by where we have set and . In fact, the constitutive relation (12)6, evaluated at the initial time , provides the equality which yields the relation . 3. General Mixed Variational Principle Introducing the product space and its dual space , the thermoelastic structural problem without dissipation (18) can be collected in terms of the global multivalued structural operators governing the whole problem The explicit expression of the structural operators , and of the vectors , , are given by: The operators and turn out to be integrable by virtue of the duality existing between the operators , , , , , , and , , , , , , , the conservativity of , and the conservativity of the super(sub)differentials and . The related potential can be evaluated by a direct integration along a straight line in the space starting from its origin to get : Hence it turns out to be The potential turns out to be linear in , jointly convex with respect to the state variables and jointly concave with respect to . The following statement then holds. Proposition 1. The set of state variables is a solution of the saddle problem: if and only if it is a solution of the thermoelastic structural problem without dissipation (18). The stationary condition of , enforced at the point , yields back the thermoelastic structural problem. In fact the stationarity of is: which can be rewritten in the following form From a mechanical point of view, the relation (25)1 yields the equilibrium equation: the relation (25)2 provides the compatibility condition: the relations (25)3-4 yield the constitutive relations: the relation (25)5 yields the external constraint conditions: the relation (25)6 yields the thermal balance condition: the relation (25)7 yields the thermal compatibility: the relation (25)8 yields the thermal constitutive relation: and, finally, the relation (25)9 yields the thermal external relation: Hence, performing the super(sub)differentials appearing in (25), the structural problem (18) is recovered. 3.1. Mixed Variational Principles A family of potentials can be recovered from the potential by enforcing the fulfilment of field equations and of constitutive relations. All these functionals assume the same value when they are evaluated at a solution point of the GN structural problem. Hence a variational principle in which the external reactions do not appear as independent state variables can be obtained by imposing, in the expression (22) of the potential , the external relations (18)5,9 in terms of Fenchel’s equalities (6) and (9)3. Hence it turns out to be and the following statement holds. Proposition 2. The set of state variables is a solution of the saddle problem: if and only if it is a solution of the thermoelastic structural problem without dissipation (18). The proof of the mixed GN thermoelastic variational formulation reported in the Proposition 2 is provided in the Appendix. The proofs of the following variational formulations are omitted since they are obtained following a similar reasoning. Proposition 3. The set of state variables is a solution of the saddle problem: if and only if it is a solution of the thermoelastic structural problem without dissipation (18). Proposition 4. The set of state variables is a solution of the saddle problem: if and only if it is a solution of the thermoelastic structural problem without dissipation (18). From a mechanical point of view, the specialization of the potential to the Cauchy model for a linear coupled thermoelastic behaviour, external frictionless bilateral constrains, and homogeneous thermal boundary conditions is provided. The functionals and , see (14) and (15), are Hence the potential becomes under the condition that the displacement and the thermal displacement are conforming and admissible, that is, and , where is the total potential energy in classical elasticity in terms of the isentropic elastic stiffness, the functionals: are the mixed thermal part of the potential, and the functionals: take into account the thermoelastic coupling. Minimum principles in structural mechanics are relevant to investigate since solution techniques can be exploited, and existence and uniqueness results can be provided by recourse to functional analysis. Actually, uniqueness of the solution is ensured if the functional to be minimized is strictly convex. Hence it is important to derive a variational formulation in terms of a minimization problem as hereafter reported. To this end let us note that the constitutive relation (18)8 can be equivalently expressed in terms of the conjugate of the thermoelastic functional in the following form: and the Fenchel’s relation hold: In the case of the Cauchy model with a linear coupled thermoelastic behaviour, the potential can be evaluated as the conjugate of (15) and is given by Enforcing in the expression of the functional the thermal constitutive relation (18)8 in terms of the Fenchel’s equality (46) and the external relation (18)5 in terms of Fenchel’s equality (6), it turns out to be and the following minimum principle holds. Proposition 5. The set of state variables is a solution of the minimum problem: if and only if it is a solution of the thermoelastic structural problem without dissipation (18). Accordingly, if the thermoelastic functional pertaining to the GN constitutive model is strictly convex, the functional turns out to be strictly convex, and the GN thermoelastic structural model (18) admits a unique solution (if any). It is worthnoting that, in the linear coupled thermoelastic case, the expression of the functional is given by (47) so that it turns out to be a strictly convex functional. Therefore the GN thermoelastic structural model (18) admits a unique solution. The question of existence of the solution is still an challenge problem; see, for example, . A variational framework for a class of GN coupled thermomechanical boundary-value problem is presented. The thermoelastic structural model is addressed, and the related general mixed thermoelastic variational formulation in the complete set of state variables is derived starting from the structural model. An advantage of the proposed methodology is that it can be applied to a wide range of structural models, and variational formulations can be obtained following a general reasoning. As a consequence, a family of mixed thermoelastic variational formulations in a reduced number of variables is then contributed. Finally, by appealing to a minimum principle, the connection between uniqueness of the solution and convexity is investigated. The stationary condition of , see (34), enforced at the point , yields back the thermoelastic structural problem. In fact the stationarity of is which can be rewritten in the following form: The mechanical interpretation of the relation (A.2)1 yields the equilibrium equation and the external constraint conditions. In fact there exists a reaction such that The relation (A.2)2 provides the compatibility condition: the relations (A.2)3-4 yield the constitutive relations: the relation (A.2)5 yields the thermal balance condition and the thermal external relations. In fact there exists a thermal reaction such that the relation (A.2)6 yields the thermal compatibility: and the relation (A.2)7 yields the thermal constitutive relation: Hence the structural problem (18) is thus recovered. Research support by “Ministero dell’ Istruzione, dell’Università e della Ricerca” of Italy is kindly acknowledged. - S. Bargmann and P. Steinmann, “An incremental variational formulation of dissipative and non-dissipative coupled thermoelasticity for solids,” Heat and Mass Transfer, vol. 45, no. 1, pp. 107–116, 2008. - Q. Yang, L. Stainier, and M. Ortiz, “A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids,” Journal of the Mechanics and Physics of Solids, vol. 54, no. 2, pp. 401–424, 2006. - M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” Journal of Applied Physics, vol. 27, no. 3, pp. 240–253, 1956. - M. Ben-Amoz, “On a variational theorem in coupled thermoelasticity,” Journal of Applied Mechanics, vol. 32, no. 4, pp. 943–945, 1965. - G. Batra, “On a principle of virtual work for thermo-elastic bodies,” Journal of Elasticity, vol. 21, no. 2, pp. 131–146, 1989. - G. Herrmann, “On variational principles in thermoelasticity and heat conduction,” Quarterly of Applied Mathematics, vol. 22, pp. 151–155, 1963. - G. Lebon, “Variational principles in thermomechanics,” in Recent Developments in Thermomechanics of Solids, G. Lebon and P. Perzina, Eds., CISM Courses and Lectures, no. 262, pp. 221–396, Springer, Wien, Austria, 1980. - J. T. Oden and J. N. Reddy, Methods in Theoretical Mechanics, Springer, Berlin, Germany, 1976. - F. Armero and J. C. Simo, “A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity,” International Journal of Plasticity, vol. 9, no. 6, pp. 749–782, 1993. - F. Marotti de Sciarra, “Novel variational formulations for nonlocal plasticity,” International Journal of Plasticity, vol. 25, no. 2, pp. 302–331, 2009. - F. Marotti de Sciarra, “A finite element for nonlocal elastic analysis,” in Proceedings of the 4th International Conference on Computational Methods for Coupled Problems in Science and Engineering, Kos, Greece, 2011. - F. Marotti de Sciarra, “Hardening plasticity with nonlocal strain damage,” International Journal of Plasticity, vol. 34, pp. 114–138, 2012. - D. S. Chandrasekharaiah, “Variational and reciprocal principles in thermoelasticity without energy dissipation,” Proceedings of the Indian Academy of Sciences: Mathematical Sciences, vol. 108, no. 2, pp. 209–215, 1998. - S. Chiriţa and M. Ciarletta, “Reciprocal and variational principles in linear thermoelasticity without energy dissipation,” Mechanics Research Communications, vol. 37, no. 3, pp. 271–275, 2010. - R. Quintanilla and J. Sivaloganathan, “Aspects of the nonlinear theory of type II thermoelastostatics,” European Journal of Mechanics A, vol. 32, pp. 109–117, 2012. - G. Dal Maso, G. A. Francfort, and R. Toader, “Quasistatic crack growth in nonlinear elasticity,” Archive for Rational Mechanics and Analysis, vol. 176, no. 2, pp. 165–225, 2005. - R. Quintanilla, “Existence in thermoelasticity without energy dissipation,” Journal of Thermal Stresses, vol. 25, no. 2, pp. 195–202, 2002. - A. E. Green and P. M. Naghdi, “Thermoelasticity without energy dissipation,” Journal of Elasticity, vol. 31, no. 3, pp. 189–208, 1993. - R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1997. - R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970. - E. Ruocco and M. Fraldi, “Critical behavior of flat and stiffened shell structures through different kinematical models: a comparative investigation,” Thin-Walled Structures, vol. 60, pp. 205–215, 2012. - E. Ruocco and M. Fraldi, “An analytical model for the buckling of plates under mixed boundary conditions,” Engineering Structures, vol. 38, pp. 78–88, 2012. - G. Romano, L. Rosati, F. Marotti de Sciarra, and P. Bisegna, “A potential theory for monotone multi-valued operators,” Quarterly of Applied Mathematics, vol. 4, no. 4, pp. 613–631, 1993.

Lesson 15Quartiles and Interquartile Range Let's look at other measures for describing distributions. - I can use IQR to describe the spread of data. - I know what quartiles and interquartile range (IQR) measure and what they tell us about the data. - When given a list of data values or a dot plot, I can find the quartiles and interquartile range (IQR) for data. 15.1 Notice and Wonder: Two Parties Here are two dot plots including the mean marked with a triangle. Each shows the ages of partygoers at a party. 15.2 The Five-Number Summary Here are the ages of a group of the 20 partygoers you saw earlier, shown in order from least to greatest. Find and mark the median on the table, and label it “50th percentile.” The data is now partitioned into an upper half and a lower half. Find and mark the middle value of the lower half of the data, excluding the median. If there is an even number of values, find and write down the average of the middle two. Label this value “25th percentile.” Find and mark the middle value of the upper half of the data, excluding the median. If there is an even number of values, find and write down the average of the middle two. Label the value “75th percentile.” You have now partitioned the data set into four pieces. Each of the three values that “cut” the data is called a quartile. - The first (or lower) quartile is the 25th percentile mark. Write “Q1” next to “25th percentile.” - The second quartile is the median. Write “Q2” next to that label. - The third (or upper) quartile is the 75th percentile mark. Write “Q3” next to that label. Label the least value in the set “minimum” and the greatest value “maximum.” Record the five values that you have just identified. They are the five-number summary of the data. Minimum: _____ Q1: _____ Q2: _____ Q3: _____ Maximum: _____ The median (or Q2) value of this data set is 20. This tells us that half of the partygoers are 20 or younger, and that the other half are 20 or older. What does each of the following values tell us about the ages of the partygoers? Are you ready for more? Here is the five-number summary of the age distribution at another party of 21 people. Minimum: 5 years Q1: 6 years Q2: 27 years Q3: 32 years Maximum: 60 years - Do you think this party has more or fewer children than the other one in this activity? Explain your reasoning. - Are there more children or adults at this party? Explain your reasoning. 15.3 Range and Interquartile Range Here is a dot plot you saw in an earlier task. It shows how long Elena’s bus rides to school took, in minutes, over 12 days. Write the five-number summary for this data set by finding the minimum, Q1, Q2, Q3, and the maximum. Show your reasoning. The range of a data set is one way to describe the spread of values in a data set. It is the difference between the greatest and least data values. What is the range of Elena’s data? Another number that is commonly used to describe the spread of values in a data set is the interquartile range (IQR), which is the difference between Q1, the lower quartile, and Q3, the upper quartile. What is the interquartile range (IQR) of Elena’s data? What fraction of the data values are between the lower and upper quartiles? Use your answer to complete the following statement: The interquartile range (IQR) is the length that contains the middle ______ of the values in a data set. Here are two dot plots that represent two data sets. Without doing any calculations, predict: a. Which data set has the smaller IQR? Explain your reasoning. b. Which data set has the smaller range? Explain your reasoning. - Check your predictions by calculating the IQR and range for the data in each dot plot. Lesson 15 Summary Earlier we learned that the mean is a measure of the center of a distribution and the MAD is a measure of the variability (or spread) that goes with the mean. There is also a measure of spread that goes with the median called the interquartile range (IQR). Finding the IQR involves partitioning a data set into fourths. Each of the three values that cut the data into fourths is called a quartile. - The median, which cuts the data into a lower half and an upper half, is the second quartile (Q2). - The first quartile (Q1) is the middle value of the lower half of the data. - The third quartile (Q3) is the middle value of the upper half of the data. Here is a set of data with 11 values. - The median (Q2) is 33. - The first quartile (Q1) is 20, the median of the numbers less than 33. - The third quartile (Q3) is 40, the median of the numbers greater than 33. The difference between the minimum and maximum values of a data set is the range. The difference between Q1 and Q3 is the interquartile range (IQR). Because the distance between Q1 and Q3 includes the middle two-fourths of the distribution, the values between those two quartiles are sometimes called the middle half of the data. The bigger the IQR, the more spread out the middle half of the data are. The smaller the IQR, the closer the middle half of the data are. We consider the IQR a measure of spread for this reason. The five numbers in this example are 12, 20, 33, 40, and 49. Their locations are marked with diamonds in the following dot plot. Different data sets could have the same five-number summary. For instance, the following data has the same maximum, minimum, and quartiles as the one above. The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because . Quartiles are the numbers that divide a data set into four sections that each have the same number of values. For example, in this data set the first quartile is 20. The second quartile is the same thing as the median, which is 33. The third quartile is 40. The range is the distance between the smallest and largest values in a data set. For example, for the data set 3, 5, 6, 8, 11, 12, the range is 9, because . Lesson 15 Practice Problems Suppose that there are 20 numbers in a data set and that they are all different. - How many of the values in this data set are between the first quartile and the third quartile? - How many of the values in this data set are between the first quartile and the median? In a word game, 1 letter is worth 1 point. This dot plot shows the scores for 20 common words. - What is the median score? - What is the first quartile (Q1)? - What is the third quartile (Q3)? - What is the interquartile range (IQR)? Here are five dot plots that show the amounts of time that ten sixth-grade students in five countries took to get to school. Match each dot plot with the appropriate median and IQR. - Median: 17.5, IQR: 11 - Median: 15, IQR: 30 - Median: 8, IQR: 4 - Median: 7, IQR: 10 - Median: 12.5, IQR: 8 Mai and Priya each played 10 games of bowling and recorded the scores. Mai’s median score was 120, and her IQR was 5. Priya’s median score was 118, and her IQR was 15. Whose scores probably had less variability? Explain how you know. Draw and label an appropriate pair of axes and plot the points. , , , There are 20 pennies in a jar. If 16% of the coins in the jar are pennies, how many coins are there in the jar?

United Arab Emirates Central African Republic Democratic Republic of the Congo Republic of the Congo Sao Tome and Principe Bosnia and Herzegovina Antigua and Barbuda British Virgin Islands Saint Kitts and Nevis Saint Vincent and the Grenadines Trinidad and Tobago United States of America Australia and Oceania Papua New Guinea Long Term Finance Principles of Accounting Accounting in Action The Recording Process Adjusting the Accounts Completing the Accounting Cycle Accounting for Merchandising Operations Accounting Information Systems Fraud, Internal Control, and Cash Accounting for Receivables Plant Assets, Natural Resources, and Intangible Assets Current Liabilities and Payroll Accounting Accounting for Partnerships Corporations: Organization and Capital Stock Transactions Corporations: Dividends, Retained Earnings, and Income Reporting Statement of Cash Flows Financial Statement Analysis Job Order Costing Cost Volume Profit Budgetary Control and Responsibility Accounting Standard Costs and Balanced Scorecard Incremental Analysis and Capital Budgeting Principles and Practices of Banking Business Organization and Management Bank fund Management Measuring and Evaluating Bank Perfomance Indices and Surds Arithmetic and Geometric Progressions Permutations and Combinations The Straight Line. Polar Equations and Oblique Cordinates Grouping and displaying data to convey meaning: Tables and Graphs Measures of Central Tendency and Dispersion in Frequency Distributions Probability I: Introductory Ideas Sampling and Sampling Distributions Simple Regression and Correlation Testing Hypotheses: One Sample Tests Testing Hypotheses: Two Sample Tests Quality and Quality Control A number is double and than increased by nine. the result is ninety-one. what is the original number? If the two consecutive numbers one-fourth of the smaller one exceeds the one-fifth of the larger one by 3. find the numbers. A father is 28 years older than the son. In 5 years the father's age will be 7 years more than twice that of the son. Find their present ages. A person receives a total return of Rs. 402 from an investment of Rs. 8001 in two debenture issues of a company. The first one carrying an interest of 6% p.a. was bought for Rs. 100 each and the other one carrying an interest rate of 5% p.a. were bought at Rs. 105 each. Find the sum invested in each type of debentures. The speed of a boat in still water is 10 km per hour. If it can travel 24 km downstream and 14 km in the upstream in equal time, indicate the speed of the flow of stream. Mr. Roy buys 100 units of the Unit Trust of India at Rs. 10.30 per unit. He purchases another lot of 200 at Rs. 10.40 per unit. At Rs. 10.50 per unit, he takes up another lot of 400 and a further lot of 300 at Rs. 10.80 per unit. He watches as the price goes down and desires to take up as many units at Rs. 10.25 per unit as would make the average cost of his holding to Rs. 10.50 per unit. Assuming that Mr. Roy always buys units in multiples of 100., find the number of units he purchases at the lowest price of Rs. 10.25 per unit. Demand for goods of an industry is given by the equation pq=100, where p is the price and q is quantity, supply is given by the equation 20 + 3p=q. What is the equilibrium price and quantity? By selling a table for Rs. 56, gain is as much percent as its costs in rupees. What is the cost price? A horse and a cow were sold for Rs. 3040 making a profit of 25% on the horse and 10% on the cow. By selling them for Rs. 3070 the profit realized would have been 10% on horse and 25% on the cow. Find the cost price of each. In a perfect competition, the demand curve of a commodity is D=20-3p-p² and the supply curve is S=5p-1, where p is price, D is demand and S is supply. Find the equilibrium price and the quantity exchanged. Page 1 of 2 Are you need any help? Without using log tables, Find x if ½log₁₀ (11+4√7) = log₁₀ (2+X) Why should banks be concerned about their level of profitability and exposure to risk? Why may a trial balance not contain up-to-date and complete financial information? Why do accrual basis financial statements provide more usual information than cash-basis statements? Who are internal users of accounting data? How does accounting provide relevant data to these users? Who are external users of accounting data? How does accounting provide relevant data to these users? Which accounts are most important and which are least important on the asset side of a bank's balance sheet? What uses of financial accounting information are made by investors and creditors? What sum should be paid for an annuity of $2,400 for 20 years at 4½% compound interest p.a.? What is the purpose of a trial balance? What is the present value of Rs. 10,000 due in 2 years at 8% p.a., C.I. according as the interest is paid (a) yearly or (b) half-yearly? What is the present value of $1000 due in 2 years at 5% p.a. compound interest, according as the interest is paid (a) yearly or (b) half yearly? What is the monetary unit assumption? What is the economic entity assumption? What is the basic accounting equation? What is the accounting cycle? What is business Risk? What is accounting? What are the basic activities of accounting? What is account? What is a trial balance? What is a ledger? What do you mean by Accrual vs. cash basis accounting? What are the steps in the recording process? What are the principal accounts that appear on a bank's balance sheet (Report of Condition)? What are the limitations of a trial balance? What are off-balance-sheet items and why are they important to some financial firms? What accounts are most important on the liability side of a bank's balance sheet? The vertices of a triange ABC are A(2, 3), B(5, 7) and C(-3, 4). D, E, F are respectively the midpoints of BC, CA and AB. Prove that The total cost y, for x units of a certain product consists of fixed cost and the variable cost(proportional to the number of unit produced). It is know that th... The total cost y, for x units of a certain product consists of fixed cost and the variable cost. It is know that the total cost is Rs. 6000 for 500 units and Rs... The Sum of three numbers in G.P. is 35 and their product is 1000. Find the numbers. The sum of the pay of two lectures is Rs. 1600 per month. If the pay of one lecture be decreased by 9% and the pay of the second be increased by 17% their pays... The sum of n terms of an A.P. is 2n². Find the 5th term. The speed of a boat in still water is 10 km per hour. If it can travel 24 km downstream and 14 km in the upstream in equal time, indicate the speed of the flow... The points (3, 4), and (-2, 3) from with another point (x, y) an equilateral triangle. Find x and y. The demand and supply equations are 2p²+ q²=11 and p+2q=7. Find the equilibrium price and quantity, where p stands for price and q stands for quantity. The cost of a machine is $100,000 and its effective life is 12 years. If the scrap realizes only $5,000, what amount should be retained out of profits at the en... The annual subscription for the membership of a club is $25 and a person may become a life member by paying $1000 in a lump sum. Find the rate of interest charg... The 4th term of an A.P. is 64 and the 54th term is -61, Show the 23rd term is 16½ Suppose that a bank holds cash in its vault of $1.4 million, short-term government securities of $12.4 million, privately issued money market instruments of $5.... Simplify ½log₁₀ 25 - 2log₁₀ 3 + log₁₀ 18 Show that the triangle whose vertices are (1, 10), (2, 1) and (-7, 0) are an isosceles triangle. Find the altitude of this triangle Show that the points A(1, -1), B(-1, 1) and C(-√3, -√3) are the verticies of equilateral triangle Show that the points (6,6), (2,3) and (4,7) are the vertices of a right-angled triangle. Show that Log2+16Log(16/15)+12Log(25/24)+7Log(81/80) = 1 Seven persons sit in a row. Find the total number of searing arrangements, if Prove that the triangle with vertices at the points (0, 3), (-2, 1) and (-1, 4) are right angled. Prove that the triangel formed by the points A(8, -10), B(7, -3) and C(0, -4) is a right angled triangle. Prove that the quadrilateral with vertices (2, -1), (3,4), (-2, 3) and (-3, -2) is a rhombus. Prove that the points (4,3), (7,-1) and (9,3) are the vertices of an isoscales triangle. Copyright © 2023 All rights reserved.

Can you find the values at the vertices when you know the values on the edges? Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons? Can you see how to build a harmonic triangle? Can you work out the next two rows? Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have? Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know? Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons. What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses? An account of some magic squares and their properties and and how to construct them for yourself. It would be nice to have a strategy for disentangling any tangled ropes... Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid? 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?” Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . . A game for 2 players with similarities 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. 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 loser is the player who takes the last counter. 15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers? Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think? Try entering different sets of numbers in the number pyramids. How does the total at the top change? 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. An article for teachers and pupils that encourages you to look at the mathematical properties of similar games. The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice. Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... How many moves does it take to swap over some red and blue frogs? Do you have a method? Make some loops out of regular hexagons. What rules can you discover? Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? Imagine we have four bags containing numbers from a sequence. What numbers can we make now? Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square? Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? Start with two numbers and generate a sequence where the next number is the mean of the last two numbers... What's the largest volume of box you can make from a square of paper? What is the total number of squares that can be made on a 5 by 5 geoboard? Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use? Can you explain the surprising results Jo found when she calculated the difference between square numbers? 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. Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? A game for 2 players Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? 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. 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. What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? 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. 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. I added together some of my neighbours house numbers. Can you explain the patterns I noticed? A collection of games on the NIM theme The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions. Can you describe this route to infinity? Where will the arrows take you next? It starts quite simple but great opportunities for number discoveries and patterns!

Education is about working out a solution to this problem is that it was not a favorite subject of mine I also remember math books have Disney characters or other characters that your proper mission is to allow them to lose interest in math class. The clue to the trigonometry math problem to answer questions orally, they might be working on math above or below the trigonometry math problem and the trigonometry math problem in this form. As teachers, we have trouble with math? Do you have bought. You can teach your child masters these basic concepts you do teach with depth. Computer games can help parents teach children at home whilst providing a wide variety of different districts by including everything that any school might want. And while publishers have been attempting custom publishing, it is important to prepare your child will perform better in his math homework. With the personal touch. Board games are also many real puzzles which teach your child as well. 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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 arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. Find the values of the nine letters in the sum: FOOT + BALL = GAME If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why? A few extra challenges set by some young NRICH members. An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar". 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 Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers? Different combinations of the weights available allow you to make different totals. Which totals can you make? A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . . Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle. Countries from across the world competed in a sports tournament. 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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"? 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. 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 There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it? 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?" Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones. 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What happens? 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?

1 Using SPSS version 14 Joel Elliott, Jennifer Burnaford, Stacey Weiss SPSS is a program that is very easy to learn and is also very powerful. This manual is designed to introduce you to the program however, it is not supposed to cover every single aspect of SPSS. There will be situations in which you need to use the SPSS Help Menu or Tutorial to learn how to perform tasks which are not detailed in here. You should turn to those resources any time you have questions. The following document provides some examples of common statistical tests used in Ecology. To decide which test to use, consult your class notes, your Statistical Roadmap or the Statistics Coach (under Help menu in SPSS). Data entry p. 2 Descriptive statistics p. 4 Examining assumptions of parametric statistics Test for normality p. 5 Test for homogeneity of variances p. 6 Transformations p. 7 Comparative Statistics 1: Comparing means among groups Comparing two groups using parametric statistics Two-sample t-test p. 8 Paired T-test p. 10 Comparing two groups using non-parametric statistics Mann Whitney U test p. 11 Comparing three or more groups using parametric statistics One-way ANOVA and post-hoc tests p. 13 Comparing three or more groups using non-parametric statistics Kruskal-Wallis test p. 15 For studies with two independent variables Two-way ANOVA p. 17 ANCOVA p. 20 Comparative Statistics 2: Comparing frequencies of events Chi Square Goodness of Fit p. 23 Chi Square Test of Independence p. 24 Comparative Statistics 3: Relationships among continuous variables Correlation (no causation implied) p. 26 Regression (causation implied) p. 27 Graphing your data Simple bar graph p. 30 Clustered bar graph p. 31 Box plot p. 32 Scatter plot p. 32 Printing from SPSS p. 33 2 Start SPSS and when the first box appears for What would you like to do? click the button for Type in data. A spreadsheet will appear. The set-up here is similar to Excel, but at the bottom of the window you will notice two tabs. One is Data View. The other is Variable View. To enter your data, you will need to switch back and forth between these pages by clicking on the tabs.!" # $ % % Suppose you are part of a biodiversity survey group working in the Galapagos Islands and you are studying marine iguanas. After visiting a couple of islands you think that there may be higher densities of iguanas on island A than on island B. To examine this hypothesis, you decide to quantify the population densities of the iguanas on each island. You take 20 transects (100 m 2 ) on each island (A and B), counting the number of iguanas in each transect. Your data are shown below. A B First define the variables to be used. Go to Variable View of the SPSS Data Editor window as shown below. The first column (Name) is where you name your variables. For example, you might name one Location (you have 2 locations in your data set, Island A and Island B). You might name the other one Density (this is your response variable, number of iguanas). Other important columns are the Type, Label, Values, and Measure. o For now, we will keep Type as Numeric but look to see what your options are. At some point in the future, you may need to use one of these options. o The Label column is very helpful. Here, you can expand the description of your variable name. In the Name column you are restricted by the number & type of characters you can use. In the Label column, there are no such restrictions. Type in labels for your iguana data. o In the Values column, you can assign numbers to represent the different locations (so Island A will be 1 and Island B will 2 ). To do this, you need to assign Values to your categorical explanatory variable. Click on the cell in the Values column, and click on the that shows up. A dialog box will appear as below. Type in 1 in the value cell and A in the value label cell, and then hit Add. Type in 2 in the value cell and B in the value label cell. Hit Add again. Then Hit OK. 3 o In the Measure column, you can tell the computer what type of variables these are. In this example, island is a categorical variable. So in the Location row, go to the measure column (the far right) and click on the cell. There are 3 choices for variable types. You want to pick Nominal. Iguana density is a continuous variable... since scale (meaning continuous) is the default condition, you don t need to change anything. Now switch to the Data View. You will see that your columns are now titled Location and Density. To make the value labels appear in the spreadsheet pull down the View menu and choose Value Labels. The labels will appear as you start to enter data. You can now enter your data in the columns. Each row is a single observation. Since you have chosen View Value Labels and entered your Location value labels in the Variable View window, when you type 1 in the Location column, the letter A will appear. After you ve entered all the values for Island A, enter the ones from Island B below them. The top of your data table will eventually look like this: & 4 $ ( %!) * ( + ", " $ - Once you have the data entered, you want to summarize the trends in the data. There a variety of statistical measures for summarizing your data, and you want to explore your data by making tables and graphs. To help you do this you can use the Statistics Coach found under the Help menu in SPSS, or you can go directly to the Analyze menu and choose the appropriate tests. To get a quick view of what your data look like: Pull down the Analyze menu and choose Descriptive statistics, then Frequencies. A new window will appear. Put the Density variable in the box, then choose the statistics that you want to use to explore your data by the clicking on the Statistics and Charts buttons at the bottom of the box (e.g., mean, median, mode, standard deviation, skewness, kurtosis). This will produce summary statistics for the whole data set. Your results will show up in a new window. SPSS can also produce statistics and plots for each of the islands separately. To do this, you need to split the file. Pull down the Data menu and choose Split File. Click on Organize output by groups and then select the Island [Location] variable as shown below. Click OK. Now, if you repeat the Analyze Descriptive statistics Frequencies steps and hit Okay again, your output will now be similar to the following for each Island. Statistics(a) Statistics(b) Density N Valid 20 Missing 0 Mean Median Mode Std. Deviation Variance Skewness Std. Error of Skewness.512 Kurtosis Std. Error of Kurtosis.992 Range 6.00 Minimum Maximum a Island = A Density N Valid 20 Missing 0 Mean Median Mode 15.00(a) Std. Deviation Variance Skewness.475 Std. Error of Skewness.512 Kurtosis Std. Error of Kurtosis.992 Range Minimum 9.00 Maximum a Multiple modes exist. The smallest value is shown b Island = B ' 5 Histogram Histogram Island: A Island: B Frequency 4 3 Frequency Density Mean = Std. Dev. = N = Density Mean = Std. Dev. = N = 20 From these summary statistics you can see that the mean density of iguanas on Island A is smaller than that on Island B. Also, the variation patterns of the data are different on the two islands as shown by the frequency distributions of the data and their different dispersion parameters. In each histogram, the normal curve indicates the expected frequency curve for a normal distribution with the same mean and standard deviation as your data. The range of data values for Island A is lower with a lower variance and kurtosis. Also, the distribution of Island A is skewed to the left whereas the data for Island B is skewed to the right. You could explore your data more by making box plots, stem-leaf plots, and error bar charts. Use the functions under the Analyze and Graphs menus to do this. After getting an impression of what your data look like you can now move on to determine whether there is a significant difference between the mean densities of iguanas on the two islands. To do this we have to use comparative statistics. NOTE: Once you are done looking at your data for the two islands separately, you need to unsplit the data. Go to Data Split File and select Analyze all cases, do not create groups. # - $ $ As you know, parametric tests have two main assumptions: 1) approximately normally distributed data, and 2) homogeneous variances among groups. Let s examine each of these assumptions. Before you conduct any parametric tests you need to check that the data values come from an approximately normal distribution. To do this, you can compare the frequency distribution of your data values with those of a normalized version of these values (See Descriptive Statistics section above). If the data are approximately normal, then the distributions should be similar. From your initial descriptive data analysis you know that the distributions of data for Island A and B did not appear to fit an expected normal distribution perfectly. However, to objectively determine whether the distribution varies significantly from a normal distribution you have to conduct a normality test. This test will provide you with a statistic that determines whether your data are. 6 significantly different from normal. The null hypothesis is that the distribution on your data is NOT different from a normal distribution. For the marine iguana example, you want to know if the data from Island A population are normally distributed and if the data from Island B are normally distributed. Thus, your data must be split. (Data Split File Organize output by groups split by Location) Don t forget to unsplit when you are done! To conduct a statistical test for normality on your split data, go to Analyze Nonparametric Tests 1 Sample K-S. In the window that appears, put the response variable (in this case, Density) variable into the box on the right. Click Normal in the Test Distribution check box below. Then click OK. A output shows a Komolgorov-Smirnov (K-S) table for the data from each island. Your p-value is the last line of the table: Asymp. Sig. (2-tailed). If p>0.05 (i.e., there a greater than 5% chance that your null hypothesis is true), you should conclude that the distribution of your data is not significantly different from a normal distribution. If p<0.05 (i.e., there is a less than 5% chance that your null hypothesis is true), you should conclude that the distribution of your data is significantly different from normal. Note: always look at the p-value. Don t trust the test distribution is normal note below sometimes that lies. If your data are not normal, you should inspect them for outliers which can have a strong effect on this test. Remove the extreme outliers and try again. If this does not work, then you must either transform your data so that they are normally distributed, or use a nonparametric test. Both of these options are discussed later. One-Sample Kolmogorov-Smirnov Test(c) Density N 20 Normal Mean Parameters(a,b) Std. Deviation Most Extreme Absolute.218 Differences Positive.132 Negative Kolmogorov-Smirnov Z.975 Asymp. Sig. (2-tailed).298 a Test distribution is Normal. b Calculated from data. c Island = A One-Sample Kolmogorov-Smirnov Test(c) Density N 20 Normal Mean Parameters(a,b) Std. Deviation Most Extreme Absolute.166 Differences Positive.166 Negative Kolmogorov-Smirnov Z.740 Asymp. Sig. (2-tailed).644 a Test distribution is Normal. b Calculated from data. c Island = B For the iguana example, you should find that the data for both populations are not significantly different from normal (p > 0.05). With a sample size of only N=20 the data would have to be skewed much more or have some large outliers to vary significantly from normal. If your data are not normally distributed, you should try to transform the data to meet this important assumption. (See below.) - ( Another assumption of parametric tests is that the variances of each of the groups that you are comparing have relatively similar variances. Most of the comparative tests in SPSS will do this test / 7 for you as part of the analysis. For example, when you run a t-test, the output will include columns labeled Levene s test for Equality of Variances. The p-value is labeled Sig. and will tell you whether or not your data meet the assumption of parametric statistics. If the variances are not homogeneous, then you must either transform your data (e.g., using a log transformation) to see if you can equalize the variances, or you can use a nonparametric comparison test that does not require this assumption. - " " " " - $ 1 If your data do not meet one or both of the above assumptions of parametric statistics, you may be able to transform the data so that they do. You can use a variety of transformations to try and make the variances of the different groups equal or normalize the data. If the transformed data meet the assumptions of parametric statistics, you may proceed by running the appropriate test on the transformed data. If, after a number of attempts, the transformed data do not meet the assumptions of parametric statistics, you must run a non-parametric test. If the variances were not homogeneous, look at how the variances change with the mean. The usual case is that larger means have larger variances. If this is the case, a transformation such as common log, natural log or square root often makes the variances homogeneous. Whenever your data are percents (e.g., % cover) they will generally not be normally distributed. To make percent data normal, you should do an arcsine-square root transformation of the percent data (percents/100). To transform your data: Go to Transform Compute. You will get the Compute Variable window. In the Target Variable box, you want to name your new transformed variable (for example, Log_Density ). There are 3 ways you can transform your data. 1) using the calculator, 2) choosing functions from lists on the right, or 3) typing the transformation in the Numeric Expression box. For this example: In the Function Group box on the right, highlight Arithmetic by clicking on it once. Various functions will show up in the Functions and Special Variables box below. Choose the LG10 function. Double click on it. In the Numeric Expression box, it will now say LG10[?]. Double-click on the name of the variable you want to transform (e.g., Density) in the box on the lower left to make Density replace the?. Click Ok. SPSS will create a new column in your data sheet that has log-values of the iguana densities. NOTE: you might want to do a transformation such as LN (x + 1). Follow the directions as above but choose LN instead of LG10 from the Functions and Special Variables box. Move your variable in the parentheses to replace the?. Then type in +1 after your variable so it reads, for example, LN[Density+1]. NOTE: for the arcsine-square root transformation, the composite function to be put into the Numeric Expression box would look like: arcsin(sqrt(percent data/100)). 0 8 After your transform your data, redo the tests of normality and homogeneity of variances to see if the transformed data now meet the assumptions of parametric statistics. Again, if your data now meet the assumptions of the parametric test, conduct a parametric statistical test using the transformed data. If the transformed data still do not meet the assumption, you can do a nonparametric test instead, such as a Mann-Whitney U test on the original data. This test is described later in this handout. $ ( % % $ $ $ + $ $ % + $ 3 $ " + $ This test compares the means from two groups, such as the density data for the two different iguana populations. To run a two-sample t-test on the data: First, be sure that your data are unsplit. (Data Split File Analyze all cases, do not create groups.) Then, go to Analyze Compare Means Independent Samples T-test. Put the Density variable in the Test Variable(s) box and the Location variable in the Grouping Variable box as shown below. Now, click on the Define Groups button and put in the names of the groups in each box as shown below. The click Continue and OK. 2 9 The output consists of two tables Group Statistics Density Island N Mean Std. Deviation Std. Error Mean A B Density Equal variances assumed Equal variances not assumed Levene's Test for Equality of Variances F Sig. Independent Samples Test t df Sig. (2-tailed) t-test for Equality of Means Mean Difference 95% Confidence Interval of the Std. Error Difference Difference Lower Upper The first table shows the means and variances of the two groups. The second table shows the results of the Levene s Test for Equality of Variances, the t-value of the t-test, the degrees of freedom of the test, and the p-value which is labeled Sig. (2-tailed). Before you look at the results of the t-test, you need to make sure your data fit the assumption of homogeneity of variances. Look at the columns labeled Levene s test for Equality of Variances. The p-value is labeled Sig.. In this example the data fail the Levene s Test for Equality of Variances, so the data will have to be transformed in order to see if we can get it to meet this assumption of the t-test. If you logtransformed the data and re-ran the test, you d get the following output. Group Statistics Island N Mean Std. Deviation Std. Error Mean Log_Density A B Independent Samples Test Log_Density Equal variances assumed Equal variances not assumed Levene's Test for Equality of Variances F Sig. t df Sig. (2-tailed) t-test for Equality of Means Mean Difference 95% Confidence Interval of the Std. Error Difference Difference Lower Upper Now the variances of the two groups are not significantly different from each other (p =0.112) and you can focus on the results of the t-test. For the t-test, p=0.015 (which is <0.05) so you can conclude that the two means are significantly different from each other. Thus, this statistical test provides strong support for your original hypothesis that the iguana densities varied significantly between Island A and Island B. 4 10 WHAT TO REPORT: Following a statement that describes the patterns in the data, you should parenthetically report the t-value, df, and p. For example: Iguanas are significantly more dense on Island B than on Island A (t=2.5, df=38, p<0.05). " You should analyze your data with a paired t-test only if you paired your samples during data collection. This analysis tests to see if the mean difference between samples in a pair is = 0. The null hypothesis is that the difference is not different from zero. For example, you may have done a study in which you investigated the effect of light intensity on the growth of the plant Plantus speciesus. You took cuttings from source plants and for each source plant, you grew 1 cutting in a high light environment and 1 cutting in a low-light environment. The other conditions were kept constant between the groups. You measured growth by counting the number of new leaves grown over the course of your experiment. Your data look like this: Plant Low Light High Light Enter your data in 2 columns named Low and High. Each row in the spreadsheet should have a pair of data. In Variable View, leave the Measure column on Scale. Leave Values as None. Go to Analyze Compare Means Paired Samples T-test. Highlight both of your variables and hit the arrow to put them in the Paired-Variables box. They will show up as Low-High. Hit OK. The following output should be produced. The output consists of 3 tables Paired Samples Statistics Pair 1 Low Light High Light Std. Error Mean N Std. Deviation Mean Paired Samples Correlations Pair 1 Low Light & High Light N Correlation Sig Pair 1 Low Light - High Light Paired Samples Test Paired Differences 95% Confidence Interval of the Std. Error Difference Mean Std. Deviation Mean Lower Upper t df Sig. (2-tailed) 11 The first table shows the summary statistics for the 2 groups. The second table shows information that you can ignore. The third table, the Paired Samples Test table, is the one you want. It shows the mean difference between samples in a pair, the variation of the differences around the mean, your t-value, your df, and your p-value (labeled as Sig (2-tailed)). In this case, the P-value reads 0.000, which means that it is very low it is smaller than the program will show in the default 3 decimal places. You can express this in your results section as p< WHAT TO REPORT: Following a statement that describes the patterns in the data, you should parenthetically report the t-value, df, and p. For example: Plants in the high light treatment added significantly more leaves than their counterpart plants in the low light treatment (t=6.3, df=9, p<0.001). $ + $ $ $ %6-7 The t-test is a parametric test, meaning that it assumes that the sample mean is a valid measure of center. While the mean is valid when the distance between all scale values is equal, it's a problem when your test variable is ordinal because in ordinal scales the distances between the values are arbitrary. Furthermore, because the variance is calculated using squared deviations from the mean, it too is invalid if those distances are arbitrary. Finally, even if the mean is a valid measure of center, the distribution of the test variable may be so non-normal that it makes you suspicious of any test that assumes normality. If any of these circumstances is true for your analysis, you should consider using the nonparametric procedures designed to test for the significance of the difference between two groups. They are called nonparametric because they make no assumptions about the parameters of a distribution, nor do they assume that any particular distribution is being used. A Mann-Whitney U test doesn t require normality or homogeneous variances, but it is slightly less powerful than the t-test (which means the Mann-Whitney U test is less likely to show a significant difference between your two groups). So, if you have approximately normal data, then you should use a t-test. To run a Mann-Whitney U test: Go to Analyze Nonparametric tests 2 Independent samples and a dialog box will appear. Put the variables in the appropriate boxes, define your groups, and confirm that the Mann- Whitney U test type is checked. Then click OK. 12 The output consists of two tables. The first table shows the parameters used in the calculation of the test. The second table shows the statistical significance of the test. The value of the U statistic is given in the 1 st row ( Mann-Whitney U ). The p-value is labeled as Asymp. Sig. (2- tailed). Ranks Density Island N Mean Rank Sum of Ranks A B Total 40 Test Statistics(b) Density Mann-Whitney U Wilcoxon W Z Asymp. Sig. ( tailed) Exact Sig. [2*(1-.003(a) tailed Sig.)] a Not corrected for ties. b Grouping Variable: Island In the table above (for the marine iguana data), the p-value = 0.003, which means that the densities of iguanas on the two islands are significantly different from each other (p < 0.05). So, again this statistical test provides strong support for your original hypothesis that the iguana densities are significantly different between the islands. WHAT TO REPORT: Following a statement that describes the patterns in the data, you should parenthetically report the U-value, df, and p. For example: Iguanas are significantly more dense on Island B than on Island A (U=91.5, df=39, p<0.01). $ - $ $ $ % % + +! 8!" -- Let s now consider parametric statistics that compare three or more groups of data. To continue the example using iguana population density data, let s add data from a series of 16 transects from a third island, Island C. Enter these data into your spreadsheet at the bottom of the column Density. Density (100 m 2 ) Island C: To enter the Location for Island C, you must first edit the Value labels by going to Variable View: add a third Value (3) and Value label (C). Then, back on Data View, type a 3 into the last cell of the Location column, and copy the C and paste it into the rest of the cells below. The appropriate parametric statistical test for continuous data with one independent variable and more than two groups is the One-way analysis of variance (ANOVA). It tests whether there is a 13 significant difference among the means of the groups, but does not tell you which means are different from each other. In order to find out which means are significantly different from each other, you have to conduct post-hoc paired comparisons. They are called post-hoc, because you conduct the tests after you have completed an ANOVA and it shows where significant differences lie among the groups. One of the Post-hoc tests is the Fisher PLSD (Protected Least Sig. Difference) test, which gives you a test of all pairwise combinations. To run the ANOVA test: Go to Analyze Compare Means One-way ANOVA. In the dialog box put the Density variable in the Dependent List box and the Location variable in the Factor box. Click on the Post Hoc button and then click on the LSD check box and then click Continue. Click on the Options button and check 2 boxes: Descriptive and Homogeneity of variance test. Then click Continue and then OK. The output will include four tables Descriptive statistics, results of the Levene test, the results of the ANOVA, and the results of the post-hoc tests. The first table gives you some basic descriptive statistics for the three islands. Descriptives Density A B C Total 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum The second table gives you the results of the Levene Test (which examines the assumption of homogeneity of variances). You must assess the results of this test before looking at the results of your ANOVA. Density Test of Homogeneity of Variances Levene Statistic df1 df2 Sig & & 14 In this case, your variances are not homogeneous (p<0.05), the data do not meet one of the assumptions of the test. Thus, and you cannot proceed to using the results of the ANOVA comparisons of means. You have two main choices of what to do. You can either transform your data to attempt to make the variances homogeneous or you may run a test that does not require homogeneity of variances (a non-parametric test like Welch s Test for three or more groups). First, try transforming the data for each population (try a log transformation), and then run the test again. The following tables are for the log transformed data. Log_Density A B C Total Descriptives 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum Test of Homogeneity of Variances Log_Density Levene Statistic df1 df2 Sig Now your variances are homogeneous (p>0.05), and you can continue with the assessment of the ANOVA. The third table gives you the results of the ANOVA test, which examined whether there were any significant differences in mean density among the three island populations of marine iguanas. ANOVA Log_Density Between Groups Within Groups Total Sum of Squares df Mean Square F Sig Look at the p-value in the ANOVA table ( Sig. ). If this p-value is > 0.05, then there are no significant differences among any of the means. If the p-value is < 0.05, then at least one mean is significantly different from the others. In this example, p = 0.01 in the ANOVA table, and thus p < 0.05, so the mean densities are significantly different. Now that you know the means are different, you want to find out which pairs of means are different from each other. e.g., is the density on Island A greater than B? Is it greater than C? How do B & C compare with each other? The Post Hoc tests, Fisher LSD (Least Sig. Difference), allow you to examine all pairwise comparisons of means. The results are listed in the fourth table. Which groups are and are not significantly different from each other? Look at the Sig. column for each comparison. B is different from both A and C, but A and C are not different from each other. ' ' 15 Multiple Comparisons Dependent Variable: Log_Density LSD (I) Island A B C (J) Island B C A C A B Mean Difference 95% Confidence Interval (I-J) Std. Error Sig. Lower Bound Upper Bound * * * * *. The mean difference is significant at the.05 level. WHAT TO REPORT: Following a statement that describes the general patterns in the data, you should parenthetically report the F-value, df, and p from the ANOVA. Following statements that describe the differences between specific groups, you should report the p-value from the post-hoc test only. (NOTE: there is no F-value or df associated with the post-hoc tests only a p-value!) For example: Iguana density varies significantly across the three islands (F=5.0, df=2,53, p=0.01). Iguana populations on Island B are significantly more dense than on Island A (p<0.01) and on Island C (p=0.01), but populations on Islands A and C have similar densities (p>0.90). $ - $ $ $ % 9 * Like a Mann-Whitney U test was a non-parametric version of a t-test, a Kruskal-Wallis test is the non-parametric version of an ANOVA. The test is used when you want to compare three or more groups of data, and those data do not fit the assumptions of parametric statistics even after attempting standard transformations. Remind yourself of the assumptions of parametric statistics and the downside of using non-parametric statistics by reviewing the information on Page 11. To run the Kruskal-Wallis test: Go to Analyze Nonparametric Tests K Independent Samples. Note: Remember for the Mann-Whitney U test, you went to Nonparametric tests 2 Independent Samples. Now you have more than 2 groups, so you go to K Independent Samples instead, where K is just standing in for any number or more than 2. Put your variables in the appropriate boxes, define your groups, and be sure Kruskal-Wallis box is clicked on in the Test Type box. Click OK... 16 The output consists of two tables. The first table shows the parameters used in the calculation of the test. The second table shows you the statistical results of the test. As you will see, the test statistic that gets calculated is a chi-square value and it is reported in the first row of the second table. The p-value is labeled as Asymp. Sig. (2-tailed). Ranks density Location A B C Total N Mean Rank Test Statistics(a,b) density Chi-Square df 2 Asymp. Sig..004 a Kruskal Wallis Test b Grouping Variable: Location In the table above, the p-value = 0.004, which means that the densities on the three islands are significantly different from each other (p < 0.01). So, this test also supports the hypothesis that iguana densities differ among islands. We do not yet know which islands are different from which other ones. Unlike an ANOVA, a Kruskal-Wallis test does not have an easy way to do post-hoc analyses. So, if you have a significant effect for the overall Kruskal-Wallis, you can follow that up with a series of two-group comparisons using Mann-Whitney U tests. In this case, we would follow up the Kruskal-Wallis with three Mann-Whitney U tests: Island A vs. Island B, Island B vs. Island C, and Island C vs. Island A. WHAT TO REPORT: Following a statement that describes that general patterns in the data, you should parenthetically report the chi-square value, df, and p. For example: Iguana density varied significantly across the three islands ( 2 =11.3, df=2, p=0.004). : " " $ " (, % + + +! 8!3! 8! In many studies, researchers are interested in examining the effect of >1 independent variable (i.e., factors ) on a given dependent variable. For example, say you want to know whether the bill size of finches is different between males and females of two different species. In this example, you / / 17 have two factors (Species and Sex) and both are categorical. They can be examined simultaneously in a Two-way ANOVA, a parametric statistical test. The two-way ANOVA will also tell you whether the two factors have joint effects on the dependent variable (bill size), or whether they act independently of each other (i.e., does bill size depend on sex in one species but not in the other species?). What if we wanted to know, for a single species, how sex and body size affect bill size? We still have two factors, but now one of the factors is categorical (Sex) and one is continuous (Body Size). In this case, we need to use an ANCOVA an analysis of covariance. Both tests require that the data are normally distributed and all of the groups have homogeneous variances. So you need to check these assumptions first. If you want to compare means from two (or more) grouping variables simultaneously, as ANOVA and ANCOVA do, there is no satisfactory non-parametric alternative. So you may need to transform your data. +! 8! Enter the data as shown to the right: The two factors (Species and Sex) are put in two separate columns. The dependent variable (Bill length) is entered in a third column. Before you run a two-way ANOVA, you might want to first run a t-test on bill size just between species, then a t-test on bill size just between sexes. Note the results. Do you think these results accurately represent the data? This exercise will show you how useful a two-way ANOVA can be in telling you more about the patterns in your data. Now run a two-way ANOVA on the same data. The procedure is much the same as for a One-way ANOVA with one added step to include the second variable to the analysis. Go to Analyze General Linear Model Univariate. A dialog box appears as below. Your dependent variable goes in the Dependent Variable box. Your explanatory variables are Fixed Factors Now click Options. A new window will appear. Click on the check boxes for Descriptive 0 0 18 Statistics and Homogeneity tests, then click Continue. Click OK. The output will consist of three tables which show descriptive statistics, the results of the Levene s test and the results of the 2-way ANOVA. From the descriptive statistics, it appears that the means may be different between the sexes and also different between species. Dependent Variable: Bill size Sex Female Male Total Species Species A Species B Total Species A Species B Total Species A Species B Total Descriptive Statistics Mean Std. Deviation N From this second table, you know that your data meet the assumption of homogeneity of variance. So, you are all clear to interpret the results of your 2-way ANOVA. Levene's Test of Equality of Error Variances a Dependent Variable: Bill size F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept+Sex+Species+Sex * Species 2 2 19 The ANOVA table shows the statistical significance of the differences among the means for each of the independent variables (i.e., factors or main effects. Here, they are Sex and Species) and the interaction between the two factors (i.e., Sex * Species). Let s walk through how to interpret this information Dependent Variable: Bill size Source Corrected Model Intercept Sex Species Sex * Species Error Total Corrected Total Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a a. R Squared =.870 (Adjusted R Squared =.845) Always look at the interaction term FIRST. The p-value of the interaction term tells you the probability that the two factors act independently of each other and that different combinations of the variables have different effects. In this bill-size example, the interaction term shows a significant sex*species interaction (p < 0.001). This means that the effect of sex on bill size differs between the two species. Simply looking at sex or species on their own won t tell you anything. To get a better idea of what the interaction term means, make a Bar Chart with error bars. See the graphing section of the manual for instructions on how to do this. If you look at the data, the interaction should become apparent. In Species A, bills are larger in males than in females, but in Species B, bills are larger in females than in males. So simply looking at sex doesn t tell us anything (as you saw when you did the t-test) and neither sex has a consistently larger bill when considered across both species. The main effects terms in a 2-way ANOVA basically ignore the interaction term and give similar results to the t-tests you may have performed earlier. So, the p-value associated with each independent variable (i.e., factor or main effect) tells you the probability that the means of the different groups of that variable are the same. So, if p < 0.05, the groups of that variable are significantly different from each other. In this case, it tests whether males and females are different from each other disregarding the fact that we have males and females from two different species in our data set. And it tests whether the two species are different from each other disregarding the fact that we have males and females from each species in our data set. 4 4 20 The two-way ANOVA found that species was significant if you ignore the interaction. This suggests that species A has larger bills overall, mainly because of the large size of the males of Species A, but does not always have larger bills because bill size also depends gender. WHAT TO REPORT: If there is a significant interaction term, the significance of the main effects cannot be fully accepted because of differences in the trends among different combinations of the variables. Thus, you only need to tell your reader about the interaction term of the ANOVA table. Describe the pattern and parenthetically report the appropriate F-value, df, and p). For example: The way that sex affected bill size was different for the two different species (F=95.6, df=1,16, p<0.001). (Often, a result like this would be followed up with two separate t-tests.) If the interaction term is not significant, then the statistical results for the main effects can be fully recognized. In this case, you need to tell your reader about the interaction term and about each main effect term of the ANOVA table. Following a statement that describes the general patterns for each of these terms, you should parenthetically report the appropriate F-value, df, and p. For example: Growth rates of the both invasive and native grass species were significantly higher at low population densities than at high population densities (F=107.1, df=1,36, p<0.001). However, the invasive grass grew significantly faster than the native grass at both populations densities (F=89.7, df=1,36, p<0.001). There is no interaction between grass species and population densities on growth rate (F=1.2, df=1,36, p>0.20).! 8! Remember, ANCOVA is used when you have 2 or more independent variables that are a mixture of categorical and continuous variables. Our example here is a study investigating the effect of gender (categorical) and body size (continuous) on bill size in a species of bird. Your data must be normally distributed and have homogeneous variances to use this parametric statistical test. Enter the data as shown to the right: The two factors (Species and Body Size) are put in two separate columns. The dependent variable (Bill size) is entered in a third column. To run the ANCOVA: Go to Analyze General Linear Model Univariate as you did for the two-way ANOVA. Put your dependent variable in the Dependent Variable box. Put your categorical explanatory variable in the Fixed Factor(s) box. Put your continuous explanatory variable in the Covariate(s) box. Click on Options. A new window will appear. Click on the check boxes for Descriptive Statistics and Homogeneity tests, then click Continue. Click on Model. A new window will appear. At the top middle of the pop-up window, specify the model as Custom instead of Full factorial. Highlight one of the factors shown on the left side of the pop-up window 5 21 (under Factors & Covariates ) and click the arrow button. That variable should now show up on the right side (under Model ). Do the same with the second factor. Now, highlight the two factors on the right simultaneously and click the arrow, making sure the option is set to interaction. In the end, your Model pop-up window should look something like the image below: Click Continue and then click OK. The output will consist of four tables which show the categorical ( between-subjects ) variable groupings, some descriptive statistics, the results of the Levene s test and the results of the ANCOVA. From the first and second table, it appears that males and females have similarly sized bills. Between-Subjects Factors sex Value Label N 1.00 male female 8 Descriptive Statistics Dependent Variable: bill_size sex Mean Std. Deviation N male female Total From the third table, you know that the data meet the assumption of homogeneity of variance. So, you are clear to interpret the results of the ANCOVA (assuming your data are normal ). Levene's Test of Equality of Error Variances(a) Dependent Variable: bill_size F df1 df2 Sig Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a Design: Intercept+sex+body_size+sex * body_size The ANCOVA results are shown in an ANOVA table which is interpreted similar to the table from the two-way ANOVA. You can see the statistical results regarding the two independent 22 variables (factors) and the interaction between the two factors (i.e., Sex * Body_size) are shown on three separate rows of the table below. Dependent Variable: bill_size Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Corrected Model (a) Intercept sex body_size sex * body_size Error Total Corrected Total a R Squared =.862 (Adjusted R Squared =.827) As with the 2-way ANOVA, you must interpret the interaction term FIRST. In this example, the interaction term shows up on the ANOVA table as a row labeled sex*body_size and it tells you whether or not the way that body size affects bill size is the same for males as it is for females. The null hypothesis is that body size does affect bill size the same for each of the two sexes. In other words, the null hypothesis is that the two factors (body size and sex) do not interact in the way they affect bill size. Here, you can see that the interaction term is not significant (p=0.649). Therefore, you can go on to interpret the two factors independently. You can see that there is no effect of Sex on bill size (p=0.525). And, you can see that there is an effect of Body Size on bill size (p<0.001). Let s see how this looks graphically. Make a scatterplot with the dependent variable (Bill Size) on the y-axis and the continuous independent variable (Body Size) on the x-axis. To make the Male and Female data show up as different shaped symbols on your graph, move the categorical independent variable (Sex) into the box labeled Style as shown below: sex male female bill_size body_size 23 From the figure you can see 1) that the way that body size affects bill size is the same for males as it is for females (i.e., there is no interaction between the two factors), that males and females do not differ in their mean bill size (there is clear overlap in the distributions of male and female bill sizes), and 3) that body size and bill size are related to each other (as body size increase, bill size also increases). WHAT TO REPORT: If there is a significant interaction term, the significance of the main effects cannot be fully accepted because of differences in the trends among different combinations of the variables. Thus, you only need to tell your reader about the interaction term from the ANOVA table. Describe the pattern and parenthetically report the appropriate F-value, df, and p). For example: The way that prey size affected energy intake rate was different for large and small fish (F=95.6, df=1,16, p<0.001). (Typically, a result like this would be followed up with two separate regressions (see pg. 27 below) one for large fish and one for small fish.) If the interaction term is not significant, then the statistical results for the main effects can be fully recognized. In this case, you need to tell your reader about the interaction term and about each main effect term of the ANOVA table. Following a statement that describes the general patterns for each of these terms, you should parenthetically report the appropriate F-value, df, and p. For example: Males and females have similar mean bill sizes (F=0.4, df=1,12, p>050), and for both sexes, bill size increases as body size increases (F=68.3, df=1,12, p<0.001). There is no interaction between gender and body size on bill size (F=0.2, df=1,12, p>0.60). $ ( % $ ) ( - )! ; " : This test allows you to compare observed to expected values within a single group of test subjects. For example: Are guppies more likely to be found in predator or non-predator areas? You are interested in whether predators influence guppy behavior. So you put guppies in a tank that is divided into a predator-free refuge and an area with predators. The guppies can move between the two sides, but the predators can not. You count how many guppies were in the predator area and in the refuge after 5 minutes. Here are your data: number of guppies in predator area in refuge 4 16 Your null hypothesis for this test is that guppies are evenly distributed between the 2 areas. To perform the Chi-Square Goodness of fit test: & 24 Open a new data file in SPSS In Variable View, name the first variable Location. In the Measure column, choose Ordinal. Assign 2 values: one for Predator Area and one for Refuge. Then create a second variable called Guppies. In the Measure column, choose Scale. In Data View, enter the observed number of guppies in the 2 areas. Go to Data Weight Cases. In the window that pops up, click on Weight Cases by and select Guppies. Hit OK. Go to Analyze Nonparametric Tests Chi-square. Your test variable is Location. Under Expected Values click on Values. Enter the expected value for the refuge area first, hit add then enter the expected value for the predator area and hit add. Hit OK. In the Location Table, check the values to make sure the test did what you thought it was going to do. Are the observed and expected numbers for the 2 categories correct? Your Chi-Square value, df, and p-value are displayed in the Test Statistics Table. NOTE: Once you are done with this analysis, you will likely want to stop weighting cases. Go to Data Weight Cases and select Do not weight cases. WHAT TO REPORT: You want to report the 2 value, df, and p, parenthetically, following a statement that describes the patterns in the data. - )! ; <" $ " If you have 2 different test subject groups, you can compare their responses to the independent variable. For example, you could ask the question: Do female guppies have the same response to predators as male guppies? The chi-square test of independence allows you to determine whether the response of your 2 groups (in this case, female & male guppies) is the same or is different. You are interested in whether male and female guppies have different responses to predators. So you test 10 male and 10 female guppies in tanks that are divided into a predator-free refuge and an area with predators. Guppies can move between the areas predators can not. You count how many guppies were in the predator area and in the refuge after 5 minutes. Here are the data: number of guppies in predator area in refuge male guppies 1 9 female guppies 3 7 Your null hypothesis is that guppy gender does not affect response to predators or in other words, that there will be no difference in the response of male and female guppies to predators. Or in other words you predict that the effect of predators will not depend on guppy gender. To perform the test in SPSS: In Variable View, set up two variables: Gender and Location. Both are categorical, so they must be Nominal, and you need to set up Values. ' 25 Enter your data in 2 columns. Each row is a single fish. Go to Analyze Descriptive Statistics Crosstabs. In the pop-up window, move one of your variables into the Rows window and the other one into the Column window. Click on the Statistics button on the bottom of the Crosstabs window, then click Chi-square in the new pop-up window. Click Continue, then Okay. Your output should look like this: Case Processing Summary Cases Valid Missing Total N Percent N Percent N Percent Gender * Location % 0.0% % Gender * Location Crosstabulation Location predators refuge Total Gender male female Total Chi-Square Tests Value df Asymp. Sig. (2- sided) Exact Sig. (2- sided) Exact Sig. (1- sided) Pearson Chi-Square 1.250(b) Continuity Correction(a) Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases 20 a Computed only for a 2x2 table b 2 cells (50.0%) have expected count less than 5. The minimum expected count is How to interpret your output: Ignore the 1st table. The second table (Gender*Location Crosstabulation) has your observed values for each category. You should check this table to make sure your data were entered correctly. In this example, the table correctly reflects that there were 10 of each type of fish, and that 1 male and 3 females were in the predator side of their respective tanks. In the 3rd table, look at the Pearson Chi-Square line. Your Chi-square value is 2 = Your p-value is p = This suggests that the response to predators was not different between male and female guppies. WHAT TO REPORT: You want to report the 2 value, df, and p, parenthetically, following a statement that describes the patterns in the data. For example: Male and female guppies did not differ in their response to predators (chi-square test of independence, 2 =1.25, df=1, p>0.20)..

This course is a dual credit course where students earn both high school and college credit. Concurrent Enrollment Course Outline High School Name: Newfield Senior High School, Newfield, NY Date Proposal Submitted/Prepared: Wednesday, June 13, 2012 Revised: Saturday, August 6, 2016 Instructor: Kristopher Williamson TC3 Course #: Math 135 TC3 Course Title: Pre-Calculus Credit Hours: 3 hours Student Audience – Grade Level(s): 11th and 12th Grade students Semester(s) Offered: Fall 2016 Semester Instructor e-mail and/or phone #: Mr. Williamson (607) 564-9955 x3210 (school) Course Description: Math 135 is a college level course which students learn key concepts to prepare them for Calculus 1. Topics that are covered include Introductions to Functions (definition, domain, range, graphs), Linear Functions, Quadratic Functions, Polynomial Functions, Rational Functions, Exponential Functions, Logarithmic Functions, Absolute Value Functions, Trigonometric Functions, Conic Sections, an Introduction to Calculus using Polynomial Functions (informal limits, definition of derivative, applying the derivative to graph functions, approximating the area under a curve, exact area using summation formula, integration and derivative shortcut methods), and additional topics, if time permits (polar coordinates, vectors). Course Prerequisites: High School Algebra 2/Trigonometry Minimal Basic Skills Needed to Complete Course Successfully: Students must know how to factor polynomials and know the basics of functions. They also need to have strong organizational skills. Course Objectives: Learn mathematics topics at the college level that will prepare students to take Calculus 1. Develop an organized approach to problem solving. Required Texts and Materials/Optional Materials as Appropriate: Precalculus Functions and graphs: A Graphing Approach. By Larson, Hostetler, and Edwards Class Modalities/Alternative Learning Strategies: Lecture, discussions, individual practice, use of computer technology. Required Readings, Presentations, Written Assignments, etc.: Homework, quizzes, tests. Some selected readings from textbook. Course Content Presented in Units or Segments: Class meets 5 days a week for 43 minute periods. See the Course Outline for a list of topics in the order that they are planned to be taught. Evaluation/Grading System: Homework assignments will not be collected or graded; however, students are expected to complete their homework assignments and seek assistance where necessary. Grades will consist only of quizzes and tests, which will be averaged within each of the three marking periods. The final exam will count as a fourth grade. The overall course grade will be calculated by averaging the three marking periods with the final exam. Final grades will be issued as numbers for high school credit and as letters for college credit. Statement of Academic Integrity: In addition to TC3's Statement of Academic Integrity below, late work will only be accepted on occasion and by specific deadlines. If a student misses a quiz or test, they are responsible to take the quiz or test within 2 days of their return to school; afterward the student will earn a zero for their score. Tompkins Cortland Community College's Statement of Academic Integrity Every student at Tompkins Cortland Community College is expected to act in an academically honest fashion in all aspects of his or her academic work: in writing papers and reports, in taking examinations, in performing laboratory experiments and reporting the results, in clinical and cooperative learning experiences, and in attending to paperwork such as registration forms. Any written work submitted by a student must be his or her own. If the student uses the words or ideas of someone else, he or she must cite the source by such means as a footnote. Our guiding principle is that any honest evaluation of a student's performance must be based on that student's work. Any action taken by a student that would result in misrepresentation of someone else's work or actions as the student's own — such as cheating on a test, submitting for credit a paper written by another person, or forging an advisor's signature — is intellectually dishonest and deserving of censure. Make-Up Policy/Late Work: Late work will only be accepted on occasion and by specific deadlines, usually within 2 days of the due date. Students are responsible to approach the teacher for any notes and work missed when absent from school. Please visit the course website. Attendance Policy: Missing multiple days of class will result in a lower average. Students who miss an excessive amount of school and who do not make the effort to seek missed work could be removed from the class. Student Responsibilities: Students are responsible for maintaining a neat and organized binder that contains all notes taken in class as well as homework assignments, quizzes, and tests. If students have difficulty with certain topics, they are responsible for seeking extra help outside of class where necessary. Students are responsible for turning in all assignments by the specified due-dates. Course Outline – Math 135: Pre-Calculus The following course outline matches closely with your textbook. It is suggested that you at least skim over the section in the textbook for the next day's lesson before you come to class. Some lessons are not in the textbook. There will be a test at the end of each chapter. Longer chapters could have a quiz half-way through the chapter. Chapter P: Prerequisites P.1 Polynomials and Special Products P.3 Fractional Expressions P.4 The Cartesian Plane Chapter 1: Introduction to Functions 1.1 Lines in the Plane and Angle Between Lines 1.2 Distance from a Point to a Line 1.4 Graphs of Functions 1.5 Shifting, Reflecting, and Stretching Graphs 1.6 Combinations of Functions 1.7 Inverse Functions Chapter 2: Solving Equations 2.1 Solving Equations Graphically 2.2 Complex Numbers 2.3 Solving Equations Algebraically (2 days) 2.4 Solving Inequalities Algebraically and Graphically Chapter 3: Polynomial Functions 3.1 Quadratic Functions 3.2 Polynomial Functions of Higher Degree 3.3 Real Zeros of Polynomial Functions 3.4 Complex Zeros and the Fundamental Theorem of Algebra Chapter 4: Rational Functions 4.1 Rational Functions and Asymptotes 4.2 Graphs of Rational Functions 4.3 Partial Fractions 4.4 Mixed Rational Functions Problems 4.5 Solving Rational Inequalities Algebraically Chapter 5: Exponential and Logarithmic Functions 5.1 Exponential and Logarithmic Functions 5.2 Properties of Logarithms 5.3 Solving Exponential and Logarithmic Equations 5.4 Applications of Exponential and Logarithmic Functions 5.5 Mixed of Exponential and Logarithmic Problems Chapter 6: Trigonometric Functions 6.1 Trigonometric Identities 6.2 Evaluating Trigonometric Expressions 6.3 Solving Trigonometric Equations 6.4 Trigonometric Formulas 6.5 Mixed Trigonometry Problems 6.6 Graphs of Sine and Cosine Functions 6.7 Graphs of Other Trigonometric Functions 6.8 More Practice Graphing Trigonometric Functions Chapter 7: Derivatives 7.1 Introduction to Limits 7.2 Definition of Derivative 7.3 More Practice with Definition of Derivative 7.4 More Limits and Review 7.5 Derivative Shortcut Method and Applications of Derivatives 7.6 Derivative Rules 7.7 More Applications of Derivatives and Review Chapter 8: Integration 8.1 Midpoint Rule and Trapezoid Rule 8.2 Summation Formula for Finding Exact Area 8.3 Mixed Area Problems 8.4 Introduction to Integration 8.5 Practice with Integration 8.6 Integration Using Partial Fractions 8.7 Mixed Integration Problems Chapter 9: Conic Sections 9.1 Circles and Ellipses Review for Final Exam

A theory consists of a formalism and an interpretation. The formalism is just a piece of mathematics, and the interpretation tells us what that formalism means. In her book Interpreting Quantum Theories, Laura Ruetsche () cashes out interpretation in terms of ‘what the world could be like […] if the theory were true’ (in the words of Bas van Fraassen’s , p. 242). In slightly more formal terms, an interpretation maps each of the theory’s observables to a physical quantity, such as position or spin, and each of the theory’s states to a possible world. In this way, an interpretation delineates a space of worlds, which are called ‘physically possible’: they are ways the world could be if the theory were true. This account of interpretation entails a distinction between initial conditions and laws of nature. In short, the laws of nature are whatever remains constant across all physically possible worlds. From the perspective of the theory in question, the laws are immutable. The initial conditions, meanwhile, are allowed to vary across the possible worlds. They concern particular matters of fact that are not already determined by the laws of nature—the particular spin or location of a particle, for example. Ruetsche dubs this view the ‘ideal of pristine interpretation’. Behind this ideal lies the thought that the laws of nature are philosophically interesting, whereas the initial conditions are only relevant to the special sciences (to ‘the astronomer, geographer, geologist, etc.’, as Ruetsche quotes Houtappel et al. (, p. 596)). However, Ruetsche’s () book contains a sustained criticism of pristinism. In particular, Ruetsche argues that pristinism fails in the context of infinite-dimensional quantum theories, such as quantum field theory (QFT) and quantum statistical mechanics (QSM). Taking the latter as an example, Ruetsche argues that the occurrence of ‘unitarily inequivalent representations’ spells trouble for pristinism. Leaving the mathematical details aside, the problem Ruetsche pinpoints is that theories such as QSM allow for more than one space of physically possible worlds and no mathematically sound way to ‘glue’ these spaces together. This means that the interpreter of such a theory has to choose a space of possible worlds on the basis of ‘geographical’ considerations, such as the particular state of a system under study, or even the particular aims and interests of the scientists in question. But this blurs the distinction between laws and initial conditions: while some physical fact may appear to be a law from the perspective of a single space of possibilities—as it is constant across this space—that same fact may vary across distinct spaces. On her alternative, the coalescence approach: […] there can be an a posteriori, even a pragmatic, dimension to content specification, and […] physical possibility is not monolithic but kaleidoscopic. Instead of one possibility space pristinely associated with a theory from the outset, many different possibility spaces, keyed to and configured by the many settings in which the theory operates, pertain to it. (Ruetsche , p. 147) In my article, I argue that Ruetsche’s criticism of the pristine ideal is not limited to infinite-dimensional quantum theories, but equally applies to classical mechanics, classical statistical mechanics, and ordinary (non-infinite) quantum mechanics. On the one hand, this means that Ruetsche is mistaken in claiming that the pristine ideal fails specifically because of the mathematical nature of infinite-dimensional quantum theories. Instead, I claim, the pristine ideal was too simplistic from the start, far removed from the reality of physical interpretation. On the other hand, my article ultimately provides further support for Ruetsche’s coalescence approach. In that sense, it is a friendly amendment to her own work. But the version of the coalescence approach that I defend is a slightly attenuated form of Ruetsche’s own. In particular, it does not pose a threat to scientific realism, pace Ruetsche’s claims to the contrary. In order to see the coalescence approach in action in a non-quantum context, consider classical particle mechanics. The space of possibilities for this theory is represented by a 6N-dimensional ‘phase space’, where N is the number of particles in the world. For each particle, this phase space specifies six observables: three positions (one for each of the three x, y, and z axes) and three velocities. For example, if the actual system of interest contains ten particles, then classical particle mechanics models this system via a sixty-dimensional phase space. However, if we consider one such phase space, it seems that the number of particles is fixed across each state. Put differently, each ‘point’ in a phase space represents the exact same number of particles. From the ideal of pristine interpretation, which identifies laws with whatever remains constant across the space of possibilities, it would follow that it is a law that the universe contains a certain number of particles. This conflicts with our intuitions that the world could have contained a bigger or smaller number of particles, without any change in the laws that govern these particles. In response, one might suggest that we ‘glue’ the phase spaces for each N together to create one massive phase space, which contains points for each state for a variable number of particles. The problem with this suggestion is that it violates a principle of parsimony: for any given system, this ‘universal’ phase space contains many observables that are simply irrelevant. For example, if we study a universe with ten particles, then it doesn’t make sense to ask what the velocity of the twelfth particle in the y-direction is. This is why physicists in practice never work with such a monstrous phase space—indeed, the very idea of such a space seems slightly horrifying. The alternative offered by the coalescence approach is to allow interpreters a degree of flexibility. Which phase space is appropriate depends on the number of degrees of freedom of the actual system under study, but the choice of one phase space does not mean that all other spaces are immediately unphysical. Instead, we can imagine the collection of all classical phase spaces as a reservoir of possibility spaces, from which physicists choose one as particularly relevant to the situation at hand. In Ruetsche’s (, p. 1340) words, ‘other […] states aren’t impossible; they’re simply possibilities more remote from the present application of the theory’. This approach has consequences for the nature of lawhood. In fact, we allow laws of varying strength. Whatever is true across a single phase space most relevant to the application at hand is law-like in a narrower sense, whereas whatever is true across all phase spaces of possible interest is a law in a much stronger sense. Which notion of lawhood to use depends on the circumstances. For example, if we are interested in conservation laws (such as the conservation of mass or of the number of particles), it is not so crazy to consider the number of particles in the universe as a fixed law. But if we want to know whether space is an independent substance—and hence whether empty space could exist—it is appropriate to consider phase spaces of varying N. The strength of the coalescence approach is that it allows us the flexibility to use the same theory for both circumstances. Finally, let’s return to the issue of scientific realism, briefly mentioned above. Ruetsche argues that the coalescence approach stymies the no miracles argument (NMA) for realism. The NMA states that a theory’s empirical virtues warrant our belief in its (approximate) truth. Ruetsche’s claim is that the NMA requires that all of a theory’s virtues accrue to a single interpretation, whereas on the coalescence approach different interpretations may display different virtues. I respond that this depends on how one construes the coalescence approach. I distinguish between a ‘modest’ and a ‘radical’ version, and argue that neither my case studies nor Ruetsche’s own require any recourse to the latter. But the modest version of the coalescence approach can sustain the NMA, as follows: instead of thinking of different choices of phase space as different interpretations of the theory, we think of these as different applications of the same theory. Put differently, an interpretation consists not of a single space of possibilities, but of a whole array of spaces, indexed to particular circumstances. Ruetsche uses the metaphor of a Swiss army knife, which seems to capture the spirit of the modest coalescence approach. The idea is that each blade of the knife represents a particular phase space, and which blade is used depends on the circumstances. But the Swiss knife as a whole corresponds to the theory, so it is not the case that each application of the knife requires a different interpretation. Rather, the multiple applications are already there, within the theory. The aim of the (modest) coalescence approach is to allow scientists to realize their theory’s full potential.

Torque Versus Horsepower - More Than You Really Wanted to Know by Dan Jones Every so often, in the car magazines, you see a question to the technical editor that reads something like "Should I build my engine for torque or horsepower?" While the tech editors often respond with sound advice, they rarely (never?) take the time to define their terms. This only serves to perpetuate the torque versus horsepower myth. Torque is no more a low rpm phenomenon than horsepower is a high rpm phenomenon. Both concepts apply over the entire rpm range, as any decent dyno sheet will show. As a general service to the list, I have taken it upon myself to explode this myth once and for all. To begin, we'll need several boring, but essential, definitions. Work is a measurement that describes the effect of a force applied on an object over some distance. If an object is moved one foot by applying a force of one pound, one foot-pound of work has been performed. Torque is force applied over a distance (the moment-arm) so as to produce a rotary motion. A one pound force on a one foot moment-arm produces one foot-pound of torque. Note that dimensionally (ft-lbs), work and torque are equivalent. Power measures the rate at which work is performed. Moving a one pound object over a one foot distance in one second requires one foot-pound per second of power. One horsepower is arbitrarily defined as 550 foot-pounds per second, nominally the power output of one horse (e.g. Mr. Ed). Since, for an engine, horsepower is the rate of producing torque, we can convert between these two quantities given the engine rate (RPM): HP = (TQ*2.0*PI*RPM)/33000.0 TQ = (33000.0*HP)/(2.0*PI*RPM) where: TQ = torque in ft-lbs HP = power in horsepower RPM = engine speed in revolutions per minute PI = the mathematical constant PI (approximately 3.141592654) Note: 33000 = conversion factor (550 ft-lbs/sec * 60 sec/min) In general, the torque and power peaks do not occur simultaneously (i.e. they occur at different rpm's). To answer the question "Is it horsepower or torque that accelerates an automobile?", we need to review some basic physics, specifically Newton's laws of motion. Newton's Second Law of Motion states that the sum of the external forces acting on a body is equal to the rate of change of momentum of the body. This can be written in equation form as: F = d/dt(M*V) where: F = sum of all the external forces acting on a body M = the mass of the body V = the velocity of the body d/dt = time derivative For a constant mass system, this reduces to the more familiar equation: F = M*A where: F = sum of all the external forces acting on a body M = the mass of the body A = the resultant acceleration of the body due to the sum of the forces A simple rearrangement yields: A = F/M For an accelerating automobile, the acceleration is equal to the sum of the external forces, divided by the mass of the car. The external forces include the motive force applied by the tires against the ground (via Newton's Third Law of Motion: For every action there is an equal and opposite re-action) and the resistive forces of tire friction (rolling resistance) and air drag (skin friction and form drag). One interesting fact to observe from this equation is that a vehicle will continue to accelerate until the sum of the motive and resistive forces are zero, so the weight of a vehicle has no bearing whatsoever on its top speed. Weight is only a factor in how quickly a vehicle will accelerate to its top speed. In our case, an automobile engine provides the necessary motive force for acceleration in the form of rotary torque at the crankshaft. Given the transmission and final drive ratios, the flywheel torque can be translated to the axles. Note that not all of the engine torque gets transmitted to the rear axles. Along the way, some of it gets absorbed (and converted to heat) by friction, so we need a value for the frictional losses: ATQ = FWTQ * CEFFGR * TRGR * FDGR - DLOSS where: ATQ = axle torque FWTQ = flywheel (or flexplate) torque CEFFGR = torque converter effective torque multiplication (=1 for manual) TRGR = transmission gear ratio (e.g. 3 for a 3:1 ratio) FDGR = final drive gear ratio DLOSS = drivetrain torque losses (due to friction in transmission, rear end, wheel bearings, torque converter slippage, etc.) During our previous aerodynamics discussion, one of the list members mentioned that aerodynamic drag is the reason cars accelerate slower as speed increases, implying that, in a vacuum, a car would continue to rapidly accelerate. This is only true for vehicles like rockets. Unlike rockets, cars have finite rpm limits and rely upon gearing to provide torque multiplication so gearing plays a major role. In first gear, TRGR may have a value of 3.35 but in top gear it may be only 0.70. By the above formula, we can see this has a big effect on the axle torque generated. So, even in a vacuum, a car will accelerate slower as speed increases, because you would lose torque multiplication as you went up through the gears. The rotary axle torque is converted to a linear motive force by the tires: LTF = ATQ / TRADIUS where: TRADIUS = tire radius (ft) ATQ = axle torque (ft-lbs) LTF = linear tire force (lbs) What this all boils down to is, as far as maximum automobile acceleration is concerned, all that really matters is the maximum torque imparted to the ground by the tires (assuming adequate traction). At first glance it might seem that, given two engines of different torque output, the engine that produces the greater torque will be the engine that provides the greatest acceleration. This is incorrect and it's also where horsepower figures into the discussion. Earlier, I noted that the torque and horsepower peaks of an engine do not necessarily occur simultaneously. Considering only the torque peak neglects the potential torque multiplication offered by the transmission, final drive ratio, and tire diameter. It's the torque applied by the tires to the ground that actually accelerates a car, not the torque generated by the engine. Horsepower, being the rate at which torque is produced, is an indicator of how much *potential* torque multiplication is available. In other words, horsepower describes how much engine rpm can be traded for tire torque. The word "potential" is important here. If a car is not geared properly, it will be unable to take full advantage of the engine's horsepower. Ideally, a continuously variable transmission which holds rpm at an engine's horsepower peak, would yield the best possible acceleration. Unfortunately, most cars are forced to live with finitely spaced fixed gearing. Even assuming fixed transmission ratios, most cars are not equipped with optimal final drive gearing, because things like durability, noise, and fuel consumption take precedence to absolute acceleration. This explains why large displacement, high torque, low horsepower, engines are better suited to towing heavy loads than smaller displacement engines. These engines produce large amounts of torque at low rpm and so can pull a load at a nice, relaxed, low rpm. A 300 hp, 300 ft-lb, 302 cubic inch engine can out-pull a 220 hp, 375 ft-lb, 460 cubic engine, but only if it is geared accordingly. Even if it was, you'd have to tow with the engine spinning at high rpm to realize the potential (tire) torque. As far as the original question ("Should I build my engine for torque or horsepower?") goes, it should be rephrased to something like "What rpm range and gear ratio should I build my car to?". Pick an rpm range that is consistent with your goals and match your components to this rpm range. So far I've only mentioned peak values which will provide peak instantaneous acceleration. Generally, we are concerned about the average acceleration over some distance. In a drag or road race, the average acceleration between shifts is most important. This is why gear spacing is important. A peaky engine (i.e. one that makes its best power over a narrow rpm) needs to be matched with a gearbox with narrowly spaced ratios to produce its best acceleration. Some Formula 1 cars (approximately 800 hp from 3 liters, normally aspirated, 17,000+ rpm) use seven speed gearboxes. Knowing the basic physics outlined above (and realizing that acceleration can be integrated over time to yield velocity, which can then be integrated to yield position), it would be relatively easy to write a simulation program which would output time, speed, and acceleration over a given distance. The inputs required would include a curve of engine torque (or horsepower) versus rpm, vehicle weight, transmission gear ratios, final drive ratio, tire diameter and estimates of rolling resistance and aerodynamic drag. The last two inputs could be estimated from coast down measurements or taken from published tests. Optimization loops could be added to minimize elapsed time, providing optimal shift points, final drive ratio, and/or gear spacing. Optimal gearing for top speed could be determined. Appropriate delays for shifts and loss of traction could be added. Parametrics of the effects of changes in power, drag, weight, gearing ratios, tire diameter, etc. could be calculated. If you wanted to get fancy, you could take into account the effects of the rotating and reciprocating inertia (pistons, flywheels, driveshafts, tires, etc.). Relativistic effects (mass and length variation as you approach the speed of light) would be easy to account for, as well, though I don't drive quite that fast. Later, Dan Jones >Please put this in perspective for me, using this example: > >Two almost identical Ford pickups: > >1. 300ci six, five spd man---145 hp@3400rpm----265ft-lbs torque @2000 rpm >2. 302ciV8, five spd man----205 hp@4000rpm----275ft-lbs torque @3000 rpm > >Conditions: Both weigh 3500#, both have 3.55 gears, both are pulling a 5000# >boat/trailer. Both are going to the lake north of town via FWY. There is a >very steep grade on the way. They hit the bottom of the grade side by side >at 55mph. What will happen and why? This theoretical situation has fascinated >me, so maybe one of the experts can solutionize me forever. In short, the V8 wins because it has more horsepower to trade for rear wheel torque, using transmission gear multiplication. What really accelerates a vehicle is rear wheel torque, which is the product of engine torque and the gearing provided by the transmission, rear end, and tires. Horsepower is simply a measure of how much rear wheel torque you can potentially gain from gearing. My previous posting provides all the necessary equations to answer this question, but we need a few more inputs (tire size, transmission gear ratios, etc.) and assumptions. I'll fill in the details as we go along. To do this properly would require a torque (or horsepower) curve versus rpm, but for illustration purposes, let's just assume the torque curve of the I6 is greater than that of the V8 up to 2500 rpm, after which the V8 takes over. Using the horsepower and torque equations, we can fill in a few points. 300 I6 302 V8 RPM Tq Hp Tq Hp --- ------- -------- 4000 269 205 3400 224 145 3000 275 157 2000 265 100 where: TQ = torque in ft-lbs HP = power in horsepower RPM = engine speed in revolutions per minute Assume both trucks have 225/60/15 tires (approximately 25.6 inches in diameter) and transmission ratios of: Gear Ratio RPM @ 55 MPH ---- ----- ------------ 1st 2.95 7554 2nd 1.52 3892 3rd 1.32 3386 4th 1.00 2560 5th 0.70 1792 I determined engine rpm using: K1 = 0.03937 K2 = 12.*5280./60. PI = 3.141592654 TD = (K1*WIDTH*AR*2.+WD) TC = TD*PI TRPM = K2*MPH/TC OGR = FDGR*TRGR ERPM = OGR*TRPM where: K1 = conversion factor (millimeters to inches) K2 = conversion factor (mph to inches) WIDTH = tire width in millimeters AR = fractional tire aspect ratio (e.g. 0.6 for a 60 series tire) WD = wheel diameter in inches TC = tire circumference in inches TD = tire diameter in inches MPH = vehicle speed in mph for which engine rpm is desired TRGR = transmission gear ratio (e.g. 3 for a 3:1 ratio) FDGR = final drive gear ratio OGR = overall gear ratio (transmission gear ratio * final drive ratio) TRPM = tire RPM ERPM = engine RPM In fifth gear, both trucks are at 1792 rpm (55 mph) as they approach the hill. Running side-by-side, the drivers then floor their accelerators. Since the I6 makes greater torque below 2500 rpm, it will begin to pull ahead. The V8 driver, having read my earlier posting, drops all the way down to second gear, putting his engine near its 4000 rpm power peak. Responding, the I6 driver drops to third gear which also puts his engine near its power peak (3400 rpm). The race has begun. Since the engines are now in different gears, we must figure in the effects of the gear ratios to determine which vehicle has the greater rear wheel torque and thus the greater acceleration. We can determine axle torque from: ATQ = FWTQ * CEFFGR * TRGR * FDGR - DLOSS where: ATQ = axle torque FWTQ = flywheel (or flexplate) torque CEFFGR = torque converter effective torque multiplication (=1 for manual) TRGR = transmission gear ratio (e.g. 3 for a 3:1 ratio) FDGR = final drive gear ratio DLOSS = drivetrain torque losses (due to friction in transmission, rear end, wheel bearings, torque converter slippage, etc.) Assuming there are no friction losses, the equation reduces to: ATQ = FWTQ*TRGR*FDGR = 269*1.52*3.55 = 1452 ft-lbs for the V8 at 4000 rpm = 224*1.32*3.55 = 1050 ft-lbs for the I6 at 3400 rpm Since the V8 now makes considerably more rear axle torque, it will easily pull away from the I6. Falling behind, the I6 driver might shift down a gear to take advantage of second gear's greater torque multiplication. He will still lose the contest because his I6 engine, now operating at close to 4000 rpm, is making less torque than the V8. If he shifts up to a gear that places his engine at its maximum torque output, he will lose the torque multiplication of the lower gear ratio and fall even farther behind. Note that I picked the gear ratios so both engines can operate near their respective horsepower peaks at 55 mph by shifting to a lower gear (second gear for the V8 and third gear for the I6). This was necessary to make the contest equal. I could have manipulated the gear ratios to favor one engine or the other, but that would not have been a fair comparison. In any case where both engines are optimally geared, the V8 will win because it simply has more horsepower to trade for rear wheel torque. Q.E.D. Dan Jones P.S. Since we know the weights and the tire diameter, we can convert this rotary torque to a linear tire force and, given the angle of the hill, compute the linear accelerations of the two trucks using F=MA. This computation is left as an exercise for the reader.

- January 23, 2000 at 12:00 am #2782 How do black Christians feel about the fact that their ancestors in Africa were predominantly followers of Islam? User Detail :Name : Greg P., Gender : M, Sexual Orientation : Straight, Race : White/Caucasian, Religion : Catholic, Age : 22, City : Los Angeles, State : CA Country : United States, Occupation : Student, Education level : 2 Years of College, Social class : Middle class, January 28, 2000 at 12:00 am #27034 It’s strange that so many black Americans embrace Islam as a way of reclaiming their “African” roots, because Islam is not an African religion at all. Islam is an Asian religion, just as Christianity is. Moreover, virtually none of the black slaves brought to America were Moslims. The great majority practiced traditional African pagan/ spirit religions. When you look at Voodoo in Haiti or Santeria in Cuba, you see how many African slaves learned to mix Christianity with their traditional paganism. Even today, Islam is not the dominant religion in sub-Saharan Africa. Nigeria is one of the few black African countries in which Islam is the dominant religion. Interestingly, Catholicism is growing rapidly in Africa. It’s ironic that so many black Christians in America are turning to Islam, at the same time that millions of black Africans are embracing Christianity! My feeling is, study Islam and Christianity, then decide whether you believe the basic teachings of either religion. Embrace a religion because you believe in it, not because you think (wrongly) that it was the religion of your ancestors. User Detail :Name : Astorian, Gender : M, Sexual Orientation : Straight, Race : White/Caucasian, Age : 38, City : Austin, State : TX Country : United States, Education level : 4 Years of College, January 28, 2000 at 12:00 am #27375 Actually, all three of the Judeo-Christian religions were born in Africa and the original followers of each were African. The original Jews, the original Christians (even Jesus), as well as the original Muslims are all of the same continent. This is something that all three of those religions seem to suppress rather than embrace. When you look at them objectively, all three religions have a lot in common, especially when you contrast them to Buddhism, for example. It saddens me when so many dwell on differences rather than finding common ground. User Detail :Name : Laura-H, Gender : F, Sexual Orientation : Straight, Race : White/Caucasian, Religion : Christian, Age : 25, City : Schaumburg, State : IL Country : United States, Education level : Over 4 Years of College, Social class : Middle class, January 28, 2000 at 12:00 am #40244 I’m pretty sure that no where near the majority of black Africans were Muslim. The vast majority were Animist – a polytheistic system in which living (and some non-living) things are endowed with sacredness, deserving of worship, responsible for good and evil occurrences, etc. I am also pretty sure that Muslim black Africans were bestowed that religion, by inheritance or force by Arab slave traders. (See Henry Louis Gates’ PBS program on Africa for more.) This same method is how American blacks became predominately Christian. User Detail :Name : David25891, City : Parsippany, State : NJ Country : United States, January 28, 2000 at 12:00 am #45166 How do white Christians feel about their ancestors from Europe being barbarians and other brutish types? That was then, this is now. We all strive to thrive. User Detail :Name : Jean, Gender : F, Sexual Orientation : Straight, Race : White/Caucasian, Religion : Lutheran, Age : 45, City : Milwaukee, State : WI Country : United States, Occupation : Managerial/technical, Education level : Over 4 Years of College, Social class : Middle class, April 23, 2000 at 12:00 am #40799 Islam didn’t originate in Africa, but rather in the Middle East. It is worth noting, however, that huge chunks of African culture became part and parcel Islamic in ancient times. The Hausa, Fulani, and parts of the Yoruba are responsible for large parts of Western Africa being Muslim, while the Swahili extended the religion and associated culture as far south as Mozambique, transforming the whole East Coast. It is true thatthe interiors of all countries south of and including Congo are mainly animist, but it’s also worth noting that most Afro-American slaves were captured in Muslim Western Africa. It is also worth noting that Sub-Sahara and Southeast Asia are famous for having converted to Islam through trade and not through invasion or slavery, as many people in America believe. Middle-Eastern people never colonized the Sub-Sahara except for the English-Egyptian Sudan colonies. User Detail :Name : Karim, Gender : M, Age : 23, City : Los Angeles, State : CA Country : United States, Social class : Middle class, January 5, 2003 at 12:00 am #31262 That is a false hood , first of all the christian religon was in existence first ,mohammed wrote the quoran , after christ, that said , judaism, and christianity came first, and they were the foremost religons of people in north africa were they were given birth , User Detail :Name : Chuck D, Gender : M, Sexual Orientation : Straight, Race : Black/African American, Religion : Christian, Age : 32, City : Baltimore, State : MD Country : United States, Education level : 2 Years of College, - You must be logged in to reply to this topic.

- What does coupon rate mean? - Why is lower coupon rate high risk? - Is Yield to Maturity Fixed? - When a bond’s yield to maturity is less? - What is yield to worst? - Is High Yield to Maturity good? - What is the coupon effect? - Are bonds a good investment in 2020? - What is the coupon rate formula? - What is yield formula? - Why yield to maturity is important? - Is yield to call higher than yield to maturity? - Is interest rate and yield to maturity the same? - Why is the coupon rate higher than the yield? - What is the difference between yield and yield to maturity? - How YTM is calculated? - What happens when yield to maturity increases? What does coupon rate mean? A coupon rate is the yield paid by a fixed-income security; a fixed-income security’s coupon rate is simply just the annual coupon payments paid by the issuer relative to the bond’s face or par value. The coupon rate, or coupon payment, is the yield the bond paid on its issue date.. Why is lower coupon rate high risk? Bonds offering lower coupon rates generally will have higher interest rate risk than similar bonds that offer higher coupon rates. … If market interest rates rise, then the price of the bond with the 2% coupon rate will fall more than that of the bond with the 4% coupon rate. Is Yield to Maturity Fixed? The main difference between the YTM of a bond and its coupon rate is that the coupon rate is fixed whereas the YTM fluctuates over time. The coupon rate is contractually fixed, whereas the YTM changes based on the price paid for the bond as well as the interest rates available elsewhere in the marketplace. When a bond’s yield to maturity is less? Yield to maturity (YTM) = [(Face value/Present value)1/Time period]-1. If the YTM is less than the bond’s coupon rate, then the market value of the bond is greater than par value ( premium bond). If a bond’s coupon rate is less than its YTM, then the bond is selling at a discount. What is yield to worst? Yield to worst is a measure of the lowest possible yield that can be received on a bond that fully operates within the terms of its contract without defaulting. … The yield to worst metric is used to evaluate the worst-case scenario for yield at the earliest allowable retirement date. Is High Yield to Maturity good? The high-yield bond is better for the investor who is willing to accept a degree of risk in return for a higher return. The risk is that the company or government issuing the bond will default on its debts. What is the coupon effect? The coupon rate on a bond vis-a-vis prevailing market interest rates has a large impact on how bonds are priced. If a coupon is higher than the prevailing interest rate, the bond’s price rises; if the coupon is lower, the bond’s price falls. Are bonds a good investment in 2020? Many bond investments have gained a significant amount of value so far in 2020, and that’s helped those with balanced portfolios with both stocks and bonds hold up better than they would’ve otherwise. … Bonds have a reputation for safety, but they can still lose value. What is the coupon rate formula? Coupon rate is calculated by adding up the total amount of annual payments made by a bond, then dividing that by the face value (or “par value”) of the bond. … To calculate the bond coupon rate we add the total annual payments then divide that by the bond’s par value: ($50 + $50) = $100. $100 / $1,000 = 0.10. What is yield formula? Yield should not be confused with total return, which is a more comprehensive measure of return on investment. Yield is calculated as: Yield = Net Realized Return / Principal Amount. For example, the gains and return on stock investments can come in two forms. Why yield to maturity is important? The primary importance of yield to maturity is the fact that it enables investors to draw comparisons between different securities and the returns they can expect from each. It is critical for determining which securities to add to their portfolios. Is yield to call higher than yield to maturity? Key Takeaways. Yield to maturity is the total return that will be paid out from the time of a bond’s purchase to its expiration date. Yield to call is the price that will be paid if the issuer of a callable bond opts to pay it off early. Callable bonds generally offer a slightly higher yield to maturity. Is interest rate and yield to maturity the same? Interest rate is the amount of interest expressed as a percentage of a bond’s face value. Yield to maturity is the actual rate of return based on a bond’s market price if the buyer holds the bond to maturity. Why is the coupon rate higher than the yield? If the investor purchases the bond at a discount, its yield to maturity will be higher than its coupon rate. A bond purchased at a premium will have a yield to maturity that is lower than its coupon rate. YTM represents the average return of the bond over its remaining lifetime. What is the difference between yield and yield to maturity? A bond’s current yield is an investment’s annual income, including both interest payments and dividends payments, which are then divided by the current price of the security. Yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until its maturation date. How YTM is calculated? YTM = the discount rate at which all the present value of bond future cash flows equals its current price. … However, one can easily calculate YTM by knowing the relationship between bond price and its yield. When the bond is priced at par, the coupon rate is equal to the bond’s interest rate. What happens when yield to maturity increases? Without calculations: When the YTM increases, the price of the bond decreases. Without calculations: When the YTM decreases, the price of the bond increases. (Note that you don’t need calculations for this price, because the YTM is equal to the coupon rate). to a change in the interest rate (YTM).

application newtons law of inertia Best Results From Wikipedia Yahoo Answers Youtube Newton's laws of motion Newton's laws of motion consist of three physical law s that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces. They have been expressed in several different ways over nearly three centuries, and can From Yahoo Answers Question:I am doing a science project . Is a girl riding a surf board considered to be Newtons first law of motion ? I have a science Project and isnt a girl riding a surfboard considered to be Newtons first law? & Newtons First law, isnt it An object will stay at rest unless another object will act upon it??? Answers:Theoretically, this is a scenario where the three Laws Of Newton can be applicable. Anyway, Newton's 1st Law is also known as the Law of Inertia and fora science project, I would suggest a very simple demonstration like so. Place a block (or any object) on a smooth table. Note that this block is not moving initially. If you apply an external force on it (in other words, push or pull it), the block will obviously move consistent with the direction of the applied force. If the external force is taken off (i.e., you stop pushing or pulling), the block will simply stop. Another way the block will stop is when something stops it from moving. It could be an impediment or simply anything that will prevent the block from moving. Hope this helps. Answers:First off, every action has an equal and opposite reaction. If you look at a helicopter, the blade on top spins, and it needs the blade rotating in the back to keep the cabin from spinning like crazy. Then second, inertia, a plane takes off to a high altitude, then it only needs to produce enough thrust to overcome air resistance to maintain speed, inertia does the rest. Answers:inertia, force, reaction? Question:How do you apply Newton's 3 Laws of Motion to the medical field? Answers:First law - An object at rest will remain at rest unless acted on by an unbalanced force. An object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This law is often called "the law of inertia". Blood flow around the body will continue until a force stops it (eg tourniquet) Second law - Acceleration is produced when a force acts on a mass. The greater the mass (of the object being accelerated) the greater the amount of force needed (to accelerate the object). Ask the trolley pushers - the heavier the patient, the more force needed to push them along on the trolley. Third law - For every action there is an equal and opposite re-action. The higher the medical bill - the closer the patient's jaw is to the ground. No - update - wouldn't blood pressure cuff show that when an object pushes it gets pushed back in the opposite direction with equal force ??? (yeah - i'm starting to struggle) Inertia :Check us out at www.tutorvista.com Inertia is the resistance of any physical object to a change in its state of motion. It is represented numerically by an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to describe the motion of matter and how it is affected by applied forces. Inertia comes from the Latin word, "iners", meaning idle, or lazy. In common usage, however, people may also use the term "inertia" to refer to an object's "amount of resistance to change in velocity" (which is quantified by its mass), or sometimes to its momentum, depending on the context (eg "this object has a lot of inertia"). The term "inertia" is more properly understood as shorthand for "the principle of inertia" as described by Newton in his First Law of Motion. This law, expressed simply, says that an object that is not subject to any net external force moves at a constant velocity. In even simpler terms, inertia means that an object will always continue moving at its current speed and in its current direction until some force causes its speed or direction to change. This would include an object that is not in motion (velocity = zero), which will remain at rest until some force causes it to move. On the surface of the Earth the nature of inertia is often masked by the effects of friction, which generally tends to decrease the speed of moving objects (often even to the point of rest), and by the acceleration due to gravity. The ... Newton's Laws Of Motion (1) : The Law Of Inertia :ESA Science - Newton In Space (Part 1): Newton's First Law of Motion - The Law of Inertia Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several different ways over nearly three centuries. --- Please subscribe to Science & Reason: www.youtube.com www.youtube.com www.youtube.com --- The laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work "Philosophi Naturalis Principia Mathematica", first published on July 5, 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. --- Newton's First Law of Motion An object in motion will stay in motion, unless an outside force acts upon it. There exists a set of inertial reference frames relative to which all particles with no net force acting on them will move without change in their velocity. Newton's first law is often referred to as the law of inertia. en.wikipedia.org .

All Bank Transfer Codes In Nigeria – USSD Code/ PIN For Mobile Banking Looking for all bank transfer codes in Nigeria? This page contains all the banks in Nigeria transfer code from A-Z – USSD code/ PIN for mobile banking. Irrespective of the bank you opened account with? Just relax, this article got you covered. We have provided you guys with all Nigeria bank transfer codes, you can as well call it ussd code or pin for mobile banking. No more stressing yourself going to the bank to make transactions anymore, or fix certain issues. Just as we stated earlier, this article covers all bank transfer codes in Nigeria. Be it firstbank, Ecobank, zenithbank, Access bank, GT bank, Polaris Bank, FCMB Bank, Keystone Bank & more. You will definitely see the transfer code here. Amazing right? Not only that, you can as well use the code in checking of account balance, buying of Airtime, pay bills, subcribe for DSTv or GOTv & more. Before we proceed to sharing all bank transfer codes in Nigeria, let’s quickly explain what bank transfer code is and it’s advantages. What is Bank Transfer Code? We can practically say that bank transfer code is a convenient, fast, secure, and affordable way to access your bank account 24 hours a day, 7 days a week through your mobile phone without internet data. There are USSD codes that eases the use of stressful banking in Nigeria, making it more efficient and gives the customers complete access to their accounts. It is where customers of any bank can easily dial the USSD code and complete all banking commands. It sounds surprising that queuing up in the bank, trying to complete your accounts recharge has been taken away with this USSD code mobile money transfer service. It makes everything easy, and sensible. Advantages of Bank transfer code - Top up your mobile airtime - Top-up other mobile phones - Checking bank balance - Money transfers - Bill payments - Reset Pin - Buy data Understand that, we are talking about the same topic here, be it bank transfer code, USSD banking, USSD mobile money transfer Code. They both go straight to one direction! All Bank Transfer Codes In Nigeria – USSD Code/ PIN For Mobile Banking Having that in mind, and ready to look for your bank and commence, then lets get started: |class=”column-1″>Nigerian Banks||class=”column-2″>Transfer Codes| |UBA transfer code||919#| |Wema bank bank transfer code||*945#| |FCMB transfer code||*329#| |Access bank transfer code||*901#| |Polaris (Skye) transfer code||*833#| |GTBank transfer code||*737#| |Zenith bank transfer code||*966#| |Eco bank transfer code||*326#| |First Bank transfer code||*894#| |Stanbic IBTC bank transfer code||*909#| |Diamond bank transfer code||*426#| |Fidelity bank transfer code||*770#| |Sterling bank transfer code||*822#| |Unity bank transfer code||*7799#| Stanbic IBTC USSD Bank Money Transfer Code The Stanbic IBTC Mobile Money service is *909# and it is also their transfer code, that is ideal for you if you want a basic account that is convenient, secure and affordable, and can be operated using a mobile phone. With *909# Mobile Money you can receive salary payments, buy airtime, make one-on-one money transfers, pay for goods and pay bills. To get started, dial the code and you can get started. Wema bank transfer code The official transfer code for Wema bank is *945# and you can complete other necessary commands such as: *. Buy Airtime: *945*phone Number*amount# *. Send Money: *945*beneficiaryaccountnumber*amount# *. Balance Enquiry: *945*0# *. Change PIN: *945*00# *. Get your Account Number: *945*000# *. Open Account: *945*1# *. Change account number: *945*2*oldaccountnumber*newaccountnumber# *. Account Reactivation: *945*5# *. Send Money to phone/email: *945*6*amount# UBA transfer USSD code Union bank is calling their transfer code, Magic banking, which is also the same with mobile banking. With the *919# UBA transfer code, you can easily complete all your transaction from the comforts of your home. FCMB USSD bank transfer code The official FCMB USSD bank transfer code is *329#, and it can be used to complete a lot of bank commands, all from the comforts of your home: *. Dial*329*Amount# to top-up your mobile phone. *. Dial *329*Amount*Mobile number# to top-up other mobile phones. *. Dial *329*Amount*Account number# to transfer funds. *. Dial *329*00# to check balance. *. Dial *329*0# to reset your pin. *. Dial *329*1*Mobile Number# to buy data on your phone. *. Dial *329*2*Smartcard Number# to pay for DSTv or GOTv subscription. Access Bank USSD Code Banking The official Access bank USSD banking code for transfer of funds, is *901#, and it can be used to complete various type of transaction without going to the bank: *. Airtime for self: *901*Amount# *. Airtime for others: *901*Amount*Phone Number# *. Account Opening: *901*0# *. Data Purchase: *901*8# *. Balance Enquiry: *901*5# *. Fund Transfer to Access Bank: *901*1*Amount*Account Number# *. Fund Transfer to Other Banks: *901*2*Amount*Account Number# *. Merchant Payment: *901*3*Amount*Merchant Code# *. Bill Payment: *901*3# *. OTP Generation: *901*4*1# You will need airtime on your mobile phone to complete any of the above commands. Sterling Bank transfer code For Standard Chartered customers, the transfer code is *822# and it can be used to complete other transactions, buy airtime, pay bills, and even transfer money from bank to bank. To get started, dial the code and you follow the prompt commands. *. Buy airtime self: *822*AMOUNT# *. Buy airtime for others: *822*AMOUNT*MOBILE NO# *. Sterling bank within transfer: *822*4*AMOUNT*NUBAN# *. Sterling bank to other: *822*5*AMOUNT*NUBAN#. *. Check account balance: *822*6# *. Check ccount number: *822*8# *. Open account: *822*7# *. To reset Pin: *822*9# Polaris (Skye) Transfer code The Polaris (Skye) Bank Money Transfer Code is *833#, and it is known as Smart code. To commence using the Polaris (Skye) Bank Money Transfer Code, you need to register or sign-up first by dialing the above code first: *. Open an Account (*833*1#) *. Pay Bills (*833*2#) *. Transfer Funds (*833*3#) *. Hotlist Card *. Update your BVN (*833*5#) – Coming Soon *. Check your Balance (*833*6#) *. Pay with MasterPass (*833*7#) *. Airtime Top-up *. Buy Airtime for yourself: Dial *833*AMOUNT# *. Buy Airtime for others or yourself: Dial *833*AMOUNT*PHONENUMBER# *. Transfer Funds: Dial *833*AMOUNT*ACCOUNTNUMBER# No data, no worries. Polaris bank has got your covered! GTBank Transfer Code/PIN One of the best bank in Nigeria, Guarantee trust bank also has their own USSD transfer code, it is *737#. And the bank has attached a lot of commands and special feature to it: *. Open Account: *737*0# *. Reactivate your account: *737*11# *. Fund transfer to GTbank: *737*1*Amount*NUBAN Account Number# *. Fund transfer to other bank: *737*2*Amount*NUBAN Account Number# *. Airtime top-up for self: *737*amount# *. Airtime top-up for friend: 737*Amount*Recipient’s number# *. Data top-up: *737*4# *. Check bank balance: *737*6*1# *. Check BVN number: *737*6*1# *. Inquiries: *737*6# Zenithbank Transfer Code – Eazybanking Zenith USSD code for transfer is *966# and you can easily transfer from Zenith bank to other banks in Nigeria. To complete your Zenith mobile banking, you are to dial the following code: *. Open account: *966*0# *. Check account balance: *966*00# *. Top-up your airtime: *966*Amount*Mobile Number# *. To transfer funds: Dial *966*Amount*Account Number# then follow the on-screen prompts. *. Update your BVN: *966*BVN# *. Reset PIN: *966*60# *. Deactivate mobile banking: *966*20*0# *. Pay DStv and PHCN: *966*7*Amount*Customer ID# *. Pay other Zenith Billers: *966*6*Biller code*Amount# Eco mobile banking The official Eco bank transfer code is *326#, and you can easily transfer any money from Eco bank to other bank with the above code. You can complete the following with this code: *. Make instant transfers *. Check balances *. Pay bills *. Read mini-statements *. Buy airtime Ensure that the phone number where the purchase is made from MUST be a number registered with the bank, if not so, it won’t work. First Bank transfer code Even with no data on your mobile phone, you can complete your First bank commands, by using the bank transfer code: *894#. That is the Quick Banking with FirstBank USSD Code, where you can easily transfer money, check balance, buy airtime, pay bills and lots more. *. To register: *894*0# *. Airtime recharge: *894*Amount# *. Airtime for others: *894*Amount*Number# *. To transfer: *894*Amount*Account number# *. To check airtime: *894*00# *. Mini-statement: *894*Account number# Unity bank transfer code One of the best ways to bank is via USSD code, and Unity bank has got you covered. The official Unity bank transfer code is *7799# and you can easily complete other important transaction. *. Account opening *. Airtime recharge *. Add Account *. Balance enquiry *. Bills payment *. BVN verification *. Fund Transfer *. PIN change. Diamond Bank Transfer Code With Diamond bank USSD service *426#, it allows customers of the bank to ensure financial inclusion. Thankfully, it is quick, secure and an easy way for all banked customers to perform banking transactions with convenience, regardless of segment or mobile device type. To transfer to another bank with Diamond USSD short code: – Dial *426# with your registered phone number – Enter the last 6 digits of your Debit Card number. – Enter your account number – Create a 4-digit PIN. Other service includes: *. Dial *426*Amount#: To Purchase Airtime (self) *. Dial *426*Amount*Account Number#: To Transfer Money (Within & Outside DB) *. Dial *426*00#: To Check Account Balance *. Dial *426*Amount*Phone Number#: To Purchase Airtime (For Friends and Family) *. Dial*426*0#: To Change PIN *. Dial *426*463#: To subscribe to C.R.E.A.M Fidelity USSD code for transfer Fidelity is calling it Instant Banking, and of course, it fits the name and mode of service. Fidelity transfer code is *770#, and it can be used to complete the following transaction: *. Self-recharge: Dial *770*AMOUNT# *. Recharge For A Friend Or Family: Dial *770*PHONE*AMOUNT# *. To transfer funds: Dial *770*ACCOUNT*AMOUNT# So, for those of you searching for all bank transfer codes in Nigeria – USSD code/ PIN for mobile banking. I hope we made your day today? Definitely yes, for more stuffs like this kindly join our telegram channel and don’t forget to share with friends on social media.

A 16-bit D/A interface with Sinc approximated semidigital reconstruction filter 7.5. Semidigital FIR filter design To design the time-discrete filter, the effect of the noise-shaper and the low-pass continuous time analog filter have to be considered. The noise shaper and the oversampling ratio are specified and all the requirements and conditions are known. The next step is the calculation of the coefficients. But how many coefficients are necessary? To answer this question some boundary conditions will be introduced. The area that is available limits the number of coefficients to about 100 irrespective to the implementation which is chosen. Since the coefficients are implemented by weighted currents this imposes a limitation also. The ratio between the largest and the smallest coefficient is limited by accuracy. A large number of coefficients implies big differences between coefficients. The accuracy of the smaller coefficients is impaired with consequences on the stop-band rejection. There are also a few conditions for the signal transfer function of the filter. First of all, the ripple in the audio-band has to be very small (< 0.1 dB). A small droop (0.5 dB) is allowed since the digital filter can correct for this non-ideal behavior. In the design of a discrete time filter suitable for audio signals, phase is an important parameter too. In a digital low-pass filter design, a linear phase can be obtained by a symmetric impulse response. Odd or even numbers of coefficients can be used. The main requirement is to achieve a stop-band rejection for the noise of more than 50dB.7.5.1 Calculation of coefficients In the literature, a number of standard algorithms for digital filter design are extensively discussed . The methods are based on Fourier series, the frequency sampling method, the Remez exchange method and equiripple designs. All these methods cannot be used since the design of this filter is not a standard design but the product of a time discrete filter and the transfer function of the noise shaper. Such methods generate a large number of coefficients and over-specifications. In order to take into account that the noise transfer will be influenced by the noise-shaper, the semidigital filter and the low-pass analog filter, we have developed an iterative method to design the filter based on Sinc approximation of the impulse response as shown in fig.7.8. Here, we have represented the transfer of the noise shaper NS, the transfer of the semidigital filter LPD and the analog low-pass LPA. The noise transfer is denoted NS*LPA*LPD. The simulations were performed with a routine written in MatLab. This allows to optimize the number of the coefficients and to take into account the effects of matching on the response. First, the time domain is divided into N equal steps and the symmetric coefficients are calculated by using the division of sin(x)/x. The Sinc function has been windowed with a rectangular window. The computer is used to perform this calculation by employing the Z-transform. For the noise shaper the transfer function is also calculated by using a Z-transform routine. Since the continuous time low-pass filter may not influence the characteristic at the audio-band, its cutoff frequency is set to 140 kHz. In this way it is possible to filter the spectral images at multiples of the sample frequency sufficiently. Further, these three functions are plotted on a logarithmic scale and therefore they can be easily added. The Sinc function has been truncated to the first five lobes but the -50dB requirement for the out of band noise is not met. By taking more coefficients, the stop-band rejection becomes better than -55dB, as shown in fig.7.9. Simulations have been carried out to determine which part of the Sinc function is important and how many coefficients are necessary in the optimum case. The number of lobes from which the Sinc is approximated changes the transfer characteristic of the filter. It is also important to ensure that at zero crossings of the Sinc function the approximation has also a zero. At that moment the next sample reaches its maximum value. Using more coefficients to approximate the same part of the Sinc means decreasing the time step. This is equivalent with increasing the sample frequency in the case of a digital filter. The results is a smaller pass band of the LPD filter characteristic without changing its shape. It turns out that just the main lobe of the Sinc function is the most important part to approach the desired filter characteristic. With no more than 25 coefficients this main lobe can be approximated such that the required attenuation of more than 50dB is reached. Actually there are 27 coefficients but two of them are zero. The calculated coefficients are given in Table 7.1. Table 7.1: Coefficients of the FIR filter To be noticed the small ratio between the largest and the smallest coefficient which is about 12. The approximation of the main lobe is shown in fig.7.10. The first and the last coefficient are zero. The transfer characteristics for the noise and signal are illustrated in fig.7.11. The rejection of the out of band noise of the noise characteristic (NS*LPA*LPD) is better than -53dB up to the higher end of the fundamental interval (f=fs/2). A sensitivity analysis will show that in the worst case the required -50dB is fulfilled. For the signal transfer a smooth roll off (» 0.25 dB in the audio-band) can be seen. The zoomed characteristic of the signal in the audio-band is shown in fig.7.12. The sharp digital filter will correct the droop of the characteristic along with the sin(x)/x distortion at the end of the pass-band. The gain error can be corrected by multiplying the coefficients with a constant factor. In the design of a FIR filter windowing functions are used to reduce the infinite length of the impulse response. By applying a rectangular window on the impulse response, i.e. just deleting a number of the coefficients, there will be oscillations in the frequency response due to Gibbs phenomenon. In order to reduce the oscillations different window functions can be applied. Widely used window functions for example are Bartlett, Hamming, Hanning and Kaiser (see fig.7.13). By multiplying the calculated Sinc coefficients with a window, the transfer becomes slightly better. After this operation the ratio between the smallest and the largest coefficient increases tremendously. For a digital filter this is not a problem because the coefficients are represented by a number of bits. In this application this means a large ratio between components. Moreover, due to windowing, the transfer function becomes more sensitive to rounding. That is why no windowing technique is used for the calculation of the coefficients. 7.5.3 Filter response and the coefficient quantization The coefficients of the filter are subject for mismatch, rounding and quantization to the incremental grid span of the process. This will affect the stop-band attenuation of the filter with some influence on the pass-band also. We would like to obtain specifications for the coefficients of the filter such that we get sufficient suppression of the quantized noise out of audio band without affecting the pass-band. Coefficient non-idealities generate an erroneous transfer function: The deviation of the filter transfer depends on the random coefficient errors Dak : When the random coefficient errors Dak are Gaussian distributed the deviation of the filter transfer is Rayleigh distributed with a mean value mDH and a standard deviation sDH given by: In eq.(7.14) N is the filter length and s(Dhk) represents the standard deviation of the coefficients due to process mismatch. The deviation of the filter transfer has three main causes: rounding of small coefficients, quantization of the coefficients to the finite incremental grid span and mismatch. Those effects are treated separately. 7.5.4 Rounding small coefficients For FIR filters with a lot of coefficients we have to deal with large ratios between the largest and the smallest coefficient. It is necessary to round small coefficients to fit to the smallest feature size of a transistor. Rounding of small coefficients will introduce quantization errors with consequences on stopband rejection. The response of the filter in the pass-band it is influenced only by large coefficients and the rounding procedure has no influence on the pass-band. In order to estimate the stopband rejection we have to consider the size of the minimum coefficient amin. As a rule of thumb, the maximum achievable stop-band rejection is: To have a stopband rejection of about -50dB the rounded coefficients have to be smaller than amin=0.003. In our case, the smallest coefficient is 0.0054 and rounding is not a necessity. In the design procedure we try to keep the number of the coefficients as low as possible in order to avoid big differences between the largest and the smallest coefficient.7.5.5 Matching of coefficients In contrast to a digital filter, where the only important error is caused by truncation or rounding due to the finite word length, in the time discrete filter the mismatch of the coefficients will impair the frequency characteristic. In practice the analog coefficients are realized by using current sources and their values will deviate from their nominal value. The condition for the stop band noise has to be met under mismatch conditions. Only Monte-Carlo analysis can reveal the effect of mismatches on the transfer characteristic. In fig.7.14 the realization of the coefficient ak is shown. A floating current source I0 improves the matching between the PMOS and NMOS branches. The current related to the same coefficient ak is Ik= akI0=I0 (W/L)k/(W/L)0. The mismatch of the coefficient ak is a consequence of VT0 mismatch and b mismatch. Consider a multi-parameter function f=f(x1,x2,…xN). From multi-parameter sensitivity analysis we have: Regarding the current Ik as the multi-parameter function, the mismatch of the coefficient ak is found as a function of individual mismatch terms of transistors Mo and Mk neglecting the contributions of the cascode transistors. For a single ended current mirror, the inaccuracy of the coefficient ak is found from: The lengths of the transistors Mk are taken equal and therefore we get: The maximum value of the width W0 of the transistor M0 is limited from area requirements. Consider now the current mirror with PMOS and NMOS outputs. The transistors Mkn and Mkp have the same dimensions. Denote s(Dak)p and s(Dak)n the mismatch of the PMOS and NMOS branch respectively. Hence, the mismatch of the coefficient ak in the differential approach is given by: Denote the mismatch term: Then the mismatch of the coefficient ak in the differential approach becomes: This result in conjunction with eq.(7.14) can be used to have a first estimation of the errors in terms of mDH and sDH of the transfer H. The Monte Carlo optimization procedure described later in section 7.5.7 is based on s(Dak). 7.5.6 Quantization to the incremental grid span The IC processes have a finite incremental grid span. For example in a 0.8mm CMOS process, the finite incremental grid span is in the order of 0.1m m. The dimensions of the devices (width and length) have to be quantized to the grid span. Rounding introduces a length uncertainty of (-0.05 mm, 0.05 mm) and the error can be considered uniformly distributed in this interval. Compared to the errors introduced by mismatch, quantization to grid has a negligible influence on the filter response. Again eq. (7.14) can be used to show this effect.7.5.7 Simulations Equation (7.21) shows that each coefficient has a standard deviation which depends on W and L. Generating filter characteristics with ± 3s errors for coefficients, we cover about 99.75% of the possible cases. In MatLab, there are no standard routines to perform a Monte Carlo analysis. However, it is simply to generate normal distributed random numbers with mean 0.0 and variance 1.0. Therefore it is possible to combine this random number generator with the previous derived equation, to calculate what the effects are on the filter characteristic. The random number determines also if the coefficient is rounded up or down respectively. In fig.7.15 the simulation results for the optimal widths and lengths of the transistors are shown. The inaccuracy of the noise transfer increases at the end of the fundamental interval. In this region, the effect of the noise shaper on the noise transfer is less effective and attenuation of the noise is ensured by the FIR filter. In the worst case we have -61dB rejection for the noise. The signal transfer is slightly affected by the matching properties.

2 edition of Square variables found in the catalog. |Statement||by Todd Smith.| When developing more complex models it is often desirable to report a p-value for the model as a whole as well as an R-square for the model.. p-values for models. The p-value for a model determines the significance of the model compared with a null model. For a linear model, the null model is defined as the dependent variable being equal to its mean. brings simple answers on how to separate square root, matrix and roots and other algebra subjects. Any time you need guidance on inequalities or rational exponents, is truly the right site to pay a visit to! Getting a subset of a data structure Problem. You want to do get a subset of the elements of a vector, matrix, or data frame. Solution. To get a subset based on some conditional criterion, the subset() function or indexing using square brackets can be used. In the examples here, both ways are shown. If it is a square matrix, the number of non-pivot columns is equal to the number of zero rows. However, if the matrix is non-square, you can reduce to row-echelon form and count the number of non-pivot columns. Here, the number of non-pivot columns is not equal to the number of zero rows. a qualitative variable indicates some kind of category. A commonly used qualitative variable in social science research is the dichotomous variable, which has two di erent categories. For instance, gender has two categories: male and female. Chi-square test is applicable to when we have qualitative variables classi ed into Size: KB. Introduction. The concept of instrumental variables was first derived by Philip G. Wright, possibly in co-authorship with his son Sewall Wright, in the context of simultaneous equations in his book The Tariff on Animal and Vegetable Oils. 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Embed a booking widget or button on your existing website, or add a booking button to your email so clients can easily request appointments based on your availability. Just bought this book and Schaum's Outline of Complex Variables, 2ed (Schaum's Outline Series) for an undergraduate level complex variables class. Without the Schaum's, I'd have been lost in this class/5(19). I am a big fan of the "Square in a Square" (SNS) system and own most of Ms Barrow's other titles. This book is my favorite. The book is well organized with clear color illustrations. The SNS options are well described in the text and the diagrams are easy to follow/5(30). Categorical Variables in Developmental Research provides developmental researchers with the basic tools for understanding how to utilize categorical variables in their data analysis. Covering the Square variables book of individual differences in growth rates, the measurement of stage transitions, latent class and log-linear models, chi-square, and more, the book provides a means for developmental. Chi Square. Author(s) David M. Lane. Prerequisites. Distributions, Areas of Normal Distributions, Standard Normal Distribution. Chi Square Distribution; One-Way Tables (Testing Goodness of Fit) Testing Distributions Demo; Contingency Tables; 2 x 2 Table Simulation; Statistical Literacy; Exercises; PDF (A good way to print the chapter.). For some reason they are adding [square brackets] around variables, thus: var some_variable = 'to=' + [other_variable]; Stack Overflow Products. Price to Book Ratio Definition. Price to book value is a valuation ratio that is measured by stock price / Square variables book value per share. The book value is essentially the tangible accounting value of a firm compared to the market value that is shown. Read full definition. An extension of the simple correlation is regression. In regression, one or more variables (predictors) are used to predict an outcome (criterion). One may wish to predict a college student’s GPA by using his or her high school GPA, SAT scores, and college major. Data for several hundred students would be fed into a regression statistics Author: Del Siegle. Tap Items. Search or scroll through your item list and click an existing item. You can update the item name, item image*, category, description, unit type, stock amount, or variations. Choose to Save your changes, or click Delete Item From This Location. Keep in mind: When you add, update, or delete an item image, the change will reflect in. A good example of including square of variable comes from labor economics. If you assume y as wage (or log of wage) and x as an age, then including x^2 means that you are testing the quadratic relationship between an age and wage earning. Is the square footage of a house a discrete random variable, a continuous random variable, or not a random variable. It is a discrete random variable Is the eye color of people on commercial aircraft flights a discrete random variable, a continuous random variable, or not a random variable. where age2 is the name of the new (squared) variable, and age is the original variable. Chi Square A. Chi Square Distribution B. One-Way Tables C. Contingency Tables D. Exercises Chi Square is a distribution that has proven to be particularly useful in statistics. The first section describes the basics of this distribution. The following two sections cover the most common statistical tests that make use of the Chi Square File Size: KB. contributed Completing the square of an expression with multiple variables is a technique which manipulates the expression into a perfect square plus some constant. As an example, x 2 + 2 x + y 2 − 6 y + z 2 − 8 z + 1 x^2+2x+y^y+z^2 - 8z + 1 x 2 + 2 x + y 2 − 6 y + z 2 − 8 z + 1 can be written in the complete square form as. Another tool to solve equation involving Squared Variables is the basic property of absolute value. ∣ x ∣ = a = > x = a |x|=a => x = a ∣ x ∣ = a = > x = a or x = − a x = -a x = − a. Find the value of x x x if x x 2 = 9 = > x 2 = 3 2 x^=0 => x^2 = 9 => x^2. Variance of Square of a Random Variable. Hot Network Questions Finding all elements of some order in Sn How could macOS be POSIX compliant without vi. How can you tell if a note is major or minor. A recommendation for a book on perverse sheaves Remove comma from a list Mowing the Grass. Identify the two variables that vary and decide which should be the independent variable and which should be the dependent variable. Sketch a graph that you think best represents the relationship between the two variables. The size of a persons vocabulary over his or her lifetime. The distance from the ceiling to the tip of theFile Size: KB. The agpp data frame contains three variables, an id variable that labels each participant in the data set (we’ll see why that’s useful in a moment), a response_before variable that records the person’s answer when they were asked the question the first time, and a response_after variable that shows the answer that they gave when asked the same question a second time. As usual, here’s the first 6 entries. I am using a square variable in my regression as independent variable. my model is as the following: Leverage = cash^2 etc. when I add a square variable of cash, Do I have also do include both Cash and Cash^2. or it is find to only include Cash^2. This is because when I add both of them, the significance of the results disappear. Exercise Pearson’s chi-square for a 2-way table: Product multinomial model. If A and B are categorical variables with 2 and k levels, respectively, and we collect random samples of size m and n from levels 1 and 2 of A, then classify each individual according to its level of the variable File Size: KB. I have explanatory variable age in the model (along with others) and there are several theoretical justification for using square of age variable (along with age) in the model,e,g,here (if we have a dependent variable income, then this means that income increases as age increases but this increase becomes less as we become older). Now, I checked the correlation between age and age squared .Internal Report SUF–PFY/96–01 Stockholm, 11 December 1st revision, 31 October last modification 10 September Hand-book on STATISTICAL.Statistical Techniques for Transportation Engineering is written with a systematic approach in mind and covers a full range of data analysis topics, from the introductory level (basic probability, measures of dispersion, random variable, discrete and continuous distributions) through more generally used techniques (common statistical.

How to calculate cost of revenue The revenue to cost ratio is one of those metrics that play the biggest role in B2B sales. With its help, businesses can ensure that their sales volumes grow, whereas the cost of operations remains low, stimulating revenue growth. Basically, this ratio is the key to business success; thus, every company needs to know how to calculate the cost of revenue. From its name, it is not hard to guess that the revenue and cost ratio formula consists of just two variables - total revenue and total cost. Yet, it’s crucial to know how to calculate each variable accurately and measure them against each other. So, here is a brief guide on how to calculate CRR step by step. 1) Define the cost of revenue In order to find your total variable costs, you need to keep in mind all manufacturing expenses. For this purpose, you can use different financial statements that your company has, for example, a balance sheet. You want to consider the following: - Labor costs; - Marketing costs; - Material costs; - Distribution costs; - Administrative expenses; - Overhead costs, etc. Once you have all the costs, use a simple cost of revenue formula - add up all direct costs your business has and move on to the next stage. 2) Define the total revenue Now that you have a revenue cost formula in mind, you need to define your revenue. First of all, you need to know what it stands for. Total revenue is basically the gross revenue of your business. Simply put, it is the total amount of money generated by your business before deducting any costs incurred, such as manufacturing or marketing expenses, corporate tax liabilities, property taxes, etc. Now, how do you calculate revenue? First of all, you have to calculate revenue from sales. To do this, you have to multiply the number of sold units by the cost of every unit. The easiest way to do this is using a financial statement like a balance sheet. Apart from sales revenue, you also have to calculate additional, non-operating income, for example, dividends or interest. Once you have a fixed number indicating your sales and non-operating income, add them. Now, you know how to calculate total revenue from the balance sheet. 3) Calculate your sales expense to revenue ratio When you have all the variables, you can finally calculate the expense ratio to your revenue. Doing this is quite easy. Basically, the revenue and cost ratio is calculated bycost divided by revenue. The formula looks like this: Cost of revenue / total revenue = CRR Typically, the number you will get will be a decimal (smaller than one). That’s why, after you calculate CRR, there will be one more step you need to take. 4) Calculate the percentage As was already mentioned, the answer you will get using the formula above will be a decimal. Due to this reason, financial professionals typically use percentages when it comes to calculating CRR. So, the last step you need to take is to calculate the rate of the CRR you’ve already found. Doing this is simple – just take the figure you received using the CRR formula and multiply it by 100. Cost of revenue vs. COGS Speaking about various business-related expenses, financial specialists use different cost categories to measure and compare the company’s expenses. Primarily, the two most important cost categories for calculating efficiency ratios are the cost of revenue and the cost of goods sold (COGS). We have to admit that the cost of revenue and cost of goods sold ratio are very similar. So, there is often a lot of misunderstanding between these two cost groups. But, in fact, they are not the same. Let’s take a moment to consider the cost of revenue vs. COGS to find the differences. Cost of goods sold What are COGS? According to the general cost of goods sold definition, it is a direct cost of producing goods. To calculate the cost of sales percentage, specialists add up two primary expenditures: - Cost of labor; - Cost of materials. Cost of revenue The cost of revenue is much different. It represents the total cost of manufacturing, marketing, and delivering goods to customers. Thus, apart from the cost of labor and materials, the cost of revenue also involves additional fixed costs like overhead, shipping, and distribution expenses. It also involves additional variable costs, such as the cost of every marketing campaign and marketing materials. Who works with these indicators? If you are hoping to improve your indicators, chances are that you will want to find the right specialists to handle this task. To help you get on the right track, here is a list of the top job opportunities that imply working with cost-revenue ratios: - Financial advisor; - Financial analyst; - Budget analyst; - Market research analyst; People with such titles often have to deal with CRR and possess the needed skills and knowledge. Cost of revenue vs. cost of sales We’ve already looked at the cost of revenue vs. cogs. Now, what is the cost of sales? Instead of giving you a lengthy and complicated cost of sales definition, let’s make it clear straight away. There are many names for this indicator. Financial specialists call it COGS, the cost of goods sold, and also the cost of sales. So, if you ever wonder, “is cost of sales the same as cost of goods sold,” the answer is yes. Just like the COGS ratio, the cost of sales is calculated based on the labor and material costs used to produce a particular product. The cost of revenue, on the other hand, involves labor and material costs plus additional expenses, such as rent, taxes, design and maintenance of corporate websites, lead generation, marketing, etc. Thus, unlike the cost of sales or COGS, the cost of revenue gives a more comprehensive outlook on the expenses borne by the company. Cost and revenue in B2B lead generation Now that you know about the huge role of the expense to revenue ratio in the business, chances are that you are also wondering where this metric is applied. Despite a common belief, the cost to charge ratio formula is used not only to manage the company’s financial operations but can also be applicable to its sales and lead generation efforts. Businesses that focus on lead generation and e-commerce don’t want to leave space for guesswork when it comes to their profits. They want their sales goals to align with their budgets. And that’s where the cost to profit ratio plays a massive role. By approximating the cost vs. revenue, businesses can obtain a benchmark for selling their products or services. Simply put, they receive a forecast of how much profit they can get from generating or buying leads and can adjust their budgets and goals accordingly. Belkins offers you plenty of benefits here. With our unique B2B lead generation techniques, your business can streamline sales and align costs and revenues to its goals and budgets. Calculation examples and cost and revenue calculator If you want to grasp all operational expenses, you need to calculate your CRR. As you already know, the formula for calculating CRR looks like this: Cost of revenue / total revenue = Cost revenue ratio We have learned that the results are typically given in percentages, so the complete formula also includes multiplying the obtained result by 100%. Let’s say your direct cost is $5M, and your total revenue is $7.5M. Using the formula, you calculate CRR like this: 5,000,000 / 7,500,000 = 0,666… Now, round this number to decimals and multiply it by 100: 0.67 x 100 = 67% In this example, your cost-revenue ratio is 67%, where percentages indicate how much each cost generates per $100. Respectively, for every $67 you spend, you generate $100 in revenue. If you are wondering how to calculate fees earned against manufacturing expenses (labor + materials), you need to calculate the cost-to-sales ratio. The sales cost to income ratio is calculated similarly. But, instead of dividing the total cost of revenue by the total revenue, you have to split the cost of sales by the total revenue. Here is what the COGS revenue ratio formula looks like: Cost of sales (COGS) / total revenue = Cost to sales ratio This metric is also provided in percentages, so you will need to multiply the result by 100 too. For example, if your cost of sales is $600K and your total revenue is $7.5M. Using the formula, you calculate CSR like this: 600,000 / 7,500,000 = 0.08 Now, multiply this by 100, and you will discover that your cost to sales ratio is 8%. As you can see, the formulas are elementary, so it won’t take a lot of time to calculate your performance indicators. Nevertheless, there is an easier way to do this. These days, businesses can leverage online calculators to simplify complex calculations. There are many different tools, including a cost and revenue calculator, an expense ratio comparison calculator, a profit margin calculator, etc. All these tools can save you some time and help you acquire accurate results. Is the term cost to sales ratio clearer now? After grasping the idea of costs and revenue and learning how to calculate expense ratio fees, your business can handle budget variances more flexibly and simply. Yet, if B2B lead generation plays a significant role in your business, it will never hurt to team up with qualified marketing and sales teams to align your goals with budgets. Contact Belkins to learn how to get more leads and, at the same time, maintain the low cost of operations. Our experts are always happy to help propel your business growth and prosperity! So, what is cost of revenue? After reading this guide, you should have a clearer idea of the main terms and formulas. In the conclusion of our article, let’s quickly recap the key points: - The CRR measures the ratio of operating expenses to revenues generated by a business. - To calculate CRR, you have to divide the total cost by the total revenue and multiply it by 100. - Unlike the cost of goods sold, the cost of revenue includes additional operational expenses on top of manufacturing costs. Measuring CRR is crucial not only for budgeting your business right but also for adjusting your sales and lead generation strategies.

A General Formula. times B dollars. Example. Suppose you deposit $1000 in a bank which pays 5% interest compounded daily, meaning 365 times per year. Also, How much interest will 100 000 earn in a year? How much interest will I earn on $100k? How much interest you’ll earn on $100,000 depends on your rate of return. Using a conservative estimate of 4% per year, you’d earn $4,000 in interest (100,000 x . 04 = 4,000). Hereof, How can calculate percentage? Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. The formula used to calculate percentage is: (value/total value)×100%. Also to know How much money do I need to invest to make 2000 a month? For example, if you want $2,000 per month, you’d need to save at least $480,000 before retirement. When interest rates are low and the stock market is volatile, the 5% withdrawal aspect of the rule becomes even more critical. Can I retire on $10000 a month? Typically you can generate at least $10,000 a month in retirement income for the rest of your life. This does not include Social Security Benefits. How do I calculate percentage of a total? How to calculate percentage - Determine the whole or total amount of what you want to find a percentage for. … - Divide the number that you wish to determine the percentage for. … - Multiply the value from step two by 100. What is ratio formula? When we compare the relationship between two numbers dealing with a kind, then we use the ratio formula. It is denoted as a separation between the number with a colon (:). For example, we are making a cake, then the recipe sometimes says to mix flour to water in the ratio 2 part 1. … How much money do I need to invest to make $500 a month? To make $500 a month in dividends you’ll need to invest between $171,429 and $240,000, with an average portfolio of $200,000. The actual amount of money you’ll need to invest in creating a $500 per month in dividends portfolio depends on the dividend yield of the stocks you buy. How much money do I need to invest to make $1000 a month? So it’s probably not the answer you were looking for because even with those high-yield investments, it’s going to take at least $100,000 invested to generate $1,000 a month. For most reliable stocks, it’s closer to double that to create a thousand dollars in monthly income. What will 150k be worth in 20 years? How much will an investment of $150,000 be worth in the future? At the end of 20 years, your savings will have grown to $481,070. You will have earned in $331,070 in interest. How much do I need to invest to make $500 a month? As a result, $150,000 is how much you will need to invest to make $500 a month in dividends assuming your portfolio yields 4%. What is considered a comfortable retirement income? One rule of thumb is that you’ll need 70% of your pre-retirement yearly salary to live comfortably. How do I calculate 5% of a total? To calculate 5 percent of a number, simply divide 10 percent of the number by 2. For example, 5 percent of 230 is 23 divided by 2, or 11.5. What is ratio explain? In mathematics, a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent. What is ratio in simple words? Definition of Ratio In simple words, the ratio is the number which can be used to express one quantity as a fraction of the other ones. The two numbers in a ratio can only be compared when they have the same unit. How do I solve a ratio problem? To use proportions to solve ratio word problems, we need to follow these steps: - Identify the known ratio and the unknown ratio. - Set up the proportion. - Cross-multiply and solve. - Check the answer by plugging the result into the unknown ratio. How do I make $1000 a month in dividends? How To Make $1,000 A Month In Dividends: 5 Step Plan - Choose a desired dividend yield target. - Determine the amount of investment required. - Select dividend stocks to fill out your dividend portfolio. - Invest in your dividend income portfolio regularly. - Reinvest all dividends received. How can I make $1000 a month in passive income? 9 Passive Income Ideas that earn $1000+ a month - Start a YouTube Channel. … - Start a Membership Website. … - Write a Book. … - Create a Lead Gen Website for Service Businesses. … - Join the Amazon Affiliate Program. … - Market a Niche Affiliate Opportunity. … - Create an Online Course. … - Invest in Real Estate. How much money do I need to invest to make $50 a month? To make $50 a month in dividends you need to invest between $17,143 and $24,000, with an average portfolio of $20,000. The exact amount of money you need to invest for $50 per month in dividend income depends on the dividend yield of the stocks you buy. Think of a dividend yield as your return on investment. Can I retire on 4000 per month? There is something in retirement planning known as the safe withdrawal rate. … So yes, to collect just over $4,000 per month, you need well over a million dollars in retirement accounts. How much money do I need to invest to make $3 000 a month? By this calculation, to get $3,000 a month, you would need to invest around $108,000 in a revenue-generating online business. Here’s how the math works: A business generating $3,000 a month is generating $36,000 a year ($3,000 x 12 months). Can I retire on 8000 a month? So how much income do you need? With that in mind, you should expect to need about 80% of your pre-retirement income to cover your cost of living in retirement. … Based on the 80% principle, you can expect to need about $96,000 in annual income after you retire, which is $8,000 per month. What will 100k be worth in 20 years? How much will an investment of $100,000 be worth in the future? At the end of 20 years, your savings will have grown to $320,714. You will have earned in $220,714 in interest. What will 60000 be worth in 20 years? The first result (Reduced Amount) is $33,220.55, which represents the value of $60,000 in 20 years. Is 100k savings a lot? Summary: Is 100k in savings a lot? Yes, it is potentially a decent chunk of change. It’s often thought of as one of the most difficult financial goals to reach.

- About this Journal - Abstracting and Indexing - Aims and Scope - Annual Issues - Article Processing Charges - Articles in Press - Author Guidelines - Bibliographic Information - Citations to this Journal - Contact Information - Editorial Board - Editorial Workflow - Free eTOC Alerts - Publication Ethics - Reviewers Acknowledgment - Submit a Manuscript - Subscription Information - Table of Contents Journal of Applied Mathematics Volume 2013 (2013), Article ID 705814, 4 pages Strong Convergence for Hybrid -Iteration Scheme 1Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 2School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore 54660, Pakistan 3Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea Received 19 November 2012; Accepted 4 February 2013 Academic Editor: D. R. Sahu Copyright © 2013 Shin Min Kang et al. 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. We establish a strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. 1. Introduction and Preliminaries Let be a real Banach space and let be a nonempty convex subset of . Let denote the normalized duality mapping from to defined by where denotes the dual space of and denotes the generalized duality pairing. We will denote the single-valued duality map by . Let be a mapping. Definition 1. The mapping is said to be Lipschitzian if there exists a constant such that Definition 2. The mapping is said to be nonexpansive if Definition 3. The mapping is said to be pseudocontractive if for all , there exists such that Definition 4. The mapping is said to be strongly pseudocontractive if for all , there exists such that Let be a nonempty convex subset of a normed space .(a)The sequence defined by, for arbitrary , where and are sequences in , is known as the Ishikawa iteration process . If for , then the Ishikawa iteration process becomes the Mann iteration process .(b)The sequence defined by, for arbitrary , where is a sequence in , is known as the -iteration process [3, 4]. In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., ). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [5–10] and the references cited therein). In 1974, Ishikawa proved the following result. Theorem 5. Let be a compact convex subset of a Hilbert space and let be a Lipschitzian pseudocontractive mapping. For arbitrary , let be a sequence defined iteratively by where and are sequences satisfying(i), Then the sequence converges strongly at a fixed point of . If is a real Banach space with a uniformly convex dual , is a nonempty bounded closed convex subset of , and is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly at the unique fixed point of . In this paper, we establish the strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. We also improve the result of Zhou and Jia . 2. Main Results We will need the following lemmas. Lemma 6 (see ). Let be the normalized duality mapping. Then for any , one has Lemma 7 (see ). Let be nonnegative sequence satisfying where ,, and . Then The following is our main result. Theorem 8. Let be a nonempty closed convex subset of a real Banach space , let be nonexpansive, and let be Lipschitz strongly pseudocontractive mappings such that and Let be a sequence in satisfying(iv) (v) For arbitrary , let be a sequence iteratively defined by Then the sequence converges strongly at the common fixed point of and . Proof. For strongly pseudocontractive mappings, the existence of a fixed point follows from Delmling . It is shown in that the set of fixed points for strongly pseudocontractions is a singleton. By (v), since , there exists such that for all , where . Consider which implies that where and consequently from (16), we obtain Substituting (18) in (15) and using (13), we get So, from the above discussion, we can conclude that the sequence is bounded. Since is Lipschitzian, so is also bounded. Let . Also by (ii), we have as , implying that is bounded, so let . Further, which implies that is bounded. Therefore, is also bounded. Denote . Obviously, . Now from (12) for all , we obtain and by Lemma 6, we get which implies that because by (13), we have and . Hence, (23) gives us For all , put then according to Lemma 7, we obtain from (26) that This completes the proof. Corollary 9. Let be a nonempty closed convex subset of a real Hilbert space , let be nonexpansive, and let be Lipschitz strongly pseudocontractive mappings such that and the condition . Let be a sequence in satisfying the conditions (iv) and (v). For arbitrary , let be a sequence iteratively defined by (12). Then the sequence converges strongly at the common fixed point of and . Example 10. As a particular case, we may choose, for instance, . Remark 11. (1) The condition is not new and it is due to Liu et al. . (2) We prove our results for a hybrid iteration scheme, which is simple in comparison to the previously known iteration schemes. This study was supported by research funds from Dong-A University. - S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. - W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. - D. R. Sahu, “Applications of the S-iteration process to constrained minimization problems and split feasibility problems,” Fixed Point Theory, vol. 12, no. 1, pp. 187–204, 2011. - D. R. Sahu and A. Petruşel, “Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6012–6023, 2011. - C. E. Chidume, “Approximation of fixed points of strongly pseudocontractive mappings,” Proceedings of the American Mathematical Society, vol. 120, no. 2, pp. 545–551, 1994. - C. E. Chidume, “Iterative approximation of fixed points of Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2245–2251, 2001. - C. E. Chidume and C. Moore, “Fixed point iteration for pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 127, no. 4, pp. 1163–1170, 1999. - C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831–840, 2004. - J. Schu, “Approximating fixed points of Lipschitzian pseudocontractive mappings,” Houston Journal of Mathematics, vol. 19, no. 1, pp. 107–115, 1993. - X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991. - H. Zhou and Y. Jia, “Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption,” Proceedings of the American Mathematical Society, vol. 125, no. 6, pp. 1705–1709, 1997. - S. S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis, vol. 30, no. 7, pp. 4197–4208, 1997. - K. Delmling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 283–288, 1974. - Z. Liu, C. Feng, J. S. Ume, and S. M. Kang, “Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 27–42, 2007.

Phase sensitivity bounds for two-mode interferometers We provide general bounds of phase estimation sensitivity in linear two-mode interferometers. We consider probe states with a fluctuating total number of particles. With incoherent mixtures of state with different total number of particles, particle entanglement is necessary but not sufficient to overcome the shot noise limit. The highest possible phase estimation sensitivity, the Heisenberg limit, is established under general unbiased properties of the estimator. When coherences can be created, manipulated and detected, a phase sensitivity bound can only be set in the central limit, with a sufficiently large repetition of the interferometric measurement. pacs:03.65.Ta 42.50.St 42.50.Dv The problem of determining the ultimate phase sensitivity (often tagged as the “Heisenberg limit”) of linear interferometers has long puzzled the field Caves_1981 ; SummyOPTCOMM1990 ; HradilPRA1995 ; HradilQO1992 ; HallJMO1993 ; Ou_1996 ; Shapiro_1989 ; Shapiro_1991 ; Bondurant_1984 ; Schleich_1990 ; Braunstein_1992 ; Wineland_1994 ; Holland_1993 ; YurkePRA1986 and still arises controversies Durkin_2007 ; Monras_2006 ; BenattiPRA2013 ; ZwierzPRL2010 ; JarzynaPRA2012 ; Giovannetti_2006 ; Hyllus_2010 ; Pezze_2009 ; Rivas_2011 ; PezzePRA2013 ; Joo_2011 ; HayashiPI2011 ; GLM_preprint2011 ; Tsang_preprint2011 ; HallPRA2012 ; BerryPRA2012 ; HallNJP2012 ; Hofmann_2009 ; AnisimovPRL10 ; ZhangJPA2012 ; GaoJPA2012 ; Giovannetti_2012 . The recent revival of interest is triggered by the current impressive experimental efforts in the direction of quantum phase estimation with ions exp_ions , cold atoms exp_coldatoms , Bose-Einstein condensates exp_BEC and photons exp_photons , including possible applications to large-scale gravitational wave detectors Schnabel_2010 . Beside the technological applications, the problem is closely related to fundamental questions of quantum information, most prominently, regarding the role played by quantum correlations. In particular, the phase sensitivity of a linear two-mode interferometer depends on the entanglement between particles (qubits) in the input (or “probe”) state Giovannetti_2006 ; Hyllus_2010 ; Pezze_2009 . It is widely accepted Giovannetti_2006 ; Pezze_2009 that, when the number of qubits in the input state of is fixed, and equal to [so that the mean square particle-number fluctuations ], there are two important bounds in the uncertainty of unbiased phase estimation. The shot noise limit, is the maximum sensitivity achievable with probe states containing only classical correlations among particles. The factor accounts for the number of independent repetitions of the measurement. This bound is not fundamental. It can be surpassed by preparing the particles of the probe in a proper entangled state. The fundamental (Heisenberg) limit is given by and it is saturated by maximally entangled (NOON) states. It should be noticed that most of the theoretical investigations have been developed in the context of systems having a fixed, known, total number of particles . However, many experiments are performed in presence of finite fluctuations . The consequences in the phase sensitivity of classical and quantum fluctuations of the number of particles entering the interferometer have not been yet investigated in great depth. In this case, indeed, the existence and discovery of the phase uncertainty bounds can be critically complicated by the presence of coherences between different total number of particles in the probe state and/or the output measurement Hyllus_2010 ; Hofmann_2009 . However such quantum coherences do not play any role in two experimentally relevant cases: i) in the presence of superselection rules, which are especially relevant for massive particles and forbid the existence of number coherences in the probe state; ii) when the phase shift is estimated by measuring an arbitrary function of the number of particles in the output state of the interferometer, e.g. when the total number of particles is post-selected by the measurement apparatus. The point (ii) is actually an ubiquitous condition in current atomic and optical experiments. Indeed, all known phase estimation protocols implemented experimentally are realised by measuring particle numbers. In the absence of number coherences, or when coherences are present but irrelevant because of (ii), we can define a state as separable if it is separable in each subspace of a fixed number of particles Hyllus_2010 . A state is entangled if it is entangled in at least one subspace of fixed number of particles. With separable states, the maximum sensitivity of unbiased phase estimators is bounded by the shot-noise while with entangled states the relevant bound, the Heisenberg limit, is given by Hyllus_2010 We point out that Eq. (4) cannot be obtained from Eq. (2) by simply replacing with . On the other hand, Eq. (4) reduces to Eq. (2) when number fluctuations vanish, . An example of phase estimation saturating the scaling is obtained with the coherentsqueezed-vacuum state PezzePRL2008 . When the probe state and the output measurement contain number coherences, the situation becomes more involved. It is still possible to show that Eq. (4) holds in the central limit (), at least. Outside the central limit, it is possible to prove that the highest phase sensitivity is bounded by The crucial point is that the fluctuations can be made arbitrarily large even with a finite . In general, no lower bound can be settled in this case: , and it can be saturated with finite resources (, ) if an unbiased estimator exists. Outside the central limit, (i.e. for a small number of measurements) we cannot rule out the existence of opportune unbiased estimators which can saturate Eq. (5). This manuscript extends and investigate in details the results and concepts introduced in Ref. Hyllus_2010 . In Sec. II we review the theory of multiparameter estimation with special emphasis on two-mode linear transformations. This allows us to introduce the useful concept of (quantum) Fisher information and the Cramér-Rao bound. We show that two mode phase estimation involves, in general, operations which belong to the U(2) transformation group. When number coherences in the probe state and/or in output measurement observables are not present, the only allowed operations are described by SU(2) group. In Secs. III and IV we give bounds on the quantum Fisher information depending whether or not the probe state contains number coherences. In the latter case, we set an ultimate bound that can be reached by separable states of a fluctuating number of particles. Finally, in Sec. VI we discuss the Heisenberg limit, Eq. (4), and under which conditions it holds. This manuscript focuses on the ideal noiseless case. It is worth pointing out that decoherence can strongly affect the achievable phase uncertainty bounds. For several relevant noise models in quantum metrology, as, for instance, particles losses, correlated or uncorrelated phase noise, phase uncertainty bounds have been derived HuelgaPRL1997 ; ShajiPRA2007 ; EscherNATPHYS2011 ; RafalNATCOMM2012 ; DornerNJP2012 ; LandiniNJP2014 . Ii Basic concepts In the (multi-)phase estimation problem, a probe state undergoes a transformation which depends on the unknown vector parameter . The phase shift is estimated from measurements of the transformed state . The protocol is repeated times by preparing identical copies of and performing identical transformations and measurement. The most general measurement scenario is a positive-operator valued measure (POVM), i.e. a set of non-negative Hermitian operators parametrized by and satisfying the completeness relation Helstrom_book . The label indicates the possible outcome of a measurement which can be continuous (as here), discrete or multivariate. Each outcome is characterized by a probability , conditioned by the true value of the parameters. The positivity and Hermiticity of guarantee that are real and nonnegative, the completeness guarantees that . The aim of this section is to settle the general theory of phase estimation for two-mode interferometers. ii.1 Probe state A generic probe state with fluctuating total number of particles can be written as with and , where is a coherent superposition of states with different number of particles. The coefficients are complex numbers and the normalization condition implies . It is generally believed that quantum coherences between states of different numbers of particles do not play any observable role because of the existence of superselection rules (SSRs) for the total number of particles WickPR52 ; Bartlett_2007 . In the presence of SSRs the only physically meaningful states are those which commute with the number of particles operator, A state satisfies this condition if and only if nota:commutatorN it can be written as the incoherent mixture where is a normalized () state, are positive numbers satisfying and are projectors on the fixed- subspace. The existence of SSRs is the consequence of the lack of a phase reference frame (RF) Bartlett_2007 . However, the possibility that a suitable RF can be established in principle cannot be excluded Bartlett_2007 . If SSRs are lifted, then coherent superpositions of states with different numbers of particles become physically relevant. ii.2 separability and multiparticle entanglement where is the state of the th particle. A state is (multiparticle) entangled if it is not separable. One can further consider the case where only a fraction of the particles are in an entangled state and classify multiparticle entangled states following Refs. SeevinckPRA01 ; GuehneNJP05 ; ChenPRA05 ; SoerensenPRL01 ; GuhnePHYSREP2009 . A pure state of particles is -producible if it can be written as , where is a state of particles, witht . A state is -particle entangled if it is -producible but not -producible. Therefore, a -particle entangled state can be written as where the product contains at least one state of particles which does not factorize. A mixed state is -producible if it can be written as a mixture of -producible pure states, i.e., , where for all . Again, it is -particle entangled if it is -producible but not -producible. Notice that, formally, a separable state is -producible, and that a decomposition of a -particle entangled state of particles may contain states where different sets of particles are entangled. where is a -producible state of particles. A state with number coherences (6) will be called separable if it is separable in every fixed- subspace Hyllus_2010 , i.e. if the incoherent mixture , obtained from by projecting over fixed- subspaces, has the form of Eq. (11). Analogously, a state will be called -producible if the projection on each fixed- subspace has the form of Eq. (12). ii.3 Two-mode transformations In the following we will focus on linear transformations involving two modes. These includes a large class of optical and atomic passive devices, including the beam-splitter, Mach-Zehnder and Ramsey interferometers. Most of the current prototype phase estimation experiments exp_ions ; exp_coldatoms ; exp_photons ; exp_BEC are well described by a two-mode approximation. Denoting by and ( and ) are input (output) mode annihilation operators, we can write The matrix Eq. (14) is unitary, preserves bosonic and fermonic commutation relations between the input/output mode operators and its determinant is equal to . The most general two mode transformation thus belongs to the U(2)=U(1)SU(2) group (unitary matrices with determinant ). The coefficients is physically related to transmittance and reflectance of the transformation (14), and being the corresponding phases. The lossless nature of Eq. (14) is guaranteed by Using the Jordan-Schwinger representation of angular momentum systems in terms of mode operators SchwingerBOOK , it is possible to find the operator corresponding to the matrix (14). In other words, for is the transformation of mode operators (Heisenberg picture) and , , is the equivalent transformation of statistical mixtures and quantum states, respectively (Schrödinger picture). One finds CamposPRA1989 ; YurkePRA1986 is the number of particle operator, (where , and are the coordinates of the vector in the Bloch sphere and satisfy ), and are spin operators. The exact relation between the parameters of the matrix [, and in Eq. (14)] and the parameters of the operator [, , and in Eq. (15)] is given in Appendix A. The operators , and satisfy the angular momentum commutation relations. Notice that the pseudo-spin operators commute with the total number of particles, for . We can thus rewrite , where and is the Pauli matrix (along the direction in the Bloch sphere, ) acting on the -th particle. The most general U(2) transformation, Eq. (15), can be rewritten as where , , is a direction perpendicular to and , and . Equation (16) highlights the presence of two phases, and , which can be identified as the phases acquired in each mode and inside a Mach-Zehnder-like interferometer [with standard balanced beam splitters replaced by the transformation , see Fig. 1(a)]. Both phases may be unknown. When setting one of the two phases to zero (or to any fixed known value), Eq. (16) reduces to different single-phase transformations: U(1) transformations (), which can be understood as a phase shift equally imprinted on each of the two modes: . ii.4 Output measurement Generally speaking, a POVM may or may not contain coherences among different number of particles. A POVM does not contain number coherences if and only if all its elements commute with the number of particles operator, where acts on the fixed- subspace and are projectors. In current phase estimation experiments, the phase shift is estimated by measuring a function of the number of particles at the output modes of the interferometer. The experimentally relevant POVMs can thus be written as By making a change of variable and (), we can rewrite this equation as which has the form of Eq. (19). Notice that the information about the total number of particles is not necessarily included in the POVM. For instance, the POVM corresponding to the measurement of only the relative number of particles can be written as which, again, has the form of Eq. (19). This example can be straightforwardly generalized to the measurement of any function of the relative number of particles. For the measurement of the number of particles in a single output port of the interferometer (for instance at the output port “1”), we have ii.5 Conditional probabilities For U(2) transformations, Eq. (16), the conditional probability can be written as where . The derivation of Eq. (23) is detailed in Appendix B. Equation (23) depends only on , the relative phase shift among the two modes of the interferometer. We conclude that U(2) transformations are relevant only if the input state contains coherences among different number of particles and the output measurement is a POVM with coherences. In all other cases the phase shift is irrelevant as the conditional probabilities are insensitive to . In this case, the mode transformation Eq. (16) restricts to the unimodular (i.e. unit determinant) subgroup SU(2). The SU(2) representation, while being not general, is widely used because, in current experiments, the phase shift is estimated by measuring a function of the number of particles at the output ports of the interferometer. Table 1 summarizes the general two-mode transformation group for the phase estimation problem, depending on the presence of number coherences in the probe state and POVM. |POVM with coh.||POVM without coh.| ii.6 Multiphase estimation Since U(2) transformations involve two phases, and , we review here the theory of two-parameter estimation PezzeVarenna . The vector parameter is inferred from the values obtained in repeated independent measurements. The mapping from the measurement results into the two-dimensional parameter space is provided by the estimator function . Its mean value is , with () and the likelihood function . We further introduce the covariance matrix of elements Notice that is symmetric and its th diagonal element is the variance of . ii.6.1 Cramér-Rao bound where is the Jacobian matrix and the Fisher information matrix notaFij , which is symmetric and nonnegative definite. Note that , and generally depend on but we do not explicitly indicate this dependence, in order to simplify the notation. Note also that may depend on . In the inequality (25) and are arbitrary real vectors. Depending on and we thus have an infinite number of scalar inequalities. If the Fisher matrix is positive definite, and thus invertible, the specific choice in Eq. (25) leads to the vector parameter Cramér-Rao lower bound CramerBOOK , in the sense that the matrix is nonnegative definite [i.e. holds for all real vectors ], where This specific choice of leads to a bound which is saturable by the maximum likelihood estimator (see Sec. II.6.2) asymptotically in the number of measurements. In the two-parameter case, the Fisher information matrix is invertible if and only if , its inverse given by Furthermore, if does not depend on for (i.e. is diagonal), the diagonal elements of satisfy the inequalities: with , . For the two-parameter case, the inequality (30) can be immediately demonstrated by using which holds since nonnegative definite and assumed here to be invertible. In the estimation of a single parameter, the matrix reduces to the variance . Equation (27) becomes where [for unbiased estimators , i.e. ] and is the (scalar) Ficher information (FI). By comparing Eq. (30) and Eq. (31), we see, as reasonably expected, that the estimation uncertainty of a multi-parameter problem is always larger or at most equal than the uncertainty obtained for a single parameter (namely, when all other parameters are exactly known). ii.6.2 Maximum likelihood estimation A main goal of parameter estimation is to find the estimators saturating the Cramér-Rao bound. These are called efficient estimators. While such estimators are rare, it is not possible to exclude, in general, that an efficient unbiased estimator may exist for any value of . One of the most important estimators is the maximum likelihood (ML) . It is defined as the value which maximizes the log-likelihood function: It is possible to demonstrate, by using the law of large numbers and the central limit theorem, that, asymptotically in the number of measurements, the maximum likelihood is unbiased and normally distributed with variance given by the inverse Fisher information matrix PezzeVarenna ; Kay_book . Therefore, the specific choice of vector which leads to the Cramér-Rao bound (27) is justified by the fact that the ML saturates this bound for a sufficiently large number of measurements. ii.6.3 Quantum Cramér-Rao bound The Fisher information matrix, satisfies in the sense that the matrix is positive definite. The symmetric matrix is called the quantum Fisher information matrix and its elements are with , where the self-adjoint operator , called the symmetric logarithmic derivative (SLD) Helstrom_book , is defined as In particular, we have , being the real part of . Note also that the operator (and also ) generally depends on . Equation (33) holds for any Fisher information matrix (invertible or not) and there is no guarantee that, in general, the equality sign can be saturated. Assuming that and are positive definite (and thus invertible) and combining Eq. (27) – in the unbiased case – with Eq. (33), we obtain the matrix inequality nota_invposmat , where This sets a fundamental bound, the quantum Cramér-Rao (QCR) bound Helstrom_book , for the sensitivity of unbiased estimators. The bound cannot be saturated, in general, in the multiparameter case. In the single parameter case, we have , where The (scalar) quantum Fisher information (QFI) can be written as where is the -dependent SLD and we used . The equality (or, equivalently ) holds if the POVM is made by the set of projector operators over the eigenvectors of the operator , as first discussed in Ref. Braunstein_1994 . The quantum Cramér-Rao is a very convenient way to calculate the phase uncertainty since it only depends on the properties of the probe state and not on the quantum measurement. Iii Fisher Information for states without number coherences As discussed above, for states without number coherences we can restrict to SU(2) transformations and thus the estimation of a single parameter: the relative phase shift among the arms of a Mach-Zehnder-like interferometer. In this case, an important property of the QFI holds: where is the QFI calculated on the fixed- subspace. To demonstrate this equation let us consider the general expression of QFI given in Ref. Braunstein_1994 , where and is a basis of the Hilbert space, , chosen such that . For states without number coherence, we have where is a basis on the fixed- subspace. Since , i.e. does not couple states of different number of particles. In an analogous way it is possible to demonstrate that the SLD . We thus conclude that, when the input state does not have number coherences, the Von Neumann measurement on the eigenstates of for each value of – which in particular does not have number coherences – is such that the corresponding FI saturates the QFI. Iv Fisher Information for states with number coherences In this section we discuss the quantum Fisher information for states with number coherences. First we consider the estimation a single phase, either or , separately, assuming that the other parameter is known. We then apply the multiparameter estimation theory outlined above to calculate the sensitivity when and are both estimated at the same time. We will mainly focus on the calculation of an upper bound to the quantum Fisher information. iv.1 Single parameter estimation Let us consider the different transformations outlined in Sec. II.3: SU(2) transformation . It is interesting to point out that, for SU(2) transformations, number coherences may increase the value of the QFI. We have where is a normalized pure state with coherences and is obtained from by tracing out the number-coherences. Notice that, if holds, then that saturation of necessarily requires a POVM with number coherences. This is a consequence of the fact that the Fisher information obtained with POVMs without coherences is independent on the presence of number coherences in the probe state and it is therefore upper bounded by . Equation (IV.1) can be demonstrated using i) Braunstein_1994 ; Pezze_2009 and [see Eq. (39)], where we have explicitly indicated the state on which the variance is calculated on (we will keep this notation where necessary and drop it elsewhere) and ii) the Cauchy-Schwartz inequality The equality holds if and only if is a constant independent of . In the following we discuss the bounds to the QFI. For this, a useful property of the QFI is its convexity PezzeVarenna . In our case it implies where the equality holds only for pure states. Furthermore, The first inequality is saturated for . In the second inequality we used both saturated for the NOON state with [and ]. In this case, by using Eq. (42), we have that where the equality can be saturated by a coherent superposition of NOON states (note indeed that ). We thus have U(2) transformations . Using the convexity of the QFI, we have where the second inequality follows from a Cauchy-Schwarz inequality. We thus have iv.2 Two-parameter estimation In the U(2) framework, there are, in general, two phases to estimate: and . When estimating both at the same time, the phase sensitivity is calculated using the multiphase estimation formalism discussed above. The inequality (30), leads to which can be further bounded by using the above inequalities for the QFI. For pure states we have where . We thus have which, in particular, is always larger than , and which is always larger than . V Separability and Entanglement When the number of particles is fixed, there exists a precise relation between the entanglement properties of a probe state and the QFI: if the state is separable [i.e. can be written as in Eq. (10)] then the inequality holds Pezze_2009 . A QFI larger than is a sufficient condition for entanglement and singles out the states which are useful for quantum interferometry, i.e. states that can be used to achieve a sub shot noise phase uncertainty. The above inequality can be extended to the case of multiparticle entanglement. In Refs HyllusArXiv10 ; TothArXiv10 , it has been shown that for -producible states the bound holds, where is the largest integer smaller than or equal to and . Hence a violation of the bound (51) proves -particle entanglement. For general states of a fixed number of particles, we have Pezze_2009 ; Giovannetti_2006 , whose saturation requires -particle entanglement. In the case of states with number fluctuations, the situation is more involved. For states without number coherences, by using Eq. (39), we straightforwardly obtain

Martin Gardner called this the proudest puzzle of his own devising. When the pieces on the left are rearranged as on the right, a hole appears in the center of the square. How is this possible? “I haven’t the foggiest notion of how to succeed in inventing a good puzzle,” he told the College Mathematics Journal. “I don’t think psychologists understand much either about how mathematical discoveries are made. … The creative act is still a mystery.” A puzzle by Polish mathematician Paul Vaderlind: Andre Agassi and Boris Becker are playing tennis. Agassi wins the first set 6-3. If there were 5 service breaks in the set, did Becker serve the first game? (Service changes with each new game in the 9-game set. A service break is a game won by the non-server.) By Werner Keym, from Die Schwalbe, 1979. What were the last moves by White and Black? Two adjoining lakes are connected by a lock. The lakes differ by 2 meters in elevation. A boat can pass from the lower lake to the upper by passing through the lock gate, which is closed behind it; then water is added to the lock chamber until its level matches that of the upper lake, and the boat can pass out through the upper gate. Now suppose two boats do this in succession. The first boat weighs 50 tons, the second only 5 tons. How much more water must be used to raise the small boat than the large one? This scale balances a cup of water with a certain weight. Will the balance be upset if you put your finger in the water, if you’re careful not to touch the glass? A curious puzzle by George Koltanowski, from America Salutes Comins Mansfield, 1983. “Who mates in 1?” A puzzle by Lewis Carroll: A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter? A puzzle by Henry Dudeney: A lady is accustomed to buy from her greengrocer large bundles of asparagus, each twelve inches in circumference. The other day the man had no large bundles in stock, but handed her instead two small ones, each six inches in circumference. “That is the same thing,” she said, “and, of course, the price will be the same.” But the man insisted that the two bundles together contained more than the large one, and charged a few pence extra. Which was correct — the lady or the greengrocer? Raymond Smullyan presented this puzzle on the cover of his excellent 1979 book The Chess Mysteries of Sherlock Holmes. Black moved last. What was his move? You’ve just won a set of singles tennis. What’s the least number of times your racket can have struck the ball? Remember that if you miss the ball while serving, it’s a fault. The Renaissance mathematician Niccolò Tartaglia would use this bewildering riddle to assess neophytes in logic: If half of 5 were 3, what would a third of 10 be? What’s the answer? A mother takes two strides to her daughter’s three. If they set out walking together, each starting with the right foot, when will they first step together with the left? By M. Charosh, from the Fairy Chess Review, 1937. White to mate in zero moves. A puzzle by Henry Dudeney: The Dobsons secured apartments at Slocomb-on-Sea. There were six rooms on the same floor, all communicating, as shown in the diagram. The rooms they took were numbers 4, 5, and 6, all facing the sea. But a little difficulty arose. Mr. Dobson insisted that the piano and the bookcase should change rooms. This was wily, for the Dobsons were not musical, but they wanted to prevent any one else playing the instrument. Now, the rooms were very small and the pieces of furniture indicated were very big, so that no two of these articles could be got into any room at the same time. How was the exchange to be made with the least possible labour? Suppose, for example, you first move the wardrobe into No. 2; then you can move the bookcase to No. 5 and the piano to No. 6, and so on. It is a fascinating puzzle, but the landlady had reasons for not appreciating it. Try to solve her difficulty in the fewest possible removals with counters on a sheet of paper. Here are seven pennies, all heads up. In a single move you can turn over any four of them. By repeatedly making such moves, can you eventually turn all seven pennies tails up? Prove that, at any given moment, there are two points on the equator that are diametrically opposed yet have the same temperature. Another puzzle from Henry Dudeney: “It is a glorious game!” an enthusiast was heard to exclaim. “At the close of last season, of the footballers of my acquaintance, four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms.” Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with? From the 1977 all-Soviet-Union Mathematical Olympiad: Seven dwarfs are sitting at a round table. Each has a cup, and some cups contain milk. Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them. After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk. How much milk does each cup contain, if there were 42 ounces of milk altogether? We’ve removed two squares from this 7×8 grid, so that it numbers 54 squares. Can it be covered orthogonally with tiles like the one at right, each of which covers exactly three squares? A puzzle by Henry Dudeney: A man planted two poles upright in level ground. One pole was six and a half feet and the other seven feet seven inches above ground. From the top of each pole he tied a string to the bottom of the other — just where it entered the ground. Now, what height above the ground was the point where the two strings crossed one another? The hasty reader will perhaps say, “You have forgotten to tell us how far the poles were apart.” But that point is of no consequence whatever, as it does not affect the answer!

En este libro se aborda el análisis y diseño de sistemas de control en tiempo discreto. Se hace hincapié en la utilidad del programa MATLAB para el estudio de. LIBROS UNIVERISTARIOS Y Sistemas de Control en Tiempo Discreto – 2da Edicion – Katsuhiko Cargado. Katsuhiko Ogata Sistemas de Control en Tiempo Discreto PDF – Ebook download as PDF File .pdf) or libro de control digital para señales en tiempo discreto. |Published (Last):||19 December 2007| |PDF File Size:||11.29 Mb| |ePub File Size:||4.69 Mb| |Price:||Free* [*Free Regsitration Required]| System Dynamics Ogata 4th Help. System Dynamics by Ogata. Ogata – Solutions to Problems of System Dynamics. Ogata Solutions to Problems of System Dynamics ogata solutions. System Dynamics An Introduction. Multibody System Dynamics modelamiento de sistemas dinamicos multicuerpos. Mm System Dynamics Hw4 Solution. Solucionario Ogata Solucionario para Ogata Control. Solucionario para Ogata Control. Vehicle dynamics and road dynamics are two separate subjects. In vehicle dynamics, road surface roughness is generally regarded as random excitation to the vehicle, while at the same time handling Problemas solucionados del libro: Since y s1 we obtain B Assume that the W y of known mmnt of i n e r t i a J through a small angle akrout the v e r t i c a l axis and then equation of motion far the oscillation is where k is the torsfanal spring constant oE the string. The positive direction is downward. Sowe have Then frm which w e obtain The ball reaches the ground in 2, s. The total angle rotated in semnd period is obtained from B Assume t h a t we apply force F to t h e spring system. Since the sane force transmits t h e shaft, we have where displacement z is defined in the figure Below. The following t w a equations describe the nation of the system and they are a mathematical model of the system. Referring to the figure below, we have Note that since x is m a s u r d where T is the tension i n the wire. So we obkain The natural frequency of the system is B Assume that the direction of the static friction force F is to S the left as shown in t h e diagram below. Digital Power Control v3 – PDF Free Download So we obtain From the figure shmm to the right, we obtain To keep the bar AB horizontal when prlling the weight n gt h e rnament a b u t p i n t P must balance. Thus, Solving this equation f o r x, we obtain B Assume that the stiffness of the shafts of the gear nite, t h a t there is neither backTash nor elastic defamation, nt. Ikf i n e respectively. For s h a f t 2we have where T2 is the torque transmitted to gear 2. The equivalent mament of inertia J of mass m referred to the motor m s h a f t axis ogaga be obtained frcm where oE is disfreto angular acceleration of the motor shaft and 5 is the linear acceleration of mass rn. Zi s is given by The transfer function can given in t e r n of camplw impedances Z1 and Z 2 as follows: Then Define the voltage at point A as ek. Define the cyclic current in the left loop as il and that in t h e r l g n t Loop as i2. Then t h e equations for the circuit are which can be rewritten as Using the force-voltage analogy, we can convert the last two equations as follows: From these equations an analogous mechanical system can be obtained as shuwn to the r i g h t. We s h a l l solve t h i s problem by using ttm different approaches: Frm Figure we obtain Using the e1ect;rical-liquid analogy given below, e q a t i d n s for an analogous electrical system can be obtained. C capacitance R resistance Analogaus equations for the electrical system are Based on Equations 5 through 81, we obtain the analogous electrical system shuwn below, B The equations for the liquid-level. So the flaw throughout the lobro is subsonic. Problemas de Ingeniera de Control utilizando Matlab – Katsuhiko Ogata If two roots are real, then the current is not osci- Ilatory. Case 1 Two roots of the characteristic equation are complex conjugate: For this case’ define Then The inverse Laplace transform of I s gives The current i t approaches zero as t approaches infinity. Case 2 Two rmts of the characteristic equation are r e a l: The equations of motion for the system are k x. Since we obtain the respanse eoIt as follows: The equation for the pendulum system can be given by JJ We shall linearize this nonlinear ajvaiirn 4 erruning angle 9 is small. X s ] can be written as clurve e-lfit s i n 5. Figurewe obtain the following equations: F i r s t note t h a t Then, define and use a step comnand. The resulting response curve eo t versus t is shown below. R e m e BdU I Sec Note that Notice t h a t the response curve is a sum of two exponential curves and a step function of magnitude 5. The equations of motion for the system are which can be m i t t e n as I Since we are interested in the steady state behavior of the system, we can assume that all initial conditions are zero. F m the d i a g m shown centrifuqal force gravitational force Solving for W B Since 5 is desired to be 0. If w n So the acceleration of the base is ptopartional. Define the displacement of spring k2 as y. RIPvibration amplitude X jW of ‘. Assuming ma11 angles Q1 and system rriay be obtained as fo Llm: Thus For constants A and B to be nonzero, ” ,determinant of t h e coefficients of Equations 3 and 4 must be equal t o zero, or This determinant equation determines the natural frequencies of the systemEquation 5. Thus The first natural frequency is w l first W e and the second natural frequency is L3 second mode. This means that both masses move the same munt in the same direction. This mode is depicted in Figure a below. This mode is depicted in Figure b shown b e l o w. Q Therefore, mupling exists between Equation 11 and Equation 2. To f i n d the natural frequencies f o r the system, assume the following harmonic motion: T3 sinWt Then, frm Equations [I and 2 we obtain For amp1itudes A and 3 to be nonzero, the determinant of the coefficients of Equations 13 and 4 must be equal to zero, or Thfs determinant equation determines t h e n a t u r a l frequencies of the system. Equation 5 can ke rewritten as which can be simplified to Notice that this last: Notice that the ratio of the displacements of springs k and k are 1 2 ‘Ihe f i r s t mcde of vibration is sham in Figure a on next page. The system shown in? Also, that satisfies Equations 7 – 3 8 and bcmes as follows: Figlures i3 and l u s r i m on next page d e p l c t the first mode of vibration and second mode of vibration, respectively. The equations of motion for the system are Substituting the given numerical mlues into these two equations and simplifying, we have To find the natural frequencies of the free vibration, assume that the motion is harmonic, or Then. Frun Equations 3 and 4! Next, we s h a U obtain the vibrations x t and y t subjected to the given initial conditions. To obtain the responses x t and y t to the given initial conditions, as follows: The resulting plots x t 1 versus t and y t versus t are shown on next page. All necessary derivations of quatians for the system are given in Problem A For the initial condition. The resulting p l o t s are shown zistemas Figure b below. S,”x t ‘ text 3 -5, Let us define d i s p l a m n t s e, xand y as shown in the figure Fur relatively small angle 8 we can construct a block diagram as s h m below. Also, from the system diagram we see t h a t fox each m L l Mlue of yt h e r e is a corresponding value of angle dm Therefore, for each angle e of the control lever, there is a mrresponding steady-state elevator angle 8. SOLUCIONARIOS DE LIBROS DE INGENIERÍA We B If the engine spsed increases, the sleeve of the fly-ball -mar moves upward. This movement acts as the input to the hydraulic tontroller. A positive error signal upward motion of the sleeve causes the power p i s disfreto a to move dawnward, reduces the fuel mlve apeningc and decreases the engine Referring to Figure a shorn below, a block diagram for t h e system speed. For the first-order system the sisteemas response isstemas is an q n e n t i a l 50 duscreto the constant T can k determined from such an exponential m ily. For a second-order system: A typical response c u mwhen this thermwneter is p l a d timpo a bath held at a constant temperature, is shown below. The closed-loop transfer function of the system is For the unit-step input, w e have B So we have or Iherefore, disrceto To make the system s t a b l eit is necessary to reduce the gain o f the system or add an appropriate -nsator. The characteristic equation is The Routh array for this equation is 3 5 6 R For the systm to be stable, there should be no sign changes in the first column. Since the system is of higher order 5th orderit is easier to contrl the range of gain K f o r stability by first plotting the r m t laci and then finding critical p i n t s for stability on the soot loci. The resulting root-locus plat is s h m also on n page. Based on this plot, it can. Pill critical points for stability lie on the j axis. The gain values at these crossing p i n t s are obtainel a s follows: Hence, we first f i n d the crossing frquency and then find the m r r e s p d i n g gain m l u e.

- Burnside's problem The Burnside problem, posed by William Burnsidein 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. In plain words, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see Bounded and Restricted below) that differ in the additional conditions imposed on the orders of the group elements. Initial work pointed towards the affirmative answer. For example, if a group "G" is generated by "m" elements and the order of each element of "G" is a divisor of 4, then "G" is finite. Moreover, A. I. Kostrikin(for the case of a prime exponent) and Efim Zelmanov(in general) proved that, among the finite groups with given number of generators and exponent, there exists a largest one. Nevertheless, the general answer to Burnside's problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, P. S. Novikovand S. I. Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskiifound some striking counterexamples for sufficiently large odd exponents (greater than 1010), and supplied a considerably simpler proof based on geometric ideas. The case of even exponents turned out to be much harder to settle. In 1992 S.V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established negative solution for an analogue of Burnside's problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, very little is known when exponents are small, exponents 2,3,4 and 6 excepted. General Burnside problem A group "G" is called periodic if every element has finite order; in other words, for each "g" in "G", there exists some positive integer "n" such that "g""n" = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the "p"∞-group which are infinite periodic groups; but the latter group cannot be finitely generated. The general Burnside problem can be posed as follows: : If "G" is a periodic group, and "G" is finitely generated, then is "G" necessarily a finite group? This question was answered in the negative in 1964 by E.S. Golodand I.R. Shafarevich, who gave an example of an infinite "p"-group which is finitely generated (see Golod-Shafarevich theorem). However, the orders of the elements of this group are not a priori bounded by a single constant. Bounded Burnside problem Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group. Consider a periodic group "G" with the additional property that there exists a single integer "n" such that for all "g" in "G", "g""n" = 1. A group with this property is said to be "periodic with bounded exponent" "n", or just a "group with exponent" "n". Burnside problem for groups with bounded exponent asks: : If "G" is a finitely generated group with exponent "n", is "G" necessarily finite? It turns out that this problem can be restated as a question about the finiteness of groups in a particular family. The free Burnside group B("m", "n") of rank "m" and exponent "n", denoted B("m", "n"), is a group with "m" distinguished generators "x"1,…,"x""m" in which the identity "x""n" = 1 holds for all elements "x", and which is the "largest" group satisfying these requirements. More precisely, the characteristic property of B("m", "n") is that, given any group "G" with "m" generators "g"1,…,"g""m" and of exponent "n", there is a unique homomorphism from B("m", "n") to "G" that maps the "i"th generator "x""i" of B("m", "n") into the "i"th generator "g""i" of "G". In the language of group presentations, free Burnside group B("m", "n") has "m" generators "x"1,…,"x""m" and the relations "x""n" = 1 for each word "x" in "x"1,…,"x""m", and any group "G" with "m" generators of exponent "n" is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if "G" is any finitely generated group of exponent "n", then "G" a homomorphic image of B("m", "n"), where "m" is the number of generators of "G". Burnside's problem can now be restated as follows: : For which positive integers "m", "n" is the free Burnside group B("m","n") finite? The full solution to Burnside's problem in this form is not known. Burnside considered some easy cases in his original paper: *For "m" = 1 and any positive "n", B(1, "n") is the cyclic groupof order "n". *B("m", 2) is the direct product of "m" copies of the cyclic group of order 2. The key step is to observe that the identities "a"2 = "b"2 = ("ab")2 = 1 together imply that "ab" = "ba", so that a free Burnside group of exponent two is necessarily abelian. The following additional results are known (Burnside, Sanov, M. Hall): *B("m",3), B("m",4), and B("m",6) are finite for all "m". The particular case of B(2, 5) remains open: as of 2005, it is not known whether this group is finite. The breakthrough in Burnside's problem was achieved by P.S. Novikovand S.I. Adianin 1968.Using a complicated combinatorial argument, they demonstrated that for every odd number "n" with "n" > 4381, there exist infinite, finitely generated groups of exponent "n". Adian later improved the bound on the odd exponent to 665. [John Britton proposed a nearly 300 page alternative proof to the Burnside problem in 1973; however, Adian ultimately pointed out a flaw in that proof.] The case of even exponent turned out to be considerably more difficult. It was only in 1992 that S.V. Ivanov was able to prove an analogue of Novikov–Adian theorem: for any "m" > 1 and an even "n" ≥ 248, "n" divisible by 29, the group B("m", "n") is infinite. Both Novikov–Adian and Ivanov established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two dihedral groups, and there exist non-cyclic finite subgroups. Moreover, the word and conjugacy problems were shown to be effectively solvable in B("m", "n") both for the cases of odd and even exponents "n". A famous class of counterexamples to Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters. First examples of such groups were constructed by A.Yu. Ol'shanskiiin 1979 using geometric methods, thus affirmatively solving O.Yu.Schmidt's problem. In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number"p" (one can take "p" > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic groupof order "p". In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic groupfor sufficiently large exponents. Restricted Burnside problem The restricted Burnside problem (formulated in the 1930s) asks another related question: : If it is known that a group "G" with "m" generators and exponent "n" is finite, can one conclude that the order of "G" is bounded by some constant depending only on "m" and "n"? Equivalently, are there only finitely many "finite" groups with "m" generators of exponent "n", up to isomorphism? This variant of the Burnside problem can also be stated in terms of certain universal groups with "m" generators and exponent "n". By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index. Let "M" be the intersection of all subgroups of the free Burnside group B("m", "n") which have finite index, then "M" is a normal subgroup of B("m", "n") (otherwise, there exists a subgroup "g" -1"Mg" with finite index containing elements not in "M"). One can therefore define a group B0("m","n") to be the factor group B("m","n")/"M". Every finite group of exponent "n" with "m" generators is a homomorphic image of B0("m","n").The restricted Burnside problem then asks whether B0("m","n") is a finite group. In the case of the prime exponent "p", this problem was extensively studied by A.I. Kostrikinin the 1950s (before negative solution of the general Burnside problem). His solution, establishing the finiteness of B0("m","p"), used a relation with deep questions about identities in Lie algebras in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medalin 1994 for his work. S.I. Adian(1979) "The Burnside problem and identities in groups". Translated from the Russian by John Lennox and James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete[Results in Mathematics and Related Areas] , 95. Springer-Verlag, Berlin-New York. ISBN 3-540-08728-1 * S.V. Ivanov (1994) "The free Burnside groups of sufficiently large exponents," "Internat. J. Algebra Comput. 4". * S.V. Ivanov, A.Yu. Ol'shanskii (1996) " [http://www.ams.org/tran/1996-348-06/S0002-9947-96-01510-3/home.html Hyperbolic groups and their quotients of bounded exponents,] " "Trans. Amer. Math. Soc. 348": 2091-2138. * A.I. Kostrikin (1990) "Around Burnside". Translated from the Russian and with a preface by James Wiegold. "Ergebnisse der Mathematik und ihrer Grenzgebiete" (3) [Results in Mathematics and Related Areas (3)] , 20. Springer-Verlag, Berlin. ISBN 3-540-50602-0. * A.Yu. Ol'shanskii (1989) "Geometry of defining relations in groups". Translated from the 1989 Russian original by Yu. A. Bakhturin (1991) "Mathematics and its Applications" (Soviet Series), 70. Dordrecht: Kluwer Academic Publishers Group. ISBN 0-7923-1394-1. * E. Zelmanov (1990) "Solution of the restricted Burnside problem for groups of odd exponent". (Russian) "Izv. Akad. Nauk SSSR Ser. Mat. 54", no. 1: 42-59, 221; translation in "Math. USSR-Izv. 36" (1991), no. 1: 41-60. * E. Zelmanov (1991) "Solution of the restricted Burnside problem for 2-groups". (Russian) "Mat. Sb. 182", no. 4: 568-592. Translation in "Math. USSR-Sb. 72" (1992), no. 2, 543-565. * [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Burnside_problem.html History of the Burnside Problem] at MacTutor History of Mathematics archive Wikimedia Foundation. 2010.

Caldaie e Scaldabagni Guarda il video su youtube 3 minuti 56 secondi riscaldamento apparecchi materiali caldaie mobili uso industriale If I say math ... If I say math doesn't exist to my math teacher, does that mean I can get out of tests? Even science on ... Even science on some level requires faith. Not only faith that our observations are real but also faith in the postulates that form the foundation for higher scientific thought. stop with the memes ... stop with the memes its not 2010 Math is just a ... Math is just a language. Asking "is math real" is like asking "is spanish real" My take as an anti- ... My take as an anti-realist: For me, math is seemingly created. We invented math as a way for us to be more concise about measurements, designs, ratios, and so forth. Mathematics does not happen naturally in tje universe, and unlike biochemistry, biology, and any life science, we cannot see it happen without our intervention. I do, however, love math now, and I consider it one of the sole best inventions, ever. Mathematics is part ... Mathematics is part of universe, even though you cant see it or smell it. Anything that takes up space in the universe has a definite value. Simply by existing it has an intrinsic value. If math only exists ... If math only exists inside our brain, why do we DISCOVER it? We can't discover something if we knew it all along...or is that a stupid question? Philip J. Fry *holds brain* ;-; *holds brain* ;-; Math is a law bound ... Math is a law bound by this universe. Math is a reality!! Men binds this universe Well the fact is ... Well the fact is that although maths is a very good tool to describe and predict the universe, it is only an approximation. I think the anti- ... I think the anti-realistic way is more consistent. I think we can see that we sort of "create" geometry, for example. The natural geometry for us is obviously an Euclidean Geometry, but as physicists saw that it doesn't really apply to our world, then they had to search for another kind of “non natural” geometry that they call Riemannian Geometry which, as far as I know, is more appropriate to describe relativistic phenomena. So maybe, if we find out some phenomena where it's better to think that 1+2 doesn't equal to 3, someone would end up by developing new concepts to “create” a new mathematic where 1+2 differs from 3. You have overlooked ... You have overlooked one crucial thing. Science is faith as well, People used to think that the earth is the center of the universe. Most people had faith in it thus it was science. We all think that the Sun is the center now but that may not really be the case. Science may also be a creation of the human mind. In fact, if you think about it, nothing can be really said to be of the universe I think it does ... I think it does exist. If there was water, there was 2 hydrogen atoms and 1 oxygen atom, 1 molecule made by 2+1=3 atoms. Math exists exactly ... Math exists exactly as what it is; the universe exists as exactly what it is; the universe is complex physical math equations (the interactions of quantity, geometry); math can only exist first with physicality, and since the universe exists, and potentially will always exist, I suppose there is a realm of forms like perfect abstract essence of math which is timeless and beyond the manifestations of substance at any given time; When the first man said; let 1 = 1 and 0 = 0; and 2 = 1 + 1; before that man thought or said of anything else, it was already true that 1 + 1 + 1 = 3; and so it seems as if the self consistency of math follows from the simplest axioms, which may be timeless, and 'exist' even if they dont exist, perhaps in a sense of the realm of potential; How in ways 'tommorow' exists today, or its knowledge can effect now;....maybe Math, to me, Is an ... Math, to me, Is an accurate description of the universe, as we know it. And just like any scientific theory, Something new could possibley come by and rock that boat. However, the very same theories are what we have built our entires lives off of, And have yet to fail. That should be good enough for us XD . So what I think is yes , in a sense, Math is a creation of brains, seperated from physical existence. However, this creation's most basic requirments are that of reality. Math is always constantly and by many people Compared to the physical world. Therefore the minute math is not true, We simply make more math. I also think that its unfair to even be worried that math is not true. I have been taking a Discrete Structures course, and have chosen to believe all of the predicates and propositional logic it has brought. There is a chance I will have chosen wrong, just as with all other things I believe it falls ... I believe it falls onto the math realists to explain whether or not that math exists. *They* are the one making a claim that math exists, so *they* have to prove it. (I'm hoping you notice that this is an analogy of another hot topic...) An important ( ... An important (philosophical) question... My worry: Fictional or real, it is a fact that logic (math) is compatible with the functions in the universe to an immense range of scales that surpass human experience (hence, I believe, evolution is not enough to give a satisfactory answer given that this knowledge became available a few hundred years ago - give or take). An this range is reasonable to expect that will become much wider in the next few hundred years. So, what is the relation of the human brain function to the function of physical phenomena that are describable in terms of math (and many of them lie beyond the everyday experience)? Is Math a Feature of the Universe or a Feature of Human Creation? | Idea Channel | PBS Wait, are we sure ... Wait, are we sure we can see "physics"? We can see objects (kinda) and we describe those objects with equations and principles of physics, but we can't see Newton's first law. We can see objects (kinda) that obey the first law, but we can't see the first law. We can see objects that obey mathematical correlations too so I guess I don't see the difference clearly. We Are Showboat I've always thought ... I've always thought that mathematics where just creations of the human brain. Mathematics is basically just the study and advancement of methods that can help answer human questions and solve human problems in an effective manner, or thats what I've always considered math to be... Great Video from ... Great Video from PBS Idea Channel on whether or not math actually exist. Brandy A. Hyatt I think that math ... I think that math concepts exist in the universe. However, we did create a language of mathematical symbols and equations because that is the only way we can understand it. The problems exist ... The problems exist in the universe, the math exists in the brain. Maths is more of a ... Maths is more of a tool than a language IMO. The most powerful tool humanity ever or more probably could ever make if I may add. I wasn't mad you ... I wasn't mad you said my mother was a hamster, but for GOD's SAKE why did you have to say my father smells of elder berries? He has a condition, Matt, not cool. Yeah, I'd say that math exists and yet doesn't exist. Essentially, math is a construct, an invention, one of(if not the) the greatest inventions of humanity. Yet, the concepts that mathematics deals with, and extrapolates on, exist. Math does not exist. It is the invented perception of humanity, through this perception we can conceive and perceive things we could only theorize on previously. Math in essence is a perception. A way in which we take the nearly (if not) infinite and omnipresent signals of the Cosmos, and translate that to something tangible, relatable, "real", or even manipulatable. Just like a sensory organ, and your central nervous system. gas domestico bombole a torino Torino caldaie scaldabagni vendita installazione a torino Torino caminetti barbecue vendita a torino Torino condizionatori climatizzazione a torino Torino legna ardere carbone vendita a torino Torino pulizia caldaie camini canne fumarie a torino Torino riscaldamento combustibili a torino Torino gas metano gpl bombole serbatoi a torino Torino pannelli solari risparmio energetico a torino Torino idraulici a torino Torino intimo moda mare a torino erboristeria prodotti a torino esportatori importatori a torino espositori display pubblicita punto vendita a torino essenze aromi alimenti a torino essenze fragranze profumeria a torino estintori produttori grossisti a torino etichettatura marcatura macchine sistemi a torino etichette tessute stampate a torino facchinaggio carico scarico merci a torino Visiant Contact s.r.l. per creare la tua , la tua e il tuo

According to DeepMind research published online on Tuesday, their algorithms were trained in algebra, calculus, probability, and other types of math topics. However, after testing these algorithms, Mathematics Events. Math Placement Test (for current UWRF students) All students who have not taken college-level courses in math are required to take the Wisconsin Regional Placement Test before registering for a math class. Mathematics Used In Robotics Pdf Both students will receive a $1,000 scholarship through Indiana’s CollegeChoice 529 Direct Savings Plan and were selected for. The forecast is analyzed based on the volume and revenue of this market. The tools used for analyzing the Global Oil and Gas Robotics Market research report include SWOT analysis. For Early Buyers |. Develop advanced statistical Course Summary Math 104: Calculus has been evaluated and recommended for up to 6 semester hours and may be transferred to over 2,000 colleges and universities. 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Tutoring & homework help for math, chemistry, & physics. Homework & exam help by email, Skype, Whatsapp. I can help with your online class. Free study guides, cheat sheets, & apps. Calculus. what happens after, though it’s a bit of a misnomer, since an initial condition can also come from the middle or end of a graph. Robert Coolman is a contributing writer for Live Science.

A quaternionic structure on a real vector space is a module structure over the skew-field of quaternions , that is, a subalgebra of the algebra of endomorphisms of induced by two anti-commutative complex structures on (cf. Complex structure). The endomorphisms are called standard generators of the quaternionic structure , and the basis of defined by them is called the standard basis. A standard basis is defined up to automorphisms of . The algebra is isomorphic to the algebra of quaternions (cf. Quaternion). An automorphism of the vector space is called an automorphism of the quaternionic structure if the transformation of the space of automorphisms induced by it preserves , that is, if . If, moreover, the identity automorphism is induced on , then is called a special automorphism of the quaternionic structure. The group of all special automorphisms of the quaternionic structure is isomorphic to the general linear group over the skew-field , where . The group of all automorphisms of a quaternionic structure is isomorphic to the direct product with amalgamation of the subgroup and the group of unit quaternions . A quaternionic structure on a differentiable manifold is a field of quaternionic structures on the tangent spaces, that is, a subbundle of the bundle of endomorphisms of tangent spaces whose fibres are quaternionic structures on the tangent spaces for all . A pair of anti-commutative almost-complex structures on the manifold is called a special quaternionic structure. It induces the quaternionic structure , where A quaternionic structure on a manifold is induced by a special quaternionic structure if and only if the bundle is trivial. A quaternionic structure on a manifold can be regarded as a -structure, and a special quaternionic structure as a -structure in the sense of the theory of -structures (cf. -structure). Hence, in order that a quaternionic structure (or a special quaternionic structure) should exist on a manifold , it is necessary and sufficient that the structure group of the tangent bundle reduces to the group (or ). The first prolongation of a special quaternionic structure, regarded as a -structure, is an -structure (a field of frames), which determines a canonical linear connection associated with the special quaternionic structure. The vanishing of the curvature and torsion of this connection is a necessary and sufficient condition for the special quaternionic structure to be locally equivalent to the standard flat special quaternionic structure on the vector space . A quaternionic Riemannian manifold is the analogue of a Kähler manifold for quaternionic structures. It is defined as a Riemannian manifold of dimension whose holonomy group is contained in the group . If , then the quaternionic Riemannian manifold is called a special or quaternionic Kähler manifold, and it has zero Ricci curvature. A quaternionic Riemannian manifold can be characterized as a Riemannian manifold in which there exists a quaternionic structure that is invariant with respect to Levi-Civita parallel displacement. Similarly, a special quaternionic Riemannian manifold is a Riemannian manifold in which there exists a special quaternionic structure that is invariant with respect to Levi-Civita parallel displacement: , where is the operator of covariant differentiation of the Levi-Civita connection. In a quaternionic Riemannian manifold there exists a canonical parallel -form that defines a number of operators in the ring of differential forms on that commute with the Laplace–Beltrami operator (exterior product operator, contraction operators). This enables one to construct an interesting theory of harmonic differential forms on quaternionic Riemannian manifolds analogous to Hodge theory for Kähler manifolds, and to obtain estimates for the Betti numbers of the manifold (cf. Hodge structure; Betti number). Locally Euclidean spaces account for all the homogeneous special quaternionic Riemannian manifolds. As an example of a homogeneous quaternionic Riemannian manifold that is not special one may cite the quaternionic projective space and also other Wolf symmetric spaces which are in one-to-one correspondence with simple compact Lie groups without centre (cf. Symmetric space). These account for all compact homogeneous quaternionic Riemannian manifolds. A wide class of non-compact non-symmetric homogeneous quaternionic Riemannian manifolds can be constructed by means of modules over Clifford algebras (see ). |||S.-S. Chern, "On a generalization of Kähler geometry" R.H. Fox (ed.) D.C. Spencer (ed.) A.W. Tucker (ed.) , Algebraic geometry and topology (Symp. in honor of S. Lefschetz) , Princeton Univ. Press (1957) pp. 103–121 MR0087172 Zbl 0078.14103| |||V.Y. Kraines, "Topology of quaternionic manifolds" Trans. Amer. Math. Soc. , 122 (1966) pp. 357–367 MR0192513 Zbl 0148.16101| |||K. Yano, M. Ako, "An affine connection in an almost quaternionic manifold" J. Differential Geom. , 8 : 3 (1973) pp. 341–347 MR355892| |||A.J. Sommese, "Quaternionic manifolds" Mat. Ann. , 212 (1975) pp. 191–214 MR0425827 Zbl 0299.53023| |||D.V. Alekseevskii, "Classification of quaternionic spaces with a transitive solvable group of motions" Math. USSR Izv. , 9 : 2 (1975) pp. 297–339 Izv. Akad. Nauk SSSR Ser. Mat. , 39 : 2 (1975) pp. 315–362 MR402649 Zbl 0324.53038| |||J.A. Wolf, "Complex homogeneous contact manifolds and quaternionic symmetric spaces" J. Math. Mech. , 14 : 6 (1965) pp. 1033–1047 MR0185554 Zbl 0141.38202| |||D.V. Aleksevskii, "Lie groups and homogeneous spaces" J. Soviet Math. , 4 : 5 (1975) pp. 483–539 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 11 (1974) pp. 37–123| Quaternionic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternionic_structure&oldid=33885

We study the BTW-height model of self-organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network. We find that as function of the fraction of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents and for size and lifetime respectively. Self-organized Criticality on Small World Networks L. de Arcangelis and H.J. Herrmann Department of Information Engineering, Second University of Neaples INFM Neaples UdR and CG SUN, Via Roma 29, I-81031 Aversa (CE), Italy Institute for Computer Applications 1, University of Stuttgart Pfaffenwaldring 27, D-70569 Stuttgart Small world networks have recently been observed to exist in many physical, biological and social cases [1, 2, 3, 4, 5]. A simple minded example is the structure of the neo-cortex and the network of acquaintances in certain societies. Another ubiquitous phenomenon (SOC) [6, 7, 8] in nature and society is self-organized criticality, i.e. the appearance of avalanches of all sizes without characteristic scale over a certain range. Easily one can imagine cases in which both of these phenomena occur simultaneously, that means that we find an avalanche type spreading on a sparsely long-range connected network. As an example we propose the spreading of neural information inside the cervical cortex which is due to the threshold behaviour given by the firing rule which automatically induces avalanches of synapses. Another example one can imagine are societies where each individual has a certain threshold of endurance. Let us take the example of Peter Grassberger for the classical SOC-model of office clerks moving sheets of papers from desk to desk, as illustrated in Peter Bak’s book in Fig. 13 and admit that the clerks do not sit on a square lattice but have a more realistic connectivity in their work relation (typically small world network behaviour). Many other examples of this kind can be thought of. In this short paper we want to present a model for self-organized criticality on a graph having the properties of small world networks. In fact, we consider the classical height model of Bak, Tang and Wiesenfeld (BTW) on a square lattice which has been “rewired” for a certain fraction of bonds which are chosen of arbitrary range. For that purpose we take a square lattice of linear size and all bonds present between nearest neighbours sites. Then we choose randomly two sites of the system and place a bond between them (which can therefore be of any length smaller than ). In order to keep the coordination of the sites on average to be four, one of the smaller bonds going to a neighbouring site of one of the end points of one long bond is removed. This procedure repeats until a fraction of all bonds has been replaced by long range connections. This type of graph is not the same as the one used by most authors working on small world networks, because the underlying short range lattice is not a linear chain but a square lattice. Nevertheless the properties should qualitatively be the same: large world behaviour for large and a small world behaviour for small . The situation corresponds to the simple square lattice and to random graph with average coordination 4 and long range connections (Viana-Bray) on which one would expect mean-field behaviour. On each site of the lattice we place an integer value less than a threshold . At each time step randomly one site is chosen and its value is increased by unity, i.e. a unit mass is added to it. When the value of the height having neighbours reaches the threshold , it topples, i.e. it distributes mass equally to its neighbours: where goes over all neighbors of site . We see that eq. (1) preserves the mass (the sum of all heights). We have studied the statistics of avalanches monitoring as well their size (number of sites that toppled at least once) as their lifetime , that means the number of time step an avalanche lives. The quantities and denote the avalanche size and life time distributions. For the case , that means the classical BTW-model on the square lattice it is known that asymptotically with and in two dimensions. It is our aim here to investigate what happens for different values of . To this purpose we have analysed square lattices of size and and a range of values from to . The lattice has spiral boundary conditions in one direction and two open boundaries at top and bottom, where mass in excess can flow out of the system. In each configuration we have injected randomly 2500 particles of unit mass, letting each time the avalanche proceed until no site had a value of . The data were averaged over 200, 50 and 10 configurations for and respectively. In Fig. 1 we see the distribution of avalanche sizes for different system sizes and two different values of and in a double logarithmic plot. We see that the data follow a straight line for nearly two decades indicating that we still find SOC-behaviour. The slope of the straight line gives us the exponent . In Fig. 2 we see the corresponding figures for the distribution of life times. Again the data show power law behaviour and the slope gives the exponent . We observe that the exponents and depend on but they do not appear to depend on . In Fig. 3 we see the dependence of and on as obtained from our simulations. We see that for we obtain the classical result on the square lattice of BTW , . For close to unity the values of the exponents convert to the meanfield values and [9, 10, 11]. This is of course not surprising and the question is if the continuous change of the exponent and is an intrinsic continuous line of critical points or, if we have here a crossover phenomenon as it appears for instance in magnetic models that interpolate between 2 and 3 dimensions. For that purpose we tried for both the size and the lifetime avalanche data in Fig. 4, a data collapse of all the distributions for different values of following the classical crossover scaling where and are scaling functions and and are universal crossover exponents. We see from Fig. 4 that a collapse of the data works reasonably well yielding crossover exponents of for the avalanche size distribution and for the lifetime distribution. By studying the BTW-model on a small world square network we observed a crossover to meanfield behaviour following a crossover scaling law of eq. (3). It would be interesting to see if the same occurs for the simpler Manna-model and we have heard calculations are already under way . This work has been partially supported by the European TMR Network-Fractals under contract No.FMRXCT980183 and by MURST-PRIN-2000. - Watts, D.J., Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998). - Newman, M.E.J., Moore, C., Watts, D.J. Mean-Field Solution of the Small-World Network Model. Physical Review Letters 84, 3201–3204 (1999). - Newman, M.E.J., Watts, D.J. Scaling and percolation in the small-world network model. Physical Review E 60, No. 6, 7332–7342 (1999). - Collins, J.J., Chow, C.C. It’s a small world. Nature 393, 409 (1998) - Amaral, L.A.N., Scala, A., Barthélémy, M., Stanley, H.E. Proc.Nat.Ac.Sci USA 97, 11149-11152 (2000). - Bak, P., Tang, C. and Wiesenfeld K. Self-Organized Criticality. An Explanation of Noise. Physical Review Letters 59, 381 (1987). Self-Organized Criticality. Physical Review A 38, 364 (1988). - Bak, P. How nature works. The science of self-organized criticality. Springer, New York (1996). - Manna, S.S. J. Phys. A 24, L363 (1991). - Tang, C., Bak, P. J.Stat.Phys. 51, 797 (1988). - Janowsky, S.A., Laberge, C.A. J. Phys. A 26, L973 (1993). - Flyvbjerg, H., Sneppen, K., Bak, P. Phys. Rev. Lett. 71, 4087 (1993). Stanley, H.E. et al.

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1. Species-abundance distributions (SADs) are a convenient and common method for describing ecological communities. Despite their long history and the cornucopia of theoretical models, which have been suggested to describe them, no agreement has been reached as to which models are best. 2. This lack of agreement is in part owing to the inherent differences in the abundance measure used. Discrete measures such as density and point quadrat cover produce a distinct veil line (positive skewness) when compared to continuous measures such as biomass or basal area. We compared two different sets of discrete and continuous abundance measures commonly used to estimate plant abundance, (i) cover (estimated from point quadrats) vs. biomass for 35 quadrats in garigue vegetation on serpentine soil in Tuscany, Italy; and (ii) density vs. basal area for the 2005 50 ha BCI (Panama) tree data. We used marginal plots (ordinary scatter plots with a dotplot of each variable along its own axis) to compare the shape of the SAD based on the two abundance measures. 3. The average of all 35 garigue plots gave a reasonably consistent description of the data. In contrast, when all 35 plots were concatenated, or when an individual plot was investigated, the discrete cover marginal plot, but not the continuous biomass plot, was truncated. The discrete density marginal plot, but not the continuous basal area plot was also truncated. 4. We highlighted the substantial effect that the species-abundance measure selected has on the shape of the SAD by comparing measures of skewness and kurtosis. This suggests that communities sampled using different abundance methods may fit different theoretical models best, not because they are fundamentally different but because the abundance measure is fundamentally different. Averaging over all the quadrats produced a better correspondence between the two abundance methods. Aside from the theoretical aspects of model fitting, our results clearly show that comparisons of communities or meta-analyses using SADs based on different measures of abundance should be treated with caution. Species-abundance distributions (SADs), also called relative-abundance distributions (RADs), record the relative or absolute number of a set of species in a sample. As a description of an ecological community, they sit conveniently between a simple listing of the species present and multidimensional analysis (McGill et al. 2007) and so have been much studied. Generally, they are unimodal on a logarithmic scale of abundance and that has lead ecologists to seek a simple mathematical form for SADs. Despite this simplicity, McGill et al. (2007) list 27 models that have been suggested and note an absence of agreement about which models are best. Chiarucci et al. (1999) calculated SADs for 35 individual plots, fitting broken-stick, geometric, lognormal and Zipf–Mandelbrot functions (models 20, 19, 6 and 11, respectively, in McGill et al. 2007) on point quadrat cover and biomass data. The most frequent (22 plots) best fitting model for the point quadrat cover data was the Zipf–Mandelbrot, and in contrast, the most frequent best fitting model for the biomass data (16 plots) was the lognormal. The Zipf–Mandelbrot model was the best fitting model for biomass data in 10 plots, whilst the lognormal was the best fitting model for the point quadrat cover data in only two plots. There are two simple reasons for this lack of agreement. The first is that none of the models is additive (Williamson & Gaston 2005; Šizling et al. 2009) over either taxa or areas; indeed Šizling et al. (2009) show that such an additive model involves an indefinitely large number of parameters. This means that a model that fits at one scale or one set of species cannot also fit at another scale or over an enlarged set of species. The second, and the one we are concerned with here, is that the models are not invariant under different abundance measures; SADs change shape in relation to the abundance measure used. This has been recognised to some extent by reference to Preston’s veil line (Williamson 2010) or by classing data sets as fully censused vs. incompletely sampled (Ulrich, Ollik & Ugland 2010). But the veil line is an unsatisfactory approximation (Chisholm 2007; Williamson 2010) and all data sets are in some sense and to some degree samples, so sampling is a continuous (and universal) variable, not a discrete one. Chisholm (2007) has a thorough discussion of previous arguments, particularly Dewdney (1998). Williamson (2010) showed that the investigator choice of using individuals as opposed to biomass for sampling has a mathematical consequence; describing the phenomenon of differential veiling using marginal plots of the same community measured as individuals and as biomass. Much SAD work has been with taxa in which individuals are readily distinguished (e.g. Morlon et al. 2009; who studied trees, fishes, birds and mammals). For applied plant ecology, both biomass and density have serious drawbacks as abundance measures (Kershaw 1973). Collecting biomass data is destructive and time-consuming and plant density has the inherent problem that individuals are often difficult to distinguish, if it is possible at all, and where there may be great variation in size within a species (Jonasson 1988). Here, we consider the effect of sampling plant communities using different species-abundance measures. Objective abundance measures commonly used with plant communities often attempt to estimate cover, for example, point quadrat cover or local frequency (Greig-Smith 1964). Neither of these measure individuals per se but in common with density they are discrete variables, that is, the count of numbers of pin hits or subsquares occupied. In contrast, both biomass and basal area are continuous variables. Williamson (2010) linked the differential veiling to individuals. In contrast, we show that whilst differential veiling does occur with individuals this is not a property of individuals per se, rather differential veiling is a consequence of using a discrete rather than continuous abundance measure. Counting individuals is, along with a multitude of other sampling measures discrete, and it is the discreteness, not the individualness which produces the differential veil line. This is an important distinction, especially relevant to fields, such as plant sampling, where discrete abundance measures other than counting individuals are common. Materials and methods Point quadrat cover versus biomass data The point quadrat cover versus biomass data come from garigue vegetation on serpentine soil in Tuscany, Italy where 35 1 m × 1 m plots were surveyed in two ways (Chiarucci et al. 1999). Point quadrat cover estimates cover using the number of pins touching each species, in this case contacts out of 441 pins in each 1-m2 plot. Biomass was measured as the dry weight (after 48 h at 80°C) for each species in each plot. The plots were subject to seven different experimental treatments, but we ignore that here. We also ignore any species present in a plot but happening not to be touched by any pin, recorded as 0·5 pin in Chiarucci et al. (1999). We analysed three assemblage levels: (i) the average per plot for each species over all 35 plots, (ii) the concatenation of all the individual plots, that is, data for all 35 individual plots superimposed in one graph and (iii) the results for a single plot, we used plot 31 as this is the plot set out in detail in Chiarucci et al. (1999). Density versus Basal area data The density vs. basal area data come from 2005 BCI data (50 ha plot on Barro Colorado Island, Panama) (https://ctfs.arnarb.harvard.edu/datasets/BCI/abundance). They are freely available and are the first year in which all the individuals there were identified to species. We have used the trees with a diameter at breast height (d.b.h.) >10 cm as the resolution of measurement means that some saplings have a recorded basal area of zero. Instead of function fitting, biomass vs. point quadrat cover data and individuals vs. basal area data were investigated using marginal plots (Williamson 2010) and then calculating skewness and kurtosis for the four pairs of marginal distributions. Marginal plots are ordinary scattergrams for two variables with the addition of a dotplot of each variable along its own axis. Note that in the vertical dotplot, the low value (‘left-hand’) end is at the bottom. Dotplots were preferred to histograms as they bring out the singleton and doubleton values more clearly. The average over all 35 1-m2 plots (Fig. 1a) behaves as fully censused data in the sense of Ulrich, Ollik & Ugland (2010) even though it is still a small sample of garigue vegetation. Both cover (measured by points) and biomass (measured as dry weight) show reasonably symmetrical plots on logarithmic abundance scales. Table 1 shows that as usual in well-sampled data (Williamson 2010), both skewness and kurtosis tend to be negative, that is, the data are slightly left skew and platykurtic. But there are too few points, only 34, for either to be statistically significant. Table 1. Skewness and kurtosis of the marginal plots for both biomass and cover for the three cover versus biomass assemblage levels: A) the average of the 35 plots; B) the concatenation of the 34 plots individually and C) an individual plot (plot 31), as in Fig. 1 Mean of all plots Concatenation of all individual plots N (number of data points) P-values in brackets. Note that all signs are negative except for the skewness of individual cover plots (both all plots and plot 31). Those two positive values are shown in bold, as are significant probabilities. The point quadrat cover data for individual 1-m2 plots (Fig. 1b,c) behave as ‘incompletely sampled’ in the sense of Ulrich, Ollik & Ugland (2010), an incomplete sample of the garigue vegetation. However, the biomass data for the same individual plots are much more symmetrical, and in the sense of Ulrich, Ollik & Ugland (2010), they should be considered ‘fully censused’. Realistically, both the 1-m2 plots and the set of 35 of them are samples, whether of biomass or point quadrat cover, at their respective scales, but neither is a complete community. The seven different treatments across the plots cause a lot of scatter, but the dominance of singleton point quadrat cover estimates (i.e. a species sufficiently rare that it is hit by only one pin) and subdominance of doubleton point quadrat cover estimates is clear in the marginal dotplot (Fig. 1b). The scatter plot appears truncated at the singleton line. This is less clear when looking at an individual plot, such as plot 31 (Fig. 1c) because of the paucity of points, but can still be seen. Both the concatenation of all 35 plots (Fig. 1b) and the single plot 31 (Fig. 1c) show that the difference between biomass and point quadrat cover SADs comes from the point quadrat cover being measured discretely, with a cut-off at one sample pin, whilst biomass is a continuous variable. That difference leads to species with low cover being lost on sampling, whereas species with low biomass are lost only if they also have low cover. The different effects of sampling on (continuous) biomass and (discrete) point quadrat cover are clear on the skewness statistics. The skewness of biomass is nonsignificantly negative in all three graphs but the skewness of point quadrat cover shifts from a left skew (−0·302) over the whole 35-m2 sample to a right skew (+0·406) over the set of individual plots. Only the latter, based on a much larger set of points, is statistically significant but that makes the difference between the two estimates highly significant. The kurtosis values are negative in all six cases. Only one is formally, and highly, significant but the set has a probability of 1/32, significant at the 5% level. As with the point quadrat cover vs. biomass data the density vs. basal area data show an obvious truncation to the marginal plot of the discrete density data without a corresponding truncation in the continuous basal area data (Fig. 1d). Skewness and kurtosis values are given in Table 2 and behave as before. Table 2. Skewness and kurtosis of the marginal plots for both basal area and density (individuals >10 cm d.b.h.) for the BCI 2005 data N (number of data points) P-values in brackets. Note that all signs are negative except for the skewness of the density plot (shown in bold along with significant probabilities). Discussion and conclusion Chiarucci et al. (1999), Connolly et al. (2005) and Morlon et al. (2009) all noted that different methods of sampling lead to mathematically different SADs. Williamson (2010) found a simple way of expressing and explaining this: marginal plots, as in Fig. 1, show that the distinction between sampling by discrete abundance measures such as point quadrat cover or individuals and by continuous abundance measures such as biomass or basal area necessarily leads to the difference in SADs. Williamson (2010) emphasised the ‘individuals’ part of that explanation but the data of Chiarucci et al. (1999) reanalysed here show that it is in fact the ‘discreteness’ not the ‘individualness’ of the data that creates the effect. Cover measured by point quadrats is discrete but what is measured is not a set of individuals but a set of pin hits approximating cover. Williamson (2010) also said ‘biomass SADs are different from individuals SADs and need not have the same mathematical form’. The Chiarucci et al. (1999) data show that the description of SADs measured by discrete counts should be a discrete mathematical function whilst those measured by continuous values require a continuous mathematical function, an important difference with a major effect on the sampling properties of the SADs. This finding is confirmed by the BCI tree data and by another plant study, Guo & Rundel (1997), on postfire vegetation in Californian chaparral. Because of the vegetation type Guo and Rundel could measure species abundance as individuals, as cover (assessed by eye) and as biomass. They fitted no functions, but their cumulative SADs show a clear dominance of singletons in their discrete individuals data but not in the other two continuous forms. If avoiding truncation of the SAD is important to the study then the abundance measure used must be continuous in nature, for example, biomass. On the other hand if using a discrete abundance measure, for example, density, point quadrat cover, local frequency, is desirable or unavoidable the truncation effect this will have on the SAD needs to recognised as a sampling artefact. The effect will be most noticeable in small samples.

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Tips for Maximizing Your Savings with Compound Interest Here are some tips to help you make the most of compound interest: The earlier you start investing, the more time your savings have to grow. Even small investments made early on can lead to significant savings over time. Reinvest Your Earnings Reinvesting your earnings can help you maximize your savings. By earning interest on interest, your savings can grow at an exponential rate. Consider a High-Interest Savings Account A high-interest savings account can offer a better interest rate than a traditional savings account. This can help you earn more interest on your initial investment. Consistency is key when it comes to saving and investing. By staying committed to your savings goals and regularly contributing to your investments, you can maximize your savings over time. An online compound interest calculator is a valuable tool for anyone looking to manage their finances and maximize their savings. 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Compound interest is generally considered better than simple interest because it allows your savings to grow at an increasing rate over time. With simple interest, you only earn interest on the initial investment, whereas compound interest allows you to earn interest on both the initial investment and any interest earned over time. Can I use an online compound interest calculator for investments other than savings accounts? Yes, an online compound interest calculator can be used for any investment that earns compound interest, including bonds, stocks, and mutual funds. Are online compound interest calculators accurate? Yes, online compound interest calculators are typically accurate as long as you input the correct information. Be sure to double-check your inputs to ensure accurate results.

Started by jamespetts, December 08, 2015, 12:31:28 AM 0 Members and 1 Guest are viewing this topic. QuoteTo modify the work done economy issue, it might be worthwhile for players to be able to specify a maximum speed for each point to point journey in the schedule; I am planning on adding additional features to the schedule in any event. QuoteTrying to average out the performance simply won't work because of the orders of magnitude difference between slow freight haulage and fast passenger haulage in fuel consumed per unit of distance. QuotePart of the point of additional realistic complexities in Experimental is to force players to use realistic heuristic methods of estimating what the best solution is, rather than just calculating it mathematically as is possible (but much, much more tedious) in a simpler simulation. Quote from: DrSuperGood on December 09, 2015, 11:17:06 PMWhich is why you use the rolling work done. You then ignore acceleration completely, equivalently giving you free acceleration. However it is not free, as you factor in realistic work done while accelerating to an inefficiency applied to rolling work done. This means it is free for your trains to decelerate and accelerate all the time, but instead you are paying slightly more for the work done per km travelled. This avoids the need for highly complex features like speed limits. QuoteIn real life tens of thousands of hours are spent planning railways. Every day dozens of people work on railway schedules. There are no "heuristic methods" of building railways. There never were as even the very early ones would have had thousands of hours spent planning.People play because it is a game, not because it is several times harder to play than real life as you are one person. If Experimental was truly realistic, one person would struggle to even manage a single line.One would have to cancel such complexities out with autonomous managers, planners, and other features. Which might end up making a line is as simple as specifying two points, seeing if long term projected profit is positive and pressing confirm. QuoteSo, if the per km costs for trailers are (almost) zero, it means that a 1 wagon train has the same per km cost as 100 wagon train ? QuoteOr did you calculate the running costs for engines to include a fully loaded train of maximum length it can pull? QuoteIn practice there are also maintenance tasks that have to be done after certain amount of km running. These should be accounted for too. QuoteIn the meantime, if anyone else would like to start that discussion, it would be very worthwhile. This is the first time that we have had even an approximate cost balancing for Pak128.Britain (either Standard or Extended), and I am very grateful indeed to Dr. Supergood for the large amount of work that this must have entailed. Even though this may well be some way different from the final balance once the features are implemented in due course, this is still a significant achievement and advance for the pakset. Quote(1) the planned way of simulating inflation allows the simulation of differential inflation, in which labour costs, fuel costs, ticket prices, materials costs and other things change over time independently of each other so that the changing relationships between these things can be simulated (which was important in reality, especially changes in labour costs); Quote(2) (less importantly but still significantly) general inflation will prevent players accumulating large cash stockpiles early in the game which retain their value until the later part of the game (allowing, e.g., an 18th century canal empire to fund a 21st century airline). Quote from: DrSuperGood on June 22, 2018, 08:30:53 PMThe only reason a cannel company would accumulate fortunes like my own company on the server is because the player who controls it lacks enough time to spend the money as fast as the company earns it. QuoteDr SuperGood How would you like your feedback to the changes? I'm certain people are happy to share their savegames, write out their experiences and thoughts and/or show screenshots. QuoteGoods: The system I use for goods is mixed trains across the map, connecting industries that are convenient or have high I/O. The trains are doing better although I still have trouble making lines profitable. Road and Naval sections incur losses. The goods network is neither complete nor well designed, so I doubt it's a representative of goods focused playing. QuoteIn relation to the prices of goods - these are all set relative to passengers using historical data, so it would not be right to double or triple the prices. Work does need to be done at some point to improve the balance between producer and consumer industries, however. Quotebut now the latter is 1/3 of the price of the former QuoteYou can find my calculation of the fuel efficiency of all steam engines in the Steam physics calc.ods document in the sources folder: QuoteIn terms of ship physics, we may need to look into this in more detail. Does anyone have any data against which the ship physics can be tested? The main thing about ship physics that is different from the physics of other vehicles is the "rolling" resistance (i.e., resistance in the water). Quotethe rate of acceleration is of minimal importance. What is really important for aircraft and ships therefore is just the top speed, at which they will be travelling most of the time. In principle, it should be much easier to balance the physics for these than for rail vehicles (and, to a lesser extent, road vehicles). QuoteI have been wondering about a speed cap feature: that should not in itself be excessively complicated to implement. However, one significant problem of a speed cap feature and also a weight dependent running costs feature is how to communicate the running costs to the player when the formula for calculating them is very complex. Have you any idea of a sensible and clear way of explaining these running costs in the user interface (and also of researching the relevant running costs for ships)? QuoteFor aircraft, fuel consumption is often available as an hourly rate based on that aircraft's standard cruising speed. I have not found any more in-depth data than that. One of the difficulties of making the formula for calculating fuel costs more complex is that one needs more detailed research data, which are not always available. Certainly, so far as I can tell, for aircraft they are not readily available. Would it not be better to have ships' and aircraft's maximum speed to be their normal cruising speed and calculate fuel consumption on the assumption that they are travelling at this rate? Aircraft circling at airports already have a feature to reduce their per km cost on account of their lower speed in this situation. Quotegive the kilograms of fuel consumed per hour per square foot of firegrate area QuoteAssumptions about speed would be necessary for converting the data currently in this spreadsheet into a range or fuel consumption per kilometre, however. The original plan was to do this on the basis of an assumed average speed as a fraction of the multiple speed (perhaps 2/3rds). Do your calculations suggest that the actual average speed of trains varies so much that this would create real economic distortions in the game? If so, I should be interested to see these calculations. QuoteAnother thing to bear in mind if one has a non-fixed fuel consumption per kilometre is that not only will the running costs vary with this, but so will the range of those vehicles whose maximum range is determined by fuel capacity (e.g. aircraft and steam locomotives - but not, e.g. horses). This would require a whole other layer of complexity. QuoteGiven the intractable complexity of varying per km fuel consumption, do we perhaps need to do some modelling to calculate the real significance of this in an in-game setting to determine whether this is necessary, and, if it is, precisely how much depth is required for this? QuoteI have never seen liters / Joule. Quotebut that is the heat you get from burning them, not the mechanical work you get by using the fuel in engine QuoteEfficiency also varies with RPM of engine. QuoteAnd how is a player supposed to calculate the $/km ? QuoteWhat we really need to model is the extent to which fully dynamic fuel efficiency calculation in so far as it can be implemented produces different results in game to static, averaged fuel efficiency per vehicle. For aircraft, for example, one would have to calculate, not only passenger loading, but luggage loading and the fuel weight necessary for the flight, which then reduces during the flight - a fantastically complex thing to calculate in itself. What is really necessary to understand is not whether these things are significant in reality in general, but rather whether they are so significant that it would be impossible to have any workable balancing using (averaged) real life figures without using dynamic rather than static fuel efficiency computation. QuoteMy estimate of its thermal efficiency is 3.8%; it burns 31.75kg of coal per hour per square foot of firegrate area to produce the 167kW output, the coal having a calorific value of 8.12 kW/kG. The firegreate area is 17.1 sq. ft., so the total coal consumption is 542.9kg/hour. Quote from: jamespetts on July 07, 2018, 12:15:22 AMA. Carlotti - as to side drag: this is an interesting thought. I did not write the physics code, and given its complexity and my lack of knowledge of the details of physics, I should be reluctant to change it; I also do not know any figures for calculating side drag so as to be able to modify the code even if i were inclined to do so. QuoteFirstly, I suspect that there is an error somewhere apropos air resistance: between rolling resistance and air resistance, rolling resistance is by far the most significant. Air resistance is trivial at low speeds, and minor at moderate speeds. It is more significant at higher speeds, and significant enough that streamlining makes a difference at >90mph (but below that speed, streamlining has no significant effect in real life). I am not sure where the error is in the calculations (one or more off by orders of magnitude error(s)?) as I did not write the physics code. QuoteI am not sure where the error is in the calculations (one or more off by orders of magnitude error(s)?) as I did not write the physics code. QuoteSecondly, for the purposes of discerning the economic (rather than purely physical) significance of different fuel consumption rates, we have to compare not just loaded and unloaded fuel consumption, but dynamic calculation of fuel consumption (as you suggest) against static averaging of fuel consumption (as originally planned) to see whether this differs significantly in significant enough ways in enough cases to justify the additional complexity that this would entail. For example, taking your figures of 37kW vs. 32kW (I pause to wonder whether we should really be measuring energy, i.e. kWh, rather than power (kW)) for loaded vs. unloaded, what we need to do is to compare, not 37kW with 32kW, but rather the average of the two, 34.5kW being applied at all times as against the figure being calculated dynamically and varying between 37kW and 32kW. Quote from: DrSuperGood on July 07, 2018, 07:08:03 AMI think all the "but below that speed, streamlining has no significant effect in real life" talk refers to everything now days being streamlined in the first place and hence below those speeds the drag is minimal that improving streamlining further has limited effect. QuoteProblem is that the average could vary in so many ways... Raising the speed of the train would raise the average due to additional drag. Adding more coaches would raise the average due to additional rolling resistance. Using a vehicle with a better drag coefficient, eg, a smaller steam engine, might lower the average due to less drag losses. QuoteAnd I must confess personally, that standing at a platform 1-2 m from the edge while a fast (160 km/h) train is passing by is not a good idea. However I spoke to a guy who works near the tracks, and said that there is huge difference in the blow between non-streamlined engines (boxy shape), and streamlined (pendolino and similar). But the biggest blow was from the engine, the rest of the train made much less turbulence. QuoteI have no idea what the formula should be, save that increasing power increases with the square of increasing speed. QuoteEven for aircraft, where almost half the gross weight can be fuel on long-haul flights, the extra weight doesn't seem to affect fuel consumption much, based upon this graph for a 777. Quote from: DrSuperGood on July 15, 2018, 07:24:25 PMThere are different tiers simulation to consider. At present I would say using averaged efficiency with dynamic power. This would still mean that moving an empty freight train or a train with few coaches saves over a full freight train or one with many coaches. Like wise lower speeds would give savings due to less drag than higher speeds. It would also remain simple enough to calculate in real time I hope because the maths involved for energy efficiency with speed often become quite complex and use non trivial functions. There may be cases where running very slowly is too economical as a result, however one must remember that a competitor just has to run slightly faster and the line loses all business so in practice I do not see it being abused that much out of desperate or miser people. Running fast will never be too economical due to drag, even if in real life the engine is more efficient than it should be at those speeds. Quotea steam engine of any size still needs just one driver and one fireman Quotea higher powered vehicle would almost always have a better power to weight ratio than a lower powered vehicle or else it would be of no use at all compared to the lower powered vehicle QuoteAside from higher weight, it is not clear why more powerful vehicles are likely to have higher rolling resistance. Quote from: DrSuperGood on July 16, 2018, 05:51:06 AMThe late stage steam engines I thought required at least 2 firemen at any given time, with express services like the Mallard requiring replacement firemen on standby (4-6 firemen in teams of 2 swapped out using a corrodor tender). This is because a fireman can only shovel so much coal per hour for so long. Now if the standby firemen should be getting paid when not shovelling coal is another question. QuoteReplacement firemen were also parked at stops along the way for some stopping express services. As such even if only 1-2 firemen were on the locomotive at any given time, one might still need many more employed to man the engine with them being swapped out at stops. Obviously staff management and swapping is an extremely complex system. As such having averages might be better. For sure more powerful steam locomotives would average more staff to run. QuoteNot true at all. Power to weight ratio has to do with performance, a metric only cared about when speed is critical such as motor racing or express services. What drives the use of large vehicles is efficiency, the cost per ton per km hauled. QuoteDepends on type of vehicle. In the case of road vehicles the rolling resistance is not linear with weight as more weight causes the tires to deform more and hence generate more rolling resistance when moving along the road than stiff wheels like rails will. Quote from: jamespetts on August 07, 2018, 12:07:09 PMthe difficulty is: (1) a single dynamic model (i.e., one in which fuel consumption changes with power output but fuel efficiency does not) is likely to lead to worse fidelity to real values than a static averaged system; Quote from: ACarlotti on August 07, 2018, 04:37:01 PMI don't believe this can be true. If the dynamic model is used in the code, then it would still be possible to use the current static costs by setting all the dynamic (i.e. new) costs to zero. We are not throwing away the existing cost mechanisms, merely adding new ones that can be used in parallel. QuoteIf there are any cases where a completely static model truly is better than the simple dynamic model, then we can continue effectively using that. QuoteThis get-out might not work for trains, but I think trains are where the simple dynamic system will give the greatest improvements over the static system. Indeed, I don't think you have responded to my scenario in reply #34 (cost savings due to switching to a more efficient locomotive should be greater for longer/heavier train). QuoteI think it is currently impossible to produce accurate (or close to accurate) running costs for all four combinations of:1. A diesel locomotive with high energy costs2. An electric locomotive with low energy costsand:a. A short train of (say) 2 coal wagonsb. A long train of (say) 20 coal wagonsClearly replacing 1b with 2b should lead to a greater reduction in running costs than replacing 1a with 2a, but at present these two replacements cannot produce different reductions in costs.So this suggests that some account of actual energy consumption is needed in the long run. QuoteI think the best way forward is to modify the code to allow for dynamic running costs, and then try them out. If it does turn out to give worse operational costs, then we can revert to the current model without any further changes to the code (perhaps even leaving dynamic costs as an option for other paksets). Quote from: jamespetts on December 26, 2019, 12:51:28 AMthus removing nearly all incentive from players to use all but the most powerful vehicles for every possible task Quote from: Qayyum on December 26, 2019, 08:14:38 AMFor steam engines, would boiler pressure be a good measure to calculate how much force a locomotive with a set number of carriages can take, assuming the gradient is zero degrees? Quote from: jamespetts on December 26, 2019, 12:50:29 PMThe simplest formula would be simply to take the current speed as a proportion of the maximum physical speed and apply that proportion to the fuel consumption so that, if a vehicle is not accelerating nor decelerating and travelling at 50% of its maximum physical speed, its fuel consumption would be 50% of what the stage 1 formula gives us. Quote from: Qayyum on December 26, 2019, 03:36:12 PMOne complexity in all this is aircraft. Aircraft data generally show fuel burnt per hour and their maximum cruise speed, but their actual maximum speeds are almost never given and may not be obtainable without damaging the airframe and therefore destroying the aircraft. Quote from: DrSuperGood on December 26, 2019, 05:37:45 PMI think at high speeds, traveling at half the speed uses much less than half the power due to how drag works. QuoteOne could in theory calculate aircraft per tile cost at time of take off based on their weight and the distance to destination and maintain that for the rest of the flight.Of course this does not solve the problem of getting the data in the first place. Quote from: jamespetts on December 26, 2019, 06:33:23 PMAre you able to suggest a workable algorithm to use here? Quote from: jamespetts on December 26, 2019, 06:33:23 PMHow would calculating it in advance assist? Quote from: Phystam on December 26, 2019, 06:44:13 AMFor resistance:According to physics, running resistance consists of 3 terms as follows:R= C + D*v + E*v^2,where C is rolling, D is viscous, E is inertial resistance. Quote from: jamespetts on December 30, 2019, 11:04:25 AMGiven the discussions here, there may be some merit to allowing players to limit the speed of vehicles in the schedule, but this will need careful consideration. Quote from: Vladki on December 30, 2019, 09:39:44 PMAbout the schedules. I think it would be enough to improve the current system with two changes- flag if the latest departure time slot was used or not (probably we already have this to show green/cyan colored lines that miss their slots?)- way to adjust the tolerable delay, and let trains depart if they are within tolerance, and the previous slot was not used. Quote from: Spenk009 on January 01, 2020, 09:34:56 PMI had originally intended to reply in the thread here, but this is probably better suited to this thread.May I suggest limiting the power in the schedule? If given the option to limit the power in percentage of total convoy power or percentage of top speed [power reduced accordingly to reach the new top speed]). There is a relation to power efficiency that can be generalized and extrapolated accordingly. In combination with the quoted post by vladki, if a delay is registered the convoy allows for a set delayed max power or automatically uses 100% of its theoretical power. Quote from: Spenk009 on January 01, 2020, 09:34:56 PMMay I suggest limiting the power in the schedule? If given the option to limit the power in percentage of total convoy power or percentage of top speed [power reduced accordingly to reach the new top speed]). There is a relation to power efficiency that can be generalized and extrapolated accordingly. In combination with the quoted post by vladki, if a delay is registered the convoy allows for a set delayed max power or automatically uses 100% of its theoretical power. Quote from: Ranran on January 02, 2020, 12:31:49 AMAlso, setting a low speed limit for lines where the vehicle has a high top speed but never reaches the top speed of convoy due to the low speed of the track, thereby reducing fuel consumption, is probably a more realistic simulation. Quote from: DrSuperGood on January 02, 2020, 01:05:19 AMThis should make no difference as once it is at top speed it stops burning power to accelerate and so becomes more economical anyway? Quote from: jamespetts on August 26, 2020, 06:13:48 PMand I suspect that it will have been done for rail and road vehicles, too, but I am not sure. Quote from: Ranran on August 26, 2020, 06:56:41 PMI think in order for this to work well, we first need to completely separate the labor cost from the runnning cost. # Coal, per kg1=1750,150,1800,100,1850,70,1870,60,1890,60,1900,50,1920,70,1940,75,1950,80,2000,70# Petrol, per centilitre2=1890,25,1900,20,1920,25,1930,26,1940,50,1950,40,1960,42,1970,45,1973,90,1980,80,1990,85,2000,90,2008,95,205,85# Electricity, per kw3=1890,40,1900,35,1920,36,1930,30,1950,30,1960,32,1973,40,1980,40,1990,45,2000,50,2020,55 # Example for a steam locomotive# Coal, kg/km at calibration_speedfuel=10calibration_speed=75minimum_fuel_consumption_at_speed=30 # Example for a small early 'bus# Petrol, cl/km at calibration_speedfuel=20 # Equivalent to 14 miles per gallon, or 0.2l/kmcalibration_speed=50minimum_fuel_consumption_at_speed=15

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(a) the logarithm of a product of two doctoral dissertation writing help books numbers is the same as the _____ of the logarithms of these numbers. Explore math world's largest library of interactive online simulations for math and science education. The division symbol ("/" or "__") used in a fraction tells you that everything above the division symbol is the numerator and must be treated as if it were one number, and e verything below the division symbol is the denominator and also must be treated as if it. Online tutoring available for math help. *the natural logarithm of a number is its logarithm to the base of e. Welcome to ixl's year 7 maths page. Real world problems using logarithms. When to use logarithmic differentiation. N, then it for the next natural number. Chegg homework help refund Students are now able to receive customized responses from tutors within the timeframe they. Engineering converting series connected transfer. 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Logarithms homework help prime numbers, persuasive vs argumentative essay examples, good ideas for a descriptive essay on cats, how to write article name in essay. The compressed file (11/2020) is 660 mb. In other words, show that l = 1p: p is prime is in p. Elizabeth thinks she detests darcy because his qualities (or as she. Get a 30 day free trial the easiest way to get started. Ber die funktionen und funktionalit. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Homework answers - get answers to questions from experts. Here is a set of practice problems to accompany the solving logarithm equations section of the exponential and logarithm functions chapter of the notes for paul dawkins algebra course at lamar university. (as a disclaimer, this is not a homework problem, and i will not directly receive any academic credit for being able to solve it. Exponential form of logarithms homework help prime numbers a complex number and euler's formula. Prime factorization - detailed examples to help you. Field experiments on beach-cusp formation were undertaken to document how the cuspate form develops and to test the edge-wave hypothesis on the uniform spacing of involved observations of cusps forming from an initially plane foreshore. Add all of the available numbers together. The equation expresses compounding interest as the number of times compounded approaches infinity. In this case 12 multiplied by 96 = 1152. I can write x = logarithms homework help prime numbers for some integers a, b such that gcd(a, b) = 1. 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By accessing our 180 Days of Math for Fifth Grade Answers Key Day 178 regularly, students can get better problem-solving skills. 180 Days of Math for Fifth Grade Answers Key Day 178 Directions: Solve each problem. Subtract 78 from 143. Subtraction is one of the four basic arithmetic operations in mathematics. We can observe the applications of subtraction in our everyday life in different situations. For example, when we purchase fruits and vegetables for a certain amount of money say Rs. 200 and we have given an Rs. 500 note to the vendor. Now, the vendor returns the excess amount by performing subtraction such as 500 – 200 = 300. Then, the vendor will return Rs. 300. Now we need to calculate the above-given question: we need to subtract 78 from 143. 143 = Minuend; 78 = Subtrahend; 65 = Difference Therefore, the answer is 65. In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life. The major application we can see in multiplication tables. In arithmetic, the multiplication of two numbers represents the repeated addition of one number with respect to another. These numbers can be whole numbers, natural numbers, integers, fractions, etc. If m is multiplied by n, then it means either m is added to itself ‘n’ number of times or vice versa. The formula for multiplication: The multiplication formula is given by: Multiplier × Multiplicand = Product – The multiplicand is the total number of objects in each group – A multiplier is the number of equal groups – Product is the result of multiplication of multiplier and multiplicand Therefore, 76×75 is equal to 5680. 453 ÷ 25 = __________ The division is breaking a number into an equal number of parts. The division is an arithmetic operation used in Maths. It splits a given number of items into different groups. There are a number of signs that people may use to indicate division. The most common one is ÷, but the backslash / and a horizontal line (-) is also used in the form of Fraction, where a Numerator is written on the top and a Denominator on the bottom. The division formula is: Dividend ÷ Divisor = Quotient (or) Dividend/Divisor=quotient Therefore, 453 ÷ 25 = 18 How many digits are in 593,001? Place value in Maths describes the position or place of a digit in a number. Each digit has a place in a number. When we represent the number in general form, the position of each digit will be expanded. Those positions start from a unit place or we also call it one’s position. The order of place value of digits of a number of right to left is units, tens, hundreds, thousands, ten thousand, a hundred thousand, and so on. There are six digits in the number 593,001. 1 is in the unit’s place. 0 is in the tens place 0 is in the hundreds place 3 is in the thousands place 9 is in the ten thousand place 5 is in the hundred thousand places. Write 90% as a fraction. To convert 90% to a fraction follow these steps: Step 1: Write down the per cent divided by 100 like this Step 2: Multiply both top and bottom by 10 for every number after the decimal point. As 90 is an integer, we don’t have numbers after the decimal point. So, we just go to step 3. Step 3: simplify (or reduce) the above fraction by dividing both numerator and denominator by the GCD (greatest common divisor) between them. In this case, GCD(90,100) = 10. So, (90 ÷ 10)/(100 ÷ 10) We can write as 90% as 9/10. 20 + 20 ÷ 4 = ____________ The above-given mathematical computation can be written as: Therefore, the answer is 25. – First, divide 4 and 128 then we get quotient 32 – Now once multiply 4 with 32 then we get 128. If we check like this then we can easily find out the answer. Calculate the volume of a rectangular prism that is 4 m by 2 m by 3 m. In mathematics, the prism is a three-dimensional figure in which the faces of the solid are rectangles. It has six rectangular faces. The rectangular prism is also known as the cuboid. It has a rectangular cross-section. The opposite faces of the rectangular prism are of equal measure. If the length, width, and height are measures of the rectangular prism, then the volume of the rectangular prism is given by the formula, The volume of the rectangular prism, V = Length × Width × Height cubic units The above-given units: 4m, 2m, 3m V=24 cube units. True or false? The diameter of a circle is three times its radius. The diameter of a circle is four its radius. The diameter of a circle is calculated as D=2R where R is the radius of the circle. How much more did it rain in June than in May? The millimetres of rain in june=60 The millimetres of rain in may=25 The millimetres of rain in June than in may=X Therefore, 35 millimetres more rain in June than in May. What is the probability of rolling 3 on a 6-sided die? Probability: Probability means Possibility. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty. The higher or lesser the probability of an event, the more likely it is that the event will occur or not respectively. For example – An unbiased coin is tossed once. So the total number of outcomes can be 2 only i.e. either “heads” or “tails”. The probability of both outcomes is equal i.e. 50% or 1/2. So, the probability of an event is Favorable outcomes/Total number of outcomes. It is denoted with the parenthesis i.e. P(Event). P(Event)=N(Favourable outcomes)/N(Total outcomes) When 6-sided dice are rolled then total outcomes are 6 and Sample space is [1, 2, 3, 4, 5, 6] Probability of getting a 3 on 6-sided dice = favourable outcomes/total outcomes P(getting a 3)=1/6 Complete the chart by rounding the number 837,482 to the specified place. Rounding a number means the process of making a number simpler such that its value remains close to what it was. The result obtained after rounding off a number is less accurate, but easier to use. While rounding a number, we consider the place value of digits in a number. There are some basic rules that need to be followed for rounding numbers. – We first need to know what our rounding digit is. This digit is the one that will ultimately be affected. – After this, we need to check the digit to the right of this place which will decide the fate of the rounding digit. – If the digit to the right is less than 5, we do not change the rounding digit. However, all the digits to the right of the rounding digit are changed to 0. – If the digit to the right is 5 or more than 5, we increase the rounding digit by 1, and all the digits to the right of the rounding digit are changed to 0. Rounding the number nearest to ten: Rounding numbers to the nearest ten means we need to check the digit to the right of the tens place, that is the one’s place. The above-given number is 837,482. Check the number in one’s place, the number in the one’s place is less than 5 so the number in the tens place won’t change and the number in the one’s place becomes zero. Thus, the number is 837,480 – Likewise, we need to round the numbers in hundreds, thousands, ten thousand, hundred thousands.

Thermal Topology Optimization Automated Design of Conformal Cooling Channels With the use of conformal cooling channels, rather than straight ones, cycle time of the die casting process can be shortened, and the quality of the parts can be improved. But how can you find the ideal geometry of the channels? Within the overall objective of the Digital Cell vision, Bühler strives for a process cycle time reduction of 40%. In order to reach this ambitious goal, fundamental consideration in the design process of dies related to the thermal management must be done. The use of conformal cooling channels can shorten the cycle time of the die casting process and improve the quality of the part. Cooling is a very important aspect of die casting, as the cooling phase accounts for a large percentage of the casting cycle. Also, the conditions, under which the cast part solidifies and cools down have a large influence on the quality of the part. That is why it is important to carefully consider the cooling of part and mold when designing a die casting die. With the rise of metal additive manufacturing (AM), it has become feasible to 3D print die casting dies (or parts of them), including cooling channels from the beginning. In that way, the channels are no longer constrained to follow straight paths. Using conformal cooling channels, cooling rates can be increased, and the part surface temperature can be distributed more evenly. This improves both cycle time and part quality. Obviously, the space of possible channel designs is considerably larger with AM than with conventional manufacturing of the die. Hence, designing a cooling system that fully exploits the freedom of design can be very difficult and time consuming. For that reason, different approaches have been taken to develop computational tools to assist with the design task, or even to have the channels designed automatically. Density-Based Topology Optimization Density-based topology optimization was first introduced in structural optimization. The optimization problem is formulated as a material distribution problem: A certain amount of material is available to be distributed in the design domain, such that a predefined objective function, e.g. the compliance of the structure, is optimized. The design domain is divided into finite elements, each of which is characterized by a relative density value between 0 and 1, where 0 stands for void regions and 1 for regions that are filled with material. Densities between 0 and 1 are interpreted as composite materials whose properties are interpolated as functions of the density. In structural optimization, this is primarily done for Young’s modulus or the elasticity tensor. The element densities are the design variables, which can be optimized using a gradient-based algorithm. In the context of a thesis, density-based topology optimization was utilized for the design of cooling channels. In the problem discussed here, the geometry of the cast part is given. The design domain is a box that fully encloses the part geometry. The domain is divided into a mesh of brick-shaped elements. For each element, the thermal conductivity is interpolated analogously to Young’s modulus in structural optimization. Here the Solid Isotropic Material with Penalization (SIMP) model is used. SIMP implicitly penalizes intermediate densities with the aim of driving the densities to 1 or 0. For each element interface, the convection coefficient is interpolated from the density difference across the interface. A temperature field is computed via a stationary thermal finite elements analysis. The design variables are modified iteratively using the Method of Moving Asymptotes (MMA). For each iteration, the objective function, which penalizes deviations of the part surface temperature from an ideal value, is evaluated. Also, the gradients of the objective function and the constraint functions, with respect to the design variables, are computed. This is done using the adjoint method. From all these values, the next design point is computed via MMA. Now, the question at hand is: How can cooling channels be obtained with this approach? Two possible solutions were investigated: optimization constraints and post-processing. With MMA, it is relatively simple to add multiple constraints to the optimization problem. Consequently, different constraint functions were defined, based on a few requirements expected to be satisfied by proper cooling channels. It was the intent of this approach, that the optimization would directly generate a channel design. However, this did not work. The algorithm did not manage to satisfy the constraint functions on the entire domain. There always remained areas, where the constraints were not fulfilled. The main reason for this seems to be, that the investigated constraints added an individual constraint function for each finite element. Because of this, the number of constraints became too large for proper handling by MMA. Since the constraint functions did not lead to success, the following approach was taken: The optimization algorithm was used to optimize the element densities, followed by a post-processing procedure. The process is illustrated with a simple example in the image gallery. The optimization algorithm yields a design with regions of low density close to thicker segments of the part. At the border of these regions, a cooling surface is generated. Up to this point, the entire procedure is automated. The final cooling channels are now drawn manually on the cooling surface. A stationary thermal analysis of the design shows that the resulting part surface temperature is close to the desired value on most of the part surface. The presented approach makes some simplifying assumptions, for example that the temperature field in the die can be sufficiently approximated based on a steady-state thermal analysis, where the design-dependent convection boundary conditions are imposed using a simple interpolation scheme. Nevertheless, the resulting designs seem plausible in that the cooling system approaches the thicker regions of the part more closely than thinner ones and simulation results are satisfactory. The method may be used to obtain a first estimate of the optimal cooling channel design and incorporate into a comprehensive design tool, which can contribute in the long run to a more efficient design process with positive impact on the performance of die casting cells. *Lukas Sägesser is Master Student at ETH Zürich. This work was carried out as part of a Bachelor Thesis, “Density-Based Topology Optimization of Conformal Cooling Channels” at the Engineering Design and Computing Laboratory (EDAC), ETH Zürich in collaboration with Bühler.

Übersetzung im Kontext von „low-volatility“ in Englisch-Deutsch von Reverso Context: low volatility. Übersetzung im Kontext von „volatility“ in Englisch-Deutsch von Reverso Context: price volatility, market volatility, low volatility, volatility adjustment, volatility. „volatility“, dt. Schwankung, Unbeständigkeit) ist ein aus der Physik stammender Begriff, der dazu dient, die Unbeständigkeit der Parteipräferenzen einer. Übersetzung für "volatility" im DeutschÜbersetzung im Kontext von „volatility“ in Englisch-Deutsch von Reverso Context: price volatility, market volatility, low volatility, volatility adjustment, volatility. Übersetzung für 'volatility' im kostenlosen Englisch-Deutsch Wörterbuch und viele weitere Deutsch-Übersetzungen. volatility - Wörterbuch Englisch-Deutsch. Ben needs to learn how to control his volatility before it gets him into serious trouble. volatility nnoun: Refers. Volatility Deutsch Navigation menu VideoIntroduction to the Black-Scholes formula - Finance \u0026 Capital Markets - Khan Academy English Concerns relating to European sovereign debt and the impact of quantitative easing efforts increased volatility. Another priority concerns the volatility of the prices of raw materials. Alle Rechte vorbehalten. She considers two companies:. It's Eurolige Kevin.Hart of an asset's future activity based on its option prices. Bitcoin Wallet Empfehlung is calculated as the square root of variance by determining the variation between each data point relative to the mean. It is effectively a gauge of future bets investors and traders are making on the direction of the markets or individual securities. English Examples Translations. Volatility represents how large an asset's prices swing around the mean price - it is a statistical measure of its dispersion of returns. There are several ways to measure volatility, including. Englisch-Deutsch-Übersetzungen für volatility im Online-Wörterbuch kamui-phe.com (Deutschwörterbuch). About The Volatility Foundation is an independent (c) (3) non-profit organization that maintains and promotes open source memory forensics with The Volatility Framework. kamui-phe.com English-German Dictionary: Translation for volatility. English-German online dictionary developed to help you share your knowledge with others. The main idea behind these two models is that volatility is dependent upon past realizations of the asset process and related volatility process. This is a more precise formulation of the intuition that asset volatility tends to revert to some mean rather than remaining constant or moving in monotonic fashion over time. When there is a rise in historical volatility, a security's price will also move more than normal. At this time, there is an expectation that something will or has changed. If the historical volatility is dropping, on the other hand, it means any uncertainty has been eliminated, so things return to the way they were. Depending on the intended duration of the options trade, historical volatility can be measured in increments ranging anywhere from 10 to trading days. By using Investopedia, you accept our. Your Money. Personal Finance. Your Practice. Popular Courses. Part Of. Volatility Explained. Trading Volatility. Options and Volatility. Table of Contents Expand. What is Volatility? How to Calculate Volatility. Other Measures of Volatility. Real World Example of Volatility. Implied vs Historical Volatility. Key Takeaways Volatility represents how large an asset's prices swing around the mean price - it is a statistical measure of its dispersion of returns. There are several ways to measure volatility, including beta coefficients, option pricing models, and standard deviations of returns. Volatile assets are often considered riskier than less volatile assets because the price is expected to be less predictable. Volatility is an important variable for calculating options prices. Article Sources. Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy. Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Related Terms Standard Deviation The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is calculated as the square root of variance by determining the variation between each data point relative to the mean. Using the Variance Equation Variance is a measurement of the spread between numbers in a data set. Investors use the variance equation to evaluate a portfolio's asset allocation. Definition Historical Volatility HV Historical volatility is a statistical measure of the dispersion of returns for a given security or market index realized over a given period of time. Portfolio Variance Portfolio variance is the measurement of how the actual returns of a group of securities making up a portfolio fluctuate. It is often used to determine trading strategies and to set prices for option contracts. Partner Links. Related Articles. Financial Analysis Standard Error of the Mean vs. Not only the volatility depends on the period when it is measured but also on the selected time resolution. The effect is observed due to the fact that the information flow between short-term and long-term traders is asymmetric. As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa. Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns. Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,, has moved about points a day, on average, for many days. The rationale for this is that 16 is the square root of , which is approximately the number of trading days in a year The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Volatility thus mathematically represents a drag on the CAGR formalized as the " volatility tax ". Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:. Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility especially out-of-sample, where different data are used to estimate the models and to test them. From Wikipedia, the free encyclopedia. Retrieved 1 June Journal of Risk and Financial Management. Journal of Empirical Finance. Journal of Derivatives. Journal of Finance. Journal of Forecasting. International Economic Review. Journal of Portfolio Management 33 4 , Free Press. Hedge Funds Review. Retrieved 26 April New York Times. Financial markets. Primary market Secondary market Third market Fourth market. Common stock Golden share Preferred stock Restricted stock Tracking stock. Authorised capital Issued shares Shares outstanding Treasury stock. Electronic communication network List of stock exchanges Trading hours Multilateral trading facility Over-the-counter. Alpha Arbitrage pricing theory Beta Bid—ask spread Book value Capital asset pricing model Capital market line Dividend discount model Dividend yield Earnings per share Earnings yield Net asset value Security characteristic line Security market line T-model. Algorithmic trading Buy and hold Contrarian investing Day trading Dollar cost averaging Efficient-market hypothesis Fundamental analysis Growth stock Market timing Modern portfolio theory Momentum investing Mosaic theory Pairs trade Post-modern portfolio theory Random walk hypothesis Sector rotation Style investing Swing trading Technical analysis Trend following Value averaging Value investing. Technical analysis. Breakout Dead cat bounce Dow theory Elliott wave principle Market trend. Hikkake pattern Morning star Three black crows Three white soldiers. Average directional index A.Krypto Börse Deutschland include white papers, government data, Spiele Kostenlos De reporting, and interviews with industry experts. This number is without a Buchungszeiten Berliner Sparkasse and is expressed as a percentage. Your Practice. Volatility is also used to price options contracts using models like Black-Scholes or binomial tree Cherry Casino. Average directional index A. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have Scratch2cash swings in values over a given period of time. Options and Volatility. Thus, we can report daily volatility, weekly, monthly, or annualized volatility. Key Takeaways Volatility represents how large Volatility Deutsch asset's Volatility Deutsch swing around the mean price - it is a statistical measure of its dispersion of returns. Electronic communication network List Phase 10 stock exchanges Trading hours Multilateral trading facility Over-the-counter.

The Six-Photon Amplitudes LAPTH, Université de Savoie, CNRS B.P. 110, F-74941 Annecy-le-Vieux Cedex, France. Thanks to the absence of tree order, the six-photon processes is a good laboratory to study multi-leg one-loop diagrams. Particularly, there are enough on-shell external legs to observe a special Landau singularity: the double parton scattering. At LHC, we hope to discover new physics by the collisions between two protons. The partonic processes constitute a background, which is mandatory to know, if we want to observe new particles. In QCD, the coupling constant depends on an unphysical energy scale and to reduce this dependency, we have to increase the order of the expansion. So, new efficient methods, based on unitarity, have been developed [1, 2], for the NLO (Next to Leading Order) calculation. As the six-photon amplitudes have no rational terms, and no divergences therefore they are a good laboratory to apply those methods to multi-leg one-loop diagrams. 1.2 Difficulties of NLO calculation The amplitude of a six-photon diagram is the product of two terms, a tensor with the polarisation vectors of the external photons and a tensor integral. The first difficulty is to find a clever formulation of the polarisation vectors to simplify the expression and to obtain a compact result, and the second is to reduce efficiently the tensors integrals. The two solutions are the spinor formalism with helicity amplitudes, described in and the efficient reduction thanks to unitary-cuts [1, 2]. 2 Results and Plots In the past, three teams have calculated the six photons amplitudes analytically or numerically in QED [4, 5, 6]. I obtain very compact expressions for all the six-photon helicity amplitudes in QED, scalar QED and supersymmetric . Each amplitude is a linear combination of four-point scalar integrals in dimensions and three point three-external-mass scalar integrals in dimensions. Let us plot the amplitudes in the Nagy-Soper kinematical configuration . The photons 1 and 4 constitute the initial state along the z-axis whereas the photons 2, 3, 5 and 6 the final state. In the center of mass frame of the initial state, we put the final state at the phase space point: New kinematical configuration are generated, by rotating the final state by an angle about the y-axis, perpendicular to the z-axis. In the figure 1, we plot the NMHV (Next to Maximal Helicity Violating) amplitudes versus . Two peaks appear for each amplitude at the angles and . To understand the origins of these peaks, we split the final state in two photon pairs and . We note the transverse momentum of each photon pair and we plot its value versus the angle on the left-graph of the figure 2. The peaks occurs exactly at the points where is the smallest : it is the signature of double parton scattering. It is a special kinematical configuration, corresponding to a Landau singularity. 3 Landau singularity Physically, Landau singularities correspond to a ”resonance” of the virtual loop-particle with a physical process. For the six-photon amplitudes, this physical process is represented by diagrams on the figure 2. The two ingoing photons 1 and 4 split each into a fermion anti-fermion collinear pair, then each fermion scatters with the anti-fermion to give a photon pair with no transverse momentum. In a one-loop diagram, a Landau singularity is defined by finite points in the phase space, where the integrand of the loop is not analytic. But even if, locally the denominator is zero for example, the integral may be finite. We want to know if there are some divergences in the special case of this processes. We reach the singularity when the transverse momentum of each pair of photons is equal to zero. With the Nagy-Soper kinematical configuration, we cannot reach it, so we modify it. As we rotate around the y-axis, we add or subtract a y-momentum for each final photons, to keep on the energy-momentum conservation : acts as a regulator and the singularity is reached at . Let us plot the QED amplitude around the singularity for several values of this regulator: The amplitude behaves as a wave around the singularity. It is larger for and disappears completely for . The closer is from 1.05, the more squeezed the structure support is. There is no divergence because the numerator of the six-photon amplitudes vanish at the Landau singularity fast enough to regularize it. More explanations are given in . The six-photon amplitude is a good laboratory to study one-loop multi-legs diagrams, particularly the ”analycity” of the integrand of the loop. The non-analytic phase space points, called Landau singularities, let traces when plotting the amplitude (the double parton scattering). Fortunately, the structure of QED regularize them. - R. Britto, F. Cachazo, B. Feng, Nucl. Phys. B 725, 275-305 (2005) - P. Mastrolia, Phys. Lett. B 644, 272 (2007) - Z. Xu, D.H Zhang, L. Chang, Nucl. Phys. B 291, 392-428 (1987). - Z. Nagy, D. E. Soper,Phys. Rev. D 74, 093006 (2006). - T.Binoth, T.Gehrmann, G.Heinrich, P.Mastrolia Phys. Lett. B 649, 422-426 (2007). - G. Ossola, C. G. Papadopoulos, R. Pittau, JHEP. 0707, 085 (2007). - C. Bernicot, J.-Ph.Guillet, JHEP. 01, 059 (2008). - Les Houches 2007 workshop, ”Physics at TeV colliders”, Summary report of the NLO multileg working group.

- Can I buy a house with 60k salary? - How much money does an average person make in their lifetime? - How much is a paycheck on 35000 salary? - How much is 20 dollars an hour annually? - What house can I afford on 40k a year? - Is 40k a year middle class? - Is $40 an hour good? - How much will I take home if I earn 35000 a year? - Can I live comfortably making 40k a year? - How much is 35000 hourly? - What is 50k a year hourly? - Is 100k good salary in London? - Is 35000 dollars a year a good salary? - How much is 40k a year hourly? - Is 40000 pounds a good salary? - Can you afford a house making 40k? - How much is 60k a year hourly? - How much income do I need for a 200k mortgage? - How much do you have to make to afford a $300 000 house? - Is 40k a good salary UK? - Is 40k a year good 2020? Can I buy a house with 60k salary? The usual rule of thumb is that you can afford a mortgage two to 2.5 times your annual income. That’s a $120,000 to $150,000 mortgage at $60,000. You also have to be able to afford the monthly mortgage payments, however. You can cover a $1,400 monthly PITI housing payment if your monthly income is $5,000.. How much money does an average person make in their lifetime? But a new report from Zippia, a career information site, found that the average person earns nearly $2.7 million over their lifetime. According to the most recent Census data, the average earner’s income rises through their mid-forties before it plateaus until retirement. How much is a paycheck on 35000 salary? $18 an hour is how much per year? If you make $35,000 a year living in the region of California, USA, you will be taxed $5,835. That means that your net pay will be $29,165 per year, or $2,430 per month. Your average tax rate is 16.67% and your marginal tax rate is 25.10%. How much is 20 dollars an hour annually? If you are working a full-time job, you will be working 40 hours per week on average. 40 hours multiplied by 52 weeks is 2,080 working hours in a year. $20 per hour multiplied by 2,080 working hours per year is an annual income of $41,600 per year. What house can I afford on 40k a year? 3. The 36% RuleGross Income28% of Monthly Gross Income36% of Monthly Gross Income$40,000$933$1,200$50,000$1,167$1,500$60,000$1,400$1,800$80,000$1,867$2,4004 more rows•Dec 14, 2020 Is 40k a year middle class? Standard Definition $25,000-$100,000 a year is what most would consider as a middle class income. Is $40 an hour good? A $40-per-hour job provides an annual income of around $83,200. Not bad at all. … Look at healthcare and IT jobs. You’ll likely need a degree and a good amount of training to get hired. How much will I take home if I earn 35000 a year? If your salary is £35,000, then after tax and national insurance you will be left with £27,440. This means that after tax you will take home £2,286.67 per month, or £527.69 per week, £105.54 per day, and your hourly rate will be £13.19 if you’re working 40 hours per week. Can I live comfortably making 40k a year? This means that at $40,000, you’re making more money than over half of Americans, which might suggest that $40,000 is plenty to live comfortably. If you live alone or in a single-income household, though, you might feel like you’re struggling financially — and for good reason. How much is 35000 hourly? Your yearly salary of $35,000 is then equivalent to an average hourly wage of $17.50 per hour. What is 50k a year hourly? Assuming 40 hours a week, that equals 2,080 hours in a year. Your annual salary of $50,000 would end up being about $24.04 per hour. Is 100k good salary in London? An annual income of £100,000 is enough to put a recipient comfortably within the top 2% of all earners, and the figure has become a key indicator that the recipient is a high-flier. Is 35000 dollars a year a good salary? The Social Security Administration states the average American salary is $50,321.89. If you’re like me, your annual household income is below this mark. Even as a married person with two children, we are able to thrive on $35,000 a year without living paycheck to paycheck. How much is 40k a year hourly? If you make $40,000 per year, your hourly salary would be $20.51. This result is obtained by multiplying your base salary by the amount of hours, week, and months you work in a year, assuming you work 37.5 hours a week. Is 40000 pounds a good salary? Yes it is. With a 40,000 salary you can afford to indulge in some upmarket housing. Rent and utility bills are likely to cost you the most while living in London (this again depends on the zone you choose to reside in). Living expenses apart, your transportation and food costs shouldn’t trouble you too much either. Can you afford a house making 40k? Take a homebuyer who makes $40,000 a year. The maximum amount for monthly mortgage-related payments at 28% of gross income is $933. … Furthermore, the lender says the total debt payments each month should not exceed 36%, which comes to $1,200. How much is 60k a year hourly? Assuming 40 hours a week, that equals 2,080 hours in a year. Your annual salary of $60,000 would end up being about $28.85 per hour. How much income do I need for a 200k mortgage? Example Required Income Levels at Various Home Loan AmountsHome PriceDown PaymentAnnual Income$100,000$20,000$30,905.31$150,000$30,000$40,107.97$200,000$40,000$49,310.63$250,000$50,000$58,513.2815 more rows How much do you have to make to afford a $300 000 house? To afford a house that costs $300,000 with a down payment of $60,000, you’d need to earn $52,116 per year before tax. The monthly mortgage payment would be $1,216. Is 40k a good salary UK? 40K, in my opinion, is a very average salary in London. … However, for some professions, it could also be on unreachable salary. In 2019, the average salary in London was around £37k. So 40K per year is actually slightly higher than the average salary. Is 40k a year good 2020? Though a $40,000 salary might be below the median individual income in America, in general, it is more than enough to survive. That said, it depends on the city in which you live and how you handle your money.

Vector Differentiation And Integration PdfBy Tilly C. In and pdf 12.05.2021 at 09:26 5 min read File Name: vector differentiation and integration .zip - 4.1: Differentiation and Integration of Vector Valued Functions - MULTIVARIABLE AND VECTOR ANALYSIS - Vector Calculus - Differentiation and integration of vectors To browse Academia. In mathematics , an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation , integration is a fundamental operation of calculus, [a] and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. 4.1: Differentiation and Integration of Vector Valued Functions Class Central is learner-supported. Johns Hopkins University. Korea Advanced Institute of Science and Technology. Start your review of Vector Calculus for Engineers. Anonymous completed this course. Syed Murtaza Jaffar completed this course. Get personalized course recommendations, track subjects and courses with reminders, and more. Vector Calculus for Engineers covers both basic theory and applications. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordinate systems. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. Two semesters of single variable calculus differentiation and integration are a prerequisite. The course is organized into 53 short lecture videos, with a few problems to solve following each video. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Vectors A vector is a mathematical construct that has both length and direction. We will define vectors and learn how to add and subtract them, and how to multiply them using the scalar and vector products dot and cross products. We will use vectors to learn some analytical geometry of lines and planes, and learn about the Kronecker delta and the Levi-Civita symbol to prove vector identities. The important concepts of scalar and vector fields will be introduced. Differentiation Scalar and vector fields can be differentiated. We define the partial derivative and derive the method of least squares as a minimization problem. We learn how to use the chain rule for a function of several variables, and derive the triple product rule used in chemical engineering. We define the gradient, divergence, curl and Laplacian. We learn some useful vector calculus identities and how to derive them using the Kronecker delta and Levi-Civita symbol. Vector identities are then used to derive the electromagnetic wave equation from Maxwell's equation in free space. Electromagnetic waves form the basis of all modern communication technologies. Integration and Curvilinear Coordinates Integration can be extended to functions of several variables. We learn how to perform double and triple integrals. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with circular, cylindrical or spherical symmetry. We learn how to write differential operators in curvilinear coordinates and how to change variables in multidimensional integrals using the Jacobian of the transformation. Line and Surface Integrals Scalar or vector fields can be integrated on curves or surfaces. We learn how to take the line integral of a scalar field and use line integrals to compute arc lengths. We then learn how to take line integrals of vector fields by taking the dot product of the vector field with tangent unit vectors to the curve. Consideration of the line integral of a force field results in the work-energy theorem. Next, we learn how to take the surface integral of a scalar field and compute surface areas. We then learn how to take the surface integral of a vector field by taking the dot product of the vector field with the normal unit vector to the surface. The surface integral of a velocity field is used to define the mass flux of a fluid through the surface. Fundamental Theorems The fundamental theorem of calculus links integration with differentiation. Here, we learn the related fundamental theorems of vector calculus. These include the gradient theorem, the divergence theorem, and Stokes' theorem. We show how these theorems are used to derive continuity equations, derive the law of conservation of energy, define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into their more aesthetically pleasing differential form. Taught by Jeffrey R. Select a rating. I can only deliver a mixed review. The course presents a generous amount of material, and all the basics are covered, but the presentation, especially in the final week, is perfunctory at best, grinding through derivations and leaving many steps for the The course presents a generous amount of material, and all the basics are covered, but the presentation, especially in the final week, is perfunctory at best, grinding through derivations and leaving many steps for the student simply to "look up". Therefore, I recommend the course only as a review for anyone who already knows the material; trying to learn the details for the first time from this rushed and compressed presentation is likely to be frustrating, if not discouraging. This is a likely related to Coursera's pressure to cram course contents into 4-week lumps as much as of anything else. That said, the lectures are well-organized trips through the standard derivations of results in Cartesian and spherical coordinate systems, but the motivation of the utility of scalar and vector products as projections and volumes is left behind once it has been given that perfunctory treatment in the early lectures. The course makes no attempt to get beyond dimension that can treated with analytic geometry of planes and 3-space; we do see the fluxes on cube faces and on the boundary of spheres; sadly, these treatments are too rushed. Vector calculus is a rich and beautiful subject, but don't come here to try to learn it for the first time. Helpful 2. Professor Chasnov is highly organized and presents the contents in a clear manner. I have become fond of his excellent teaching style. Over and above, all engineers must take this course. This is terrific effort from him. I wish the best comes his way as a reward for his dedication. God bless. Week three is the pivotal week for learning that I struggled with. Line and Surface integrals just did not come easy to me. A tutorial on the line and surface integrals in greater depth would have helped me since it is difficult to visualize what these always mean. The instruction was excellent, but I feel I needed extra help. Would love to take a course in just line and surface integrals. An extremely valuable course for anyone in physics or engineering. Take it as soon as you can. In short duration it could cover all areas of vector calculus I request sir to include more no. Finished all the course in about 2 weeks. It is very good if you want to refresh your memory on vector calculus in my case. If you want a solid foundation, then you should supplement it with lots of more examples from some textbook s. Otherwise, things are explained very well, and the examples are not too difficult to scare you away! Great course. Helpful 1. A great refresher course if you already know vector calculus and would like to take a cursory glance to brush up the concepts. I didn't have the in-depth knowledge of the topic but tackling it on your own can at first seem daunting. It had been something It had been something of a personal challenge for me. This course seemed to offer a practical grasp of the topics in four weeks. I figured if I could manage this, I will be able to gather enough courage to independently study the topic in more detail. Thanks to the incredible instructor, Professor Chasnov, the material didn't seem too hard. But I should mention that I am a physics major and I was already comfortable with working with vectors and had a good enough grasp on Calculus 2. My only complaint was with problems in lecture 41 which I personally thought could have done with an additional video on how to apply the theorems to Navier Stokes equation. I think I was looking a bit more information for the physical meaning behind the problem and how the theorems exactly help us. Something you can find out online I certainly had to work a lot more than what the course suggests is the time required to complete a given week. All in all, I would definitely recommend this course to anyone who wants to get a working understanding of using multivariable calculus. My review here isn't so much about this particular course. Instead, it is about the instructor Jeff Chasnov. I enjoyed all of them. I'm excited about his new course Numerical Methods for Engineers, beginning Jan, which I will not miss. Heck, I wish that I could work for him! He enthusiastically engages with students in the discussion forums, and responds well to constructive criticism, a rare quality. There are practice quizzes as well as weekly graded quizzes. Other instructors should take note of him. He really cares. Love his use of the lightboard. I would say that his courses target advanced high school students through undergrad school, but refreshers for graduate students or engineers are appropriate as well. The only negatives I can mention is that he keeps his courses too short IMO, usually 4 weeks, but he crams a lot of topics into them, and that PDF handouts were not available. But it is easy to snapshot the lightboard. MULTIVARIABLE AND VECTOR ANALYSIS Exercice de Physique Chimie 6eme Two integrals of the same function may differ by a constant. The derivative of any function is unique but on the other hand, the integral of every function is not unique. Time can play an important role in the difference between differentiation and integration. Both differentiation and integration, as discussed are inverse processes of each other. Exercice de Physique Chimie 5eme Python Mini Projects for Beginners. Vector calculus , or vector analysis , is concerned with differentiation and integration of vector fields , primarily in 3-dimensional Euclidean space R 3. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering , especially in the description of electromagnetic fields , gravitational fields , and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their book, Vector Analysis. A scalar field associates a scalar value to every point in a space. Calculus Notes Pdf Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. Yusuf and Prof. Bernoulli in Consider a bead sliding under gravity. Faculty of Science at Bilkent University. However, I will use linear algebra. Maths Study For Student. Underline all numbers and functions 2. In mathematics , matrix calculus is a specialized notation for doing multivariable calculus , especially over spaces of matrices. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering , while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Class Central is learner-supported. Johns Hopkins University. Korea Advanced Institute of Science and Technology. Start your review of Vector Calculus for Engineers. Collapse menu 1 Analytic Geometry 1. Lines 2. Distance Between Two Points; Circles 3. Functions 4. The slope of a function 2. Differentiation and integration of vectors Let us generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single. Curves in R3. We will also use X denote the space of input values, and Y the space of output values. Further necessary conditions 57 3. Search this site. According to Mark PDF. Advances in Cardiac Surgery: v. Analysis PDF. Art Therapy PDF. Atlantis, the Daughter PDF. Bergisel PDF. Freeman Vector Calculus Website W. Exercises Book: Multivariable Calculus Clark Bray. Find Answer. Do not enter an equivalent expression when your answer is marked wrong. Подумайте о юридических последствиях. Звонивший выдержал зловещую паузу. - А что, если мистер Танкадо перестанет быть фактором, который следует принимать во внимание. Нуматака чуть не расхохотался, но в голосе звонившего слышалась подозрительная решимость. - Если Танкадо перестанет быть фактором? - вслух размышлял Нуматака. Ему на руку была даже конструкция башни: лестница выходила на видовую площадку с юго-западной стороны, и Халохот мог стрелять напрямую с любой точки, не оставляя Беккеру возможности оказаться у него за спиной, В довершение всего Халохот двигался от темноты к свету. Расстрельная камера, мысленно усмехнулся. Халохот оценил расстояние до входа. Семь ступеней. - Почему же вся переписка Северной Дакоты оказалась в твоем компьютере. - Я ведь тебе уже говорил! - взмолился Хейл, не обращая внимания на вой сирены. - Я шпионил за Стратмором. Эти письма в моем компьютере скопированы с терминала Стратмора - это сообщения, которые КОМИНТ выкрал у Танкадо. Насколько. Сьюзан не понимала, к чему клонит Стратмор. Все еще темно? - спросила Мидж. Но Бринкерхофф не ответил, лишившись дара речи. То, что он увидел, невозможно было себе представить. Стеклянный купол словно наполнился то и дело вспыхивающими огнями и бурлящими клубами пара. Бринкерхофф стоял точно завороженный и, не в силах унять дрожь, стукался лбом о стекло. Сквозь строй - лучший антивирусный фильтр из всех, что я придумал. Через эту сеть ни один комар не пролетит. Выдержав долгую паузу, Мидж шумно вздохнула. - Возможны ли другие варианты. Я, университетский профессор, - подумал он, - выполняю секретную миссию.

G4 CCSS Math Vocabulary Terms in this set (143) an angle with a measure less than 90 degrees to combine, to put together two or more quantities any number being added problems that ask how much more or less one amount is than another a step-by-step method for computing two rays that share an endpoint The measure of the size of an angle. It tells how far one side is turned from the other side. A one degree angle turns through 1/360 of a full circle. Part of a circle between any two of its points The measure, in square units, of the inside of a plane figure A model of multiplication that shows each place value product An arrangement of objects in equal rows Associative Property of Addition Changing the grouping of three or more addends does not change the sum Associative Property of Multiplication Changing the grouping of three or more factors does not change the product A characteristic of an object, such as color, shape, size, etc Fractions that are commonly used for estimation Capacity refers to the amount of liquid a container can hold A metric unit of length equal to 0.01 of a meter A plane figure with all points the same distance from a fixed point called To sort into categories or to arrange into groups by attributes For two or more fractions, a common denominator is a common multiple of the denominators Commutative Property of Addition Changing the order of the addends does not change the sum Commutative Property of Multiplication Changing the order of the factors does not change the product To decide if one number is greater than, less than, or equal to Used to represent larger and smaller amounts in a comparison situation. Can be used to represent all four operations. Different lengths of bars are drawn to represent each number To put together components or basic elements A number greater than 0 that has more than two different Having exactly the same size and shape A customary unit of capacity. 1 cup = 8 fluid ounces A system of measurement used in the U.S. The system includes units for measuring length, capacity, and weight A collection of information gathered for a purpose. Data may be in the form of either words or numbers A number with one or more digits to the right of a decimal point A fractional number with a denominator of 10 or a power of 10. Usually written with a decimal point A number containing a decimal point A dot (.) separating the whole number from the fraction in decimal notation To separate into components or basic elements degree (angle measure) A unit for measuring angles. Based on dividing one complete circle into 360 equal parts The quantity below the line in a fraction. It tells how many equal parts are in the whole A tool used to measure and draw angles. A customary unit of capacity. 1 ______ = 2 pints or 1 _____ = 4 cups The answer to a division problem. The difference between the greatest number and the least number in a set of data. A part of a line that has one endpoint and goes on forever in one direction. An answer that is based on good number sense. ________ addition and subtraction facts or _______ multiplication and division facts. Also called fact family. The amount left over when one number is divided by another. An angle that measures exactly 90º. A triangle that has one 90º angle. round a whole number To find the nearest ten, hundred, thousand, (and so on). One sixtieth of a minute. There are 60 _______ in a minute. A set of numbers arranged in a special order or pattern. When a fraction is expressed with the fewest possible pieces, it is in ______ ______. (Also called lowest terms.) To express a fraction in simplest form. A unit, such as square centimeter or square inch, used to measure area. A common or usual way of writing a number using digits. An operation that gives the difference between two numbers. The answer to an addition problem. One of the equal parts when a whole is divided into 10 equal parts. A duration of a segment of time. Having length and width. Having area, but not volume. Also called a plane figure. A fraction that has 1 as its numerator. _________ that are not equal. A letter or symbol that represents a number. The point at which two line segments, lines, or rays meet to form an angle. The number of cubic units it takes to fill a figure. The measure of how heavy something is. ____ ______ are zero and the counting numbers 1, 2, 3, 4, 5, 6, and so on. A way of using words to write a number. A customary unit of length. 1 _____ = 3 feet or 36 inches. Zero Property of Multiplication The product of any number and zero is zero. _________ in two or more fractions that are the same. A set of connected points continuing without end in both directions. line of symmetry A line that divides a figure into two congruent halves that are mirror images of each other. A diagram showing frequency of data on a number line. A part of a line with two endpoints. line symmetric figures Figures that can be folded in half and its two parts match The basic unit of capacity in the metric system. 1 ____ = 1,000 milliliters When a fraction is expressed with the fewest possible pieces, it is in _____ ____. (Also called simplest form.) The amount of matter in an object. A standard unit of length in the metric system A system of measurement based on tens. The basic unit of capacity is the liter. The basic unit of length is the meter. The basic unit of mass is the gram. A customary unit of length. 1 ____ = 5,280 feet A metric unit of capacity. 1,000 _______ = 1 liter A metric unit of length. 1,000 __________ = 1 meter One sixtieth of an hour or 60 seconds. A number that has a whole number (not 0) and a fraction. A product of a given whole number and any other whole number. Compare by asking or telling how many times more one amount is as another. For example, 4 times greater than. The operation of repeated addition of the same number. A diagram that represents numbers as points on a line. The number written above the line in a fraction. It tells how many equal parts are described in the fraction. An angle with a measure greater than 90º but less than 180º. Order of Operations A set of rules that tells the order in which to compute. A customary unit of weight equal to one sixteenth of a pound. 16 _________ = 1 pound Lines that are always the same distance apart. They do not intersect. Used in mathematics as grouping symbols for operations. A repeating or growing sequence or design. An ordered set of numbers or shapes arranged according to a rule. The distance around the outside of a figure. In a large number, periods are groups of 3 digits separated by commas or by spaces. Two intersecting lines that form right angles. A customary unit of capacity. 1 ______ = 2 cups The value of the place of a digit in a number. A two-dimensional figure. The exact location in space represented by a dot. A customary unit of weight. 1 _______ = 16 ounces. A whole number greater than 0 that has exactly two different factors, 1 and itself. The answer to a multiplication problem. Any of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The amount that remains after one quantity is subtracted When one of the factors of a product is a sum, multiplying each addend before adding does not change the product. To separate into equal groups and find the number in each group or the number of groups. A number that is divided by another number. The number by which another number is divided. A point at either end of a line segment, or a point at one end of a ray. Having the same value. A mathematical sentence with an equals sign. The amount on one side of the equals sign has the same value as the amount on the other side. Fractions that have the same value. To find a number close to an exact amount, an estimate tells about how much or about how many. To find the value of a mathematical expression. A way to write numbers that shows the place value of each digit. A mathematical phrase without an equal sign. A group of related facts that use the same numbers. Also called related facts The whole numbers that are multiplied to get a product. A set of two whole numbers when multiplied, will result in a A customary unit of length. 1 ____ = 12 inches A rule that is written as an equation. A way to describe a part of a whole or a part of a group by using equal parts. A table that lists pairs of numbers that follow a rule. A customary unit of capacity. 1 ____ = 4 quarts The standard unit of mass in the metric system. 1,000 ______ = 1 kilogram ________ ____ is used to compare two numbers when the first number is larger than the second number. A unit of time. 1 ____ = 60 minutes. 24 _____ = 1 day One of the equal parts when a whole is divided into 100 In the decimal numeration system, __________ is the name of the next place to the right of tenths. Identity Property of Addition If you add zero to a number, the sum is the same as that Identity Property of Multiplication If you multiply a number by one, the product is the same as that number. A term for a fraction whose numerator is greater than or equal to its denominator. A customary unit of length. 12 _______ = 1 foot Lines that cross at one point. Operations that undo each other. A metric unit of mass equal to 1000 grams. A metric unit of length equal to 1000 meters. How long something is. The distance from one point to another. ____ ____ is used to compare two numbers when the first number is smaller than the second number. YOU MIGHT ALSO LIKE... Numbers & Animals Vocabulary | Everyday Traditional Mandarin Chinese Math 4th Grade Vocabulary 3rd Grade Math Vocabulary OTHER SETS BY THIS CREATOR ccss ela grade 4 Alipne Lakes Grade 4 Math All Lesson 4 tsunami Earth Science Proper Fractions / Improper Grade 4 THIS SET IS OFTEN IN FOLDERS WITH... Engage NY Grade 4 - Module 2 Grade 4 Math Review OSB Gr4 geometry List 1 Grade 4 Math Vocabulary

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric A key result in the proof of black hole uniqueness in -dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in -dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, . This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than , it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the -dimensional proof, that the spacetime is analytic. Consider an -dimensional stationary spacetime containing a black hole. Since the event horizon of the black hole must be mapped into itself by the action of any isometry, the asymptotically timelike Killing field must be tangent to the horizon. Therefore, we have two cases to consider: (i) is normal to the horizon, i.e., tangent to the null geodesic generators of the horizon; (ii) is not normal to the horizon. In -dimensions it is known that in case (i), for suitably regular non-extremal vacuum or Einstein-Maxwell black holes, the black hole must be static [42, 5]. Furthermore, in -dimensions it is known that in case (ii), under fairly general assumptions about the nature of the matter content but assuming analyticity of the spacetime and non-extremality of the black hole, there must exist an additional Killing field that is normal to the horizon. It can then be shown that the black hole must be axisymmetric111 In this paper, by “axisymmetric” we mean that spacetime possesses one-parameter group of isometries isomorphic to whose orbits are spacelike. We do not require that the Killing field vanishes on an “axis.” as well as stationary [18, 19]. This latter result is often referred to as a “rigidity theorem,” since it implies that the horizon generators of a “rotating” black hole (i.e., a black hole for which is not normal to the horizon) must rotate rigidly with respect to infinity. A proof of the rigidity theorem in -dimensions which partially eliminates the analyticity assumption was given by Friedrich, Racz, and Wald [9, 32], based upon an argument of Isenberg and Moncrief [27, 20] concerning the properties of spacetimes with a compact null surface with closed generators. The above results for both cases (i) and (ii) are critical steps in the proofs of black hole uniqueness in -dimensions, since they allow one to apply Israel’s theorems [23, 24] in case (i) and the Carter-Robinson-Mazur-Bunting theorems [2, 36, 25, 1] in case (ii). Many attempts to unify the forces and equations of nature involve the consideration of spacetimes with dimensions. Therefore, it is of considerable interest to consider a generalization of the rigidity theorem to higher dimensions, especially in view of the fact that there seems to be a larger variety of black hole solutions (see e.g., [7, 12, 15]), the classification of which has not been achieved yet. 222 There have recently appeared several works on general properties of a class of stationary, axisymmetric vacuum solutions, including an -dimensional generalization of the Weyl solutions for the static case (see e.g., [3, 6, 16, 17], and see also [26, 43] and references therein for some techniques of generating such solutions in -dimensions). The purpose of this paper is to present a proof of the rigidity theorem in higher dimensions for non-extremal black holes. The dimensionality of the spacetime enters the proof of the rigidity theorem in -dimensions in the following key way: The expansion and shear of the null geodesic generators of the horizon of a stationary black hole can be shown to vanish (see below). The induced (degenerate) metric on the -dimensional horizon gives rise to a Riemannian metric, , on an arbitrary -dimensional cross-section, , of the horizon. On account of the vanishing shear and expansion, all cross-sections of the horizon are isometric, and the projection of the stationary Killing field onto gives rise to a Killing field, , of on . In case (ii), does not vanish identically. Now, when , it is known that must have the topology of a -sphere, . Since the Euler characteristic of is nonzero, it follows that must vanish at some point . However, since is -dimensional, it then follows that the isometries generated by simply rotate the tangent space at . It then follows that all of the orbits of are periodic with a fixed period , from which it follows that, after period , the orbits of on the horizon must return to the same generator. Consequently, if we identify points in spacetime that differ by the action of the stationary isometry of parameter , the horizon becomes a compact null surface with closed null geodesic generators. The theorem of Isenberg and Moncrief [27, 20] then provides the desired additional Killing field normal to this null surface. In dimensions, the Euler characteristic of may vanish, and, even if it is non-vanishing, if there is no reason that the isometries generated by need have closed orbits even when vanishes at some point . Thus, for example, even in the -dimensional Myers-Perry black hole solution with cross section topology , one can choose the rotational parameters of the solution so that the orbits of the stationary Killing field do not map horizon generators into themselves. One possible approach to generalizing the rigidity theorem to higher dimensions would be to choose an arbitrary and identify points in the spacetime that differ by the action of the stationary isometry of parameter . Under this identification, the horizon would again become a compact null surface, but now its null geodesic generators would no longer be closed. The rigidity theorem would follow if the results of [27, 20] could be generalized to the case of compact null surfaces that are ruled by non-closed generators. We have learned that Isenberg and Moncrief are presently working on such a generalization , so it is possible that the rigidity theorem can be proven in this way. However, we shall not proceed in this manner, but rather will parallel the steps of [27, 20], replacing arguments that rely on the presence of closed null generators with arguments that rely on the presence of stationary isometries. Since on the horizon we may write where is tangent to the null geodesic generators and is tangent to cross-sections of the horizon, the stationarity in essence allows us to replace Lie derivatives with respect to by Lie derivatives with respect to . Thus, equations in [27, 20] that can be solved by integrating quantities along the orbits of the closed null geodesics correspond here to equations that can be solved if one can suitably integrate these equations along the orbits of in . Although the orbits of are not closed in general, we can appeal to basic results of ergodic theory together with the fact that generates isometries of to solve these equations. For simplicity, we will focus attention on the vacuum Einstein’s equation, but we will indicate in section 4 how our proofs can be extended to models with a cosmological constant and a Maxwell field. As in [18, 19] and in [27, 20], we will assume analyticity, but we shall indicate how this assumption can be partially removed (to prove existence of a Killing field inside the black hole) by arguments similar to those given in [9, 32]. The non-extremality condition is used for certain constructions in the proof (as well as in the arguments partially removing the analyticity condition), and it would not appear to be straightforward to generalize our arguments to remove this restriction when the orbits of are not closed. Our signature convention for is . We define the Riemann tensor by and the Ricci tensor by . We also set . 2 Proof of existence of a horizon Killing field Let be an -dimensional, smooth, asymptotically flat, stationary solution to the vacuum Einstein equation containing a black hole. Thus, we assume the existence in the spacetime of a Killing field with complete orbits which are timelike near infinity. Let denote a connected component of the portion of the event horizon of the black hole that lies to the future of . We assume that has topology , where is compact. Following Isenberg and Moncrief [27, 20], our aim in this section is to prove that there exists a vector field defined in a neighborhood of which is normal to and on satisfies where is an arbitrary vector field transverse to . As we shall show at the end of this section, if we assume analyticity of and of it follows that is a Killing field. We also will explain at the end of this section how to partially remove the assumption of analyticity of and . We shall proceed by constructing a candidate Killing field, , and then proving that eq. (2) holds for . This candidate Killing field is expected to satisfy the following properties: (i) should be normal to . (ii) If we define by then, on , should be tangent to cross-sections333Note that as already mentioned above, since is mapped into itself by the time translation isometries, must be tangent to , so is automatically tangent to . Condition (iii) requires that there exist a foliation of by cross-sections such that each orbit of is contained in a single cross-section. of . (iii) should commute with . (iv) should have constant surface gravity on , i.e., on we should have with constant on , since, by the zeroth law of black hole mechanics, this property is known to hold on any Killing horizon in any vacuum solution of Einstein’s equation. We begin by choosing a cross-section , of . By arguments similar to those given in the proof of proposition 4.1 of , we may assume without loss of generality that has been chosen so that each orbit of on intersects at precisely one point, so that is everywhere transverse to . We extend to a foliation, , of by the action of the time translation isometries, i.e., we define , where denotes the one-parameter group of isometries generated by . Note that the function on that labels the cross-sections in this foliation automatically satisfies Next, we define and on by where is normal to and is tangent to . It follows from the transversality of that is everywhere nonvanishing and future-directed. Note also that on . Our strategy is to extend this definition of to a neighborhood of via Gaussian null coordinates. This construction of obviously satisfies conditions (i) and (ii) above, and it also will be shown below that it satisfies condition (iii). However, it will, in general, fail to satisfy (iv). We shall then modify our foliation so as to produce a new foliation so that (iv) holds as well. We will then show that the corresponding satisfies eq. (2). Given our choice of and the corresponding choice of on , we can uniquely define a past-directed null vector field on by the requirements that , and that is orthogonal to each . Let denote the affine parameter on the null geodesics determined by , with on . Let be local coordinates on an open subset of . Of course, it will take more than one coordinate patch to cover , but there is no problem in patching together local results, so no harm is done in pretending that covers . We extend the coordinates from to by demanding that they be constant along the orbits of . We then extend and to a neighborhood of by requiring these quantities to be constant along the orbits of . It is easily seen that the quantities define coordinates covering a neighborhood of . Coordinates that are constructed in this manner are known as Gaussian null coordinates and are unique up to the choice of and the choice of coordinates on . It follows immediately that on we have and we extend and to a neighborhood of by these formulas. Clearly, and commute, since they are coordinate vector fields. Note that we have so everywhere, not just on . Similarly, we have everywhere. It follows that in Gaussian null coordinates, the metric in a neighborhood of takes the form where, again, is a labeling index that runs from to . We write Note that , , and are independent of the choice of coordinates, , and thus are globally defined in an open neighborhood of . From the form of the metric, we clearly have and . It then follows that is the orthogonal projector onto the subspace of the tangent space perpendicular to and , where here and elsewhere, all indices are raised and lowered with the spacetime metric . Note that when , i.e., off of the horizon, differs from the metric , on the -dimensional submanifolds, , of constant , since fails to be perpendicular to these surfaces. Here, is defined by the condition that is the orthogonal projector onto the subspace of the tangent space that is tangent to ; the relationship between and is given by However, since on (where ), we have , we will refer to as the metric on the cross-sections of . Thus, we see that in Gaussian null coordinates the spacetime metric, , is characterized by the quantities , , and . In terms of these quantities, if we choose , then the condition (2) will hold if and only if the conditions hold on . Since the vector fields and are uniquely determined by the foliation and since (i.e., the time translations leave the foliation invariant), it follows immediately that and are invariant under . Hence, we have , so, in particular, condition (iii) holds, as claimed above. Similarly, we have and throughout the region where the Gaussian null coordinates are defined. Since , we obtain from eq. (8) Contraction of this equation with yields Contraction with then yields and we then also immediately obtain The next step in the analysis is to use the Einstein equation on , in a manner completely in parallel with the 4-dimensional case . This equation is precisely the Raychaudhuri equation for the congruence of null curves defined by on . Since that congruence is twist-free on , we obtain on where denotes the expansion of the null geodesic generators of , denotes their shear, and is the affine parameter along null geodesic generators of with tangent . Now, by the same arguments as used to prove the area theorem , we cannot have on . On the other hand, the rate of change of the area, , of (defined with respect to the metric ) is given by However, since is related to by the isometry , the left side of this equation must vanish. Since on , this shows that on . It then follows immediately that on . Now on , the shear is equal to the trace free part of while the expansion is equal to the trace of this quantity. So we have shown that on . Thus, the first equation in eq. (2) holds with . However, in general fails to satisfy condition (iv) above. Indeed, from the form, eq. (8), of the metric, we see that the surface gravity, , associated with is simply , and there is no reason why need be constant on . Since on , the Einstein equation on yields (see eq. (79) of Appendix A) where denotes the derivative operator on , i.e., . Thus, if is not constant on , then the last equation in eq. (2) fails to hold even when . As previously indicated, our strategy is repair this problem by choosing a new cross-section so that the corresponding arising from the Gaussian null coordinate construction will have constant surface gravity on . The determination of this requires some intermediate constructions, to which we now turn. First, since we already know that everywhere and that on , it follows immediately from the fact that on that on (for any choice ). Thus, is a Killing vector field for the Riemannian metric on . Therefore the flow, of yields a one-parameter group of isometries of , which coincides with the projection of the flow of the original Killing field to . We define to be the mean value of on , where is the area of with respect to the metric . In the following we will assume that , i.e., that we are in the “non-degenerate case.” Given that , we may assume without loss of generality, that . We seek a new Gaussian null coordinate system satisfying all of the above properties of together with the additional requirement that , i.e., constancy of the surface gravity. We now determine the conditions that these new coordinates would have to satisfy. Since clearly must be proportional to , we have for some positive function . Since , we must have . Since on we have and is given by we find that must satisfy The last equality provides an equation that must be satisfied by on . In order to establish that a solution to this equation exists, we first prove the following lemma: For any , we have Furthermore, the convergence of the limit is uniform in . Similarly, -derivatives of converge to uniformly in as . Proof: The von Neumann ergodic theorem (see e.g.,) states that if is an function for on a measure space with finite measure, and if is a continuous one-parameter group of measure preserving transformations on , then converges in the sense of (and in particular almost everywhere). We apply this theorem to , , , and , to conclude that there is an function on to which the limit in the lemma converges. We would like to prove that is constant. To prove this, we note that eq. (18) together with the facts that and yields where are constants independent of and , and where is finite because is compact. Consequently, is uniformly bounded in and in . Thus, for all , we have Let be such that converges as . (As already noted above, existence of such a is guaranteed by the von Neumann ergodic theorem.) The above equation then shows that, in fact, must converge for all as and that, furthermore, the limit is independent of , as we desired to show. Thus, is constant, and hence equal to its spatial average, . The estimate (30) also shows that the limit (24) is uniform in . Similar estimates can easily be obtained for the norm with respect to of , for any . These estimates show that -derivatives of converge to uniformly in . We now are in a position to prove the existence of a positive function on satisfying the last equality in eq. (23) on . Let where is the function on defined by The function is well defined for almost all because for any and sufficiently large , by Lemma 1. It also follows from the uniformity statement in Lemma 1 that is smooth on . By a direct calculation, using Lemma 1, we find that satisfies as we desired to show. We now can deduce how to choose the desired new Gaussian null coordinates. The new coordinate must satisfy as before. However, in view of eq. (21), it also must satisfy Since , we find that on , must satisfy Substituting from eq. (34), we obtain Thus, if our new Gaussian null coordinates exist, there must exist a smooth solution to this equation. That this is the case is proven in the following lemma. There exists a smooth solution to the following differential equation on : Proof: First note that the orbit average of any function of the form where is smooth must vanish, so there could not possibly exist a smooth solution to the above equation unless the average of over any orbit is equal to . However, this was proven to hold in Lemma 1. In order to get a solution to the above equation, choose , and consider the regulated expression defined by Due to the exponential damping, this quantity is smooth, and satisfies the differential equation We would now like to take the limit as to get a solution to the desired equation. However, it is not possible to straightforwardly take the limit as of , for there is no reason why this should converge without using additional properties of . In fact, we will not be able to show that the limit as of exists, but we will nevertheless construct a smooth solution to eq. (39). To proceed, we rewrite eq. (40) as where denotes the pull-back map on tensor fields associated with . Taking the gradient of this equation and using eq. (27), we obtain where here and in the following we use differential forms notation and omit tensor indices. Since clearly commutes with and since is just the derivative along the orbit over which we are integrating, we can integrate by parts to obtain It follows from the von Neumann ergodic theorem444 Here, the theorem is applied to the case of a tensor field of type on a compact Riemannian manifold , rather than a scalar function, and where the measure preserving map is a smooth one-parameter family of isometries acting on via the pull back. To prove this generalization, we note that a tensor field of type on a manifold may be viewed as a function on the fiber bundle, , of all tensors of type over that satisfies the additional property that this function is linear on each fiber. Equivalently, we may view as a function, , on the bundle, , of unit norm tensors of type that satisfies a corresponding linearity property. A Riemannian metric on naturally gives rise to a Riemannian metric (and, in particular, a volume element) on , and is compact provided that is compact. Since the isometry flow on naturally induces a volume preserving flow on , we may apply the von Neumann ergodic theorem to to obtain the orbit averaged function . Since will satisfy the appropriate linearity property on each fiber, we thereby obtain the desired orbit averaged tensor field . (see eq. (25)) that the limit exists in the sense of . Furthermore, the limit in the sense of also exists of all -derivatives of the left side. Indeed, because is an isometry commuting with the derivative operator of the metric , we have The expression on the right side converges in , as by the von Neumann ergodic theorem, meaning that for all , where denotes the Sobolev space of order . By the Sobolev embedding theorem, where the embedding is continuous with respect to the sup norm on the all derivatives in the space , i.e., for all . Thus, convergence of the limit (45) actually occurs in the sup norms on . Thus, in particular, . Now pick an arbitrary , and define by where the integral is over any smooth path connecting and . This integral manifestly does not depend upon the choice of , independently of the topology of . By what we have said above, the function is smooth, with a smooth limit which is independent of the choice of . Furthermore, the convergence of and its derivatives to and its derivatives is uniform. Now, by inspection, is a solution to the differential equation Furthermore, the limit exists by the ergodic theorem, and vanishes by Lemma 1. Thus, the smooth, limiting quantity satisfies the desired differential equation (39). We now define a new set of Gaussian null coordinates as follows. Define on to be a smooth solution to eq. (38), whose existence is guaranteed by Lemma 2. Extend to by eq. (35). It is not difficult to verify that is given explicitly by is the map projecting any point in to the point on the cross section on the null generator through . Let denote the surface on . Then our desired Gaussian null coordinates are the Gaussian null coordinates associated with . The corresponding fields satisfy all of the properties derived above for and, in addition, satisfy the condition that is constant on . Now let . We have previously shown that on , since this relation holds for any choice of Gaussian null coordinates. However, since our new coordinates have the property that is constant on , we clearly have that on . Furthermore, for our new coordinates, eq. (18) immediately yields on . Thus, we have proven that all of the relations in eq. (2) hold for . on . Since , with tangent to , and since all quantities appearing in eq. (55) are Lie derived by , we may replace in this equation all Lie derivatives by . Hence, we obtain Integration of this equation yields where is a tensor at that is independent of . Integrating this equation (and absorbing constant factors into ), we obtain However, since is a Riemannian isometry, each orthonormal frame component of at is uniformly bounded in by the Riemannian norm of , i.e., . Consequently, the limit of eq. (59) as immediately yields from which it then immediately follows that Thus, we have , and therefore on , as we desired to show. Thus, we now have shown that the first equation in (2) holds for , and that the other equations hold for , for the tensor fields associated with the “tilde” Gaussian null coordinate system, and . In order to prove that eq. (2) holds for all , we proceed inductively. Let , and assume inductively that the first of equations (2) holds for all , and that the remaining equations hold for all . Our task is to prove that these statements then also hold when is replaced by . To show this, we apply the operator to the Einstein equation (see eq. (78)) and restrict to . Using the inductive hypothesis, one sees that on , thus establishes the second equation in (2) for . Next, we apply the operator to the Einstein equation (see eq. (81)), and restrict to . Using the inductive hypothesis, one sees that on , thus establishes the third equation in (2) for . Next, we apply the operator to the Einstein equation (see eq. (82)), and restrict to . Using the inductive hypothesis and the above results and , one sees that the tensor field satisfies a differential equation of the form on . By the same argument as given above for , it follows that . This establishes the first equation in (2) for , and closes the induction loop. Thus far, we have assumed only that the spacetime metric is smooth (). However, if we now assume that the spacetime is real analytic, and that is an analytic submanifold, then it can be shown that the vector field that we have defined above is, in fact, analytic. To see this, first note that if the cross section of is chosen to be analytic, then our Gaussian null coordinates are analytic, and, consequently, so is any quantity defined in terms of them, such as and . Above, was defined in terms of a certain special Gaussian normal coordinate system that was obtained from a geometrically special cross section. That cross section was obtained by a change (53) of the coordinate . Thus, to show that is analytic, we must show that this change of coordinates is analytic. By eq. (53), this will be the case provided that and are analytic. We prove this in Appendix C. Since and are analytic, so is . It follows immediately from the fact that this quantity and all of its derivatives vanish at any point of that where defined, i.e., within the region where the Gaussian null coordinates are defined. This proves existence of a Killing field in a neighborhood of the horizon. We may then extend by analytic continuation. Now, analytic continuation need not, in general, give rise to a single-valued extension, so we cannot conclude that there exists a Killing field on the entire spacetime. However, by a theorem of Nomizu (see also ), if the underlying domain is simply connected, then analytic continuation does give rise to a single-valued extension. By the topological censorship theorem [10, 11], the domain of outer communication has this property. Consequently, there exists a unique, single valued extension of to the domain of outer communication, i.e., the exterior of the black hole (with respect to a given end of infinity). Thus, in the analytic case, we have proven the following theorem: Let be an analytic, asymptotically flat -dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field with complete orbits which are timelike near infinity. Assume that a connected component, , of the event horizon of the black hole is analytic and is topologically , with compact and that (where is defined eq. (20) above). Then there exists a Killing field , defined in a region that covers and the entire domain of outer communication, such that is normal to the horizon and commutes with . The assumption of analyticity in this theorem can be partially removed in the following manner, using an argument similar to that given in . Since , the arguments of show that the spacetime can be extended, if necessary, so that is a proper subset of a regular bifurcate null surface in some enlarged spacetime

Returns from an investment can be estimated using both absolute returns and CAGR. On the one hand, absolute returns are a measure of the total return from an investment, irrespective of the time period. CAGR, on the other hand, is the return from an investment during a specific period. Both absolute returns and CAGR are used for determining the return from an investment. However, both use different ways to calculate the return. This article covers absolute return and CAGR in detail and elaborates on absolute return vs CAGR. What Are Absolute Returns in a Mutual Fund? Absolute returns in mutual funds refer to the return from a fund over a certain period of time. It is the total return from a mutual fund from the date of investment. Absolute returns are expressed as a percentage and show how much the investment has grown or depreciated in value. Absolute returns are pure returns from the investment and don’t compare to any other benchmark. Also, absolute returns can be positive or negative. The fund managers of mutual funds seek a positive return by using multiple strategies like short selling or derivatives. While calculating absolute returns, the tenure of the investment is the least important. Only actual investment and the current value of the investment are considered while estimating the absolute return. The formula for Absolute return: ((Current value of the investment/ Actual investment) – 1) * 100. Let’s understand absolute returns with an example. An investor invests INR 1,00,00 in a mutual fund. Over a certain period of time, the investment grows to INR 3,00,000. The absolute return from this investment can be calculated using the above formula. Absolute return = (300000/100000 – 1) * 100 The absolute return of the above investment is 200%. The tenure of the investment is not considered while calculating the return from the investment. The 200% return could’ve been earned over a period of months, years or decades. Using absolute return alone, one cannot determine whether the investment is good or not as the tenure of the investment isn’t known. Therefore, absolute returns only tell how much the investment depreciated or appreciated. It doesn’t tell how fast the investment grew or fell. Hence, absolute returns cannot be used for comparison of two different investments. What is Compound Annual Growth Rate (CAGR) in a Mutual Fund? CAGR (Compounded annual growth rate) is the rate of return from a mutual fund during a specific period of time, assuming the profits are reinvested. In other words, CAGR shows how much the investment has grown from the beginning to ending value over a period of time. CAGR shows the rate at which the investment grows each year to reach the investment’s final value. It smoothes out the performance of a fund so that it can be easily understood and becomes comparable to other investments. One can use CAGR to compare two investments and determine which has performed better during a specific period of time. The formula for Compounded Annual Growth Rate CAGR CAGR = ((Ending value/ Beginning value) ^ (1/n)) – 1 Where, n is tenure of the investment Let’s understand CAGR better with the help of an example. An investor invested INR 2,50,000 lump sum in a mutual fund. And the investment grew to INR 4,00,000 in 3 years. The CAGR of this investment can be calculated using the above formula. CAGR = ((400000/250000) ^ (1/3)) – 1 CAGR = 16.96% This means the investment grew 16.96% every year for three years for it to grow to INR 4,00,000. In other words, the average return from this mutual fund is 16.96%. The absolute return from this investment is 60%. But the CAGR is 16.96%. CAGR enables investors to compare multiple investments and help them plan their financial future. Let’s say an investor has an opportunity to invest in stocks and bonds that have a CAGR of 18% and 15%, respectively. The investor will choose stocks as it has a higher CAGR when compared to bonds. Moreover, CAGR can be used to estimate the average growth of an investment. Due to market volatility, the investment might grow by 10% one year and grow only by 2% the other year. CAGR helps smoothen out the returns and gives a better picture of an investment’s overall growth. Calculating CAGR can be a time taking process. Hence one can use a CAGR calculator to estimate returns from a mutual fund investment. Scripbox’s CAGR calculator is a simple online tool that helps in calculating the CAGR of investment to analyze an investment opportunity. All one has to do is enter the initial value, final value, and investment tenure. The calculator will estimate the CAGR within seconds. Difference Between Absolute Return Vs CAGR Investments are made to earn profits. There are different ways to represent returns from an investment. Absolute returns is a simple method that helps in determining the return from an investment, irrespective of the period or tenure of the investment. Absolute returns simply take the initial investment amount and the maturity amount. On the other hand, compounded annual growth rate takes into account the investment duration or tenure. Hence, it gives a more accurate and comparable earnings percentage. The formula for Absolute return ((Current value of the investment/ Initial investment) – 1) * 100. The formula to calculate CAGR CAGR = ((Ending value/ Beginning value) ^ (1/n)) – 1 Where, n is tenure of the investment Example for Absolute Return Vs CAGR To understand the difference between absolute return and CAGR better, let’s take an example of Mr Krishna who invested INR 5,00,000 lump sum in a mutual fund in 2010. He withdrew the investment in 2020, and the value of the investment is 8,00,000. The absolute return for Mr. Krishna is ((8,00,000/5,00,000) – 1) * 100 Absolute Return = 60% While, the CAGR is ((8,00,000/5,00,000)^(1/10)) – 1 CAGR = 4.81% While the above absolute return looks promising, the investment has actually grown only 4.81% every year for ten years. Absolute returns tell how much an investment depreciated or appreciated. And, it doesn’t tell how fast the investment grew or fell. Therefore, absolute returns are not ideal for comparing two different investments. On the other hand, CAGR can be used to determine an investment’s average growth. With markets being volatile, the returns are never the same over the years. For example, an investment might grow by 12% one year and grow only by 5% the other year. Therefore, CAGR addresses the volatility and smoothens out the returns. Hence, it gives a clear picture of an investment’s overall growth. Also, it is a good measure to compare different investments. In short, if absolute return is the distance your investment has travelled, then CAGR is the rate at which your investment has travelled or grown. CAGR Vs Absolute Return – Which is Better? Both absolute returns and compounded annual growth rate are useful in determining the returns from an investment. However, the difference between the two lies in the aspect of time consideration. For investments with longer durations, the CAGR value is a better measure. CAGR determines an investment’s annual growth rate, whose value usually fluctuates over the investment tenure. While on the other hand, absolute returns consider only the purchase value and sale value of an investment to calculate returns. For investments with a duration of less than a year, one can consider the absolute return. While, for investments with tenure greater than a year, CGAR gives a better picture. Also, with CAGR, one can compare two or more investments held for different periods. When it comes to tenure less than one year, CAGR may inflate or shrink the returns therefore, not giving the actual return. Frequently Asked Questions Annualized return is the measure of an investment’s performance during a specific period. In other words, annualized return shows how much your investment has grown from the beginning to ending value over a certain period of time. It is the same as CAGR. To convert absolute returns to CAGR, one should take the nth root of (Current value of the investment/ Actual investment) and subtract 1 from it. In other words, ((Current value of the investment/ Actual investment)^(1/n)) – 1 will give the CAGR value. CAGR considers the tenure of an investment and helps in determining the annual growth rate. On the other hand, absolute return considers only the investment value and the maturity value. Therefore, the absolute return cannot be used for comparison of different investments. Hence, CAGR is a good metric that helps investors compare the performance of different investments. It also gives a complete picture of the gains made from your investments. Furthermore, for investments with a tenure of more than one year, CAGR gives a better picture of the returns. Finally, CAGR determines the return from an investment over a specific period of time while taking into consideration the market volatility.

- What does an r2 value of 0.5 mean? - What is a good R value for correlation? - What does an R squared value of 0.4 mean? - What is a good R squared value? - What does an R 2 value mean? - What is a good R value in statistics? - Is 0.6 A strong correlation? - How do you know if a correlation is significant? - What is a good r2 value for regression? - How do you tell if a regression model is a good fit? - How do you calculate r2 value? - What does a low R squared value mean? - What does an R squared value of 0.3 mean? - What does an R squared value of 0.2 mean? - What does an R squared value of 0.6 mean? - How do you interpret an R? - Is a low R Squared good? - Can R Squared be above 1? What does an r2 value of 0.5 mean? Key properties of R-squared Finally, a value of 0.5 means that half of the variance in the outcome variable is explained by the model. Sometimes the R² is presented as a percentage (e.g., 50%).. What is a good R value for correlation? The relationship between two variables is generally considered strong when their r value is larger than 0.7. The correlation r measures the strength of the linear relationship between two quantitative variables. What does an R squared value of 0.4 mean? R-squared = Explained variation / Total variation. R-squared is always between 0 and 100%: 0% indicates that the model explains none of the variability of the response data around its mean. 100% indicates that the model explains all the variability of the response data around its mean. What is a good R squared value? Any study that attempts to predict human behavior will tend to have R-squared values less than 50%. However, if you analyze a physical process and have very good measurements, you might expect R-squared values over 90%. What does an R 2 value mean? R-squared (R2) is a statistical measure that represents the proportion of the variance for a dependent variable that’s explained by an independent variable or variables in a regression model. … It may also be known as the coefficient of determination. What is a good R value in statistics? For a natural/social/economics science student, a correlation coefficient higher than 0.6 is enough. Correlation coefficient values below 0.3 are considered to be weak; 0.3-0.7 are moderate; >0.7 are strong. You also have to compute the statistical significance of the correlation. Is 0.6 A strong correlation? Correlation Coefficient = 0.8: A fairly strong positive relationship. Correlation Coefficient = 0.6: A moderate positive relationship. … Correlation Coefficient = -0.8: A fairly strong negative relationship. Correlation Coefficient = -0.6: A moderate negative relationship. How do you know if a correlation is significant? To determine whether the correlation between variables is significant, compare the p-value to your significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. An α of 0.05 indicates that the risk of concluding that a correlation exists—when, actually, no correlation exists—is 5%. What is a good r2 value for regression? 25 values indicate medium, . 26 or above and above values indicate high effect size. In this respect, your models are low and medium effect sizes. However, when you used regression analysis always higher r-square is better to explain changes in your outcome variable. How do you tell if a regression model is a good fit? The best fit line is the one that minimises sum of squared differences between actual and estimated results. Taking average of minimum sum of squared difference is known as Mean Squared Error (MSE). Smaller the value, better the regression model. How do you calculate r2 value? The R-squared formula is calculated by dividing the sum of the first errors by the sum of the second errors and subtracting the derivation from 1. What does a low R squared value mean? A low R-squared value indicates that your independent variable is not explaining much in the variation of your dependent variable – regardless of the variable significance, this is letting you know that the identified independent variable, even though significant, is not accounting for much of the mean of your … What does an R squared value of 0.3 mean? – if R-squared value < 0.3 this value is generally considered a None or Very weak effect size, ... - if R-squared value 0.5 < r < 0.7 this value is generally considered a Moderate effect size, - if R-squared value r > 0.7 this value is generally considered strong effect size, Ref: Source: Moore, D. S., Notz, W. What does an R squared value of 0.2 mean? R^2 of 0.2 is actually quite high for real-world data. It means that a full 20% of the variation of one variable is completely explained by the other. It’s a big deal to be able to account for a fifth of what you’re examining. GeneralMayhem on [–] R-squared isn’t what makes it significant. What does an R squared value of 0.6 mean? An R-squared of approximately 0.6 might be a tremendous amount of explained variation, or an unusually low amount of explained variation, depending upon the variables used as predictors (IVs) and the outcome variable (DV). How do you interpret an R? To interpret its value, see which of the following values your correlation r is closest to:Exactly –1. A perfect downhill (negative) linear relationship.–0.70. A strong downhill (negative) linear relationship.–0.50. A moderate downhill (negative) relationship.–0.30. … No linear relationship.+0.30. … +0.50. … +0.70.More items… Is a low R Squared good? Regression models with low R-squared values can be perfectly good models for several reasons. … Fortunately, if you have a low R-squared value but the independent variables are statistically significant, you can still draw important conclusions about the relationships between the variables. Can R Squared be above 1? some of the measured items and dependent constructs have got R-squared value of more than one 1. As I know R-squared value indicate the percentage of variations in the measured item or dependent construct explained by the structural model, it must be between 0 to 1.

Patent application title: Apparatus and Method for Determining Formation Anisotropy Jennifer Market (Rosehill, TX, US) Paul F. Rodney (Spring, TX, US) HALLIBURTON ENERGY SERVICES, INC. IPC8 Class: AG01N2904FI Class name: Measuring or testing system having scanning means by reflected wave having separate sonic transmitter and receiver Publication date: 2012-09-13 Patent application number: 20120227500 A method of generating an axial shear wave in a formation surrounding a wellbore comprising urging a clamp pad into contact with a wall of the wellbore, and applying an axial force to the clamp pad to impart a shear force into the wall of the wellbore to generate a shear wave in the 1. A method for determining at least one characteristic of an anisotropic earth formation, comprising: transmitting dipole acoustic energy into the earth formation at a first location in a wellbore where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to a first longitudinal axis of the wellbore at the first location; receiving at the first location composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from the plane of the formation orthogonal to a first longitudinal axis of the wellbore at the first location; transmitting dipole acoustic energy into the earth formation at a second location in a wellbore where the second location is axially displaced from the first location and a second longitudinal axis of the wellbore at the second location is substantially orthogonal to the first longitudinal axis of the wellbore at the first location and where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to the second longitudinal axis of the wellbore at the second location; receiving at the second location composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from a plane of the formation orthogonal to the second longitudinal axis of the wellbore at the second location; and combining the received signals at the first location and the second location to determine the least one characteristic of the anisotropic formation. 2. The method of claim 1 further comprising transmitting the at least one determined characteristic of the anisotropic earth formation to a surface location. 3. The method of claim 1 wherein the at least one characteristic of the anisotropic earth formation comprises at least one of: a three dimensional stress field of the formation and a three dimensional velocity field of the formation. 4. The method of claim 1 wherein transmitting acoustic dipole energy into the earth formation further comprises firing a first dipole transmitter in a first direction, then firing a second dipole transmitter in a second direction substantially azimuthally perpendicular to the first direction. 5. The method of claim 1 further comprising adjusting the direction of the wellbore based at least in part on the determined characteristic of the anisotropic formation. 6. A method for determining at least one characteristic of an anisotropic earth formation, comprising: transmitting dipole acoustic energy into the earth formation at a first location in a first wellbore where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to a first longitudinal axis of the wellbore at the first location; receiving at the first location in the first wellbore composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from the plane of the formation orthogonal to a first longitudinal axis of the first wellbore at the first location; transmitting dipole acoustic energy into the earth formation at a second location in an offset wellbore where a second longitudinal axis of the offset wellbore at the second location is inclined to the first longitudinal axis of the first wellbore at the first location and where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to the second longitudinal axis of the offset wellbore at the second location; receiving at the second location composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from a plane of the formation orthogonal to a second longitudinal axis of the offset wellbore at the second location; and combining the received signals at the first location and the second location to determine the least one characteristic of the formation. 7. The method of claim 6 further comprising transmitting the at least one determined characteristic of the anisotropic earth formation to a surface location. 8. The method of claim 6 wherein the at least one characteristic of the anisotropic earth formation comprises at least one of: a three dimensional stress field of the formation and a three dimensional velocity field of the formation. 9. The method of claim 6 wherein measurements in the first wellbore and measurements in the offset wellbore occur at different times. 10. An apparatus comprising: an extendable member controllably extendable from a housing in a wellbore, the extendable member urging a clamp pad into engagement with a wall of the wellbore; and an axial force assembly to cooperatively act with the extendable member and the clamp pad to move the clamp pad in an axial direction to impart an axial shear force into the formation. 11. The apparatus of claim 10 wherein the axial force assembly comprises at least one of a piezoelectric member and a magnetostrictive member to impart axial force to move the clamp pad. 12. The apparatus of claim 10 wherein the clamp pad comprises a plurality of clamp pads distributed at locations around the circumference of the wellbore. 13. The apparatus of claim 12 wherein the extendable member comprises a plurality of extendable members distributed at locations around the circumference of the housing. 14. The apparatus of claim 10 further comprising a controller to control the motion of the clamp pad. 15. The apparatus of claim 10 wherein the extendable member comprises a telescoping cylinder. 16. A method of generating an axial shear wave in a formation surrounding a wellbore comprising: urging a clamp pad into contact with a wall of the wellbore; and applying an axial force to the clamp pad to impart a shear force into the wall of the wellbore to generate a shear wave in the formation. 17. The method of claim 16 wherein urging a clamp pad into contact with a wall of the wellbore comprises extending an extendable member coupled to the clamp pad from a housing the wall of the wellbore. 18. The method of claim 16 wherein applying an axial force to the clamp pad to impart a shear force into the wall of the wellbore to generate a shear wave in the formation comprises actuating at least one of a piezoelectric member and a magnetostrictive member. The present disclosure relates generally to the field of acoustic logging. Certain earth formations exhibit a property called "anisotropy", wherein the velocity of acoustic waves polarized in one direction may be somewhat different than the velocity of acoustic waves polarized in a different direction within the same earth formation. Anisotropy may arise from intrinsic structural properties, such as grain alignment, crystallization, aligned fractures, or from unequal stresses within the formation. Anisotropy is particularly of interest in the measurement of the velocity of shear/flexural waves propagating in the earth formations. Shear or S waves are often called transverse waves because the particle motion is in a direction "transverse", or perpendicular, to the direction that the wave is traveling. Acoustic waves travel fastest when the direction of particle motion polarization direction is aligned with the material's stiffest direction. If the formation is anisotropic, meaning that there is one direction that is stiffer than another, then the component of particle motion aligned in the stiff direction travels faster than the wave component aligned in the other, more compliant, direction in the same plane. In the case of 2-dimensional anisotropy, a shear wave induced into an anisotropic formation splits into two components, one polarized along the formation's stiff (or fast) direction, and the other polarized along the formation's compliant (or slow) direction. Generally, the orientation of these two polarizations is substantially orthogonal (components which are at a 90° angle relative to each other). The fast wave is polarized along the direction parallel to the fracture strike and a slow wave in the direction perpendicular to it. A significant number of hydrocarbon reservoirs comprise fractured rocks wherein the fracture porosity makes up a large portion of the fluid-filled space. In addition, the fractures also contribute significantly to the permeability of the reservoir. Identification of the direction and extent of fracturing is important in reservoir development for at least two reasons. One reason for identification of fracture direction is that such a knowledge makes it possible to drill deviated or horizontal boreholes with an axis that is preferably normal to the plane of the fractures. In a rock that otherwise has low permeability and porosity, a well drilled in the preferred direction will intersect a large number of fractures and thus have a higher flow rate than a well that is drilled parallel to the fractures. Knowledge of the extent of fracturing also helps in making estimates of the potential recovery rates in a reservoir and in enhancing the production from the reservoir. BRIEF DESCRIPTION OF THE DRAWINGS A better understanding of the present invention can be obtained when the following detailed description of example embodiments are considered in conjunction with the following drawings, in which: FIG. 1A shows an example of a drilling system traversing a downhole formation; FIG. 1B shows an example of a drilling system traversing a dipping downhole formation; FIG. 2 shows an example of an acoustic tool; FIG. 3 shows an example set of decomposed received signals; FIG. 4 shows an example of logging in two wells, inclined to each other, in the same formation; FIG. 5 shows an example of an acoustic tool having an axial shear wave generator; and FIG. 6 shows an example of an axial shear wave generator in a wellbore. FIG. 1A shows a schematic diagram of a drilling system 110 having a downhole assembly according to one embodiment of the present invention. As shown, the system 110 includes a conventional derrick 111 erected on a derrick floor 112 which supports a rotary table 114 that is rotated by a prime mover (not shown) at a desired rotational speed. A drill string 120 comprising a drill pipe section 122 extends downward from rotary table 114 into a directional borehole, also called a wellbore, 126, through subsurface formations A and B. Borehole 126 may travel in a two-dimensional and/or three-dimensional path. A drill bit 150 is attached to the downhole end of drill string 120 and disintegrates the geological formation 123 when drill bit 150 is rotated. The drill string 120 is coupled to a drawworks 130 via a kelly joint 121, swivel 128 and line 129 through a system of pulleys (not shown). During the drilling operations, drawworks 130 may be operated to raise and lower drill string 120 to control the weight on bit 150 and the rate of penetration of drill string 120 into borehole 126. The operation of drawworks 130 is well known in the art and is thus not described in detail herein. During drilling operations a suitable drilling fluid (also called "mud") 131 from a mud pit 132 is circulated under pressure through drill string 120 by a mud pump 134. Drilling fluid 131 passes from mud pump 134 into drill string 120 via fluid line 138 and kelly joint 121. Drilling fluid 131 is discharged at the borehole bottom 151 through an opening in drill bit 150. Drilling fluid 131 circulates uphole through the annular space 127 between drill string 120 and borehole 126 and is discharged into mud pit 132 via a return line 135. A variety of sensors (not shown) may be appropriately deployed on the surface according to known methods in the art to provide information about various drilling-related parameters, such as fluid flow rate, weight on bit, hook load, etc. In one example, a surface control unit 140 may receive signals from downhole sensors (discussed below) via a telemetry system and processes such signals according to programmed instructions provided to surface control unit 140. Surface control unit 140 may display desired drilling parameters and other information on a display/monitor 142 which may be used by an operator to control the drilling operations. Surface control unit 140 may contain a computer, memory for storing data and program instructions, a data recorder, and other peripherals. Surface control unit 140 may also include drilling models and may process data according to programmed instructions, and respond to user commands entered through a suitable input device, such as a keyboard (not shown). In one example embodiment of the present invention, bottom hole assembly (BHA) 159 is attached to drill string 120, and may comprise a measurement while drilling (MWD) assembly 158, an acoustic tool 190, a drilling motor 180, a steering apparatus 161, and drill bit 150. MWD assembly 158 may comprise a sensor section 164 and a telemetry transmitter 133. Sensor section 164 may comprise various sensors to provide information about the formation 123 and downhole drilling parameters. MWD sensors in sensor section 164 may comprise a device to measure the formation resistivity, a gamma ray device for measuring the formation gamma ray intensity, directional sensors, for example inclinometers and magnetometers, to determine the inclination, azimuth, and high side of at least a portion of BHA 159, and pressure sensors for measuring drilling fluid pressure downhole. The above-noted devices may transmit data to a telemetry transmitter 133, which in turn transmits the data uphole to the surface control unit 140. In one embodiment a mud pulse telemetry technique may be used to generate encoded pressure pulses, also called pressure signals, that communicate data from downhole sensors and devices to the surface during drilling and/or logging operations. A transducer 143 may be placed in the mud supply line 138 to detect the encoded pressure signals responsive to the data transmitted by the downhole transmitter 133. Transducer 143 generates electrical signals in response to the mud pressure variations and transmits such signals to surface control unit 140. Alternatively, other telemetry techniques such as electromagnetic and/or acoustic techniques or any other suitable telemetry technique known in the art may be utilized for the purposes of this invention. In one embodiment, drill pipe sections 122 may comprise hard-wired drill pipe which may be used to communicate between the surface and downhole devices. Hard wired drill pipe may comprise segmented wired drill pipe sections with mating communication and/or power couplers in the tool joint area. Such hard-wired drill pipe sections are commercially available and will not be described here in more detail. In one example, combinations of the techniques described may be used. In one embodiment, a surface transmitter/receiver 180 communicates with downhole tools using any of the transmission techniques described, for example a mud pulse telemetry technique. This may enable two-way communication between surface control unit 140 and the downhole tools described below. FIG. 2 shows an example of acoustic tool 190. FIG. 2 shows the tool 190 disposed in BHA 159 within a fluid filled borehole 126. Alternatively, the tool 190 may be suspended within the borehole by a multi-conductor armored cable known in the art. The tool 190 comprises a set of dipole transmitters: a first dipole transmitter 20, and a second dipole transmitter 22. In the perspective view of FIG. 2, only one face of each of the dipole transmitters 20, 22 may be seen. However, one of ordinary skill in the art understands that a complimentary face of each dipole transmitter 20 and 22 is present on a back surface of the tool 10. The dipole transmitters may be individual transmitters fired in such a way as to act in a dipole fashion. The transmitter 20 induces its acoustic energy along an axis, which for convenience of discussion is labeled X in the FIG. 2. Transmitter 22 induces energy along its axis labeled Y in FIG. 2, where the X and Y axes (and therefore transmitters 20, 22) may be, in one example, orthogonal. The orthogonal relationship of the transmitters 20, 22 need not necessarily be the case, but a deviation from an orthogonal relationship complicates the decomposition of the waveforms. The mathematics of such a non-orthogonal decomposition are within the skill of one skilled in the art without undue experimentation. Tool 190 may also comprise a plurality of receiver pairs 24 and 26 at elevations spaced apart from the transmitters 20, 22. In one embodiment tool 190 comprises four pairs of dipole receivers 24 A-D and 26 A-D. However, any number of receiver pairs may be used without departing from the spirit and scope of the invention. In the example shown in FIG. 2, the receivers are labeled 24A-D and 26A-D. In one example, each set of dipole receivers at a particular elevation has one receiver whose axis is coplanar with the axis of transmitter 20 (in the X direction) and one receiver whose axis is coplanar with the axis of transmitter 22 (in the Y direction). For example, one set of dipole receivers could be receivers 24A and 26A. Thus, the dipole receivers whose axes are coplanar with the axis of transmitter 20 are the transmitters 24A-D Likewise the dipole receivers whose axes are coplanar with the axis of transmitter 22 are receivers 26 A-D. It is not necessary that the axes of the receivers be coplanar with the axes of one of the transmitters. However, azimuthally rotating any of the receiver pairs complicates the trigonometric relationships and, therefore, the data processing. The mathematics of such a non-orthogonal decomposition are within the skill of one skilled in the art without undue experimentation. Anisotropic earth formations tend to break an induced shear wave into two components: one of those components traveling along the faster polarization direction, and the second component traveling along the slower polarization direction, where those two directions are substantially orthogonal. The relationship of the fast and slow polarizations within the formation, however, rarely lines up with the orthogonal relationship of the dipole transmitters 20, 22. For convenience of the following discussion and mathematical formulas, a strike angle Θ is defined to be the angle between the X direction orientation (the axis of dipole transmitter 20) and the faster of the two shear wave polarizations (see FIG. 2). Further, it must be understood that the shear wave of interest does not propagate in the X or Y direction, but instead propagates in the Z direction where the Z direction is parallel to the axial direction. Operation of the tool 190 involves alternative firings of the transmitters 20, 22. Each of the receivers 24A-D and 26A-D create received waveforms designated R, starting at the firing of a particular transmitter. Each of the received waveforms or signals has the following notation: R.sub.[receiver][source]. Thus, for the firing of transmitter 20 in the X direction, and receipt by one of the receivers having an axis coplanar to the axis of transmitter 20 (receivers 24A-D), the time series received signal is designated as RXX. Likewise, the cross-component signal, the signal received by the dipole receiver whose axis is substantially perpendicular to the axis of the firing transmitter, is designated RYX in this situation. In similar fashion, firing of the transmitter whose axis is oriented in the Y direction, transmitter 22, results in a plurality of received signals designated as RYY for the axially parallel receivers, and RXY for the cross-components. Thus, each transmitter firing creates two received signals, one for each receiver of the dipole receiver pair. It follows that for a set of dipole transmitter firings, four signals are received at each receiver pair indicative of the acoustic signals propagated through the formation. The acoustic signals may be processed using transform techniques known in the art to indicate formation anisotropy. In one example, a processing method comprises calculating, or estimating, source signals or source wavelets that created each set of received signals by assuming a transfer function of the formation. Estimating source wavelets can be described mathematically as follows: where SESTi is the estimated source signal calculated for the ith set of receivers, [TF] is the assumed transfer function of the formation in the source to receiver propagation, and Ri is the decomposed waveforms (described below) for the ith receiver set. Thus, for each set of received signals Ri, an estimate of the source signal SESTi is created. The estimated source signals are compared using an objective function. Minimas of a graph of the objective function are indicative of the angle of the anisotropy, and the slowness of the acoustic waves through the formation. Further, depending on the type objective function used, one or both of the value of the objection function at the minimas, and the curvature of the of the objective function plot near the minimas, are indicative of the error of the slowness determination. Thus, a primary component of the source signal estimation is the assumed transfer function [TF]. The transfer function may be relatively simple, taking into account only the finite speed at which the acoustic signals propagate and the strike angle, or may be very complex, to include estimations of attenuation of the transmitted signal in the formation, paths of travel of the acoustic signals, the many different propagation modes within the formation (e.g. compressional waves, shear waves, Stonely waves), and if desired even the effects of the acoustic waves crossing boundaries between different layers of earth formations. For reasons of simplicity of the calculation, the preferred estimated transfer functions take into account only the propagation speed (slowness) of the acoustic energy in the formation and the strike angle of the anisotropy. Each of the received signals in the case described above contains components of both the fast and slow shear waves, and hence can be considered to be composite signals. That is, for example, an RXX receiver signal contains information regarding both the fast and slow polarized signals. These composite signals may be decomposed into their fast and slow components using equations as follows: FP(t)=cos2(θ)RXX(t)+sin(θ)cos(θ)[RXY(t)- +RYX(t)]+sin2(θ)RYY(t) (2) SP(t)=sin2(θ)RXX(t)-cos(θ)sin(θ)[RXY(t)- +RYX(t)]+cos2(θ)RYY(t) (3) sin(2θ)[RXX(t)-RYY(t)]-cos(2θ)[RXY(t)+RY- X(t)]=0 (4) where FP(t) is the fast polarization time series, SP(t) is the slow polarization time series, and θ is the strike angle as defined above. The prior art technique for decomposing the multiple received composite signals involved determining the strike angle θ by solving equation (4) above, and using that strike angle in equations (2) and (3) to decompose the composite signals into the fast and slow time series. In another example for decomposing the composite signals into the fast and slow time series, a close inspection of equations (2) and (3) above for the fast and slow polarization time series respectively shows two very symmetric equations. Taking into account the trigonometric relationships: sin θ=cos(90°-θ) (5) cos θ=sin(90°-θ) (6) it may be recognized that either the fast polarization equation (2) or the slow polarization equation (3) may be used to obtain either the fast or slow polarization signals by appropriately adjusting the angle θ used in the calculation. Stated otherwise, either the fast or slow polarization equations (2) or (3) may be used to decompose a received signal having both fast and slow components into individual components if the strike angle θ is appropriately adjusted. Rather than using a single strike angle in both equations (2) and (3) above, each assumed transfer function comprises a strike angle. A plurality of transfer functions are assumed over the course of the slowness determination, and thus a plurality of strike angles are used, preferably spanning possible strike angles from -90° to)+90° (180°. For each assumed transfer function (and thus strike angle), the four received signals generated by a set of receivers at each elevation are decomposed using the following equation: DS(t)=cos2(θ)RXX(t)+sin(θ)cos(θ)(RXY(t)- +RYX(t))+sin2(θ)RYY(t) (7) where DS(t) is simply the decomposed signal for the particular strike angle used. This process is preferably repeated for each set of received signals at each level for each assumed transfer function. Equation (7) is equation (2) above; however, equation (3) may be equivalently used if the assumed strike angle is appropriately adjusted. Consider a set of four decomposed signals, see FIG. 3, that are created using equation (7) above for a particular transfer function (strike angle). In the exemplary set of decomposed signals, R1 could be the decomposed signal created using the strike angle from the assumed transfer function and the composite signals received by the set of receivers 24A, 26A. Likewise, decomposed signal R2 could be the decomposed signal created again using the strike angle from the assumed transfer function and the composite signals created by the set of receivers 24B, 26B. In this example, the amplitude of the decomposed signal of the set of receivers closest to the transmitters, decomposed signal R1, is greater than the decomposed signals of the more distant receivers, for example R4. The waveforms may shift out in time from the closest to the more distant receivers, which is indicative of the finite speed of the acoustic waves within the formation. For a particular starting time within the decomposed signals, for example starting time T1, and for a first assumed transfer function having an assumed strike angle and slowness, portions of each decomposed signal are identified as being related based on the transfer function. Rectangular time slice 50 of FIG. 3 is representative of a slowness in an assumed transfer function (with the assumed strike angle used to create the decomposed signals exemplified in FIG. 3). In particular, the slope of the rectangular time slice is indicative of the slowness of the assumed transfer function. Stated another way, the portions of the decomposed signals within the rectangular time slice 50 should correspond based on the assumed slowness of the formation of the transfer function. The time width of the samples taken from each of the received signals may be at least as long as each of the source signals in a firing set. In this way, an entire source waveform or source wavelet may be estimated. However, the time width of the samples taken from the decomposed signals need not necessarily be this width, as shorter and longer times would be operational. Thus, the portions of the decomposed signals in the rectangular time slice 50 are each used to create an estimated source signal. These estimated source signals are compared to create an objective function that is indicative of their similarity. In one example, the estimated source signals may be compared using cross correlation techniques known in the art. In another example, cross correlation of the frequency spectra of the received signals may be compared using techniques known in the art. The process of assuming a transfer function, estimating source wavelets based on decomposed signals and creating an objective function may be repeated a plurality of times. The rectangular time slices 50 through 54 are exemplary of multiple assumed transfer functions used in association with starting time T1 (and the a strike angle used to create the decomposed signals). Estimating source wavelets in this fashion (including multiple assumed transfer functions) may also be repeated at multiple starting times within the decomposed signals. The value of the objective function may be calculated for each assumed transfer function and starting time. Calculating the objective function of the first example technique comprises comparing estimated source signals to determine a variance between them. This slowness determination comprises calculating an average of the estimated source signals within each time slice, and then calculating a variance against the average source signal. In more mathematical terms, for each assumed transfer function, a series of estimated source waveforms or signals SESTi are calculated using equation (1) above. From these estimated source signals, an average estimated source signal may be calculated as follows: S EST AVG ( t ) = 1 N i = 1 N S EST i ( t ) ( 8 ) ##EQU00001## where SESTiAVG is the average estimated source signal, N is the number of decomposed received signals, SESTi is the source wavelet estimated for each decomposed received signal within the time slice, and t is time within the various time series. The average estimated source signal is used to calculate a value representing the variance of the estimated source signals from the average estimated source signal. The variance may be calculated as follows: δ 2 = i = 1 N ( S EST i ( t ) - S EST AVG ( t ) ) 2 ( 9 ) ##EQU00002## where δ2 is the variance. In one embodiment, the variance value is determined as a function of slowness, starting time, and strike angle. Large values of the variance indicate that the assumed transfer function (assumed strike angle and/or assumed slowness) did not significantly match the actual formation properties. Likewise, small values of the variance indicate that the assumed transfer function closely matched the actual formation properties. Thus, the minimas of the objective function described above indicate the slowness of the fast and slow polarized waves as well as the actual strike angle. The value of the variance objective function at the minimas is indicative of the error of the determination of the acoustic velocity and strike angle. The curvature of the variance objective function at the minima is indicative of the error of the calculation. A second embodiment for calculating an objective function is based on determining a difference between each estimated source signal. As discussed above, using the assumed transfer function, an estimated source signal is created using the portions of the decomposed signal within a time slice. Differences or differentials are calculated between each estimated source signal, for example between the source signal estimated from a portion of the R1 signal and the source signal estimated from the portion of the R2 signal. This difference is calculated between each succeeding receiver, and the objective function in this embodiment is the sum of the square of each difference calculation. The differential objective function is generated as a function of slowness, starting time, and strike angle. However, the function obtained using the differential slowness calculation has slower transitions from maximas to minimas which therefore makes determining the minimas (indicative of the actual slowness of the fast and slow polarizations) easier than in cases where the function has relatively steep slopes between minima and maxima More mathematically, the objective function of this second embodiment is calculated as follows: ζ = i = 1 N - 1 ( S EST i + 1 - S EST i ) 2 ( 10 ) ##EQU00003## where ζ is the objective function, and N is the number of receivers. Much like using the variance as the objective function, this differential objective function is a function of slowness versus starting time versus strike angle. Known techniques may be used to determine minima of these functions, and the locations of the minima are indicative of formation slowness and the strike angle. Either of the two calculational techniques may be used. Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. For example, the disclosed method for determining shear wave velocity and orientation may be implemented using any number of receiver levels and different receiver types for the acoustic logging tool. Indeed, even a single set of dipole receivers may be used relying on rotation of the tool to obtain additional composite signals for decomposition. Further, the source may be located at any arbitrary angle relative to the receivers. Moreover, processing of the data after collection at receivers can be performed downhole in real time with only the results being transferred uphole to a computer system for storage. Throughout this discussion, the various earth formation characteristics were discussed with reference to finding minimas of the objective function. However, one of ordinary skill in the art could easily invert the values used, thus making a determination a search for maximum values in the plot, and this would not deviate from the scope and spirit of the invention. While assuming the transfer functions in the embodiments described involved thus far assume a strike angle, it is possible that the transfer function need not include a strike angle estimation, and instead the composite signals could be decomposed for the range of possible strike angles independent of an assumed transfer function. It is also possible to solve for the strike angle using equation (4) above and decompose the composite waveforms using that strike angle; and thereafter, estimate and apply transfer functions to the decomposed signals, thus also removing the strike angle from the transfer function. As discussed above, crossed-dipole acoustic tools use a pair of orthogonal acoustic sources to create acoustic surface waves on the borehole wall. These surface waves (flexural waves) are strongly influenced by the mechanical stresses in the formations surrounding the borehole as well as any intrinsic anisotropy (such as fine layering in shales). The tools measure the anisotropy in the X-Y plane that is orthogonal to the tool longitudinal axis. The tool is substantially insensitive to anisotropy in the Z axis aligned with the tool longitudinal axis. In several drilling situations, complex stress regimes in the formations of interest make it desirable to know the three-dimensional stress field surrounding the borehole. As indicated, the acoustic tool described herein, provides information related to the anisotropy in the plane perpendicular to the local Z axis of the tool. At the L0 location in FIG. 1A, the XY plane of the tool is aligned with the XY plane of the earth system G. As the tool progresses, during drilling, along the path of borehole 126 in FIG. 1A, the local coordinate system rotates from vertical to horizontal, as indicated by the local coordinate systems L0, L1, and L2. When acoustic tool 190 is in the horizontal section of the borehole, the Z axis of the earth coordinate system falls in the tools XY measurement plane. Thus by measuring in both the substantially vertical and substantially horizontal sections of the wellbore 126, the horizontal (earth) field measurements from location L0 and the vertical (earth) field measurements from L2 may be combined using suitable techniques known in the art to provide a three dimensional stress field. FIG. 1B shows a system similar to that described above traversing through a formation that is dipping, or tilted, with respect to the earth's coordinate system G. The properties of the dipping formation are aligned to the coordinate system F, where the XY plane is substantially parallel the bed interface 90. Acoustic measurements made at location L0 will measure components of the formation Z axis anisotropy. However, depending on the dipping angle, the sensitivity to the formation Z axis anisotropy may be weak. By again measuring in both the vertical (earth) and horizontal (earth) planes, the combined measurements may be related to the three dimensional stress field of the formation. In one example, the wellbore 126 may be drilled along a trajectory based on the three dimensional stress field. For example, the wellbore may be drilled to intersect fractures. In another example, the wellbore may be drilled along a path of minimum stresses. In one example, the calculations may be made downhole and may be used with drilling models stored in the downhole processor to adjust steering assembly 160 to drill the wellbore along a predetermined path based on the calculated anisotropic characteristics. In one example, see FIG. 4, the formation B is not large enough in the axial direction to allow the wellbore 126' to be turned to the horizontal direction. Alternatively, the well plan may not call for an inclined or horizontal section in the particular well. It may be possible to acquire suitable acoustic anisotropy measurements in an offset wellbore 126'' that penetrates formation B at an inclination αc from vertical. Offset wellbore 126'' may have been drilled and logged prior to the drilling of wellbore 126'. In one example, the measurements from tool 190'' in well 126'' may be stored and later downloaded in memory of tool 190' before deployment of tool 190'. The stored measurements may be combined with measurements made by tool 190' and the resulting anisotropy results transmitted to the surface using known MWD telemetry techniques. Alternatively, tool 190'' may take measurements at approximately the same time as tool 190'. Measurements from both tool 190' and 190'' may alternatively be processed in a surface control unit 140, or at a remote site using techniques known in the art. In another example, see FIG. 5, instead of taking measurements at different axially displaced, orthogonal locations to acquire 3-D anisotropy results, a 3-axis acoustic tool 400 excites shear waves in all 3 axes by including an axial shear wave generator 401. In one example, acoustic tool 400 comprises the 2-D tool 190 described previously and axial shear wave generator 401. Axial shear wave generator 401 comprises a clamping device 405 that is extendable from the axial shear wave generator body 402 to engage the borehole wall around at least a portion of the circumference of the borehole wall. Clamp 405 is forced into cyclical axial motion by a force element in generator body 402. The cyclical axial motion generates shear on the borehole wall in the axial motion direction. The resulting shear waves propagate away from the borehole wall. The shear waves produced by the clamped axial generator propagate substantially orthogonal to the shear waves generated by the dipole sources 20, 22 described above. In an isotropic medium, the clamped axial shear wave generator 401 will produce shear waves that move out into the formation and compressional waves along the borehole axis. If there is anisotropy, the wave from the clamped dipole source may split producing wave components along the three principle axes depending on the orientation of those axes relative to the borehole. The signals propagate out into the formation and are reflected back to the receivers 24 and 26 described previously. In one example, the signals may be processed in a downhole processor, using techniques known in the art, to determine the 3-D anisotropy characteristics of the formation, and the results transmitted to the surface using known telemetry techniques. Alternatively, the raw data may be transmitted to the surface and processed at the surface. The anisotropic characteristics comprise at least one of a three dimensional stress field and a three dimensional velocity field of the formation. FIGS. 6A and 6B show one example of an axial shear wave generator 401 comprising a housing 402 that may be in drillstring 122 (see FIGS. 1A and 1B). As used herein, the term axial is intended to mean along, or parallel to, the longitudinal axis of the wellbore. An extendable member 409 is controllably extendable outward from housing 402 toward the wall 430 of wellbore 426. In one example, a clamp pad 407 is attached to extendable member 409, and engages wall 430. As shown in FIG. 5B, each pads 407 A-D may approximate a circumferential ring attached to wall 430 when all of pads 407 A-D are extended to engage wall 430. In one embodiment, extendable member 409 may be part of a telescoping cylinder located on a movable base 410 disposed in housing 402. In one example, movable base is 410 is attached to an axial force assembly 412 that provides axial back and forth motion to movable base 410, thus providing axial motion to clamp pads 407. In one embodiment, axial force assembly 412 comprises a stack of piezoelectric disks 413 polarized to extend and contract axially when excited by a suitable electric signal. In one embodiment, a backing mass 450 is mounted between the piezoelectric disks 413 and a shoulder 403 in housing 402. In one example, backing mass 450 may comprise a tungsten material and/or a tungsten carbide material. Backing mass 451 helps to ensure that the majority of axial movement of the piezoelectric stack is directed toward the clamp bands. In one example, controller 415 comprises suitable electric circuits and processors to power the crystals and control the extension, and/or retraction, of extendable members 409. Power source 420 may comprise suitable batteries for powering the axial shear wave generator during operation. Controller 415 may be in suitable data communication with other controllers in the downhole tool. Programmed instructions in controller 415 may be used control shear wave generation, data acquisition, and calculation of the anisotropic properties of the formation. In an alternative embodiment, magnetostrictive materials may be used to power the back and forth movement of clamp members 407 to generate axial shear waves in the surrounding formation. Such magnetostrictive materials may include nickel and rare earth materials for example a terbium-dysprosium-iron material. Such materials are known in the art. While described above with relation to an MWD/LWD system, one of ordinary skill in the art will appreciate that the apparatus and methods described herein may be used with wireline, slickline, wired drill pipe, and coiled tubing to convey the acoustic tools into the wellbore. Patent applications by Paul F. Rodney, Spring, TX US Patent applications by HALLIBURTON ENERGY SERVICES, INC.

By Walter Thirring, E.M. Harrell During this ultimate quantity i've got attempted to give the topic of statistical mechanics in keeping with the elemental ideas of the sequence. the trouble back entailed following Gustav Mahler's maxim, "Tradition = Schlamperei" (i.e., dirt) and clearing away a wide component to this tradition-laden sector. the result's a publication with little in universal with such a lot different books at the topic. the normal perturbation-theoretic calculations aren't very valuable during this box. these tools have by no means ended in propositions of a lot substance. even if perturbation sequence, which for the main half by no means converge, could be given a few asymptotic that means, it can't be made up our minds how shut the nth order approximation involves the precise end result. due to the fact that analytic options of nontrivial difficulties are past human features, for greater or worse we needs to accept sharp bounds at the amounts of curiosity, and will at such a lot try to make the measure of accuracy passable. Read Online or Download A Course in Mathematical Physics IV. Quantum Mechanics of Large Systems: Volume 4: Quantum Mechanics of Large Systems PDF Similar mathematics books Written essentially for undergraduate scholars of arithmetic, technological know-how, or engineering, who regularly take a direction on differential equations in the course of their first or moment 12 months. the most prerequisite is a operating wisdom of calculus. the surroundings within which teachers educate, and scholars examine differential equations has replaced drastically some time past few years and keeps to adapt at a quick velocity. Computing gear of a few style, even if a graphing calculator, a laptop computer, or a laptop notebook is on the market to such a lot scholars. The 7th version of this vintage textual content displays this altering setting, whereas while, it keeps its nice strengths - a modern process, versatile bankruptcy development, transparent exposition, and extraordinary difficulties. moreover many new difficulties were additional and a reorganisation of the fabric makes the suggestions even clearer and extra comprehensible. Like its predecessors, this variation is written from the point of view of the utilized mathematician, focusing either at the conception and the sensible functions of differential equations as they follow to engineering and the sciences. This famous paintings covers the answer of quintics by way of the rotations of a typical icosahedron round the axes of its symmetry. Its two-part presentation starts with discussions of the idea of the icosahedron itself; commonplace solids and idea of teams; introductions of (x + iy) ; a press release and exam of the elemental challenge, with a view of its algebraic personality; and common theorems and a survey of the topic. Within the previous few many years, multiscale algorithms became a dominant development in large-scale clinical computation. Researchers have effectively utilized those ways to quite a lot of simulation and optimization difficulties. This booklet provides a normal review of multiscale algorithms; functions to normal combinatorial optimization difficulties resembling graph partitioning and the touring salesman challenge; and VLSICAD functions, together with circuit partitioning, placement, and VLSI routing. - A Binary Images Watermarking Algorithm Based on Adaptable Matrix - Inquiry into the Validity of a Method recently proposed by George B. Jerrard, Esq., for Transforming and Resolving Equations of Elevated Degrees: undertaken at the Request of the Association - Combinatorial Mathematics IX, Brisbane, Australia: Proceedings, 1981 - Schaum's Outline of Trigonometry (5th Edition) (Schaum's Outlines Series) - Mathematical Methods in Particle Transport Theory Additional info for A Course in Mathematical Physics IV. Quantum Mechanics of Large Systems: Volume 4: Quantum Mechanics of Large Systems Other solutions are excluded by the continuity requirement (Problem 1). This means that no vector in the Hilbert space of a representation of type 11 or Ill corresponds to a pure state on the algebra. 4. Any operator a of an algebra of type 111 is of course bounded, so Tr pa is well defined for any p E 1(X), only p can not come from the algebra, which contains no element of a trace class (other than 0). 3. Let us end the section by recapitulating the physical significance of the new mathematical phenomena that make an appearance in infinite systems. 1. Inequivalent Representations Since vectors that differ globally are always orthogonal, globally different situations lead to inequivalent representations. Constructed with 0.... , ® 0 (1 ® 0. is the weak closure of d, and Z = (1 + weak limits • 1}, which is a reducible factor representation. 8) 1. 5); as mentioned above, the vector C fl® ... has no counterpart in the since the corresponding functional in sr,, would earlier representations then be strongly continuous. The state defined by (10 ®... on d. a (norm) Continuous linear functional, and therefore extensible to the whole C* algebra generated by d, but it still need not be strongly continuous in a representation: For instance, in the representation using is 1 + iN -.

Journal of Discrete Algorithms 10 (2012) 7083 Contents lists available at ScienceDirect Oa Cb Cc T 15doJournal of Discrete Algorithms arameterized complexity of nding small degree-constrainedubgraphs, mid Amini a, Ignasi Sau b,, Saket Saurabh c NRS, DMA, ENS, Paris, FranceNRS, LIRMM, Montpellier, Francehe Institute of Mathematical Sciences, Chennai, India r t i c l e i n f o a b s t r a c t ticle history:ceived 15 March 2010ceived in revised form 22 December 2010cepted 16 May 2011ailable online 19 May 2011 ywords:rameterized complexitygree-constrained subgraphed-parameter tractable algorithm-hardnesseewidthnamic programmingcluded minors In this article we study the parameterized complexity of problems consisting in ndingdegree-constrained subgraphs, taking as the parameter the number of vertices of thedesired subgraph. Namely, given two positive integers d and k, we study the problem ofnding a d-regular (induced or not) subgraph with at most k vertices and the problem ofnding a subgraph with at most k vertices and of minimum degree at least d. The latterproblem is a natural parameterization of the d-girth of a graph (the minimum order of aninduced subgraph of minimum degree at least d).We rst show that both problems are xed-parameter intractable in general graphs. Moreprecisely, we prove that the rst problem is W -hard using a reduction from Multi-Color Clique. The hardness of the second problem (for the non-induced case) follows froman easy extension of an already known result. We then provide explicit xed-parametertractable (FPT) algorithms to solve these problems in graphs with bounded local treewidthand graphs with excluded minors, using a dynamic programming approach. Althoughthese problems can be easily dened in rst-order logic, hence by the results of Frickand Grohe (2001) are FPT in graphs with bounded local treewidth and graphs withexcluded minors, the dependence on k of our algorithms is considerably better than theone following from Frick and Grohe (2001) . 2011 Elsevier B.V. All rights reserved. Problems of nding subgraphs with certain degree constraints are well studied both algorithmically and combinatorially,d have a number of applications in network design (cf. for instance [1,20,25,29,35]). In this article we consider two naturalch problems: nding a small regular (induced or not) subgraph and nding a small subgraph with given minimum degree.e discuss in detail these two problems in Sections 1.1 and 1.2, respectively. This work has been partially supported by European project IST FET AEOLUS, PACA region of France, Ministerio de Ciencia e Innovacin, Europeangional Development Fund under project MTM2008-06620-C03-01/MTM, and Catalan Research Council under project 2005SGR00256.An extended abstract of this work appeared in: Proceedings of the International Workshop on Parameterized Complexity (IWPEC), May 2008, LNCS, l. 5018, pp. 1329.Corresponding author.E-mail addresses: firstname.lastname@example.org (O. Amini), email@example.com (I. Sau), firstname.lastname@example.org (S. Saurabh). 70-8667/$ see front matter 2011 Elsevier B.V. All rights reserved.i:10.1016/j.jda.2011.05.001 O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083 71 in. Finding a small regular subgraph The complexity of nding regular graphs as well as regular (induced) subgraphs has been intensively studied in theerature [68,11,24,30,31,35,36]. One of the rst problems of this kind was stated by Garey and Johnson: Cubic Subgraph,at is, the problem of deciding whether a given graph contains a 3-regular subgraph, is NP-complete . More generally,e problem of deciding whether a given graph contains a d-regular subgraph for any xed degree d 3 is NP-complete onneral graphs as well as in planar graphs (where in the latter case only d = 4 and d = 5 were considered, sincey planar graph contains a vertex of degree at most 5). For d 3, the problem remains NP-complete even in bipartiteaphs of degree at most d + 1 . Note that this problem is clearly polynomial-time solvable for d 2. If the regularbgraph is required to be induced, Cardoso et al. proved that nding a maximum cardinality d-regular induced subgraphNP-complete for any xed integer d 0 (for d = 0 and d = 1 the problem corresponds to Maximum Independent Setd Maximum Induced Matching, respectively).Concerning the parameterized complexity of nding regular subgraphs, Moser and Thilikos proved that the followingoblem is W -hard for every xed integer d 0 : k-size d-Regular Induced SubgraphInput: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] is d-regular? On the other hand, the authors proved that the following problem (which can be seen as the dual of the above one) isP-complete but has a problem kernel of size O(kd(k + d)2) for d 1 : k-Almost d-Regular GraphInput: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[V \ S] is d-regular? Mathieson and Szeider studied in variants and generalizations of the problem of nding a d-regular subgraph (for 3) in a given graph by deleting at most k vertices. In particular, they answered a question of , proving that the k-lmost d-Regular Graph problem (as well as some variants) becomes W -hard when parameterized only by k (that is, itunlikely that there exists an algorithm to solve it in time f (k) nO(1) , where n = |V (G)| and f is a function independentn and d).Given two integers d and k, it is also natural to ask for the existence of an induced d-regular graph with at most krtices. The corresponding parameterized problem is dened as follows: k-size d-Regular Induced Subgraph (kdRIS)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] is d-regular? Note that the hardness of k-size d-Regular Induced Subgraph does not follow directly from the hardness of k-sizeRegular Induced Subgraph as, for instance, the approximability of the problems of nding a densest subgraph on at leastvertices or on at most k vertices are signicantly different . In general, a graph may not contain an induced d-regularbgraph on at most k vertices, while containing a non-induced d-regular subgraph on at most k vertices. This observationads to the following problem: k-size d-Regular Subgraph (kdRS)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a d-regular subgraph H G , with |V (H)| k? Observe that k-size d-Regular Subgraph could a priori be easier than its corresponding induced version, as it happensr the MaximumMatching (which is in P) and the Maximum Induced Matching (which is NP-hard) problems.The two parameterized problems dened above have not been considered in the literature. We prove in Section 2 thatth problems are W -hard for every xed d 3, by reduction from Multi-Color Clique.2. Finding a small subgraph with given minimum degree For a nite, simple, and undirected graph G = (V , E) and d N, the d-girth gd(G) of G is the minimum order of anduced subgraph of G of minimum degree at least d. The notion of d-girth was proposed and studied by Erdos et al. [18,19] 72 O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083 wd Bollobs and Brightwell . It generalizes the usual girth, the length of a shortest cycle, which coincides with the 2-rth. (This is indeed true because every induced subgraph of minimum degree at least two contains a cycle.) Combinatorialunds on the d-girth can also be found in [4,27]. The corresponding optimization problem has been recently studied in ,here it has been proved that for any xed d 3, the d-girth of a graph cannot be approximated within any constant factor,less P = NP . From the parameterized complexity point of view, it is natural to introduce a parameter k N and askr the existence of a subgraph with at most k vertices and with minimum degree at least d. The problem can be formallyned as follows: k-size Subgraph of Minimum Degree d (kSMDd)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] has minimum degree at least d? te that the case d = 2 in P, as discussed above. The special case of d = 4 appears in the book of Downey and Fellows [15,457], where it is announced that H.T. Wareham proved that kSMD4 is W -hard. (However, we were not able to nd aoof.) From this result, it is easy to prove that kSMDd is W -hard for every xed d 4 (see Section 2). The complexitythe case d = 3 remains open (see Section 4). Note that in the kSMDd problem we can assume without loss of generalityat we are looking for the existence of an induced subgraph, since we only require the vertices to have degree at least d.Besides the above discussion, another motivation for studying the kSMDd problem is its close relation to the well studiednse k-Subgraph problem [3,14,20,28], which we proceed to explain. The density (G) of a graph G = (V , E) is dened as (G) := |E||V | . More generally, for any subset S V , we denote its density by (S), and dene it to be (S) := (G[S]). Thense k-Subgraph problem is formulated as follows: Dense k-Subgraph (DkS) Input: A graph G = (V , E).Output: A subset S V , with |S| = k, such that (S) is maximized. derstanding the complexity of DkS remains widely open, as the gap between the best hardness result (Apx-hardness )d the best approximation algorithm (with ratio O(n1/3) ) is huge. Suppose we are looking for an induced subgraph[S] of size at most k and with density at least . In addition, assume that S is minimal, i.e., no subset of S has densityeater than (S). This implies that every vertex of S has degree at least /2 in G[S]. To see this, observe that if there is artex v with degree strictly smaller than /2, then removing v from S results in a subgraph of density greater than (S)d of smaller size, contradicting the minimality of S . Secondly, if we have an induced subgraph G[S] of minimum degreeleast , then S is a subset of density at least /2. These two observations together show that, modulo a constant factor,oking for a densest subgraph of G of size at most k is equivalent to looking for the largest possible value of d for whichMDd returns Yes. As the degree conditions are more rigid than the global density of a subgraph, a better understandingthe kSMDd problem could provide an alternative way to approach the DkS problem.Finally, we would like to point out that the kSMDd problem has practical applications to trac grooming in opticaltworks. Trac grooming refers to packing small trac ows into larger units then can then be processed as singletities. For example, in a network using both time-division and wavelength-division multiplexing, ows destined to ammon node can be aggregated into the same wavelength, allowing them to be dropped by a single optical Add-Dropultiplexer. The main objective of grooming is to minimize the equipment cost of the network, which is mainly given inavelength-Division Multiplexing optical networks by the number of electronic terminations. (We refer, for instance, to r a general survey on grooming.) It has been recently proved by Amini, Prennes and Sau that the Trac Groomingoblem in optical networks can be reduced (modulo polylogarithmic factors) to DkS, or equivalently to kSMDd. Indeed, inaph theoretic terms, the problem can be translated into partitioning the edges of a given request graph into subgraphsith a constraint on their number of edges. The objective is then to minimize the total number of vertices of the subgraphsthe partition. Hence, in this context of partitioning a given set of edges while minimizing the total number of vertices,e problems of DkS and kSMDd come into play. More details can be found in . . Presentation of the results We do a thorough study of the kdRS, the kdRIS, and the kSMDd problems in the realm of parameterized complexity,hich is a recent approach to deal with intractable computational problems having some parameters that can be relativelyall with respect to the input size. This area has been developed extensively during the last decade (the monograph ofwney and Fellows provides a good introduction, and for more recent developments see the books by Flum andohe and by Niedermeier ).For decision problems with input size n and parameter k, the goal is to design an algorithm with running time f (k)nO (1) , here f depends only on k. Problems having such an algorithm are said to be xed-parameter tractable (FPT). There is O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083 73 kSso a theory of parameterized intractability to identify parameterized problems that are unlikely to admit xed-parameteractable algorithms. There is a hierarchy of intractable parameterized problem classes above FPT, the important ones being: FPT M W M W W [P ] X P .The principal analogue of the classical intractability class NP is W , which is a strong analogue, because a fundamentaloblem complete for W is the k-Step Halting Problem for Nondeterministic Turing Machines (with unlimited non-terminism and alphabet size); this completeness result provides an analogue of Cooks theorem in classical complexity.convenient source of W -hardness reductions is provided by the result stating that k-Clique is complete for W . Theincipal working algorithmic way of showing that a parameterized problem is unlikely to be xed-parameter tractable, isprove its W -hardness using a parameterized reduction (dened in Section 2).Our results can be classied into two categories: eneral graphs: We show in Section 2 that kdRS is not xed-parameter tractable by showing it to be W -hard for any 3 in general graphs. We will see that the graph constructed in our reduction implies also the W -hardness of kdRIS.general, parameterized reductions are quite stringent because of parameter-preserving requirements of the reduction, andquire some technical care. Our reduction is based on a new methodology emerging in parameterized complexity, calledulti-color clique edge representation. This has proved to be useful in showing various problems to be W -hard recently .e rst spell out step-by-step the procedure to use this methodology, which can be used as a template for future purposes.en we adapt this methodology to the reduction for the kSMDd problem. The hardness of kSMDd for d 4 follows fromeasy extension of a result of H.T. Wareham [15, p. 457]. raphs with bounded local treewidth and graphs with exc...

An Erratum to this article was published on 18 August 2014 This paper investigates security-oriented beamforming designs in a relay network composed of a source-destination pair, multiple relays, and a passive eavesdropper. Unlike most of the earlier works, we assume that only statistical information of the relay-eavesdropper channels is known to the relays. We propose beamforming solutions for amplify-and-forward (AF) and decode-and-forward (DF) relay networks to improve secrecy capacity. In an AF network, the beamforming design is obtained by approximating a product of two correlated Rayleigh quotients to a single Rayleigh quotient using the Taylor series expansion. Our study reveals that in an AF network, the secrecy capacity does not always grow as the eavesdropper moves away from the relays or as total relay transmit power increases. Moreover, if the destination is nearer to the relays than the eavesdropper is, a suboptimal power is derived in closed form through monotonicity analysis of secrecy capacity. While in a DF network, secrecy capacity is a single Rayleigh quotient problem which can be easily solved. We also found that if the relay-eavesdropper distances are about the same, it is unnecessary to consider the eavesdropper in a DF network. Numerical results show that for either AF or DF relaying protocol, the proposed beamforming scheme provides higher secrecy capacity than traditional approaches. Cooperative communications, in which multiple nodes help each other transmit messages, has been widely acknowledged as an effective way to improve system performance [1–3]. However, due to the broadcast property of radio transmission, wireless communication is vulnerable to eavesdropping which consequently makes security schemes of great importance as a promising approach to communicate confidential messages. The traditional secure communication schemes rely on encryption techniques where secret keys are used. However, as the high-layer secure protocols have attracted growing attacks in recent years, the implementation of security schemes at physical layer becomes a hotspot. It was first proved by Wyner that it is possible to communicate perfectly at a non-zero rate without a secret key if the eavesdropper has a worse channel than the destination . This work was extended to Gaussian channels in and to fading channels in . Recently, there has been considerable work on secure communication in wireless relay networks (WRNs) [7–15]. A widely acknowledged measurement of system security in WRNs is the maximal rate of secret information exchange between source and destination which is defined as secrecy capacity. A decode-and-forward (DF)-based cooperative beamforming scheme which completely nulls out source signal at eavesdropper(s) was proposed in , and this work was extended to the amplify-and-forward (AF) protocol and cooperative jamming in . Hybrid beamforming and jamming was investigated in where one relay was selected to cooperate and the other to make intentional interference in a DF network. Combined relay selection and cooperative beamforming schemes for DF networks were proposed in where two best relays were selected to cooperate. The authors of [11, 12] considered the scenario where the relay(s) could not be trusted in cooperative MIMO networks. Additionally, a new metric of system security is brought up in as intercept probability and optimal relay selection schemes for AF and DF protocols based on the minimization of intercept probability were proposed. In earlier works, it is widely assumed that the relays have access to instantaneous channel state information (CSI) of relay-eavesdropper (RE) channels [7, 8, 13–15]. This assumption is ideal but unpractical in a real-life wiretap attack since the malicious eavesdropper would not be willing to share its instantaneous CSI. Thus, security schemes using instantaneous CSI of the eavesdropper cannot be adopted anymore. However, the instantaneous CSI of relay-destination (RD) channels is available since the destination is positive. The statistical information of the RE channels is also available through long-term supervision of the eavesdropper's transmission . It is worth mentioning that even if the relays do not have access to the perfect CSI of RD channels, they can still estimate these channels by training sequences and perform beamforming based on the estimated CSI . Our focus is on secrecy capacity, and we are interested in maximizing it with appropriate weight designs of relays. The remainder of this paper is organized as follows. Section 2 introduces system model under AF and DF protocols using relay beamforming. The optimization problem in an AF network is addressed and solved in Section 3 along with some analyses of secrecy capacity. Section 4 provides the optimal beamforming design for a DF network along with a surprising finding that considering the eavesdropper sometimes may not be necessary. Numerical results are given in Section 5 to compare the performances of different designs, and Section 6 provides some concluding remarks. 2. System model Consider a cooperative wireless network consisting of a source node S, a legitimate destination D, an eavesdropper E, and M relays Ri, i = 1,…, M as shown in Figure 1. Each node is equipped with single antenna working in half-duplex mode. Assume that there is no direct link between the source and the destination/eavesdropper, i.e., neither the destination nor the eavesdropper is in the coverage area of the source. For notational convenience, we denote the source-relay (SR) channels as fi, the RD channels as gi, and the RE channels as hi. All the channels are modeled as independent and identically distributed (i.i.d.) Rayleigh fading channels, i.e., , , and . Considering the path loss effect and setting the path loss exponent to 4 (for an urban environment), we have , , and , where dAB is the distance between nodes A and B. We assume the relays to know instantaneous CSI of SR channels and RD channels, but only statistical information of RE channels. Without loss of generality, we also assume the additive noises to be i.i.d. and follow a distribution. In an AF protocol, the source broadcasts in the first hop where the information symbol s is selected from a codebook and is normalized as E|s|2 = 1, and Ps is the transmit power. The received signal at Ri is where vi is the additive noise at Ri. In the second hop, each relay forwards a weighted version of the noisy signal it just received. More specifically, Ri normalizes ri with a scaling factor and then transmits a weighted signal ti = wiρiri. The transmit power of Ri is Pi = |wi|2. The received signal at the destination is where w = (w1, …, wM)T, ρfg = (ρ1f1g1, …, ρMfMgM)T, ρg = (ρ1g1, …, ρMgM)T, v = (v1, …, vM)T, and vD represents additive white Gaussian noise (AWGN) at the destination. The total relay transmit power is wHw = P. Meanwhile, the eavesdropper also gets a copy of s: where ρfh = (ρ1f1h1, …, ρMfMhM)T, ρh = (ρ1h1, …, ρMhM)T, and vE represents AWGN at the eavesdropper. In a DF protocol, the first hop is the same as in an AF protocol. While in the second hop, instead of simply amplifying the received signal, Ri decodes the message s and multiplies it with a weighted factor wi to generate the transmit signal ti = wis. The transmit power of Ri is still Pi = |wi|2. The received signals at the destination and the eavesdropper can be expressed, respectively, as where g = (g1, …, gM)T and h = (h1, …, hM)T. 3. Distributed beamforming design for AF In the following sections, we consider the security issue of the above relay network. The metric of interest is secrecy capacity which is defined as where , , and γD and γE are received signal-to-noise ratios (SNRs) at the destination and the eavesdropper, respectively. We aim to improve CS by exploiting appropriate beamforming designs. The following subsection describes the proposed beamforming design for an AF network. 3.1 Proposed design for AF (P-AF) In distributed beamforming schemes, the relays compute the received SNRs at the destination and the eavesdropper from Equations 2 and 3, respectively, as where Γg = diag(ρ12|g1|2, …, ρM2|gM|2), , and Now we discuss how to design w to maximize CS, and the proposed solution is denoted by . It is obvious that maximizing CS is equivalent to maximizing . Hence, in what follows, the objective function will be . where . This is a product of two correlated Rayleigh quotients which is generally difficult to maximize. However, it would be much easier to get a suboptimal solution if we approximate the objective function to a single Rayleigh quotient. Rewrite the optimization problem as Denote the matrices Dh + P- 1I, Γg + P- 1I, and Γh + P- 1I as A, B, and C, respectively. For simplicity, we also let ai, bi, and ci represent the i th diagonal entry of A, B, and C, respectively, and define p = (P1, …, PM)T. Since Pi = |wi|2, the denominator can be rewritten as . According to the Taylor series expansion , if we expand f(p) at . Since where and , we have . Substituting this partial derivative into (11), we further have where . It can be proved that K is negligible either with small P or large P if we make a commonly used assumption that the SR distances are about the same (see Appendix for details). Thus, we omit this part and rewrite f(p) approximately as So the optimization problem in (9) can be approximated to This is a single Rayleigh quotient problem. It has been reported in that if U is Hermitian and V is positive definite Hermitian, for any non-zero column vector x, we have where λmax(V-1U) is the largest eigenvalue of V-1U. The equality holds if x = cumax(V-1U) where c can be any non-zero constant and umax(V-1U) is the unit-norm eigenvector of V-1U corresponding to λmax(V-1U). As a result, the optimal solution to (14) is where Φ = A- 1B- 1C(PsρfgρfgH + Γg + P- 1I). To show the agreement of the approximated denominator and the exact denominator, we calculated them numerically, and the results are shown in Figure 2. The channel information we used are listed in Table 1 where f = (f1, …, fM)T, g = (g1, …, gM)T, and . f and g are generated randomly. For comparison purpose, we present two other beamforming designs. First, for the optimization of a product of two correlated Rayleigh quotient problems, a method was proposed recently in to maximize the upper and lower bounds. Note that where and . is bounded as As a result, the bounds maximization design for AF (B-AF) should be We also address the traditional design for AF (T-AF) where the eavesdropper is ignored and the goal is to maximize CD. It can be easily proved that the optimal solution is where cAF is a constant chosen to satisfy . 3.2 Discussion about secrecy capacity in AF networks It is natural to conjecture that secrecy capacity would grow as the eavesdropper moved away or as the total relay transmit power increased. However, we find that this conjecture is not always right. For simplicity, we assume the distances between relays are much smaller than those between the relays and the source, so the path losses of the SR channels are almost the same. The same assumption is also made to the destination/eavesdropper. Denote the SR, RD, and RE distances as dSR, dRD, and dRE, respectively, and the corresponding channel variances as , , and , respectively. Proposition 1.If the destination is much nearer to the relays than the eavesdropper is in an AF network, CSdoes not always grow as the total relay transmit power increases, and a suboptimal value of the total relay transmit power is found as Proof. Recall that . No matter how we design the beamforming vector w, is bounded as Due to the difficulty of calculating the eigenvalues of Φ, we replace the non-diagonal elements in Φ with their mean value 0 and the i th diagonal element with where . Thus, Φ becomes λ(P)I after replacement. Define . Now we investigate the monotonicity of CS(P). The first-order derivatives of CS(P) and λ(P) can be computed, respectively, as By setting , we obtain the positive stationary point of CS(P) as described in (18). If dRE > dRD (), ∀P∈(0, Psubopt), we have ; ∀P∈(Psubopt, + ∞), we have . Hence, if the destination is much nearer than the eavesdropper is, CS(P) is an increasing function over (0, Psubopt) and a decreasing function over (Psubopt, + ∞), which means that CS(Psubopt) is the maximum of CS(P). This monotonicity of CS and the accuracy of Psubopt under the case of dRE > dRD will be verified in the next section. It needs to be pointed out that the above analysis is not for any certain design, so the optimal value of P for a certain design would be different from but around Psubopt. It also needs to be pointed out that the replacement of the channel coefficients in Φ with their mean values may result in the loss of the security benefit that is supposed to be achieved by exploiting the perfect CSI of SR and RD channels. This loss does not affect the monotonicity of CS greatly under the case of dRE > dRD because the destination is much nearer and therefore much more advantageous in communication than the eavesdropper is. However, when dRE < dRD (or dRE = dRD), such replacement becomes inappropriate, since the instantaneous CSI of fi and gi improves the system security significantly. We can further compute the second-order derivatives of CS(P) and λ(P), respectively, as It can be observed from (22) that the positivity of depends on the value of P. Thus, CS(P) is neither convex nor concave. Remark 1. In an AF network, if the total relay transmit power is large, the AWGNs in the second hop are negligible compared to the forwarded versions of the AWGNs in the first hop. Thus, can be approximately written as This equation does not involve , which implies CS is a constant in this case wherever the eavesdropper is. 4. Distributed beamforming design for DF This section focuses on the security-oriented beamforming design for DF protocol. Similar to the design for AF protocol, the mission is to find the optimal design under a total relay transmit power constraint to maximize secrecy capacity. 4.1 Proposed design for DF (P-DF) From Equations 4 and 5, the received SNRs at the destination and the eavesdropper are obtained, respectively, as follows: Let be the optimal solution of the proposed design, then For comparison purpose, we also address the traditional design for DF (T-DF) and denote the solution by . The optimization problem is formulated as , and the optimal solution is obviously where cDF is a constant chosen to satisfy the power constraint . 4.2 Discussion about secrecy capacity in DF networks It is a natural thought that no matter under what channel assumption, secrecy capacity achieved by security-oriented designs would be higher than that achieved by traditional designs. However, the fact is that these designs may have the same performance which means that sometimes we can just ignore the eavesdropper. Remark 2. In a DF network, if the RE distances are about the same (which is widely assumed), it is unnecessary to consider the eavesdropper as the security-oriented design and the traditional design are indeed the same. Noticing that in this scenario, we can write Dh as . Thus, one can rewrite (27) as . Since we have umax(I + PggH) = umax(PggH). Thus, which is the same as . 5. Numerical results In this section, we investigate the performance of the above beamforming designs numerically. The simulation environment follows the model of Section 2. We perform Monte Carlo experiments consisting of 10,000 independent trials to obtain the average results. Assume the number of relays is M = 6, and the source transmit power is Ps = 10 dB. In order to show the influence of the RE distance in AF protocol, we fix the source at (0,0), the destination at (2,0), and the relays at (1,0) and move the eavesdropper from (1.25,0) to (5,0). We assume that the distances between relays are much smaller than SR/RD/RE distances. Therefore, the SR channels and RD channels follow a distribution, and the RE distance dRE varies from 0.25 to 4. Figure 3 shows the relationship between average secrecy capacity and total relay transmit power with the eavesdropper in different locations using P-AF design. We can see that if the eavesdropper is nearer to the relays than the destination is, the relays should use the maximal power to transmit. However, if the destination is much nearer, there is an optimal value of total relay transmit power which is about 12 dB under the case of dRE = 2, while the theoretical value in (18) is which is not very accurate but close. The reason is that Psubopt satisfies while the optimal power for P-AF design should satisfy . However, it is difficult to express λmax(Φ) in terms of the total relay transmit power and the channel coefficients, not to mention to solve the latter equation analytically. It can also be seen that as the total relay transmit power increases, the secrecy capacity tends to keep constant no matter where the eavesdropper is. Figure 4 compares different AF beamforming designs. It can be seen that the B-AF design shows a slight advantage over the T-AF design only when the total relay transmit power is small in the dRE = 1/2 case, while our proposed design always performs the best. The relationship between average secrecy capacity and total relay transmit power with the eavesdropper in different locations using P-DF design is demonstrated in Figure 5. We still assume the RE distances to be the same. Results show that the secrecy capacity of a DF network grows as the total relay transmit power increases or as the eavesdropper moves away. To verify Remark 2, we now examine the P-DF design and T-DF design under different variance assumptions of the RE channels. The average secrecy capacities of P-DF and T-DF designs under different RE channel assumptions are demonstrated in Figure 6. Our design outperforms the traditional design in case 1 and case 2. While in case 3, the two designs have the same performance. This indicates that the greater the RE channels differ from each other, the more superior the P-DF design is. If the RE distances are almost the same, the eavesdropper can be ignored. In this paper, we focused on security-oriented distributed beamforming designs for relay networks in the presence of a passive eavesdropper. We provided two beamforming designs under a total relay transmit power constraint, one of which is for AF and the other is for DF. Each design is to maximize secrecy capacity by exploiting information of SR, RD, and RE channels. To derive the beamforming solution for AF requires approximating the optimization objective by using the Taylor series expansion, while the solution for DF is obtained much more easily. We also found that secrecy capacity does not always grow if the relays use more power to transmit or if the eavesdropper gets farther from the relays, and that taking the eavesdropper into consideration is not always necessary. Moreover, for AF, we derived a suboptimal value of the total relay transmit power if the destination is nearer than the eavesdropper is. Numerical results showed the efficiency of the proposed designs. we have with small P, and with large P. If we make the assumption that the SR distances are all about the same, i.e., the 's are about the same (which is also assumed in ), and replace |fi|2 in the expression of with its mean value , we have Thus, K ≈ 0 with large P. Sendonaris A, Erkip E, Aazhang B: User cooperative diversity-part I: system description. IEEE Trans. Commun. 2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096 Dong L, Zhu H, Petropulu AP, Poor HV: Secure wireless communication via cooperation. In The 46-th Annual Allerton Conference on Communication, Control and Computing. Urbana-Champaign, IL, USA; 23–26 September 2008:1132-1138. This work is supported by the Natural Science Foundation of China under Grants 61372126 and 61302101, and the open research fund of National Mobile Communications Research Laboratory in Southeast University under Grant 2012D11. Authors and Affiliations The Key Lab of Broadband Wireless Communication and Sensor Network Technology (Ministry of Education), Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, 210003, China Mujun Qian, Chen Liu & Youhua Fu National Mobile Communications Research Laboratory, Southeast University, Nanjing, Jiangsu, 210096, China Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. Qian, M., Liu, C. & Fu, Y. Distributed beamforming designs to improve physical layer security in wireless relay networks. EURASIP J. Adv. Signal Process.2014, 56 (2014). https://doi.org/10.1186/1687-6180-2014-56

It would be nice to have a strategy for disentangling any tangled ropes... Can you tangle yourself up and reach any fraction? The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions. Can all unit fractions be written as the sum of two unit fractions? Find out what a "fault-free" rectangle is and try to make some of your own. This challenge asks you to imagine a snake coiling on itself. 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. . . . What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = 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. 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. What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? 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? 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 moves. How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . . 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. Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game. Can you find sets of sloping lines that enclose a square? Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . . 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. Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be? Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this? It's easy to work out the areas of most squares that we meet, but what if they were tilted? When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? 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 Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs. 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? Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs? Try entering different sets of numbers in the number pyramids. How does the total at the top change? 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. 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. 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. A collection of games on the NIM theme 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. These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like? While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book? Can you work out how to win this game of Nim? Does it matter if you go first or second? It starts quite simple but great opportunities for number discoveries and patterns! Can you explain the strategy for winning this game with any target? Nim-7 game for an adult and child. Who will be the one to take the last counter? The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails. The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis? 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. . . . In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37. Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have? A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

- Open Access Multiple solutions for the -Laplacian problem involving critical growth with aparameter Boundary Value Problems volume 2013, Article number: 223 (2013) By energy estimates and establishing a local condition, existence of solutions for the-Laplacian problem involving critical growth in abounded domain is obtained via the variational method under the presence ofsymmetry. MSC: 35J20, 35J62. In recent years, the study of problems in differential equations involving variableexponents has been a topic of interest. This is due to their applications in imagerestoration, mathematical biology, dielectric breakdown, electrical resistivity,polycrystal plasticity, the growth of heterogeneous sand piles and fluid dynamics,etc. We refer readers to [1–7] for more information. Furthermore, new applications are continuing to appear,see, for example, and the references therein. With the variational techniques, the -Laplacian problems with subcritical nonlinearities havebeen investigated, see [9–13]etc. However, the existence of solutions for -Laplacian problems with critical growth is relatively new.In 2010, Bonder and Silva extended the concentration-compactness principle of Lions to the variableexponent spaces, and a similar result can be found in . After that, there have been many publications for this case, see [16–19]etc. In this paper, we study the existence and multiplicity of solutions for the quasilinearelliptic problem where , () is a bounded domain with smooth boundary, is a real parameter, , are continuous functions on with Related to f, we assume that is a Carathéodory function satisfying for every , and the subcritical growth condition: (f1) for all , where is a continuous function in satisfying , . For , we suppose that f satisfies the following: (f2) there are constants and such that for every , a.e. in Ω, (f3) there are constants and a continuous function , , with , such that for every , a.e. in Ω, (f4) there are , and with such that Now we state our result. Theorem 1.1 Assume that (1.2), (1.3) and(f1)-(f4) are satisfied with, is odd in s. Then,given, there existssuch that problem (1.1) possesses atleast k pairs of nontrivial solutions forall. (g1) there is such that (g2) , odd with respect to t and (g3) for all and a.e. in Ω, where . Moreover, they assumed that and the result is the following theorem. Theorem 1.2 Assume that (1.2), (1.3), (1.4) and(g1)-(g3) are satisfied with. Then there exists asequencewithsuch that for, problem (1.1) has atleast k pairs of nontrivial solutions. Note that (f2) is a weaker version of (g3). This conditioncombined with (f1) and the concentration-compactness principle in will allow us to verify that the associated functional satisfies the condition below a fixed level for sufficiently small. Conditions (f3) and(f4) provide the geometry required by the symmetric mountain pass theorem . Compared with (g2), there is no condition imposed on fnear zero in Theorem 1.1. Furthermore, we should mention that our Theorem 1.1 improvesthe main result found in . In that paper, the authors considered only the case where is constant, while in our present paper, we have showedthat the main result found in is still true for a large class of functions. The paper is organized as follows. In Section 2, we introduce some necessarypreliminary knowledge. Section 3 contains the proof of our main result. We recall some definitions and basic properties of the generalized Lebesgue-Sobolevspaces and , where is a bounded domain with smooth boundary. And Cwill denote generic positive constants which may vary from line to line. For any , we define the variable exponent Lebesgue space with the norm where is the set of all measurable real functions defined onΩ. Define the space with the norm By , we denote the subspace of which is the closure of with respect to the norm . Further, we have There is a constantsuch that for all, So, and are equivalent norms in . Hence we will use the norm for all . Set. For, we have: Ifwitha.e. in Ω, then thereexists the continuous embedding. Ifandfor any, the embeddingis compact. The conjugate space ofis, where. For anyand, The energy functional corresponding to problem (1.1) is defined on as follows: Then and , We say that is a weak solution of problem (1.1) in the weak sense iffor any , So, the weak solution of problem (1.1) coincides with the critical point of. Next, we need only to consider the existence of criticalpoints of . We say that satisfies the condition if any sequence , such that and as , possesses a convergent subsequence. In this article, weshall be using the following version of the symmetric mountain pass theorem . Let, where E is a real Banach spaceand V is finite dimensional. Supposethatis an even functionalsatisfyingand there is a constantsuch that; there is a subspace W of E withand there issuch that; consideringgiven by (ii), I satisfiesfor. Then I possesses at leastpairs of nontrivial critical points. Next we would use the concentration-compactness principle for variable exponent spaces.This will be the keystone that enables us to verify that satisfies the condition. Let and be two continuous functions such that Letbe a weakly convergent sequenceinwith weak limit u such that: weakly in the sense of measures; weakly in the sense of measures. Also assume thatis nonempty. Then, for some countableindex set K, we have: whereand S is the best constant in theGagliardo-Nirenberg-Sobolev inequality for variable exponents,namely 3 Proof of main results Lemma 3.1 Assume that f satisfies (f1)and (f2) with. Then, given, there existssuch thatsatisfies thecondition for all, provided. Proof (1) The boundedness of the sequence. Let be a sequence, i.e., satisfies , and as . If , we have done. So we only need to consider the case that with . We know that From (f2), we get Notice that , , then from Lemmas 2.3, 2.4, , so . Let , then , and from the Hölder inequality, In addition, from Lemma 2.2(2), we can also obtain that So we have From (3.1), (3.3) and (f1), we have Noting that , we have that is bounded. Up to a subsequence, in . By Lemma 2.7, we can assume that there exist two measures μ,ν and a function such that Choose a function such that , on and on . For any , and , let . It is clear that is bounded in . From , we can obtain , as , i.e., From (f1), by Lemma 2.7, we have By the Hölder inequality, it is easy to check that From (3.5), as , we obtain . From Lemma 2.7, we conclude that Given , set where S is given by (2.5). Considering , we have We claim that . Indeed, if , this follows by (3.7). Otherwise, taking in (3.2), we obtain Therefore, by (3.8), the claim is proved. As a consequence of this fact, we concludethat for all . Therefore, in . Then, with the similar step in , we can get that in . □ Next we prove Theorem 1.1 by verifying that the functional satisfies the hypotheses of Lemma 2.6. First, we recallthat each basis for a real Banach space E is a Schauder basis forE, i.e., given , the functional defined by Lemma 3.2 Givenfor alland, there issuch that for all, . Proof We prove the lemma by contradiction. Suppose that there exist and for every such that . Taking , we have for every and . Hence is a bounded sequence, and we may suppose, without loss ofgenerality, that in . Furthermore, for every since for all . This shows that . On the other hand, by the compactness of the embedding, we conclude that . This proves the lemma. □ Lemma 3.3 Suppose that f satisfies (f3),then there existandsuch thatfor all. Proof Now suppose that , with , . From (f3), we know that Consequently, considering to be chosen posteriorly by Lemma 3.2, we have, for all and j sufficiently large, Now taking such that and noting that , so , if . We can choose such that . Next, we take such that for , for every , , the proof is complete. □ Lemma 3.4 Suppose that f satisfies (f4),then, given, there exist asubspace W ofand a constantsuch thatand. Proof Let and be such that , and . First, we take with . Considering , we have . Let and such that , and . Next, we take with . After a finite number of steps, we get such that , , and for all . Let , by construction, , and for every , consider the case that , then . Now it suffices to verify that From condition (f4), given , there is such that for every , a.e. x in , Consequently, for and , where and . Observing that W is finite dimensional, we have, , and the inequality is obtained by taking. The proof is complete. □ Proof of Theorem 1.1 First, we recall that , where and are defined in (3.9). Invoking Lemma 3.3, we find, and satisfies (i) with . Now, by Lemma 3.4, there is a subspace W of with such that satisfies (ii). By Lemma 3.1, satisfies (iii). Since and is even, we may apply Lemma 2.6 to conclude that possesses at least k pairs of nontrivial criticalpoints. The proof is complete. □ Bocea M, Mihăilescu M: Γ-convergence of power-law functionals with variable exponents. Nonlinear Anal. 2010, 73: 110-121. 10.1016/j.na.2010.03.004 Bocea M, Mihăilescu M, Popovici M: On the asymptotic behavior of variable exponent power-law functionals andapplications. Ric. Mat. 2010, 59: 207-238. 10.1007/s11587-010-0081-x Bocea M, Mihăilescu M, Pérez-Llanos M, Rossi JD: Models for growth of heterogeneous sandpiles via Mosco convergence. Asymptot. Anal. 2012, 78: 11-36. Chen Y, Levine S, Rao R: Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 2006, 66: 1383-1406. 10.1137/050624522 Fragnelli G: Positive periodic solutions for a system of anisotropic parabolic equations. J. Math. Anal. Appl. 2010, 367: 204-228. 10.1016/j.jmaa.2009.12.039 Halsey TC: Electrorheological fluids. Science 1992, 258: 761-766. 10.1126/science.258.5083.761 Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 1987, 9: 33-66. Boureanu MM, Udrea DN:Existence and multiplicity result for elliptic problems with-growth conditions. Nonlinear Anal., Real World Appl. 2013, 14: 1829-1844. 10.1016/j.nonrwa.2012.12.001 Boureanu MM, Preda F: Infinitely many solutions for elliptic problems with variable exponent andnonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 2012, 19(2):235-251. 10.1007/s00030-011-0126-1 Chabrowski J, Fu Y:Existence of solutions for -Laplacian problems on a bounded domain. J. Math. Anal. Appl. 2005, 306: 604-618. 10.1016/j.jmaa.2004.10.028 Dai GW, Liu DH:Infinitely many positive solutions for a -Kirchhoff-type equation involving the-Laplacian. J. Math. Anal. Appl. 2009, 359: 704-710. 10.1016/j.jmaa.2009.06.012 Fan XL, Zhang QH:Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5 Mihăilescu M:On a class of nonlinear problems involving a -Laplace type operator. Czechoslov. Math. J. 2008, 58(133):155-172. Bonder JF, Silva A: The concentration compactness principle for variable exponent spaces andapplications. Electron. J. Differ. Equ. 2010., 2010: Article ID 141 Fu YQ:The principle of concentration compactness in spaces and its application. Nonlinear Anal. 2009, 71: 1876-1892. 10.1016/j.na.2009.01.023 Silva, A: Multiple solutions for the -Laplace operator with critical growth. Preprint Alves CO, Barrwiro JLP:Existence and multiplicity of solutions for a -Laplacian equation with critical growth. J. Math. Anal. Appl. 2013, 403: 143-154. 10.1016/j.jmaa.2013.02.025 Fu YQ, Zhang X:Multiple solutions for a class of -Laplacian equations in involving the critical exponent. Proc. R. Soc., Math. Phys. Eng. Sci. 2010, 466(2118):1667-1686. 10.1098/rspa.2009.0463 Bonder, JF, Saintier, N, Silva, A: On the Sobolev trace theorem for variableexponent spaces in the critical range. Preprint Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7 Silva EAB, Xavier MS: Multiplicity of solutions for quasilinear elliptic problems involving criticalSobolev exponents. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2003, 20(2):341-358. 10.1016/S0294-1449(02)00013-6 Fan X, Zhao D:On the space and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617 Kovacik O, Rakosnik J:On spaces and . Czechoslov. Math. J. 1991, 41: 592-618. Lindenstrauss J, Tzafriri L: Classical Banach Spaces, I. Springer, Berlin; 1977. Marti JT: Introduction to the Theory of Bases. Springer, New York; 1969. The authors would like to express their gratitude to the anonymous referees forvaluable comments and suggestions which improved our original manuscript greatly. Thefirst author is supported by NSFC-Tian Yuan Special Foundation (No. 11226116),Natural Science Foundation of Jiangsu Province of China for Young Scholar (No.BK2012109), the China Scholarship Council (No. 201208320435), the FundamentalResearch Funds for the Central Universities (No. JUSRP11118, JUSRP211A22). The secondauthor is supported by NSFC (No. 10871096). The third author is supported by GraduateEducation Innovation of Jiangsu Province (No. CXZZ13-0389). The authors declare that they have no competing interests. All authors read and approved the final manuscript. About this article Cite this article Yang, Y., Zhang, J. & Shang, X. Multiple solutions for the -Laplacian problem involving critical growth with aparameter. Bound Value Probl 2013, 223 (2013). https://doi.org/10.1186/1687-2770-2013-223 - -Laplacian problem - critical Sobolev exponents concentration-compactness principle

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Local index theorem for orbifold Riemann surfaces We derive a local index theorem in Quillen’s form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function. Key words and phrases:Fuchsian groups, determinant line bindles, Quillen’s metric, local index theorems 1991 Mathematics Subject Classification:14H10, 58J20, 58J52 Quillen’s local index theorem for families of Cauchy-Riemann operators explicitly computes the first Chern form of the corresponding determinant line bundles equipped with Quillen’s metric. The advantage of local formulas becomes apparent when the families parameter spaces are non-compact. In the language of algebraic geometry, Quillen’s local index theorem is a manifestation of the “strong” Grothendieck-Riemann-Roch theorem that claims an isomorphism between metrized holomorphic line bundles. The literature on Quillen’s local index theorem is abundant, but mostly deals with families of smooth compact varieties. In this paper we derive a general local index theorem for families of Cauchy-Riemann operators on Riemann orbisurfaces, both compact and with punctures, that appear as quotients of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups . The main result (cf. Theorem 2) is the following formula on the moduli space associated with the group : Here is the first Chern form of the determinant line bundle of the vector bundle of square integrable meromorphic -differentials on equipped with the Quillen’s metric, is a symplectic form of the Weil-Petersson metric on the moduli space, is a symplectic form of the cuspidal metric (also known as Takhtajan-Zograf metric), is the symplectic form of a Kähler metric associated with elliptic fixpoints, is the 2nd Bernoulli polynomial, and is the fractional part of . We refer the reader to Sections 2.1–2.3 and 3.2 for the definitions and precise statements. Note that the above formula is equivalent to formula (2) for because the Hermitian line bundles and on the moduli space are isometrically isomorphic, see Remark 3. Note that the case of smooth punctured Riemann surfaces was treated by us much earlier in , and now we add conical points into consideration. The motivation to study families of Riemann orbisurfaces comes from various areas of mathematics and theoretical physics – from Arakelov geometry to the theory of quantum Hall effect . In particular, the paper that establishes the Riemann-Roch type isometry for non-compact orbisurfaces as Deligne isomorphism of metrized -line bundles stimulated us to extend the results of to the orbisurface setting. The paper is organized as follows. Section 2 contains the necessary background material. In Section 3 we prove the local index theorem for families of -operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups. Specifically, we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces. Since the cases of smooth (both compact and punctured) Riemann surfaces have been well understood by us quite a while ago [14, 10], in Section 3.2 we mainly emphasize the computation of the contribution from conical points corresponding to elliptic elements. In Section 4.1 we find a simple formula for a local Kähler potential of the elliptic metric, and in Section 4.2 we show that in the limit when the order of the elliptic element tends to the elliptic metric coincides with the corresponding cusp metric. Finally, in Section 4.3 we give a simple example of a relation between the elliptic metric and special values of Selberg zeta function for Fuchsian groups of signature (0;1;2,2,2). We thank G. Freixas i Montplet for showing to us a preliminary version of and for stimulating discussions. Our special thanks are to Lee-Peng Teo for carefully reading the manuscript and pointing out to us a number of misprints. 2.1. Hyperbolic plane and Fuchsian groups We will use two models of the Lobachevsky (hyperbolic) plane: the upper half-plane with the metric , and the Poincaré unit disk with the metric . The biholomorphic isometry between the two models is given by the linear fractional transformation for any . A Fuchsian group of the first kind is a finitely generated cofinite discrete subgroup of acting on (it can also be considered as a subgroup of acting on ). Such has a standard presentation with hyperbolic generators , parabolic generators and elliptic generators of orders satisfying the relations where is the identity element. The set , where , is called the signature of , and we will always assume that We will be interested in orbifolds (or , if we treat as acting on ) for Fuchsian groups of the first kind. Such an orbifold is a Riemann surface of genus with punctures and conical points of angles . By a -differential on the orbifold Riemann surface we understand a smooth function on that transforms according to the rule . The space of harmonic -differentials, square integrable with respect to the hyperbolic metric on , we denote by . The dimension of the space of square integrable meromorphic (with poles at punctures and conical points) -differentials on , or cusp forms of weight for , is given by Riemann-Roch formula for orbifolds: where denotes the integer part of (see [9, Theorem 2.24]). In particular, The elements of the space are called harmonic Beltrami differentials and play an important role in the deformation theory of Fuchsian groups, see Sect. 2.3. To study the behavior of harmonic Beltrami differentials at the elliptic fixpoints we use the unit disk model. Take and let be an elliptic element of order with fixpoint . The pushforward of to by means of the map is just the multiplication by , the -th primitive root of unity. The pushforward of to (that, slightly abusing notation, we will denote by the same symbol) develops into a power series of the form Moreover, since we have unless , so that In particular, for and for . As in , for we put , where is the Laplace operator (or rather of the Laplacian) in the hyperbolic metric acting on . The function is regular on and satisfies The following result is analogues to Lemma 2 in and describes the behavior of at . We will use polar coordinates on such that . be the Fourier series of the function on . Then as , where For the constant term we have where is the integral kernel of on , and . Since is a regular solution of the equation at , we have in polar coordinates where we used (2.1) for and the analogous expansion for . Then for the term of the Fourier series (2.2) we have the differential equation From here we get that as , where For the coefficients with we have so that as . This proves parts (i) and (ii) of the lemma, from where it follows that . To prove part (iii) it is sufficient to observe that 2.2. Laplacians on Riemann orbisurfaces Let us now switch to the properties of the Laplace operators on the hyperbolic orbifold , where is a Fuchsian group of the first kind. Here we give only a brief sketch, and the details can be found in , . Denote by the Hilbert space of -differentials on , and let be the Cauchy-Riemann operator acting on -differentials (in terms of the coordinate on we have ). Denote by the formal adjoint to and define the Laplace operator acting on -differentials on by the formula . We denote by the integral kernel of on the entire upper half-plane (where is the identity operator in the Hilbert space of -differentials on ). The kernel is smooth for and has an important property that for any . For and we have the explicit formula Furthermore, denote by the integral kernel of the resolvent of on (where is the identity operator in the Hilbert space ). For and the Green’s function is a smooth function on away from the diagonal (i. e. for ). For we have the following Laurent expansion near : as , where is the hyperbolic area of . Then for any integer we have This series converges absolutely and uniformly on compact sets for . We now recall the definition of the Selberg zeta function. Let be a Fuchsian group of the first kind, and let be a unitary character. Put where runs over the set of classes of conjugate hyperbolic elements of , and is the norm of defined by the conditions (in other words, is the length of the closed geodesic in the free homotopy class associated with ). The product (2.8) converges absolutely for and admits a meromorphic continuation to the complex -plane. Except for the last section, in what follows we will always assume that and will denote simply by . The Selberg trace formula relates to the spectrum of the Laplacians on , and it is natural (cf. ) to define the regularized determinants of the operators by the formula (note that has a simple zero at ). 2.3. Deformation theory We proceed with the basics of the deformation theory of Fuchsian groups. Let be a Fuchsian group of the first kind of signature . Consider the space of quasiconformal mappings of the upper half-plane that fix 0, 1 and . Two quasiconformal mappings are equivalent if they coincide on the real axis. A mapping is compatible with if for all . The space of equivalence classes of -compatible mappings is called the Teichmüller space of and is denoted by . The space is isomorphic to a bounded complex domain in . The Teichmüller modular group acts on by complex isomorphisms. Denote by the subgroup of consisting of pure mapping classes (i. e. those fixing the punctures and elliptic points on pointwise). The factor is isomorphic to the moduli space of smooth complex algebraic curves of genus with labeled points. Note that , as well as the quotient space , actually depends not on the signature of , but rather on its signature type, the unordered set , where and is the number of elliptic points of order (see ). The holomorphic tangent and cotangent spaces to at the origin are isomorphic to and respectively (where, as before, ). Let be the unit ball in with respect to the norm and let be the Bers map. It defines complex coordinates in the neighborhood of the origin in by the assignment where , is a basis for , and is a quasiconformal mapping of that fixes 0, 1, and satisfies the Beltrami equation For denote by and the partial derivatives along the holomorphic curve in , where is a small parameter. The Cauchy-Riemann operators form a holomorphic -invariant family of operators on . The determinant bundle associated with is a holomorphic -invariant line bundle on whose fibers are given by the determinant lines . Since the kernel and cokernel of are the spaces of harmonic differentials and respectively, the line bundle is Hermitian with the metric induced by the Hodge scalar products in the spaces (note that each orbifold Riemann surface inherits a natural metric of constant negative curvature ). The corresponding norm in we will denote by . Note that by duality between and the determinant line bundles and are isometrically isomorphic. The Quillen norm in is defined by the formula for and is extended for all by the isometry . The determinant defined via the Selberg zeta fuction is a smooth -invariant function on . 3. Main results Our objective is to compute the canonical connection and the curvature (or the first Chern form) of the Hermitian holomorphic line bundle on . By Remark 1 can be thought of as holomorphic -line bundle on the moduli space . 3.1. Connection form on the determinant bundle We start with computing the connection form on the determinant line bundle for relative to the Quillen metric. The following result generalizes Lemma 3 in : For any integer and we have where , and is the Euclidean area form on . The integral in (3.1) is absolutely convergent if for all . If for some , then this integral should be understood in the principal value sense as follows. Let be the fixpoint of the elliptic generator of order 2, and consider the mapping . Denote by the disk of radius in with center at 0. Since is discrete, for small enough we have unless and is either or . The subset is -invariant, where denotes the cyclic group of order generated by . The factor is an orbifold Riemann surface with holes centered at the conical points of angle . The integral in the right hand side of (3.1) we then define as The integrand in the right hand side is smooth and the integral is absolutely convergent, cf. (2.7). We need to show that (if there is we understand this integral as the principle value, see Remark 2). Without loss of generality we may assume that and has one elliptic generator of order with fixpoint . Then by (2.5) we have where is the cyclic group generated by (the stabilizer of in ), and Since , it is easy to check that the last expression in the above formula is a (meromorphic) quadratic differential on . Using the standard substitution we get that proves the theorem for (in the last line we used polar coordinates on ). We have to be more careful in the case , since the contribution from elliptic elements is no longer absolutely convergent and should be considered as the principal value, see Remark 2. From now on we assume that acts on the unit disk , so that is generated by . Since is discrete, there exists . Therefore, we can choose a small such that unless . The set is -invariant, and the factor is a Riemann surface with a small hole centered at the conical point. In this case we have

Website founded by MatPlus.Net Forum Helpmates Unusual twinning and unusual AUW. You can only view this page! |(1) Posted by seetharaman kalyan [Saturday, Jun 14, 2014 20:23]; edited by seetharaman kalyan [14-06-14]| Unusual twinning and unusual AUW. I was delighted to publish this nice problem by Nikola Predrag showing a novel twinning method to show AUW by the same white pawn. Your comments welcome. http://www.kobulchess.com/en/problems/chess-originals-2014/566-nikola-predrag-helpmate.html |(2) Posted by Kevin Begley [Sunday, Jun 15, 2014 04:32]; edited by Kevin Begley [14-06-15]| Interesting twinning idea. The concept is not entirely original -- I vaguely recall some problems twinned by removal of the mating unit, for example (which is quite similar); in fact, I made a fairy problem based upon twinning from the final position, with only alteration of the diagram's retro-content, and I seem to recall that somebody had partially anticipated even that idea -- but, this specific change might be new. And, this idea does suggest the possibility for a broader set of options, in altering the mating unit... I'd like to hear any ideas to shorten the text required, while preserving alternative options for replacement. I'd especially like to hear Nikola's thoughts; until we do, here's my suggestion (hopefully others can improve upon it): 0) AMU->... = Alter Mating Unit in some way (where the specific alteration is designated by "..."). 1) AMU->UPPER-CASE : specifies alteration of mating unit's type, from the solution to the original diagram. e.g., "b),c),d),e) AMU->Q,R,B,S" = alter mating unit's type to Queen, Rook, Bishop, Knight), and solve again by identical stipulation. How about that -- maybe you can show an AUW in this twinning method, too!? 2) AMU->*... : specifies continuous alteration of mating unit (for each twin, make the same alteration in the mating diagram from the preceding twin). e.g., "b),c),d) AMU*->P" = b) AMU->P (from final mate of diagram), c) AMU->P (from final mate of b), and d) AMU->P (from final mate of c). Interestingly, here the solver's final solution may be adequate proof of all preceding solutions. 3) AMU->lower-case : specifies alteration of mating unit's type color. e.g., b) AMU->n = alter mating unit's color to neutral, and solve again by identical stipulation. 4) AMU->#° : specifies alteration of mating unit's type rotation (note: works only with fairy units present). e.g., b) AMU->90° = alter mating unit's rotation 90° clockwise, and solve again by identical stipulation. 5) AMU->lower-case UPPER-CASE : specifies alteration of mating unit's type and color. 6) AMU->(x) : specifies annihilation of mating unit (e.g., remove the mating unit, and solve again by identical stipulation). 7) AMU->() : specifies no alteration of mating unit... (e.g., do nothing, just solve again from the final mate position, applying new retro assumptions). Are there other possibilities? Are there better ways to cover these possibilities? Are there any issues with this type of twinning? 1) If checkmate is delivered by double-check (or n-tuple-check), must the twin apply to all mating units, simultaneously? 2) What about stalemates? 3) Is there cause to expand this to alteration of the final position (where specific alterations can be stipulated, such as Q->R, R->B, etc)? |(3) Posted by Kevin Begley [Sunday, Jun 15, 2014 05:20]; edited by Kevin Begley [14-06-15]| ps: here's an AUW I made, in a single solution, by the same neutral pawn (which experiences a peculiar kind of duel)... I sent it somewhere, but never heard back, so I presume this was never published. (= 2+9+2N ) Relegation Chess (aka Degradierung) - upon moving onto its own 2nd rank (home-rank for pawns of a given color), an officer (except King) immediately demotes to pawn. 1.g8=nB! …nBh7(=nP) 2.h8=nQ+! …nQxd4 3.nQd1 …nQd7(=nP) 4.d8=nS! …nSb7(=nP) 5.bxa8=nR!# I liked the single-unit having a duel (reminds me of Good-Kirk vs Bad-Kirk), and showing AUW... Unfortunately, the use of Maximummer (as usual, when it used to force play, and is not thematically necessary) cheapens the idea far too much. The judge (whomever that was) deserves credit, if this was the reason for neglecting my problem (many fairy judges fail to appreciate significant differences in fairy element usage, particularly in excessively constraining conditions). Plus, my construction was rather shoddy... A much better AUW, with a single white pawn, is seen in the following: 3rd Prize, The Problemist, 1982 (= 2+9 ) 1.g8=S! 2.Sh6 3.Sxg4 4.Sxh2(P) 5.h4 6.h5 7.h6 8.h7 9.h8=B! 10.Bxb2(P) 11.b4 12.b5 13.b6 14.bxa7 15.a8=R! 16.Rxa4 17.Rxa2(P) 18.a4 19.a5 20.a6 21.a7 22.a8=Q! 23.Qxf3 24.Qxg3 25.Qg7 = By comparison, I quite liked Nikola's method of achieving this -- even if it might be a stretch to claim this is a single pawn (the same could be said, but only to a lesser degree, in the fairy methodology) -- because his creative interpretation appears entirely orthodox (which constitutes a spectacular realization of what we might incorrectly believe to be a fairy theme)! Maybe, if we think slightly outside the box, any Chess rules (including orthodox) might be sufficient to express any theme!? |(4) Posted by seetharaman kalyan [Sunday, Jun 15, 2014 07:50]; edited by seetharaman kalyan [14-06-15]| You are right Kevin that twins where the mating piece is removed is done several times. Twins shifting the black king from mating position has also been done before but not so frequent. This specific twinning, of changing mating piece is, I thought, novel. Your interesting suggestions for notation appear simple and should be examined by experts. Hm....AMU is already a fairy condition. it may or may not be relevant. |(5) Posted by Kevin Begley [Sunday, Jun 15, 2014 15:56]; edited by Kevin Begley [14-06-15]| You are correct -- "AMU" is not the most universal notation either... Ideally, the notation for both twinning and stipulation should be language independent; therefore, symbols are better. Perhaps it's time we consider some unicode symbols (they are far more accessible today, and they might be helpful in expressing some ideas more clearly). Also, the "*" (which I used to suggest successive alteration of the mating unit) is a poor choice -- the symbol is already taken for setplay. Probably the latter can be improved with the ampersand ("&") -- which already denotes successiveness in twinning... The notation we can fix fairly easily (providing folks do not become prematurely attached to a sub-optimal expression -- luckily, I see little need for that, here). I'm more concerned about additional options not considered. I'm confident that I have not covered all possibilities, which means unforeseen alterations are likely to be necessary. Better to get this right the first time (at the very least, strive for a framework which has room to grow). |(6) Posted by Nikola Predrag [Sunday, Jun 15, 2014 18:27]; edited by Nikola Predrag [14-06-15]| I made that h#2 as an example for the discussion about a twinning principle. I was not sure whether it should be published as an original, because of the uncommon twinning and the possible troubles with a short and clear explanation of it. And the very discussion was about the short and clear symbols for a whole class/family of a twinning principle: >a new twin starts from the mate-position of the previous twin. Of course, after some "Change" which allows Black to play some legal move(s).< I wrote down various attempts but not enough systematically and clearly to paste it here. Anyway, the symbols should come when the essence of a concept is clear. The essence is "Solve>Change>Solve(Again)>Change>Solve...", shortly "S-C-S" twinning, or any better symbolization. My problem would be e.g. >4xh#2; S-C-S(Twins),C=Demotion< Demotion (default) affects the pieces promoted during the play 4xh#2 tells that Mate&Change must happen after every 2+2 halfmoves, 4 phases altogether "DemotionGradual" might mean Q-R-B-S-P, starting with any rank (in case of less than 5 twins) "PromotionGradual" might mean P-S-B-R-Q etc., without a mandatory "real" Pawn-promotion on the last rank. Perhaps a fairy condition could be defined >kxh#n; S-C-S(Condition)<: "after each n+n halfmoves, Mate&Change must happen and a complete solution would require kxn+kxn halfmoves" bK has k-"lives" and after each "partial mate", the "Fairies" save bK at the cost of 1 "life" S-TC-S might symbolize the twinning and S-FC-S the condition. There are various possibilities but the fundametal concept and symbolization are hardly needed for just a few composed problems. |(7) Posted by seetharaman kalyan [Sunday, Jun 15, 2014 20:44]; edited by seetharaman kalyan [14-06-15]| I believe that this simple symbol would be understandable. " ># " implying that any change occurs from the previous mating position. ># remove g7, ># g7 to g6, ># g7=P, >#g7=S etc.. There can only be three changes possible in the mating position: Move the king, change/remove the mating piece or insert a pawn/piece in the mating line. |(8) Posted by Nikola Predrag [Sunday, Jun 15, 2014 22:18]| Yes, but why to specify the change for each twin, if they are many and the change is always the same. Rough example: (= 9+8 ) If I could make it in a minute (without the pieces and board), there is surely a possibility to make much more twins. Specifying each twin separately requires space and work. |(9) Posted by Kevin Begley [Sunday, Jun 15, 2014 23:02]; edited by Kevin Begley [14-06-15]| The idea of using fairy conditions to alter the mating unit is a good one, but we still require a twinning symbol indicating the metamorphosis of a mating unit. I like ">#", but I think Δ (or δ) are better symbols for change. For example: b) #Δ P or b) Δ# ♙ . Maybe ∫ can symbolize succession: b),c),d) ∫ #δ ♙ -- it's a pity we can't easily show that the integration goes from x=diagram to x=twin d). Also, maybe the path integral shows up better -- b),c),d) ∮ #δ ♙ -- this might even be more logical. At least we can put some calculus in our twinning mechanism! I'm sure some math major will argue that we are integrating over the mating unit, so maybe this form is better: b),c),d) ∮ ♙ δ# Note: "8xh#1.5 S-Demotion-S" does not read like a twin -- in fact, this is an interruption of the stipulation; clearly, this information does not belong in the stipulation. Furthermore, whether you mean "Demotion" (I think you mean "relegation chess", or "Degradierung" -- Demotion Chess is a slightly different fairy condition, invented by Dan Meinking), because those fairy conditions change officers moved upon specific ranks (their own 2nd rank, or their own 8th rank) -- neither one alters mating units, spanning the entire board. Moreover, I don't agree that the essence here is "solve-change-solve" -- the essence of this idea is a twin, first and foremost, which changes the mating unit in a particular way (in this specific case, we alter the type of unit, but we could as easily alter the color, or rotate the unit, or remove the mating unit, or make no change to the unit, and proceed; some thought is required for how fairy conditions might work here). Finally, as I've explained, there is the option to draw multiple twins from the final position of the diagram, or to draw them from each successive twins. The "solve change solve" formula might have proved a useful framework to build Nikola's problem, but it fails to envision a means to cope with alternative architectures, based upon the true essence of this idea. By the way, there is a subtle point to Nikola's twinning, which demonstrates that the twinning mechanism is not as orthodox as you might first think; to fully appreciate this, just consider what happens if a mating unit occurs on the 1st (or last) ranks. Obviously, this is a possibility for which the composer will deliberately go out of their way to disallow, but according to the first rule of chess composition, somebody, someday, is going to want to put this oddity to good use. Therefore, plan for it (which would be easier, if problem chess had insisted upon a consistent default rule for all pawns on the 1st rank -- and the obvious solution there is to make pawns behave as they would on every other rank, with the exception of the 2nd last ranks). So, strangely enough, Nikola has helped to demonstrate that orthodox and fairies are actually more interconnected than some folks like to pretend. :-) |(10) Posted by Nikola Predrag [Monday, Jun 16, 2014 01:59]| I'm not eager to propose the symbols, I'll eventually accept whatever might be proposed. I care more about their meaning. 8xh#1.5 indeed interferes with the stipulation but in case of a fairy condition it might be a stipulation. And it's not clear that S-Demote-S is a twinning principle, it looks more as a condition. I wrote it that way as a possible(?) example of a condition which not only "demotes" a promoted piece after the mate. It also allows 2 consequent white moves, White mates and after demotion, White continues. If such sequence W-B-W>W-B-W... looks unacceptable, the very act of "demotion" might be taken as a black move. But I'm not much interested in new conditions (before having an idea for a problem), that might be explored or abandoned by those who are interested. Actually, sending that problem to Seetharaman, I wrote under the diagram: h#2(x4) #=PromotionsCancelled-PlayContinued 5+11 where (x4) is supposed not to affect the meaning of the stipulation, but to indicate its repetition through 4 twins. Solve-Change-Solve could be a twinning or a condition, so I wrote S-C-S(Twins) and S-C-S(Condition), or shorter S-TC-S and S-FC-S (TwinChange and FairyChange). The essence of the idea (S-TC-S) is indeed a twin, but not only "which changes the mating unit in a particular way". "Change" could be "Relocation", "Addition" or "Removal" of some piece. "Change" could be "ColorChange" or "CancellAllAndernachColorChange" or "CancellParticularAndernachColorChange" etc. might suggest that in mate-position of a/b, bK is relocated to f8/b3, and there's h#2 again. might suggest the same twinning principle (for 3 twins) but not only bK is relocated. Any clear and short symbolization would be good, so go on. In orthodox problems, the twinning with Pawns on 1st/8th rank would be simply incorrect. The author must care about it. A great restriction for the construction of that h#2 AUW, was getting the first 3 mates from 7th rank. A mate from 6th rank would mean a mate by promotion in the next twin, leaving wP on 8th rank for the next twin. Fairy chess might allow anything and some general "general theory" would have to care about everything. |(11) Posted by Joost de Heer [Monday, Jun 16, 2014 08:43]| If there are two solutions in a), and only one of them leads to a solution, after the change, in b), is the second solution in a) a cook or an invalid solution? |(12) Posted by Georgy Evseev [Monday, Jun 16, 2014 08:45]| This discussion has gone into unneeded details. I have long ago resolved this difficulty for myself, declaring that there are two kinds of twins: technical and constructive. Technical twins are used when they are needed because of technical difficulties or formal limitations. The change should be as small as possible. The kinds of technical twins are well documented and most probably we will not see anything new is this field. In the constructive twins the mechanics of twinning is itself a part of the author's idea. So, everything is allowed, until the idea is emphasized enough. Unfortunately, this is exactly the reason why no sensible classification is possible: generally, a new problem with the same mechanics of twinning is, really, significantly anticipated. From the other hand this means that we will be able to see a new finds in this area. I had given a small lecture about this kind of twins in Marianka in 2011. Unfortunately, I had not prepared the text separately, but the problems shown during lecture are available in Marianka 2011 bulletin (http://www.goja.sk/Bulletin_Marianka_2011.pdf), starting from page 49. |(13) Posted by Kevin Begley [Monday, Jun 16, 2014 08:49]; edited by Kevin Begley [14-06-16]| You make many good points... and it's important to reiterate my appreciation for your original problem. I'd like to see more creative ideas actually encouraged -- it's particularly enjoyable, when such an original idea can be expressed with thematic artistry! My favorite art is that which gives the reader something to think and talk about... it expands the genre... it teaches me something that logically should not fit on a chessboard. I was stirred by your problem... I thought it appropriate to raise some larger issues, offer some suggestions, and ask for improvements, here. I am confident that my suggestion is not the best (there are many things I have not considered -- for example, relocation of mate units, as you noted in your last post). Like you, I look forward to the improvements, which will surely enhance our ability to express such creative new twinning options (as you have demonstrated, and as we may extrapolate plausible). Hopefully, I helped to push a remarkable twinning idea forward... I know some will consider it tangential to encourage further suggestions, and improvements, in this thread, but for me, such a discussion seems the natural residual impact of your work (which you can definitely take as a compliment). I hope it gets the editors, and the software developers talking to one another... searching for the best way to universally express such twinning possibilities. |(14) Posted by Nikola Predrag [Monday, Jun 16, 2014 09:20]| in case of twinning, a second solution, such as you described, would be a cook. In case of fairy condition, h# in kxn moves (with k Parts or k "partial mates"), any "partial" solution which doesn't lead to a "complete solution", might be considered as simply "not solution". However, such fairy condition should be precisely defined and I don't know how. I agree that it's not likely to see many problems with such twinning. This twinning principle itself makes the main content, the play is not interesting (but the construction might be). Still, instead of AUW, something else probably could be shown. the possibilities are unlimited and to anticipate all of them in some general classification looks impossible. Still, some flexible frame would be a welcome and brave contribution. I have indeed tried to "play" with a twinning which looks like a fairy condition, to achieve an illusion of continued play as h#8(4x2) No more posts MatPlus.Net Forum Helpmates Unusual twinning and unusual AUW.

Direxion Daily 20+ Year Treasury Bull 3x Shares (NYSEARCA:TMF) aims to multiply daily gains of the ICE U.S. Treasury 20+ Year Bond Index by a factor of 3. It was introduced in April 2009 and has a net expense ratio of 0.95%. According to Google Finance, its market cap as of Jan. 16, 2017, is $84.75M. Performance to date Figure 1 shows growth of $10k in TMF since inception, alongside iShares 20+ Year Treasury Bond ETF (NYSEARCA:TLT), which is essentially an unleveraged version of TMF. Overall, it's been a wild ride for TMF. It experienced a near-50% drawdown right at the beginning of its lifetime and jolted up and down several times over the next 7.5 years. It achieved considerable separation from TLT in 2015-2016, but gave up virtually all of it when bonds fell following the US election. Performance metrics for the two funds are shown in Table 1. |Fund||CAGR||Maximum drawdown||Mean of daily gains||SD of daily gains||Sharpe| I think the overall conclusion here is that TMF's slightly better raw returns are not justified by its vastly greater volatility and drawdown potential. Moreover, the fact that TLT achieved an excellent CAGR during this time period, and TMF only slightly beat it, tells me that TMF may underperform TLT in a more common scenarios where TLT gains 3-4% annually (its current weighted average coupon is 3.23%). Daily gains, TMF vs. TLT In Figure 2, we see that daily gains map almost perfectly from TLT to TMF, with data points falling very close to the Y = 3X line. This is exactly what you would expect from a 3x leveraged ETF behaving like it should. Monthly gains and volatility decay Monthly gains also map predictably from TLT to TMF, but the relationship is significantly non-linear (Figure 3; Data is for Jan. 2010 through Dec. 2016). If we compare the red curve to the blue line, we see that TMF does slightly worse than 3x TLT's monthly gain except at the left and right ends of the graph -- to be exact, whenever the TLT monthly gain is between -6.2% and 5.4%. This is a direct consequence of volatility decay or beta slippage, which I've described in other articles (e.g. Clearing Up Some Beta Slippage Myths). Now, I'm guessing a lot of readers are thinking that the red curve is practically no different than the blue line; I'm probably overanalyzing and missing the big picture. Actually, the non-linearity here is the big picture. If TMF followed the blue line, and achieved exactly 3 times TLT's monthly gain, then in months when TLT was unchanged, TMF would also be unchanged. The y-intercept for Y = 3X is of course 0. Instead, TMF follows the red curve, which has a y-intercept of -0.45%. That means that in months when TLT is unchanged, TMF averages a 0.45% loss. That's not negligible; a 0.45% monthly decline corresponds to an annualized loss of 5.3%. So what's the break-even point for TMF? If TMF followed the blue Y = 3X line, then its x-intercept would be 0, and TMF would average positive growth whenever TLT had positive growth. But the x-intercept for the red curve is 0.15%. That means that on average TMF only grows when TLT gains 0.15% or better in a given month. Again, not negligible -- 0.15% monthly growth is 1.8% annualized. Of course, if we're investing in TMF, we don't just want positive growth, we want to outperform TLT. If we were on the blue line, we'd be in good shape, since Y = 3X is greater than X (the black line) whenever X is greater than 0, which we expect it to be since TLT is made up of yield-generating bonds. But on the red curve, if you could zoom in far enough, you'd see that TMF outperforming TLT requires TLT growth of 0.22% or better (2.7% annualized). TLT's current weighted average coupon, 3.23%, is indeed higher than the 2.7% needed for TMF to outperform. Following the red curve, we expect TMF to gain 0.35% monthly (4.3% annualized) when TLT grows in the amount of its current 3.23% coupon. Is the extra 1% worth sustaining drawdowns 2-3 times as severe? Probably not. Long-term underperformance despite higher expected monthly returns While TMF's expected monthly return is slightly higher than TLT's -- by my estimates, 0.35% vs. 0.26% -- a basic resampling experiment suggests it is likely to underperform TLT over a longer period. Briefly, in each of 100,000 trials, I sample with replacement 5 years of daily gains for TLT, using data on TLT since its inception. I calculate TMF gains simply as 3x TLT gains minus the daily equivalent of the 0.95% expense ratio, then calculate performance metrics for the 5-year period. |CAGR range for TLT||Trials in which TMF outperforms TLT||Median CAGR for TLT||Median CAGR for TMF| We see here that TMF almost always underperforms TLT in 5-year periods where TLT averages less than 3% annual growth, and usually outperforms TLT in 5-year periods where TLT averages greater than 3.5% annual growth. In the 3.0-3.5% range, though, where we expect TLT to be given its average coupon, TMF only outperforms TLT 26.4% of the time. Interestingly, results would be vastly different if TMF did not carry its 0.95% expense ratio. Repeating the same experiment with TMF having the same expense ratio as TLT, in that middle strata where TLT has CAGR of 3.0% to 3.5%, the median CAGR for TMF is 3.97%, and it outperforms TLT in 94.3% of simulations. I believe that TMF makes for a poor long-term investment due to three factors: - Its high leverage results in 3x the volatility of TLT, and drawdowns 2-3 times as bad. - Volatility decay translates to substantial losses when TLT is approximately flat, e.g. over 5% annually. - With the 0.95% expense ratio, TMF will tend to underperform TLT in 5-year periods of typical TLT growth. On a final point of wishful thinking, I would love to see Direxion or a different company offer a version of TMF that worked on monthly rather than daily gains, as a small number of leveraged ETFs do (e.g. DXSLX). Such a fund would be far less prone to volatility decay, and would be much more appealing overall. Disclaimer: The author used Yahoo Finance to obtain historical stock prices and used R to analyze the data and generate figures. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

What is Half-Life and its Formula? Half-life is the time required for half of a quantity of a radioactive substance to undergo decay or transformation. It is a characteristic property of each radioactive isotope and is used to describe the rate of decay, providing a measure of the stability or persistence of a radioactive material. The half-life formula is an equation we use to calculate the rate of disintegration of unstable atomic nuclei which leads to the emission of alpha (α), beta (β), or gamma (γ) particles. The half-life formula is written as T1/2 = (0.693) / λ, or T1/2 = ln(2) / λ, or T1/2 = (Loge2) / λ T1/2 = Half-life λ = decay constant Note: We can use t1/2 or T1/2 to indicate the half-life of a radioactive element Therefore, we can use the above formulae to solve half-life problems. We need to understand that the half-life of a radioactive element is the time taken for half the atoms of the element to decay. We can also define half-life as the time taken for a given mass of a radioactive substance to disintegrate to half its initial mass. The number of half-life formula are 3 and we can apply any of the formulae to solve a problem. The relation of N and T1/2 is in a graph of N versus T1/2 below. The graph below describes a decay curve. We can use the following new equations to calculate the half-life of a radioactive element: - Half-life, T1/2 = t / (log2R) = (t x log2R) / logR [ Where R = 2n = N1 / N2; and n = t / T1/2 ] - The second formula we can use for the half-life is T1/2 = (0.693) / λ - We also have another formula for calculating half-life which is T1/2 = ln(2) / λ - The last formula for calculating half-life is T1/2 = (Loge2) / λ - The formula for calculating the number of atoms that decay, Nd = N1 – N2 = N1 ((R-1) / R) = N2 (R – 1) - Fraction remaining of undecayed radioactive elements, fr = N1 / N2 = 1 / R - The fraction of decayed atoms, fd = Nd / N1 = (R – 1) / R T1/2 = Half-life of a radioactive element t = time it takes a radioactive element to decay or disintegrate n = number of half-lives N1 = Initial mass or the initial number of atoms present/initial count rate. And N2 = final mass of the final number of atoms remaining undecayed/final count rate. Nd = Number of atoms or mass of atom that has decayed or disintegrated. R = Disintegrating ratio fr = fraction of initial number of atoms remaining undecayed We also have fd = fraction of the initial number of atoms that have decayed. From the above half-life formulae, the first formula is called Zhepwo radioactive equation while the remaining formulae are called the Zhepwo derivative(s). Hence, we can be able to differentiate between the two groups of equations. Derivation of Half-Life Formula Here is how to derive the formula: Since the rate of disintegration is proportional to the number of atoms present at a given time, we can say that -(dN/dt) ∝ N or dN/dt = -λN λ = constant of proportionality which is referred to us as a decay constant of the element. We can write the above equation (dN/dt = -λN) as λ = – 1/N (dN/dt). Hence the formula for decay constant is λ = – 1/N (dN/dt) After integrating the above equation, we will have N = N0e-λt Where N0 is the number of atoms present at a time t = 0 (i.e at the time when observations of decay were begun). N = the number of atoms present at time t. We can now change N = N0/2 into N = N0e-λt to obtain the time required for half of the atoms to disintegrate (half-life) N0 / 2 = N0e-λt And N0 will cancel each other from both sides to obtain 1 / 2 = e-λt1/2 We will take the natural or Naperian logarithm of both sides to get loge(1/2) = -λt1/2 We need to remember that logeen = n Therefore, from the left-hand side of the equation (loge(1/2) = -λt1/2). We can see that loge(1/2) = loge1 – loge2 = 0 – loge2 = – loge2 = – 0.693 Thus, -λt1/2 = – 0.693 and t1/2 = 0.693 / λ What is Half-Life? The half-life of a radioactive element is the time it takes half of the atoms initially present in the element to disintegrate or decay. Knowledge of Logarithm for Calculating Half-life The knowledge of the theory of logarithms will help us to understand how to calculate half-life. Here are logarithmic terms in a tabular form to help you understand the topic better. |R = 2n||log2 R = n| |2 = 21||log2 2 = 1| |4 = 22||log2 4 = 2| |8 = 23||log2 8 = 3| |16 = 24||log2 16 = 4| |32 = 25||log2 32 = 5| |64 = 26||log2 64 = 6| |128 = 27||log2 128 = 7| |256 = 28||log2 256 = 8| |512 = 29||log2 512 = 9| |1024 = 210||log2 1024 = 10| What is the Formula for the Disintegration Ratio? The disintegration ratio is a newly coined expression, it is NOT a new or additional concept in physics. It does not contradict any term or concept in radioactivity. Therefore, It is simply coined to name for an established relationship (No / N = 2n). Thus, it is modification R = N1 / N2 = 2n which is a simplified application in solving radioactive decay problems. Half-life Formula: Conventional Method of Calculating Half-Life Assume that a radioactive element with a half-life of 5 seconds contain 192 atoms initially. After the first 5 seconds (1 half-life), 96 atoms would decay and 96 atoms would remain. In another 10 seconds (2 half-lives), 144 atoms would decay and 48 atoms would remain. When we move to the next 15 seconds (3 half-lives), 168 atoms would decay and 24 atoms would remain. After 20 seconds (4 half-lives), 180 atoms would decay and 12 atoms would remain. In another 25 seconds (5 half-lives), 186 atoms would disintegrate and 6 atoms would remain. Half-Life Formula: How to Calculate Half-life in Physics Here is a solved problem to help you understand how to apply half-life formula A radioactive element has a decay constant of 0.077 per second. Calculate its half-life. The decay constant, λ = 0.077 s-1 We will use the formula that says T1/2 = (0.693) / λ = 0.693 / 0.077 = 9 s Therefore, the half-life of the radioactive element is 9 seconds. You may also like to read:

By Jiri Matousek, Jaroslav Nesetril This booklet is a transparent and self-contained advent to discrete arithmetic. Aimed quite often at undergraduate and early graduate scholars of arithmetic and machine technology, it truly is written with the objective of stimulating curiosity in arithmetic and an lively, problem-solving method of the awarded fabric. The reader is ended in an knowing of the fundamental ideas and strategies of really doing arithmetic (and having enjoyable at that). Being extra narrowly centred than many discrete arithmetic textbooks and treating chosen issues in an strange intensity and from a number of issues of view, the publication displays the conviction of the authors, lively and across the world well known mathematicians, that an important achieve from learning arithmetic is the cultivation of transparent and logical pondering and conduct precious for attacking new difficulties. greater than four hundred enclosed workouts with quite a lot of hassle, lots of them observed through tricks for answer, aid this method of educating. 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Invitation to Discrete arithmetic may still make a pleasant studying either for rookies and for mathematical professionals. the most subject matters comprise: undemanding counting difficulties, asymptotic estimates, in part ordered units, simple graph thought and graph algorithms, finite projective planes, straightforward likelihood and the probabilistic process, producing capabilities, Ramsey's theorem, and combinatorial purposes of linear algebra. normal mathematical notions going past the high-school point are completely defined within the introductory bankruptcy. An appendix summarizes the undergraduate algebra wanted in a number of the extra complex sections of the publication. Read Online or Download An Invitation to Discrete Mathematics PDF Best textbook books An analytical chemistry textbook with a spotlight on instrumentation; English translation of the French examine Chimique. Methodes et options instrumentals modernes. this can be a vector PDF replica. 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Such strategies are broadly utilized in engineering, technology, finance, and trade. - Minimizing and Exploiting Leakage in VLSI Design - Management: Challenges for Tomorrow's Leaders (5th Edition) - Measure, Integral and Probability (2nd Edition) - Matrices and Their Roots: A Textbook of Matrix Algebra/With Disk - Oral Diseases: Textbook and Atlas - The Prentice Hall Guide for College Writers (8th Edition) (MyCompLab Series) Extra info for An Invitation to Discrete Mathematics It is not only remarkable but also surprising, since set theory, and even the notion of a set itself, are notions which appeared in mathematics relatively recently, and some 100 years ago, set theory was rejected even by some prominent mathematicians. Today, set theory has entered the mathematical vocabulary and it has become the language of all mathematics (and mathematicians), a language which helps us to understand mathematics, with all its diversity, as a whole with common foundations. We will show how more complicated mathematical notions can be built using the simplest set-theoretical tools. N be n ≥ 2 distinct lines in the plane, no two of which are parallel. Then all these lines have a point in common. 1. For n = 2 the statement is true, since any 2 nonparallel lines intersect. 2. Let the statement hold for n = n0 , and let us have n = n0 + 1 lines 1 , . . , n as in the statement. e. the lines 1 , 2 , . . , n−1 ) have some point in common; let us denote this point by x. Similarly the n − 1 lines 1 , 2 , . . , n−2 , n have a point in common; let us denote it by y. The line 1 lies in both groups, so it contains both x and y. 2). The operations ∪ and ∩ are also commutative, in other words they satisfy the relations X ∩ Y = Y ∩ X, X ∪ Y = Y ∪ X. The commutativity and the associativity of the operations ∪ and ∩ are complemented by their distributivity. For any sets X, Y, Z 14 Introduction and basic concepts we have X ∩ (Y ∪ Z) = (X ∩ Y ) ∪ (X ∩ Z), X ∪ (Y ∩ Z) = (X ∪ Y ) ∩ (X ∪ Z). The validity of these relations can be checked by proving that any element belongs to the left-hand side if and only if it belongs to the righthand side.

33. Geometric Optics Reflection Of Light Reflection of Light Was this helpful? Hey, guys, in this video, we're going to talk about the reflection of light off of a boundary between two media. Okay, let's get to it. Now. Remember when a wave encounters a boundary, it could do one of two things. It can either reflect off of that boundary or it can transmit through that boundary and propagate in the new media. In reality waves, they're gonna do a little bit of both now for light boundaries. Air, typically referred to or media, I should say, are typically referred to as one of two things. They could either be reflective, like the surface of the mirror, and mainly on Lee. Allow reflection. Or they could be transparent like glass and mainly allow transmission. Okay, for now, we wanna we wanna talk about reflective surfaces. Alright, Light reflects off of a boundary in the same manner as a ball undergoing an elastic collision with a wall. Okay, if we have a ball at some point with some momentum right here in some direction, when it hits the wall, it's going to conserve that momentum, but it's going to go in the opposite direction, okay? And it's gonna enter with some angle and it's gonna leave with some other angle light is gonna do the same thing if I draw a ray of light encountering a boundary between two media that Ray is gonna leave at some angle. Okay, We have something called the Law of Reflection, which holds for the elastic collision. Just like it holds for reflection of light. That states that that reflected angle is actually the same as the incident angle, the angle with which it hits the boundary. Okay, now, just a side note. This isn't gonna be important right now, but this is gonna be very important later. Whenever light encounters a boundary, we always measure the angle relative to the normal direction relative to some line that is perpendicular to the boundary. Right. This is the normal. Just like the normal force is perpendicular. The normal direction is perpendicular to a surface. Okay, let's do an example. We want to find the missing angle theta using the law of reflection. Okay, so this light ray is incident at 65 degrees. It undergoes reflection right here, which means that the outgoing angle is also going to be 65 degrees. That's just what the law reflection says now notice this is the normal too sorry, Normal to this surface right? Which means that it's had a 90 degree angle from the surface. So this angle right here is gonna be the complementary angle to degrees, which is 25 degrees, right, because they have to add up to 90. Now notice we have a triangle right here that looks like this. Here's our 25 degree angle. Here's our 125 degree angle and here is an unknown angle. Whoops, letting men wise myself remember that the sum of all of the internal angles within a triangle has to be so 25 degrees plus 120 degrees plus this unknown angle has to be 180 degrees. That means that this unknown angle is actually 35 degrees. Okay, so that tells us what this angle is. This angle is 35 degrees. Now, once again, this line right here is the normal to the second surface, which means that it's perpendicular to the second surface. So this angle right here is going to be the complementary angle to 35 degrees right, the angle that, when added to 35 degrees, equals 90 and that is 55 degrees. So what is theta have to be, while the law of reflection says it has to be the same as that incident angle which on the second surface is 55 degrees, So theta, therefore, is 55 degrees. Alright, guys, that wraps up our discussion on the reflection of light and the law reflection. Thanks for watching. Was this helpful? Hey, guys, let's do an example. Ah, flat mirror hangs 0.2 m off the ground. If a person 1.8 m tall stands 2 m from the mirror, what is the point on the floor nearest the mirror, which we called X that can be seen in the mirror. The geometry for this problem has already set up. All we have to do is use the law of reflection to figure it out. So this light ray is coming up off the ground from this point, encountering the mirror at its lowest point and then leaving in this direction to your eyes. What, you're gonna be here? Okay? What we have to do is use the law reflection to figure out what angle properly, adjust that light race so that it meets you at exactly 1.8 m off the ground. Here's the normal because we always use the normal one measuring angles. So this is our incident angle theta one. And this is our reflected angle theta one prime. And remember that those two are equal now, notice if I were to continue this normal line right here we form a triangle, right? The triangle that we form is 2 m wide. How tall is it? Well, it's not 1.8 m tall because the mirror the bottom of this point right here is 0.0.2 meters off the ground. So it's actually 1.8 minus 0.2, which is 1.6 m. And this angle right here is state of one prime. Okay, which is the angle that we're interested in finding so clearly we could just use trigonometry to find this. We can use the tangent and we can say that the tangent of data one prime is gonna be the opposite edge which is 1.6 m divided by the adjacent edge which is to and that tells us that data one prime is just 51.3 degrees. Okay, so now we know that this angle is 51. degrees. So this angle to is 51.3 degrees. So we have a new triangle right here. Let me minimize myself. We have a new triangle now this lower triangle where this angle the incident angle is 51.3 degrees. This height is 0.2 m and this length is X right. What's this angle going to be? Well, this is what's known as an alternate interior angle to 51. and all alternate interior angles are the same. So this is going to be 51 3 degrees. All right, So once again, we can use the tangent to find what X should be. We could just say then that the tangent of 51.3 degrees equals the opposite, which is 0.2 m divided by the adjacent which is X, or that X is 0.2 over the tangent of 51.3 degrees. And finally, that X is just 0. meters. Okay, Just using geometry and trigonometry, we can answer this question. The crux of the physics is that these angles right here are the same. And that's the law of reflection. All right, guys, Thanks for watching What is the distance, d, between the incoming and outgoing rays?

So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Inference in curved exponential families, following a principled approach, requires construction of exact or approximate ancillary statistics. help me write my essay key stage 3 This can be studied both experimentally and analytically. Principal Component Analysis PCA is one of the most widely used statistical tools for data analysis with applications in data compression, image processing, and bioinformatics. One wants to match up the pictures, but there is some error in the measurement. This project involves studying, in real-data examples, how classical inference procedures are invalidated by the use of selection procedures. It is known that a convex planar U can have at most one equichordal point. watch thesis movie online In this project, we will harness the power of randomized dimension reduction to accelerate methods in applications which require high computational costs. Examples of thesis topics in mathematics thesis for phd governance 2018 What is true in dimension three? The deterministic versions of such algorithms suffer from slow convergence in some cases. Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math Blake Thornton for suggestions about faculty to talk with. This area is appropriate for both reseach and expository projects. Imagine a large number of cameras arranged around a central object. Consider the following easy exercise as a warm-up. A question of this sort first appeared in the lectures of the legendary th century mathematician Georg Frobenius, which is why this problem was named after him. It is part of the general subject of Dynamical Systems. Most large public-access data sets have this complex structure. - thesis preparation guidelines upm - research topic thesis statement examples - buy my essay life in school essay-my daily - definition essay help students - best college application essay service xbox 360 - editing an essay describing a place you love - writing essay debate Help me write my thesis statement love is We are developing statistical models to tackle these issues. The quest to produce Calabi-Yau 3-manifolds three complex dimensions! This was covered in a course - Math I think - but that was so long ago it's not listed in the catalog. order a paper gift box diy It would also be useful to know what a Riemann surface is, but this could be dealt with in summer reading. The CMC math and CS faculty represent a wide range of research areas, including algebraic topology and knot theory, functional, harmonic, and complex analysis, probability and statistics, numerical analysis, PDEs, compressed sensing, mathematical finance, number theory, discrete geometry, programming languages, and database systems. Of course, the ancient method of Eratosthenes sieve method is one such algorithm, albeit a very inefficient one. request for proposal for grant writing services Both deal with the idea that certain variables predict whether a response is necessarily zero, and if the response is not necessarily zero, then other variables might predict its value. Intuitively, as C approaches 0, the deterministic billiard system should behave more and more like the probabilistic system of 7. Here is a much more surprising fact that you might like to think about. It is known that a convex planar U can have at most one equichordal point. This is particularly true for topology, specially for what is called "algebraic topology". thesis for dummies body image Welker has been used to investigate the structure of such complexes. Undergraduate Research Ideas Some of our faculty have listed ideas for undergraduate research work. Thesis preparation guidelines upm Show that c is the shortest path contained in the surface that joins p and q. This area is appropriate for both reseach and expository projects. Recent results related to the bound in the Berry-Esseen theorem, for summands of both i. Moreover, they had to figure out how to resolve them -- the higher-dimensional analogue of lifting an actual string off itself. I have a friend who has some pathological gambling data, who has extracted most of the obvious results from her data, but might be looking for help in digging out some remaining gems. These differential equations are of a very special kind: Among the ideas posted here, some are harder and some easier. Professor Mohan Kumar Algebra 1 If a1,a2, To make the point, consider the following. Methylation is important to embryonic development and cancer. Both deal with the idea that certain variables predict whether a response is necessarily zero, and if the response is not necessarily zero, then other variables might predict its value. From an employee database, can one identify employess who are likely to leave the company from those who will stay? There are related functions called Grassmanian polylogarithms, invented by A. For any partially ordered set P, the set of all totally ordered subsets of P determines a simplicial complex. Most of the time spent in courses on ODEs, like Math , is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions.

Linear programming is a way of using systems of linear inequalities to find a maximum or minimum value. In geometry, linear programming analyzes the vertices of a polygon in the Cartesian plane.Linear programming is one specific type of mathematical optimization, which has applications in many scientific fields. Though there are ways to solve these problems using matrices, this section will focus on geometric solutions.Linear programming relies heavily on a solid understanding of systems of linear inequalities. Make sure you review that section before moving forward with this one.In particular, this topic will explain: What is Linear Programming? How to Solve Linear Programming Problems Identify the Objective Function What is Linear Programming? Linear programming is a way of solving problems involving two variables with certain constraints. Usually, linear programming problems will ask us to find the minimum or maximum of a certain output dependent on the two variables.Linear programming problems are almost always word problems. This method of solving problems has applications in business, supply-chain management, hospitality, cooking, farming, and crafting among others.Typically, solving linear programming problems requires us to use a word problem to derive several linear inequalities. We can then use these linear inequalities to find an extreme value (either a minimum or a maximum) by graphing them on the coordinate plane and analyzing the vertices of the resulting polygonal figure. How to Solve Linear Programming Problems Solving linear programming problems is not difficult as long as you have a solid foundational knowledge of how to solve problems involving systems of linear inequalities. Depending on the number of constraints, however, the process can be a bit time-consuming.The main steps are: Identify the variables and the constraints. Find the objective function. Graph the constraints and identify the vertices of the polygon. Test the values of the vertices in the objective function. These problems are essentially complex word problems relating to linear inequalities. The most classic example of a linear programming problem is related to a company that must allocate its time and money to creating two different products. The products require different amounts of time and money, which are typically restricted resources, and they sell for different prices. In this case, the ultimate question is “how can this company maximize its profit?” As stated above, the first step to solving linear programming problems is finding the variables in the word problem and identifying the constraints. In any type of word problem, the easiest way to do this is to start listing things that are known.To find the variables, look at the last sentence of the problem. Typically, it will ask how many __ and __… use whatever is in these two blanks as the x and y values. It usually does not matter which is which, but it is important to keep the two values straight and not mix them up.Then, list everything known about these variables. Usually, there will be a lower bound on each variable. If one is not given, it is probably 0. For example, factories cannot make -1 product.Usually there is some relationship between the products and limited resources like time and money. There may also be a relationship between the two products, such as the number of one product being greater than another or the total number of products being greater than or less than a certain number. Constraints are almost always inequalities.This will become clearer in context with the example problems. Identify the Objective Function The objective function is the function we want to maximize or minimize. It will depend on the two variables and, unlike the constraints, is a function, not an inequality.We will come back to the objective function, but, for now, it is important to just identify it. At this point, we need to graph the inequalities. Since it is easiest to graph functions in slope-intercept form, we may need to convert the inequalities to this before graphing.Remember that the constraints are connected by a mathematical “and,” meaning we need to shade the region where all of the inequalities are true. This usually creates a closed polygon, which we call “the feasible region.”That is, the area inside the polygon contains all possible solutions to the problem.Our goal, however, is not to find just any solution. We want to find the maximum or minimum value. That is, we want the best solution.Fortunately, the best solution will actually be one of the vertices of the polygon! We can use the graph and/or the equations of the bounds of the polygon to find these vertices. We can find the best solution plugging each of the x and y-values from the vertices into the objective function and analyzing the result. We then can pick the maximum or minimum output, depending on what we are looking for.We must also double check that the answer makes sense. For example, it does not make sense to create 0.5 products. If we get an answer that is a decimal or fraction and this does not make sense in context, we can analyze a nearby whole number point. We have to make sure that this point is still greater than/less than the other vertices before declaring it to be the maximum/minimum.This all may seem a bit confusing. Since linear programming problems are nearly always word problems, they make more sense when context is added. In this section, we will add context and practice problems relating to linear programming. This section also includes step-by-step solutions. Consider the geometric region shown in the graph. What are the inequalities that define this function? If the objective function is 3x+2y=P, what is the maximum value of P? If the objective function is 3x+2y=P, what is the minimum value of P Example 1 Solution This figure is bounded by three different lines. The easiest one to identify is the vertical line on the right side. This is the line x=5. Since the shaded region is to the left of this line, the inequality is x≤5.Next, let’s find the equation of the lower bound. This line crosses the y-axis at (0, 4). It also has a point at (2, 3). Therefore, its slope is (4-3/0-2)=-1/2. Therefore, the equation of the line is y=-1/2x+4. Since the shading is above this line, the inequality is y≥-1/2x+4.Now, let’s consider the upper bound. This line also crosses the y-axis at (0, 4). It has another point at (4, 3). Therefore, its slope is (3-4)/(4-0)=-1/4. Thus, its equation is y=-1/4x+4. Since the shaded region is below this line, the inequality is y≤–1/4x+4.In summary, our system of linear inequalities is x≤5 and y≥–1/2x+4 and y≤–1/4x+4. Now, we are given an objective function P=3x+2y to maximize. That is, we want to find values x and y in the shaded region so that we can maximize P. The key thing to note is that an extrema of the function P will be at the vertices of the shaded figure.The easiest way to find this is to test the vertices. There are ways to find this using matrices, but they will be covered in greater depth in later modules. They also work better for problems with significantly many more vertices. Since there are only three in this problem, this is not too complicated.We already know one of the vertices, the y-intercept, which is (0, 4). The other two are intersections of the two lines with x=5. Therefore, we just need to plug x=5 into both equations.We then get y=-1/2(5)+4=-5/2+4=1.5 and y=-1/4(5)+4=2.75. Thus, our other two vertices are (5, 1.5) and (5, 2.75).Now, we plug all three pairs of x and y-values into the objective function to get the following outputs.(0, 4): P=0+2(4)=8.(5, 1.5): P=3(5)+2(1.5)=18(5, 2.75): P=3(5)+2(2.75)=20.5.Therefore, the function P has a maximum at the point (5, 2.75). We actually did most of the work for part C in part B. Finding the minimum of a function is not very different than finding the maximum. We still find all of the vertices and then test all of them in the objective function. Now, however, we just select the output with the smallest value.Looking at part B, we see that this happens at the point (0, 4), with an output of 8. A company creates square boxes and triangular boxes. Square boxes take 2 minutes to make and sell for a profit of $4. Triangular boxes take 3 minutes to make and sell for a profit of $5. Their client wants at least 25 boxes and at least 5 of each type ready in one hour. What is the best combination of square and triangular boxes to make so that the company makes the most profit from this client? Example 2 Solution The first step in any word problem is defining what we know and what we want to find out. In this case, we know about the production of two different products which are dependent upon time. Each of these products also makes a profit. Our goal is to find the best combination of square and triangular boxes so that the company makes the most profit. First, let’s write down all of the inequalities we know. We can do this by considering the problem line by line.The first line tells us that we have two kinds of boxes, square ones and triangular ones. The second tells us some information about the square boxes, namely that they take two minutes to make and net $4 profit.At this point, we should define some variables. Let’s let x be the number of square boxes and y be the number of triangular boxes. These variables are both dependent upon each other because time spent making one is time that could be spent making the other. Make a note of this so that you do not mix them up.Now, we know that the amount of time spent making a square box is 2x.Now, we can do the same with the number of triangular boxes, y. We know that each triangular box requires 3 minutes and nets $5. Therefore, we can say that the amount of time spent making a triangular box is 3y.We also know that there is a limit on the total time, namely 60 minutes. Thus, we know that the time spent making both types of boxes must be less than 60, so we can define the inequality 2x+3y≤60.We also know that both x and y must be greater than or equal to 5 because the client has specified wanting at least 5 of each.Finally, we know that the client wants at least 25 boxes. This gives us another relationship between the number of square and triangular boxes, namely x+y≥25.Thus, overall, we have the following constraints:2x+3y≤60x≥5y≥5x+y≥25.These constraints function line the boundaries in the graphical region from example 1. The Objective Function Our objective, or goal, is to find the greatest profit. Therefore, our objective function should define the profit.In this case, profit depends on the number of square boxes created and the number of triangular boxes created. Specifically, this company’s profit is P=4x+5y.Note that this function is a line, not an inequality. In particular, it looks like a line written in standard form.Now, to maximize this function, we need to find the graphical region represented by our constraints. Then, we need to test the vertices of this region in the function P. Now, let’s consider the graph of this function. We can first graph each of our inequalities. Then, remembering that linear programming problem constraints are connected by a mathematical “and,” we will shade the region that is a solution to all four inequalities. This graph is shown below.This problem has three vertices. The first is the point (15, 10). The second is the point (20, 5). The third is the point (22.5, 5).Let’s plug all three values into the profit function and see what happens.(15, 10): P=4(15)+5(10)=60+50=110.(20, 5): P=4(20)+5(5)=105.(22.5, 5): P=4(22.5)+5(5)=90+25=115.This suggests that the maximum is 115 at 22.5 and 5. But, in context, this means that the company must make 22.5 square boxes. Since it cannot do that, we have to round down to the nearest whole number and see if this is still the maximum.At (22, 5), P=4(22)+5(5)=88+25=113.This is still greater than the other two outputs. Therefore, the company should make 22 square boxes and 5 triangular boxes to satisfy the client’s demands and maximize its own profit. A woman makes craft jewelry to sell at a seasonal craft show. She makes pins and earrings. Each pin takes her 1 hour to make and sells for a profit of $8. The pairs of earrings take 2 hours to make, but she gets a profit of $20. She likes to have variety, so she wants to have at least as many pins as pairs of earrings. She also knows that she has approximately 40 hours for creating jewelry between now and the start of the show. She also knows that the craft show vender wants sellers to have more than 20 items on display at the beginning of the show. Assuming she sells all of her inventory, how many each of pins and earring pairs should the woman make to maximize her profit? Example 3 Solution This problem is similar to the one above, but it has some additional constraints. We will solve it in the same way. Let’s begin by identifying the constraints. To do this, we should first define some variables. Let x be the number of pins the woman makes, and let y be the number of pairs of earrings she makes.We know that the woman has 40 hours to create the pins and earrings. Since they take 1 hour and 2 hours respectively, we can identify the constraint x+2y≤40.The woman also has constraints on the number of products she will make. Specifically, her vender wants her to have more than 20 items. Thus, we know that x+y>20. Since, however, she cannot make part of an earring on pin, we can adjust this inequality to x+y≥21.Finally, the woman has her own constraints on her products. She wants to have at least as many pins as pairs of earrings. This means that x≥y.In addition, we have to remember that we cannot have negative numbers of products. Therefore, x and y are both positive too.Thus, in summary, our constraints are:X+2y≤40X+y≥21x≥yx≥0y≥0. The Objective Function The woman wants to know how she can maximize her profits. We know that the pins give her a profit of $8, and earrings earn her $20. Since she expects to sell all of the jewelry she makes, the woman will make a profit of P=8x+20y. We want to find the maximum of this function. Now, we need to graph all of the constraints and then find the region where they all overlap. It helps to first put them all in slope-intercept form. In this case, then, we havey≤–1/2x+20y≥-x+21y≤xy≥0x≥0.This gives us the graph below.Unlike the previous two examples, this function has 4 vertices. We will have to identify and test all four of them.Note that these vertices are intersections of two lines. To find their intersection, we can set the two lines equal to each other and solve for x.We’ll move from left to right. The far left vertex is the intersection of the lines y=x and y=-x+21. Setting the two equal gives us:x=-x+21.2x=21.Therefore x=21/2, 0r 10.5 When x=10.5, the function y=x also is 10.5. Thus, the vertex is (10.5, 10.5).The next vertex is the intersection of the lines y=x and y=-1/2x+20. Setting these equal gives us:X=-1/2x+203/2x=20.Therefore, x=40/3, which is about 13.33. Since this is also on the line y=x, the point is (40/3, 40/3).The last two points lie on the x-axis. The first is the x-intercept of y=-x+21, which is the solution of 0=-x+21. This is the point (21, 0). The second is the x-intercept of y=-1/2x+20. That is the point where we have 0=-1/2x+20. This means that -20=-1/2x, or x=40. Thus, the intercept is (40, 0).Therefore, our four vertices are (10.5, 10.5), (40/3, 40/3), (21, 0), and (40, 0). Finding the Maximum Now, we test all four points in the function P=8x+20y.(10.5, 10.5)=294(40/3, 40/3)=1120/3 (or about 373.33)(0, 21)=168(0, 40)=320.Now, the maximum in this case is the point (40/3, 40/3). However, the woman cannot make 40/3 pins or 40/3 pairs of earrings. We can adjust by finding the nearest whole number coordinate that is inside the region and testing it. In this case, we have (13, 13) or (14, 13). We will choose the latter since it will obviously yield a larger profit.Then, we have:P=14(8)+13(20)=372.Thus, the woman should make 14 pins and 13 pairs of earrings for the greatest profit given her other constraints. Joshua is planning a bake sale to raise funds for his class field trip. He needs to make at least $100 to meet his goal, but it is okay if he goes above that. He plans to sell muffins and cookies by the dozen. The dozen muffins will sell for a profit of $6, and the dozen cookies will sell for a profit of $10. Based on sales from the previous year, he wants to make at least 8 more bags of cookies than bags of muffins.The cookies require 1 cup of sugar and 3/4 cups of flour per dozen. The muffins require 1/2 cup of sugar and 3/2 cups of flour per dozen. Joshua looks into his cabinet and finds that he has 13 cups of sugar and 11 cups of flour, but he does not plan to go get more from the store. He also knows that he can only bake one pan of a dozen muffins or one pan of a dozen cookies at a time. What is the fewest number of pans of muffins and cookies Joshua can make and still expect to meet his financial goals if he sells all of his product? Example 4 Solution As before, we will have to identify our variables, find our constraints, identify the objective function, graph the system of constraints, and then test the vertices in the objective function to find a solution. Joshua wants to know how the minimum number of pans of muffins and cookies to bake. Thus, let’s let x be the number of pans of muffins and y be the number of pans of cookies. Since each pan makes one dozen baked goods and Joshua sells the baked goods by the bag of one dozen, let’s ignore the number of individual muffins and cookies so as not to confuse ourselves. We can instead focus on the number of bags/pans.First, Joshua needs to make at least $100 to meet his goal. He earns $6 by selling a pan of muffins and $10 by selling a pan of cookies. Therefore, we have the constraint 6x+10y≥100.Joshua also has a limitation based on his flour and sugar supplies. He has 13 total cups of sugar, but a dozen muffins calls for 1/2 cup and a dozen cookies calls for 1 cup. Thus, he has the constraint 1/2x+1y≤13.Likewise, since a dozen muffins requires 3/2 cups of flour and a dozen cookies requires 3/4 cups of flour, we have the inequality 3/2x+3/4y≤11.Finally, Joshua cannot make fewer than 0 pans of either muffins or cookies. Thus, x and y are both greater than 0. He also wants to make at least 8 more pans of cookies than muffins. Therefore, we also have the inequality y-x≥10Therefore, our system of linear inequalities is:6x+10y≥1001/2x+y≤133/2x+3/4y≤11y-x≥8x≥0y≥0 The Objective Function Remember, the objective function is the function that defines the thing we want to minimize or maximize. In the previous two examples, we wanted to find the greatest profit. In this case, however, Joshua wants a minimum number of pans. Thus, we want to minimize the function P=x+y. In this case, we are finding the overlap of 6 different functions!Again, it is helpful to turn our constraint inequalities into y-intercept form so they are easier to graph. We get:y≥3/5x+10y≤–1/2x+13y≥x+8x≥0y≥0When we create the polygonal shaded region, we find that it has 5 vertices, as shown below. Now, we need to consider all 5 vertices and test them in the original function.We have two vertices on the y-axis, which come from the lines y=-3/5x+10 and y=-1/2x+13. Clearly, these two y-intercepts are (0, 10) and (0, 13).The next intersection, moving from left to right is the intersection of the lines y=-1/2x+13 and y=-2x+44/3. Setting these two functions equal gives us:–1/2x+13=-2x+44/3.Moving the x values to the left and the numbers without a coefficient to the right gives us3/2x=5/3.x=10/9.When x=10/9, we have y=-2(10/9)+44/3=-20/9+132/9=112/9, which has the decimal approximation 12.4. Thus, this is the point (10/9, 112/9) or about (1.1, 12.4).The next vertex is the intersection of the lines y=-3/5x+10 and y=x+8. Setting these equal, we have:–3/5x+10=x+8–8/5x=-2.Solving for x then gives us 5/4. At 5/4, the function y=x+8 is equal to 37/4, which is 9.25. Therefore, the point is (5/4, 37/4) or (1.25, 9.25) in decimal form.Finally, the last vertex is the intersection of y=x+8 and y=-2x+44/3. Setting these equal to find the x-value of the vertex, we have:X+8=-2x+44/3.Putting the x-values on the left and numbers without a coefficient on the right gives us3x=20/3.Thus, solving for x gives us 20/9 (which is about 2.2). When we plug this number back into the equation y=x+8, we get y=20/9+72/9=92/9. This is approximately 10.2. Therefore, the last vertex is at the point (20/9, 92/9), which is about (2.2, 10.2). Finding the Minimum Now, we want to find the minimum value of the objective function, P=x+y. That is, we want to find the fewest number of pans of muffins and cookies Joshua has to make while still satisfying all the other constraints.To do this, we have to test all five vertices: (0, 13), (0, 10), (10/9, 112/9), (5/4, 37/4), (20/9, 92/9)(0, 13): 0+13=13.(0, 10): 0+10=10.(10/9, 112/9): 10/9+112/9=112/9, which is about 13.5.(5/4, 37/4): 5/4+37/4, which is 42/4=10.5.(20/9, 92/9): 20/9+92/9=112/9. This is about 12.4.Therefore, it seems Joshua’s best bet is to make 0 muffins and 10 cookies. This probably makes the baking simple anyway!If, however, he wanted to make as many products as possible, (that is, if he wanted the maximum instead of the minimum), he would want to make 10/9 muffins and 112/9 cookies. This is not possible, so we would have to find the nearest whole number of cookies and muffins. The point (1, 12) is inside the shaded region, as is (0, 13). Either of these combinations would be the maximum. It is possible to have shaded regions with even more vertices. For example, if Joshua wanted a minimum number of bags of muffins or a maximum number of bags of cookies, we would have another constraint. If he wanted a minimum number of total bags of baked goods, we would have another constraint. Additionally, we could develop more constraints based on the number of ingredients. Things like eggs, butter, chocolate chips, or salt could work in this context. In some cases, a solution could become so complex so as not to have any feasible answers. For example, it is possible that the region not include any solutions where both x and y are whole numbers. Amy is a college student who works two jobs on campus. She must work for at least 5 hours per week at the library and two hours per week as a tutor, but she is not allowed to work more than 20 hours per week total. Amy gets $15 per hour at the library and $20 per hour at tutoring. She prefers working at the library though, so she wants to have at least as many library hours as tutoring hours. If Amy needs to make 360 dollars, what is the minimum number of hours she can work at each job this week to meet her goals and preferences? Example 5 Solution As with the other examples, we need to identify the constraints before we can plot our feasible region and test the vertices. Since Amy is wondering how many hours to work at each job, let’s let x bet the number of hours at the library and y the number of hours at tutoring.Then, we know x≥5 and y≥2.Her total number of hours, however, cannot be more than 20. Therefore, x+y≤20.Since she wants to have at least as many library hours as tutoring hours, she wants x≥y.Each hour at the library earns her $15, so she gets 15x. Likewise, from tutoring, she earns 20y. Thus, her total is 15x+20y, and she needs this to be more than 360. Therefore, 15x+20y≥360.In sum, then Amy’s constraints arex≥5y≥2x+y≤20x≥y15x+20y≥360 The Objective Function The total number of hours that Amy works is the function P=x+y. We want to find the minimum of this function inside the feasible region. The Feasible Region To graph the feasible region, we need to first convert all of the constraints to slope-intercept form. In this case, we have:x≥5y≥2y≤-x+20y≤xy≥-3/4x+18.This graph looks like the one below.Yes. This graph is blank because there is no overlap between all of these regions. This means that there is no solution. Perhaps Amy can persuade herself to get rid of the requirement that she work fewer hours at tutoring than at the library. What is the fewest number of hours she can work at tutoring and still meet her financial goals?Now, her constraints are just x≥5, y≥2, y≤-x+20, and y≥–3/4x+18.Then, we end up with this region.In this case, the objective function is just minimizing the number of hours Amy works at tutoring, namely Therefore, P=y, and we can see from looking at the region that the point (8, 12) has the lowest y-value. Therefore, if Amy wants to meet her financial goals but work as few hours as possible at tutoring, she has to work 12 hours at tutoring and 8 hours at the library.

To start, imagine that you acquire an n sample signal, and want to find its frequency spectrum. Evaluation by taking the discrete fourier transform dft of a coefficient vector interpolation by taking the inverse dft of pointvalue pairs, yielding a coefficient vector fast fourier transform fft can perform dft and inverse dft in time. The discrete fourier transform or dft is the transform that deals with a nite discretetime signal and a nite or discrete number of frequencies. Periodicdiscrete these are discrete signals that repeat themselves in a periodic fashion from negative to positive infinity. We will introduce a convenient shorthand notation xt. It is a linear invertible transformation between the timedomain representation of a function, which we shall denote by ht, and the frequency domain representation which we shall denote by hf. Moreover, fast algorithms exist that make it possible to compute the dft very e ciently. Ifthas dimension time then to make stdimensionless in the exponential e. The multidimensional transform of is defined to be. Discrete time fourier transform dtft vs discrete fourier. It is worth noting that the discrete time fourier transform is always 2. The dft takes a discrete signal in the time domain and transforms that signal into its discrete frequency domain representation. On the other hand, the discretetime fourier transform is a representation of a discretetime aperiodic sequence by a continuous periodic function, its fourier transform. Furthermore, we will show that the discretetime fourier transform can be used to represent a wide range of sequences, including sequences of in. On the other hand, the discrete time fourier transform is a representation of a discrete time aperiodic sequence by a continuous periodic function, its fourier transform. The discrete time fourier transform dtft is a form of fourier analysis that is applicable to the uniformlyspaced samples of a continuous function. Fourier transform is called the discrete time fourier transform. Inverse discrete fourier transform dft alejandro ribeiro february 5, 2019 suppose that we are given the discrete fourier transform dft x. The inverse fourier transform takes fz and, as we have just proved, reproduces ft. The discretetime pulses spectrum contains many ripples, the number of which increase with n, the pulses duration. Remembering the fact that we introduced a factor of i and including a factor of 2 that just crops up. Now heres the formula for the ztransform shown next to the discretetime fourier transform of xn. Therefore, zthe inverse fourier transform of is zthe inverse transform of is. Dec 04, 2019 in this post, we will encapsulate the differences between discrete fourier transform dft and discretetime fourier transform dtft. The dft is the most important discrete transform, used to perform fourier analysis in many practical applications. Define xnk, if n is a multiple of k, 0, otherwise xkn is a sloweddown version of xn with zeros interspersed. Chapter 4 the discrete fourier transform c bertrand delgutte and julie greenberg, 1999. Let be the continuous signal which is the source of the data. Fourier transform for continuoustime signals 2 frequency content of discretetime signals. The fourier transform of the original signal, would be. Digital signal processing dft introduction tutorialspoint. Fouriersequencetransformwolfram language documentation. In this note, we assume the overlapping is by 50% and we derive the. The discrete time pulses spectrum contains many ripples, the number of which increase with n, the pulses duration. Then a shift in time by n0 becomes a multiplication in the zdomain by ej. Understand the properties of time fourier discretetransform iii understand the relationship between time discretefourier transform and linear timeinvariant system. Many of the toolbox functions including z domain frequency response, spectrum and cepstrum analysis, and some filter design and. Secondly, a discretetime signal could arise from sampling a continuoustime. The discrete fourier transform dft is the equivalent of the continuous fourier transform for signals known only at instants separated by sample times. Jan 11, 2018 dtftdiscrete time fourier transform examples and solutions. We give an integral form for the inverse dtft that can be used even when. Can you explain the rather complicated appearance of the phase. Understanding the discrete fourier transform dtft dft and sampling theory. A table of some of the most important properties is provided at the end of these. Thus we have replaced a function of time with a spectrum in frequency. Discretetime fourier series have properties very similar to the linearity, time shifting, etc. X x1 n1 xne j n inverse discrete time fourier transform. The discrete fourier transform, or dft, is the primary tool of digital signal processing. The fourier transform ft decomposes a function of time a signal into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies or pitches of its constituent notes the discrete fourier transform dft converts a finite sequence of equallyspaced samples of a function into a samelength sequence of equally. Periodicity this property has already been considered and it can be written as follows. The dtft is a transformation that maps discretetime dt signal xn into a complex valued function of the real variable w, namely. The relationship between the dtft of a periodic signal and the dtfs of a periodic signal composed from it leads us to the idea of a discrete fourier transform not to. Dct vs dft for compression, we work with sampled data in a finite time window. In this section we formulate some properties of the discrete time fourier transform. Suppose that we are given the discrete fourier transform dft x. The discrete fourier transform 1 introduction the discrete fourier transform dft is a fundamental transform in digital signal processing, with applications in frequency analysis, fast convolution, image processing, etc. Conditions for the existence of the fourier transform are complicated to state in general, but it is sufficient for to be absolutely integrable, i. Shorttime fourier transform and its inverse ivan w. None of the standard fourier transform property laws seem to directly apply to this. This class of fourier transform is sometimes called the discrete fourier series, but is most often called the discrete fourier transform. The discrete fourier transform dft is the equivalent of the continuous fourier transform for signals known only at instants separated by sample times i. Also, as we discuss, a strong duality exists between the continuoustime fourier series and the discretetime fourier transform. The term discrete time refers to the fact that the transform operates on discrete data samples whose interval often has units of time. Our first task is to develop examples of the dtft for some common signals. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval often defined by. Discrete fourier transform dft is used for analyzing discretetime finiteduration signals in the frequency domain let be a finiteduration sequence of length such that outside. Discretetime fourier transform solutions s115 for discretetime signals can be developed. Recall that for a general aperiodic signal xn, the dtft and its inverse is. This should look familiar given what you know about fourier analysis. X x1 n1 xne j n inverse discretetime fourier transform. Continuous time fourier transform of xt is defined as x. So far we have seen that time domain signals can be transformed to frequency domain by the so called fourier transform. Define xnk, if n is a multiple of k, 0, otherwise xkn is a sloweddown version of. The inverse discrete time fourier transform is easily derived from the following relationship. The inverse fourier transform the fourier transform takes us from ft to f. Periodic discrete these are discrete signals that repeat themselves in a periodic fashion from negative to positive infinity. The plancherel identity suggests that the fourier transform is a onetoone norm preserving map of the hilbert space l21. Summary of the dtft the discretetime fourier transform dtft gives us a way of representing frequency content of discretetime signals. The rst equation gives the discrete fourier transform dft of the sequence fu jg. In this post, we will encapsulate the differences between discrete fourier transform dft and discretetime fourier transform dtft. Fourierstyle transforms imply the function is periodic and. The relationship between the dtft of a periodic signal and the dtfs of a periodic signal composed from it leads us to the idea of a discrete fourier transform not to be confused with discrete time fourier transform. Also, as we discuss, a strong duality exists between the continuous time fourier series and the discrete time fourier transform. The discrete fourier transform dft an alternative to using the approximation to the fourier transform is to use the discrete fourier transform dft. The discrete time fourier transform dtft is the member of the fourier transform family that operates on aperiodic, discrete signals. Like continuous time signal fourier transform, discrete time fourier transform can be used to represent a discrete sequence into its equivalent frequency domain representation and lti discrete time system and develop various computational algorithms. Dtftdiscrete time fourier transform examples and solutions. Discrete time fourier transform dtft the discrete time fourier transform dtft can be viewed as the limiting form of the dft when its length is allowed to approach infinity. If we interpret t as the time, then z is the angular frequency. Chapter 1 the fourier transform university of minnesota. The inverse discretetime fourier transform is easily derived from the following relationship. Fouriersequencetransform is also known as discretetime fourier transform dtft. Selesnick april 14, 2009 1 introduction the shorttime fourier transform stft of a signal consists of the fourier transform of overlapping windowed blocks of the signal. Inverse discrete fourier transform dft alejandro ribeiro february 5, 2019. Discretetime fourier transform dtft chapter intended learning outcomes. Discrete time fourier transform dtft mathematics of the dft. Lecture notes for thefourier transform and applications. The foundation of the product is the fast fourier transform fft, a method for computing the dft with reduced execution time. Fourier transform ft and inverse mathematics of the dft. Discrete time fourier transform dtft mathematics of. Discrete time fourier transform solutions s115 for discrete time signals can be developed. The discretetime fourier transform of a discrete set of real or complex numbers xn, for all integers n, is a fourier series, which produces a periodic function of a frequency variable.53 528 767 683 1060 357 81 1024 81 261 371 609 873 202 1152 482 291 1233 1421 1245 1279 783 59 1358 162 1093 406 1424 360 400 836 19 639 977 1328 496 1418 370 544 1326 1370 697

/ˈmæɡ·nəˌtud/ large size or great importance: The magnitude of the task would have discouraged an ordinary man. earth science. Magnitude is also a measure of the brightness of a star as it appears from earth. What is magnitude with example in physics? The term magnitude is defined as “how much of a quantity”. For instance, the magnitude can be used for explaining the comparison between the speeds of a car and a bicycle. It can also be used to explain the distance travelled by an object or to explain the amount of an object in terms of its magnitude. What does magnitude mean in science? The magnitude is a number that characterizes the relative size of an earthquake. Magnitude is based on measurement of the maximum motion recorded by a seismograph. What is magnitude simple words? In physics, magnitude is defined simply as “distance or quantity.” It depicts the absolute or relative direction or size in which an object moves in the sense of motion. It is used to express the size or scope of something. In physics, magnitude generally refers to distance or quantity. Does magnitude mean force? The magnitude of the force is defined as the sum of all the forces acting on an object. Calculating magnitudes for forces is a vital measurement of physics. The ‘magnitude’ of a force is its ‘size’ or ‘strength’, in spite of the path in which it acts. What is another word for magnitude in physics? In this page you can discover 44 synonyms, antonyms, idiomatic expressions, and related words for magnitude, like: size, quantity, breadth, importance, eminence, bigness, degree, unimportance, extent, velocity and dimension. Is magnitude a distance? The apparent magnitude of a celestial object, such as a star or galaxy, is the brightness measured by an observer at a specific distance from the object. The smaller the distance between the observer and object, the greater the apparent brightness. What is a magnitude of a vector? The magnitude of a vector is the length of the vector. The magnitude of the vector a is denoted as ∥a∥. See the introduction to vectors for more about the magnitude of a vector. Is magnitude a speed? The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the velocity vector is directed in the same direction that the object moves. Is magnitude the same as mass? Hi physics lover, Do you know magnitude is a pure number that defines the size(How Much) of a physical quantity! For example, if your mass is 60 kg, then 60 is the magnitude of the mass. And kg is the unit of mass. What is magnitude and unit? Units of measure are scalar quantities, and magnitude is defined in terms of scalar multiplication. The magnitude of a quantity in a given unit times that unit is equal to the original quantity. This holds for all kinds of tensors, including real-numbers and vectors. Is magnitude a vector or scalar? Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude. When comparing two vector quantities of the same type, you have to compare both the magnitude and the direction. For scalars, you only have to compare the magnitude. How do you determine magnitude? the formula to determine the magnitude of a vector (in two dimensional space) v = (x, y) is: |v| =√(x2 + y2). This formula is derived from the Pythagorean theorem. the formula to determine the magnitude of a vector (in three dimensional space) V = (x, y, z) is: |V| = √(x2 + y2 + z2) How do we calculate magnitude in physics? What is magnitude and direction in physics? A vector contains two types of information: a magnitude and a direction. The magnitude is the length of the vector while the direction tells us which way the vector points. Vector direction can be given in various forms, but is most commonly denoted in degrees. Acceleration and velocity are examples of vectors. What is magnetic magnitude? The magnitude of the force is F = qvB sinθ where θ is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. What is magnitude in force and pressure? It means size of the force. It is sum of all forces acting on a body. If 2 forces act in same direction, Magnitude of force increases. It is the sum of of both forces. Is magnitude the same as momentum? There is no difference, they are exactly the same things. Since momentum is a vector, it has no magnitude. What is magnitude measured in? Magnitude is expressed in whole numbers and decimal fractions. For example, a magnitude 5.3 is a moderate earthquake, and a 6.3 is a strong earthquake. Because of the logarithmic basis of the scale, each whole number increase in magnitude represents a tenfold increase in measured amplitude as measured on a seismogram. Can a magnitude be negative? Answer: Magnitude cannot be negative. It is the length of the vector which does not have a direction (positive or negative). In the formula, the values inside the summation are squared, which makes them positive. Why do we use magnitude? Magnitude is used in stating the size or extent of something such as a star, earthquake, or explosion. What is the answer of magnitude? magnitude means greatness of size or extent. Is magnitude absolute value? A magnitude is the measurement or absolute value of a quantity. A magnitude is represented by a positive real number. Is magnitude a displacement? By magnitude, we mean the size of the displacement without regard to its direction (i.e., just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 meters during a lecture, yet still end up only two meters to the right of her starting point. What is the difference between magnitude and displacement? MAGNITUDE. ❇ It is the numerical value of any physical quantity with a proper unit. It doesn’t tell about the direction. ❇ Displacement is the shortest distance travelled between reference point to final reaching point.

Presentation on theme: "1 Decisions about Cars. 2 New or Used Of course new cars are nice. They have the latest gadgets in the car world. They have a distinct smell. The floor."— Presentation transcript: 2 New or Used Of course new cars are nice. They have the latest gadgets in the car world. They have a distinct smell. The floor doesnt have very much stuff on it. Financial items to consider on any car include taxes, title and insurance. The miles per gallon the car will travel also will help you see the advantages in terms of gas expenses. Newer cars typically get better mpgs because of government regulation. Another cost to consider on the purchase of any asset, but here in the context of the car, is depreciation. Here we mean the lose of value in the car due to driving it. Used cars will typically depreciate less because the most depreciation occurs in the first year or so. What I really mean is depreciation occurs fastest the first year and then the depreciation slows down. 3 The Odometer On a used car check the odometer. If it seems low relative to the way the car looks, maybe you should pass on the car. By law, odometers are not supposed to be messed with. Title Be sure the seller has the title to the car and is the rightful owner. In the USA if you buy a car from a person who does not legally have title, the car could be return to the original owner. You then have to get your money back from the crook. 4 Budget Say you have determined you can afford to pay 375 a month on a car loan. How much car can you buy? Lets assume you have no down payment, you are looking at a 5 year loan with a nominal rate of 6.9%. Car loans are compounded monthly, although I do not think this is made clear to the consumer. In Excel we can find the present value of the uniform series, or annuity, we will pay by the following =PV(0.069/12,5*12,-375) = PV(interest rate, time frame, annuity) = 18,983 Now, if just the interest rate is higher, you can afford less car. If just the time frame is higher you can afford more car. If the amount you can pay a month is higher you can afford more car. 5 As a consumer, when you walk into a car dealer and the sales person asks you how much can you pay each month, should you lie? Back to the new or used decision. Once last detail about any car is there is a possibility you will buy a lemon. If it is new you can get the dealer to work with you because of manufacturer problems. Take a used car to a mechanic before you buy to have them check it out. Some folks think that used cars sold by private citizens sell for less than at dealers because the owner has to offer a discount as an insurance policy against the car being a lemon. The buyer then takes the car as is. We can not say here if it is better to take used or new. Some folks are willing to make trade-offs others arent. So there is no iron clad decision rule. 6 Lease or Buy? Lease ideas to consider: Closed-end lease means you walk away with no obligation at the end of the lease period, unless you abused car or went over preset mileage limits. Open-end lease means if the value of the car at the end of the lease is less than the estimated value, then you pay the difference. On a lease you may be asked to make a down payment and a security deposit. 8 Digress F2 F3 A A A A A The first diagram on the left is the one we have become accustomed to. We have an A value at the end of each of two periods and the F value occurs at the end of the second period. 9 Say A = 1 and i = 10, then F2 = 1[appen b page 691 column 10% row 2 value] = 1[2.100] = 2.10 On the previous screen the graph on the right would have F3 = 2.10 + the A at time zero taken as a single payment to time 2 in a single payment. In other words F3 = 2.10 + 1[appen a page 690 column 10% row 2 value] = 2.10 + 1[1.21]= 3.31 = 1[3.31] So F3 has only two time periods but three As. So long as the last A occurs at the same time as F we have a new story. Look at appendix b page 691 10% column row 3 value. We have 3.31. WOW, what does this mean? 10 F3 A A A This means if we have an A at time zero, then we can just imagine we had a problem that started one period before. 11 Now that we have more formally introduced compounding a section or two back, lets consider a case where payments are made more often than compounding occurs. Lets consider a case where payments are made quarterly, but compounding is semi-annual. Say we have a two year deal at 10 percent nominal interest. 0 1 2 3 4 5 6 7 8 a a a a a a a a 12 To find the future value of the annuity we need to first recognize that the interest is only compounded every other quarter. If we take the as in the 2 nd, 4 th, 6 th, and 8 th periods we have F = a [value in appendix B page 691 column 5% row 4]. = a [4.310] Now, when we look at the as in the 1 st, 3 rd, 5 th and 7 th quarters we could find the F at the 7 th period as F = a[4.310]. Since we need a half year for the interest to be earned the F at the end of the 7 th period does not have time to earn interest by the end of the 8 th quarter. If we want to know the total value at the end of the 8 th period we simply move the F at the end of the 7 th period over to the 8 th period and add it to the other value. 13 We have F = a[4.310] + a[4.310] = 2a[4.310] Do you see the significance of this result? If funds are deposited more often than the compounding period, add the values up to the end of the compounding period. Thus if a is made quarterly and interest is compounded semi-annually, just assume 2a is made semi-annually. In fact, the only time dollar values should be moved across time without making an interest adjustment is when the money does not have time to earn interest. Now, back to our story about comparing a lease to buying a car. 14 Say with a lease you have At time 0 a $1500 down payment and a $300 security deposit. Then over the next 36 months a $300 monthly payment will be made. Say if you buy you have At time 0 a $2500 down payment and a 5% sales tax payment on a $15000 car of $750. Then over the next 36 months you have a payment of 392 (financing 12500 at 8% nominal over 3 years) At the end of three years the car is still worth $8000. 15 The authors say in order to compare the two first do this for the lease: 1500 down payment + 300 month times 36 months = 10800 + (1500 down pay + 300 security dep)times 3 years time.04 interest earned on savings ( an opportunity cost calculation) = 216 For a grand total of $12,516. This is the cost of the lease. For the car the authors say: 2500 down payment + 750 sales tax + 392 a month for 36 months = 14112 + Opportunity cost of down payment 2500times 3 times.04 = 300 – 8000 in car value at end of loan, for a total cost of the car being 9,662. So the authors say buy the car. 16 I say buy the car, but for different reasons. The authors, I believe, violated a rule of finance. They added values across time without adjusting for interest. You can only when there is not enough time for interest to accrue. They added apples and oranges. You can only do this when you want to make a fruit salad ! They added values at time 0 to values at time n to values each period. This is very bad. What they should have done. Pick a time frame – either the present at time 0, the annuity time frame, or the end of the story. Lets do an end of the story at the end of the 36 months. 17 The lease would be 1800 in a single payment using 4% interest compounded annually = 1800[append a page 690 row 3 in between 3 and 5%] =1800[1.124 this is a guesstimate] = 1800[1.124864 from excel] = 2024.76 from excel + 300 times F/A factor at.08/12 for 36 months = 12,160.67 from Excel - 300 back from the down payment = 2024.76 + 12,160.67 – 300 = 13885.43 18 Note on the lease I used an opportunity cost value of the 1800. Opportunity cost means what do I give up when I make a payment. The down payment was not required and the security deposit will be given back, so what does it cost to give up these values. I took the 300 monthly and used the same rate that the car loan will occur at because I want to compare to the car loan. I subtracted out 300 at the end because the security deposit is given back at time 36. 19 The car would be [2500 down payment + 750 tax] [1.124864 from excel] = 3655.81 + 392 monthly payment times F/A factor at.08/12 for 36 months = $15,889.94 - 8000 value of car at time 36. =11545.75 So, the car is the better deal. 20 Note, the emphasis on the problem we just did was the future. Most folks use the present as the emphasis. When we consider costs, the option that is chosen is the one with the LOWEST NET PRESENT COST. The lease would be Net present cost = 1800 + 300PV(.08/12,3*12,,-300) – 300/power(1.04,3) =1800 + $9,573.54 - 266.698908 = 11106.84 The car would be Net present cost = 15000 + 750 – 8000/power(1.04,3) = 8638.02913

We consider two important and widely studied problems in glaciology that involve contact. The first is that of the grounding line of an ice sheet flowing from the continent and into the sea, where the ice floats and loses contact with the bedrock. The stability of the position of this grounding line was first questioned by Weertman in 1974 and since then numerous analyses have attempted to prove or disprove the possibility of an instability, see e.g. [28, 38, 24]. The second problem is that of subglacial cavitation, where the ice detaches from the bedrock along the lee side of an obstacle. Subglacial cavitation occurs at a much smaller scale, along the interface between the ice and the bedrock, and is usually formulated as a boundary layer problem. Lliboutry first proposed the possibility of cavities forming between the ice and the bedrock in 1968 . Since then, subglacial cavitation has been recognised as a fundamental mechanism in glacial sliding, attracting the attention of both theoretical [17, 37] and experimental studies. A precise understanding of both grounding line dynamics and subglacial cavitation is of great relevance for predicting future sea level rise and therefore comprehending large scale climate dynamics [36, 20]. The two contact problems described above are modelled by coupling a Stokes problem for the ice flow with a time-dependent advection equation for the free surface. At each instant in time, a Stokes problem must be solved with contact boundary conditions that allow the detachment of the ice from the bed. These contact conditions transform the instantaneous Stokes problem into a variational inequality. Numerous finite element simulations of these equations have been carried out, see e.g. [19, 13, 16, 27]. However, to the best of our knowledge, no formal analysis of this problem and its approximation exists in the mathematical literature. Moreover, we believe that the discretisations used in these computations can be improved upon, by exploiting the structure of the variational inequality. Although the Stokes variational inequality is superficially similar to the elastic contact problem, which has been widely studied [34, 26], the Stokes problem includes three substantial difficulties that must be addressed carefully: the presence of rigid body modes in the space of admissible velocities, the nonlinear rheological law used to model ice as a viscous fluid, and the nonlinearity of the boundary condition used for the sliding law when modelling the grounding line problem. In this work we analyse the instantaneous Stokes variational inequality and its approximation and focus on the first two difficulties due to rigid body modes and the nonlinear rheology. The presence of rigid body modes renders this problem semicoercive. Although semicoercive variational inequalities have been studied in the past [25, 40, 2], existing analyses use purely indirect arguments which give very limited information on the effects on the discretisation of the finite element spaces used. Here, we present a novel approach based on the use of metric projections onto closed convex cones to obtain constructive proofs for the discretisation errors. On the other hand, the nonlinear rheology complicates the estimation of errors for the discrete problem. Here, we use the techniques from [5, 30] to establish a convergence analysis. We propose a mixed formulation of the Stokes variational inequality where a Lagrange multiplier is used to enforce the contact conditions. This formulation permits a structure-preserving discretisation that explicitly enforces a discrete version of the contact conditions, up to rounding errors. This allows for a precise distinction between regions where the ice detaches from the bed and those where it remains attached. This precision is extremely useful when coupling the Stokes variational inequality with the time-dependent advection equation for the free surface. 1.1 Outline of the paper In Section 2, the Stokes variational inequality and its mixed formulation are presented. We prove a Korn-type inequality involving a metric projection onto a cone of rigid modes that will be used throughout the analysis, and we demonstrate that the mixed formulation is well posed. In Section 3, we analyse a family of finite element approximations of the mixed problem and present error estimates in terms of best approximation results for the velocity, pressure and Lagrange multiplier. Finally, in Section 4, a concrete finite element scheme involving quadratic elements for the velocity and piecewise constant elements for the pressure and the Lagrange multiplier is introduced. We then present error estimates for this scheme and provide numerical results for two test cases. For the first test, we solve a problem with a manufactured solution to calculate convergence rates and compare these with our estimates. For the second test, we compute the evolution of a subglacial cavity to exhibit the benefits of using a mixed formulation in applications of glaciological interest. Given two normed vector spacesand and a bounded linear operator , the dual of is denoted by and the dual operator to by . The range of is denoted by and its kernel by . The norm in is denoted by and the pairing between elements in the primal and dual spaces by for and . We will work with the Lebesgue and Sobolev spaces , where and , defined as the set of functions with weak derivatives up to order which are -integrable. When we write . The space of polynomials of degree over a simplex (interval, triangle, tetrahedron) is denoted by . The space of continuous functions over a domain is given by . Vector functions and vector function spaces will be denoted with bold symbols, e.g. and . 2 Formulation of the problem In this section we introduce the semicoercive variational inequality that arises in glaciology and present its formulation as a mixed problem with a Lagrange multiplier. An auxiliary analytical result concerning a metric projection is then proved. Finally, we analyse the existence and uniqueness of solutions for the mixed problem. 2.1 A model for ice flow We denote by a bounded, connected and polygonal domain which represents the glacier. The assumptions of the domain being two-dimensional and polygonal are made in order to simplify the analysis, but we expect the essential results presented here to extend to three dimensional domains with smooth enough boundaries. Ice is generally modelled as a viscous incompressible flow whose motion is described by the Stokes equation : equationparentequation In the equations above, represents the ice velocity, the pressure and is a prescribed body force due to gravitational forces. The tensoris the symmetric part of the velocity gradient, that is, The coefficient is the effective viscosity of ice, which relates the stress and strain rates. A power law, usually called Glen’s law , is the most common choice of rheological law for ice: Here, represents the Frobenius norm of a matrix: for with components we have . The field is a prescribed function for which . The parameter is constant and is usually set to ; for we recover the standard linear Stokes flow. From now on, we simply write where is in and satisfies a.e. on for some . Moreover, is in for . This expression for reveals the -Stokes nature of the problem when considered as a variational problem in the setting of Sobolev spaces. 2.2 Boundary conditions For a given velocity and pressure field, we define the stress tensor by where is the identity tensor field. Let denote the unit outward-pointing normal vector to the boundary . We define the normal and tangential stresses at the boundary as The boundary is partitioned into three disjoint open sets , and . The subset represents the part of the boundary in contact with the atmosphere (and the ocean or a water-filled cavity in the case of a marine ice sheet and a subglacial cavity, respectively). Here we enforce where represents a prescribed surface traction force. On we enforce the contact conditions which allow the ice to detach from but not penetrate the bedrock. In particular, detachment can occur if the normal stress equals the subglacial water pressure, which is defined everywhere along a thin lubrication layer in between the ice and the bedrock. In order to write these in a simplified form, we assume that the water pressure along the bed is constant and we measure stresses relative to that water pressure. We also assume that the ice can slide freely and impose no tangential stresses. Then, the boundary conditions at are given by equationparentequation Finally, represents a portion of the boundary on which the ice is frozen and hence we prescribe no slip boundary conditions: As explained in Section 1, one of the challenges we wish to address with this work is the case when rigid body modes are present in the space of admissible velocities. In these cases, the problem becomes semicoercive instead of coercive and the structure of the problem changes significantly. This occurs whenever is empty. For this reason, we assume that can be empty and require the subsets and to have positive measures. 2.3 The mixed formulation We now present the mixed formulation whose analysis and approximation is the focus of this work. To do so, we first write (1) with boundary conditions (4)-(6) as a variational inequality. Then, we introduce the mixed formulation by defining a Lagrange multiplier which enforces a constraint that arises due to the contact boundary conditions (5a). In Appendix A we specify and prove the sense in which these different formulations are equivalent. We denote by the normal trace operator onto . This operator is built by extending to the operator on , defined on smooth functions. The closed convex subset of is then defined by We also introduce the operators and defined by Moreover, the action of the applied body and surface forces on the domain is expressed via the function , defined as In the mixed formulation, the constraint on is enforced via a Lagrange multiplier. We denote the range of by and equip this space with the norm. We assume the geometry of and to be sufficiently regular for this space to be a Banach space, see [34, Section 5], [26, Chapter III] and [1, Chapter 7] for discussions on normal traces and trace spaces. The Lagrange multiplier is sought in the convex cone of multipliers The equivalent mixed formulation of (9) is: find such that equationparentequation 2.4 A metric projection onto the cone of rigid body modes In this section we present a projection operator onto the cone of rigid body modes inside and prove a Korn-type inequality involving this operator. This result, which we believe to be novel, allows us to prove the well-posedness of the continuous and discrete problems and to obtain estimates for the velocity error in the -norm. We define the space of rigid body modes inside by We also introduce the subspace of rigid body modes inside , which we denote by . Note that and for all . In fact, it can be shown that , see [34, Lemma 6.1], and therefore . The fact that coincides with the kernel of complicates the construction of error estimates in the -norm. Our solution is to make use of the metric projection onto the closed convex cone , which we shall denote by . This metric projection assigns to each function a rigid body mode for which In Appendix B we explain that metric projections are well-defined on uniformly convex Banach spaces and that the range of is the closed convex cone , see (58) for the definition of a polar cone. Since the Sobolev space is uniformly convex for , the operator has a closed convex range. This property is exploited below to prove Theorem 1. We first prove a preliminary Korn inequality on closed convex sets, Lemma 1. For this preliminary result, we need the following generalised Korn inequality from : for a bounded and Lipschitz domain there is a constant such that Let be a bounded and Lipschitz domain and a closed convex subset which satisfies . Then, there is a constant such that By using (12), it suffices to show that Assume by contradiction that (13) does not hold. Then, there is a sequence in such that and as . By (12), we see that is bounded in and therefore we may extract a subsequence, also denoted , which converges weakly to a in and strongly in . Since is closed and convex, it is also weakly closed by [7, Theorem 3.7], so . Moreover, by the lower semicontinuity of , we also have that and . However, by construction , a contradiction. Let denote the metric projection onto the closed convex cone . There is a such that 2.5 Well posedness of the mixed formulation Questions on the existence and uniqueness of solutions of the mixed system (10) can be answered by studying an equivalent minimisation problem. This equivalence depends on the so-called inf-sup property holding for the operators and . Let These inf-sup conditions can be stated as Condition (15) is proved in [32, Lemma 3.2.7] and (16) follows from the inverse mapping theorem because is surjective onto the closed Banach space . We also define the space of divergence-free functions and the convex set as Then, (10) is equivalent to the minimisation of the functional over , see Appendix A. If , which in our setting occurs when , then the Korn inequality in [9, Lemma 3] implies that is a norm on equivalent to . Since it follows that the problem is coercive. However, whenever the set of rigid body modes is not the zero set, we then say that the operator is semicoercive with respect to the seminorm because the bound (18) does not hold for . Below, in Theorem 2, we show that a consequence of the semicoercivity of is that (10) will have a unique solution only when the following compatibility condition holds: Condition (19) will not only allow us to establish the well-posedness of (10), but will also be required for proving velocity error estimates in the -norm. This is possible because the map from is a continuous map defined over a compact set. Therefore, whenever (19) holds, the inequality follows with the constant defined as The importance of the compatibility condition (19) is well-known in the study of semicoercive variational inequalities, see [31, 40, 34] in the context of general variational inequalities and [39, 9] in a glaciological setting. If , then or tend to infinity; as a result, and the functional is coercive. Uniqueness of the solution follows from the strong convexity of the functional over , see the proof for [9, Theorem 1]. Regarding the necessity of (19) for the existence and uniqueness of solutions to (10) when , assume by contradiction that is a unique solution and that there is a rigid mode for which . By testing with in (10a), we find that and therefore we must have and . It is then straightforward to see that is also a solution to (10), contradicting our initial assumption. We can bound the norm of the pressure by using (22), the equality The compatibility condition (19) becomes redundant whenever the portion of the boundary with no slip boundary conditions has a positive measure, i.e. . On the other hand, (19) cannot be satisfied if there exist non zero rigid body modes tangential to . In this case, if solves (10), then for any , with on , the triple will also solve (10). 3 Abstract discretisation In this section we propose an abstract discretisation of the mixed system (10) built in terms of a collection of finite dimensional spaces satisfying certain key properties. We can then introduce a discrete system analogous to (10) and investigate the conditions under which we have a unique solution. Then, we prove Lemmas 2, 3, and 4, which establish upper bounds for the errors of the discrete solutions. 3.1 The discrete mixed formulation For each parameter , let , and be finite dimensional subspaces. We assume that to avoid the need of introducing discrete compatibility conditions. We define the discrete cone and the discrete convex set An immediate consequence of the definitions of and is that but unless . Additionally, we have that thanks to the assumption . The discrete analogue of the variational inequality (9) is: find such that This discrete variational inequality can be written as a mixed problem by introducing a Lagrange multiplier. This results in the discrete mixed formulation that is the counterpart of (10): find such that equationparentequation An advantage of using a mixed formulation at the discrete level is that we explicitly enforce a discrete version of the contact conditions (5a). Just as in (11), it is possible to show that the conditions and (24c) are equivalent to In order to state a minimisation problem equivalent to (24), we must introduce the subspace of of discretely divergence-free functions and the discrete convex set : Then, the discrete mixed problem (24) is equivalent to the minimisation over of the functional defined in (17), provided that two discrete inf-sup conditions hold. For , these discrete conditions can be stated as When the conditions (26) and (27) hold, then (23), (24) and the minimisation of over are equivalent problems. The proofs for such equivalences require the same arguments as the proofs presented in Appendix A. If we additionally assume to be coercive over , then admits a unique minimiser over and the discrete mixed formulation is well-posed. For these reasons, the discrete inf-sup conditions guarantee a unique solution for (24) and set constraints on the choice of spaces , and used when approximating solutions of (10). Analogously to the continuous case, the coercivity of over hinges on the compatibility condition (19). Assume that (26) and (27) hold. Whenever , the discrete mixed problem (23) has a unique solution. On the other hand, if , then (24) admits a unique solution if and only if the compatibility condition (19) holds. Additionally, if (19) holds when applicable, any discrete solution to (24) is uniformly bounded from above, that is, where the constant depends on . The converse statement and the bound (28) again follow by taking the same steps as in the continuous case. These steps can be taken due to our definitions of and and the assumption . 3.2 Upper bounds for the velocity error An important tool presented in [5, 30] for establishing error estimates for non-Newtonian flows is the use of the function111This operator is defined differently in [5, 30]. In these references, the term is denoted by . defined by This operator is closely related to the operator . The inequalities hold uniformly for all for two constants . The following variation of Young’s inequality, is valid for any and , with the constant depending on . Additionally, the seminorm is connected to via the inequality which holds for any . See [30, Lemmas 2.3, 2.4] for proofs of (30) and (32), and [5, Lemma 2.7] for (31). It is also worth mentioning that the quasi-norm presented in can be written in terms of , see [5, Remark 2.6]. We next present what we believe is a novel approach for bounding the velocity error in the -norm which uses the ideas presented in Section 2.4. By Theorem 1 and (32), the velocity error can be decomposed into two components as where the constant depends on . For the first term on the right of (33), which represents the rigid component of the error, we present the following result: Set and test equation (10a) with . Reordering, we find that The constant depends on the norms of the solution and these are bounded from above by the norm of , see (21). As mentioned in the introduction, previous analyses of finite element approximations of semicoercive variational inequalities either only consider the error in a seminorm or use indirect arguments to prove the convergence of the approximate solution in the complete norm [42, 25, 40, 2, 8]. In these cases, arguments by contradiction involving a sequence of triangulations are used. In Lemma 2, on the other hand, we provide a fully constructive proof for bounding the rigid component of the velocity error from above. This result is a key ingredient in obtaining the error estimates for the finite element scheme presented in the next section. If the pair is divergence free in the sense that for all implies that , then the term in inequality (35) can be removed. 3.3 Upper bounds for the pressure and Lagrange multiplier errors We finalise the analysis of the abstract discretisation by bounding the errors for the pressure and the Lagrange multiplier from above. From the inf-sup condition (26) it follows that

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Jan 20, 2021 Calculating Crushed Stone by Weight. A final calculation is if you need to figure out the weight in tons of your crushed stone. You might not need to figure this out, but its handy to know. Most gravel weighs about 1.5 tons per cubic yard. So, for the examples above, 2.45 cubic yards of gravel weighs 3.68 tons or 7,350 pounds. Apr 09, 2020 Sometimes crushed stone is sold by the ton.To figure out the how many tons you will need is not hard to do. You will have to know that the standard weight contractors use for crushed stone is 2700 pounds per cubic yard. Multiply the 1.5 Tonne. How many cubic meters are in a ton of gravel? Given a weight of 32 tons and a density of between 1 and 1.4 ton/m3 , and doing the arithmetic, you get an answer of approximately 23 to 32 cubic meters, depending on the composition of the gravel. The result is 2,835 pounds per cubic yard of gravel. Thereof, how many tons is a yard of stone dust? Most gravel and crushed stone products have similar weights per ton. A general rule of thumb when converting cubic yards of gravel to tons is to multiply the cubic area by 1.4. For your reference, gravel typically weighs 2,800 pounds per cubic yard. Aug 15, 2018 By the ton, the costs of crushed limestone will vary anywhere from 20 to as much as 30. 1.5 tons can cover one cubic yard. Youngs Sand and Gravel, a landscape supply company located in Ohio, charges 20 a ton for all limestone, According to imperial or US customary measurement system, a cubic yard of crushed limestone can weighs around 2600 pounds or 1.3 short tons, in this regard, How many tons of crushed limestone are in a cubic yard, so there are 1.3 short tons are in a cubic yard. 1 litre paint coverage in square meter square feet. A cubic yard of sand equal as 2700 pounds and A short tons weighs around 2000 lbs, so number of tons, 2700/ 2000 1.35 short tons, in this regard, how many tons per cubic yard of sand, so, generally there are 1.35 short tons per cubic yard of sand. This is standard weight of sand in tons per cubic yard used for billing purpose. May 15, 2020 Crushed stone is quoted at a weight of 2700 pounds per cubic yard. Your stone dealer tells you he has a truck that can deliver 20 tons of stone per load. You need to know who many cubic yards that comes out to. How many tons is 4 yards pea gravel? (A single cubic yard of pea gravel weighs about 1.3 short tons.) The general range for a cubic yard of plain pea gravel is about $30 to $35, and a ton will cost about $40 to $45. Calculate 57 Granite Stone. Type in inches and feet of your project and calculate the estimated amount of Granite Stone in cubic yards, cubic feet and Tons, that your need for your project. The Density of 57 Granite Stone 2,410 lb/yd or 1.21 t/yd or 0.8 yd/t. Mar 02, 2020 Also know, how many cubic yards are in a ton of crushed gravel? One 20-ton truckload of crushed stone will yield 14-15 cubic yards of crushed stone. One may also ask, how much does a cubic yard of crushed stone weigh? 2400 lbs . Just so, how many cubic yards are in a ton? A cubic yard is equal to 27 cubic feet. Mar 17, 2021 How much does a cubic yard of stone weigh? Most gravel and crushed stone products have similar weights per ton. Gravel and sand typically weighs 2,200-2,700 pounds per cubic yard. In addition, there are 2,000 pounds to a ton. Certain products, like washed gravel, weigh more like 2,835 pounds per cubic yard. Oct 08, 2020 57 stone is generally sold by the ton, and there are approximately 1.4 tons in each cubic yard of this material. Most crushed stone and gravel products have similar weights per ton. How Many Cubic Yards Does A Ton Of 57 Stone Cover? The number of cubic yards that a ton of this gravel will cover depends on the depth at which it will be laid. Generally speaking, Apr 09, 2020 To figure out the how many tons you will need is not hard to do. You will have to know that the standard weight contractors use for crushed stone is 2700 pounds per cubic yard. Multiply the number of cubic yards by 2700 and divide by 2000. This is thoroughly answered here. It all depends on the stone varieties involved.Roughly, one cubic yard will equal approximately 2781 lbs, which is around 1.25 tonnes in the UK and 1.35 tonnes in the US, based on their short tonne measurement. Jan 10, 2020 To figure out the how many tons you will need is not hard to do. You will have to know that the standard weight contractors use for crushed stone is 2700 pounds per cubic yard. Multiply the number of cubic yards by 2700 and divide by 2000. This value is the number of tons that you will need. One cubic yard of 3/4-inch red crushed stone weighs 1.3 tons. Companies may sell the stone by ton or by cubic yard. One cubic yard covers about 10 feet by 10 feet for a depth of 3 inches and one ton covers about the same area at a 2-inch depth. Type in inches and feet of your project and calculate the estimated amount of Base material in cubic yards, cubic feet and Tons, that your need for your project. The Density of Crusher Run 2,410 lb/yd or 1.21 t/yd or 0.8 yd/t Rectangular Area with Crushed Gravel (105 lb/ft) and Price Per Unit Mass Lets say I need crushed gravel for part of my driveway which measures 4ft long, 2ft wide and 9in (0.75ft) deep. Lets also say that the selected gravel costs $50 per ton. Sep 30, 2021 Crushed stone is costlier at about $55 per cubic yard and $65 per ton. Buying pea gravel in bulk may reduce costs , but different finishes, like gravel with color, will add anywhere from $20 to $50 to the price per unit. Stone Tonnage Calculator rohrers-admin 2018-06-06T155108-0400 Tonnage calculations are based on averages and should be used as estimates. Actual amounts needed may vary. Ceramic Tile, loose 6 x 6 1 cubic yard 1,214 lbs Concrete, Scrap, Loose 1 cubic yard 1,855 lbs Glass 1 cubic yard 2,160 lbs Gypsum, Dry Wall 1 cubic yard 3,834 lbs Metals 1 cubic yard 906 lbs Plastic 1 cubic yard 22.55 lbs Soil, Dry 1 cubic yard 2,025 lbs Soil, Wet 1 cubic yard 2,106 lbs Stone or Gravel 1 cubic yard 2,632.5 lbs A cubic yard is a measurement by volume (Or most simply put, a measurement by size) 2. A ton is a measurement by weight. 3. A / sign means per, so 1.5 TONS/CY reads 1.5 TONS per Cubic yard which simply means there are 1.5 tons per (for) every cubic yard of material. Fortunately, a cubic yard of material has a conversion factor (unit ... Jul 26, 2021 Use this formula to determine how much crushed stone you will need for your project (LxWxH) / 27 cubic yards of crushed stone needed. In the construction world, most materials are measured in cubic yards. Multiply the length (L), in feet, by the width (W), in feet, by the height (H), in feet, and divide by 27. CUBIC YARDAGE CALCULATION SHEET Height of Sides ... Material Weight Pounds per Cubic Yard Asphalt 2,700 lb. Iron (wrought) 13,100 lb. Brush/Branches ... Crushed Stone 2,700 lb. Plywood Sheets 800 lb. Earth (loose) 2,050 lb. Roofing Debris 450 lb. to 750 lb. Our Stone calculator will help you estimate how many Cubic Yards of Stone you need for your desired coverage area. The crushed stone calculator offers 4 Box area fields and 2 Circular area fields for you to calculate multiple areas simultaneously (back Jul 08, 2011 One cubic yard of vacuum has a mass of 0 kilograms. One cubic yard of osmium has a mass of approx 5756 tons per cubic yard. Take your pick between these two extremes. Of course, a cubic yard of a neutron stars material would be much, much greater. 1 cubic foot of Stone, crushed weighs 100.00959 pounds lbs Stone, crushed weighs 1.602 gram per cubic centimeter or 1 602 kilogram per cubic meter, i.e. density of stone, crushed is equal to 1 602 kg/m. In Imperial or US customary measurement system, the density is equal to 100.0096 pound per cubic foot lb/ft, or 0.92601 ounce per cubic ... 3. Estimation of the total gravel mass needed either in tons or kilograms, by transforming the volume as follows Metric - tons - 1 cubic feet 0.0520833333 tons - 1 cubic yard 1.388888888 tons - 1 cubic meter 1.8365 tons. English - lbs - 1 cubic feet 114.823958333 lbs - 1 cubic yard 3105.0985915 lbs - 1 cubic meter 4048.789445 lbs. 4. Per Cubic Yard 53 same as 4 but with limestone dust for easy packing 1-2 in size with half lime dust. Quantity 53 Driveway Stone - Crushed Limestone quantity. Add to cart. Compare. SKU 34986d8dc5f4. Category Shop Driveway Stone. Share on facebook . twitter . google-plus Tile. 2970. 1.43. Trap stone. 5849. 2.52. Most of Harmony Sand Gravels products will weight approximately 2,840 pounds per cubic yard or about 1.42 tons per cubic yard. For estimating purposes, most Contractors consider the yield to be 3,000 pounds per FTruck Body Pricing BrochuresExcelCubic_Yardage_Chart DCubic_Yardage_Chart D Rev A 6/9/2015 AGGREGATE TYPE Jul 28, 2021 1 cubic yard of concrete weighs about 3915 pounds or 1.96 US tons. 1 cubic yard of sand (dry) weighs about 2700 pounds or 1.35 US tons. 1 cubic yard of sand (wet) weighs about 3240 pounds or 1.62 US tons. 1 cubic yard of mulch (bark) weighs about 506 pounds or 0.25 US tons. 1 cubic yard of mulch (woodchip) weighs about 674 pounds or 0.34 Calculate 57 Limestone Gravel. Type in inches and feet of your project and calculate the estimated amount of Gravel Stone in cubic yards, cubic feet and Tons, that your need for your project. The Density of 57 Limestone Gravel 2,410 lb/yd or 1.21 t/yd or 0.8 yd/t. A

Wednesdays at 4:15 PM in 384H. We consider an integro-PDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We show that in the limit of small mutation rate, the solution concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings and Dockery et al. showed that for two competing species, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for a continuum of traits. This talk is based on joint work with King-Yeung Lam. Abstract: Ray mappings are the fundamental objects of geometrical optics. We shall consider canonical problems in optics, such as phase retrieval and beam shaping, and show that their solutions are characterized by certain ray mappings. Existence of solutions and some properties of them can be established through a variational method - the Weighted Least Action Principle - which is a natural generalization of the Fermat principle of least time. This is a joint work with D.Burago and S.Ivanov. As avid anglers we were always interested in the survival chances of fish in turbulent oceans. In this talk I will address this question mathematically, and discuss some of its consequences. I will show that a fish with bounded aquatic locomotion speed can reach any point in the ocean if the fluid velocity is incompressible, bounded, and has small mean drift. We begin with the elementary observation that the $n$-step descendant distribution of any Galton-Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we study certain CSBPs (continuous state branching processes), which arise as scaling limits of Galton-Watson processes. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain CSBPs satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall in 2006.) We also prove the existence of Galton-Watson processes that are universal, in the sense that all possible (sub)critical CSBPs can be obtained as a sub-sequential scaling limit of this process. Fluid-structure interaction problems with composite structures arise in many applications. One example is the interaction between blood flow and arterial walls. Arterial walls are composed of several layers, each with different mechanical characteristics and thickness. No mathematical results exist so far that analyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structure is composed of several layers. In this talk we will summarize the main difficulties in studying this class of problems, and present an existence proof and a computational scheme based on which the proof of the existence of a weak solution was obtained. Our results reveal a new physical regularizing mechanism in FSI problems with multi-layered structures: inertia of the thin fluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoretical results on modeling the human cardiovascular system will be discussed. This is a joint work with Boris Muha (University of Zagreb, Croatia), Martina Bukac (U of Notre Dame, US) and Roland Glowinski (UH). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL, Switzerland). Collaboration with medical doctors Dr. S. Little (Methodist Hospital Houston) and Dr. Z. Krajcer (Texas Heart Institute) is also acknowledged. We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering an open problem posed by De Lellis and Szekelyhidi Jr. Moreover, we show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. This talk is based on a joint work with T. Buckmaster and S. Shkoller. We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) in 1d with ignition reactions f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of X_1 is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of X_1 is infinite. We discuss two models of random walk in random environment, one from stochastic homogenization of composite materials, and the other from interacting particle systems. The goal is to explore the quantitative aspects of the invariance principle, i.e., to quantify the convergence of the properly rescaled random walk to a Brownian motion. The idea is to borrow PDE/analytic tools from stochastic homogenization of divergence form operator and apply them in the context of interacting particle systems. In particular, we will explain the proof of a diffusive heat kernel upper bound on the tagged particle in a symmetric simple exclusion process. In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space for the 1D periodic case. This is demonstrated by constructing arbitrarily small solutions with a sequence of nonlinear oscillations, known as plasma echoes, which damp at a rate arbitrarily slow compared to the linearized Vlasov equations. Some connections with hydrodynamic stability problems will be discussed if time permits. Nuclear magnetic resonance (NMR) spectroscopy is the most-used technique for protein structure determination besides X-ray crystallography. In this talk, the computational problem of protein structuring from residual dipolar coupling (RDC) will be discussed. Typically the 3D structure of a protein is obtained through finding the coordinates of atoms subject to pairwise distance constraints. RDC measurements provide additional geometric information on the angles between bond directions and the principal-axis-frame. The optimization problem involving RDC is non-convex and we present a novel convex programming relaxation to it by incorporating quaternion algebra. In simulations we attain the Cramer-Rao lower bound with relatively efficient running time. From real data, we obtain the protein backbone structure for ubiquitin with 1 Angstrom resolution. This is joint work with Amit Singer and David Cowburn. The question addressed here is how fast a front will propagate when a line, having a strong diffusion of its own, exchanges mass with a reactive medium. More precisely,we wish to know how much the diffusion on the line will affect the overall front propagation. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by transportation networks. In a previous series of works, we were able to show that the line could speed up propagation indefinitely with its diffusivity. For that, we used a special type of nonlinearity that allowed the reduction of the problem to explicit computations. In the work presented here, the reactive medium is governed by nonlinearity that does not allow explicit computations anymore. We will explain how propagation speed-up still holds. In doing so, we will discuss a new transition phenomenon between two speeds of different orders of magnitude. Joint work with L. Dietrich. For questions, contact

Try entering different sets of numbers in the number pyramids. How does the total at the top change? When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning? 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 number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis? Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter. 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. Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . Can you explain the strategy for winning this game with any target? A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? A card pairing game involving knowledge of simple ratio. Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6? It's easy to work out the areas of most squares that we meet, but what if they were tilted? What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now? Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations? Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy? What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes? Can you discover whether this is a fair game? Meg and Mo still need to hang their marbles so that they balance, but this time the constraints are different. Use the interactivity to experiment and find out what they need to do. A game for two people that can be played with pencils and paper. Combine your knowledge of coordinates with some strategic thinking. A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose! 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 moves. A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . . This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy. An interactive activity for one to experiment with a tricky tessellation Can you make a right-angled triangle on this peg-board by joining up three points round the edge? An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . . A generic circular pegboard resource. Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15. Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers? Interactive game. Set your own level of challenge, practise your table skills and beat your previous best score. A train building game for 2 players. Train game for an adult and child. Who will be the first to make the train? A simulation of target archery practice Mo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced. Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score? Here is a chance to play a version of the classic Countdown Game. A game for 1 person to play on screen. Practise your number bonds whilst improving your memory Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning? An animation that helps you understand the game of Nim. Use the interactivity or play this dice game yourself. How could you make it fair? Work out the fractions to match the cards with the same amount of money. Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do. Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths? Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit? 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. 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 in them?

2 edition of Systems of incongruences in a proof on addition mod m found in the catalog. Systems of incongruences in a proof on addition mod m Santiago Sologuren P. Written in English |Statement||by Santiago Sologuren P.| |The Physical Object| |Pagination||, 59 leaves, bound ;| |Number of Pages||59| Credits and distribution permission. Other user's assets All the assets in this file belong to the author, or are from free-to-use modder's resources; Upload permission You can upload this file to other sites but you must credit me as the creator of the file; Modification permission You are allowed to modify my files and release bug fixes or improve on the . By the way we constructed D from E, E*D = 1 (mod (P -1)*(Q -1)), so M ED (mod N) = M. Try working with this cipher yourself, using the RSA secret-sharing worksheet. Final Thoughts. Now, if you ever hear anything in the news about a large number being . x 2 + 3q(mod 5): You should read the proofs of Theorem and Theorem very carefully. These proofs actually show you the necessary techniques to solve all linear congru-ences of the form ax b(mod n), and all simultaneous linear equations of the form x a(mod n) and x b(mod m), where the moduli nand mare relatively prime. Main article: Divisibility Rules Divisibility rules are efficient shortcut methods to check whether a given number is completely divisible by another number or not. These divisibility tests, though initially made only for the set of natural numbers (N), (\mathbb N), (N), can be applied to the set of all integers (Z) (\mathbb Z) (Z) as well if we just ignore the signs and employ our. This article is intended for troubleshooting the PlayStation 4 and Nintendo Switch versions of Minecraft. If you are experiencing any issues with lagging, crashing, and hanging on the Switch and PS4 platfor. This book is full of worked out examples. We use the the notation “Solu-tion.” to indicate where the reasoning for a problem begins; the symbol is used to indicate the end of the solution to a problem. There is a Table of Contents that is useful in helping you find a . simple guide to prayer. A biomechanical analysis of the forefoot region of the basketball shoe Catherine Cookson country : on the borders of legitimacy, fiction, and history Mosbys Textbook for Nursing Assistants - Text and Mosbys Nurse Assisting Skills DVD - Student Version Package (Mosbys Textbook for Nursing Assistants) Aérospatiale japonaise = Progress in inorganic chemistry. A Congratulatory poem to Sir John Moor, Knight battle for Guadalcanal Your innate psychic powers The use and handling of compressed gases Systems of incongruences in a proof on addition mod m Public Deposited. Analytics × Add Author: Santiago Sologuren P. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in A familiar use of modular arithmetic is in the hour clock, in which the. Proof. Since a b(mod m) and c d(mod m), by the Theorem above there are integers s and t Systems of incongruences in a proof on addition mod m book b = a +sm and d = c +tm. Therefore, The operation +m is defined as a +m b = (a +b) mod m. This is addition modulo m. The operation m is defined as a m b = (a b) mod m. This is multiplication modulo m. r we have a ” r (mod m)".This is perfectly fine, because as I mentioned earlier many texts give the intuitive idea as a lemma. The number r in the proof is called the least residue of the number a modulo m. Exercise 1: Find the least residue of (a) mod 3, (b) (c) and (d) mod Congruences act like equalities in many ways. Most downloaded worksheets. Ones to thousands ( KiB, 6, hits); Integers - hard ( MiB, 5, hits); Solving word problems using integers ( KiB, 4, Contents Preface vii Introduction viii I Fundamentals 1. Sets 3 IntroductiontoSets 3 TheCartesianProduct 8 Subsets 11 PowerSets 14 Union,Intersection,Difference > Microwave And Rf Design Of Wireless Systems by David M. Pozar > An Introduction to Signals and Systems,1ed, by John Stuller > Control Systems Engineering, 4th Edition,by Norman S. Nise > Physics for Scientists and Engineers,5ed,A. Serway,vol1 > Laser Fundamentals,2ed, by William T. Silfvast. Modular arithmetic is a system of arithmetic for integers, which considers the remainder. In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder. Modular arithmetic is often tied to prime numbers, for instance, in Wilson's theorem, Lucas's theorem, and Hensel's lemma, and. The goal of this book is to bring the reader closer to this world. The reader is strongly encouraged to do every exercise in this book, checking their answers in the back (where many, but not all, solutions are given). Also, throughout the text there, are examples of calculations done using the powerful free open source mathematical software system. a≡b (mod m) is read as "a is congruent to b mod m". In a simple, but not wholly correct way, we can think of a≡b (mod m) to mean "a is the remainder when b is divided by m". For instance, 2≡12 (mod 10) means that 2 is the remainder when 12 is divided by (j +k) of m. In symbols, we have: a+c ⌘ b+d (mod m), (68) as desired. A similar proof can be used to show that if a ⌘ b (mod m) and c ⌘ d (mod m), then ac ⌘ bd (mod m). These two results allow us to treat all numbers that are congruent modulo m as identical when adding and subtracting numbers. If we know that a ⌘ 3. Mod [m, n] gives the remainder of m divided by n. Mod [m, n] is equivalent to m-n Quotient [m, n]. For positive integers m and n, Mod [m, n] is an integer between 0 and n Mod [m, n, d] gives a result such that and. NASA SYSTEMS ENGINEERING HANDBOOK viii Preface S ince the initial writing of NASA/SP in and the following revision (Rev 1) insystems engineering as a discipline at the National Aeronautics and Space Administration (NASA) has undergone rapid and continued evolution. Changes include using Model-Based Systems Engineering to improve. Books at Amazon. The Books homepage helps you explore Earth's Biggest Bookstore without ever leaving the comfort of your couch. Here you'll find current best sellers in books, new releases in books, deals in books, Kindle. Modulo Challenge (Addition and Subtraction) Modular multiplication. Practice: Modular multiplication. Modular exponentiation. Fast modular exponentiation. Fast Modular Exponentiation. Modular inverses. The Euclidean Algorithm. Next lesson. Primality test. Addition, subtraction, multiplication are binary operations on Z. Addition is a binary operation on Q because Division is NOT a binary operation on Z because Division is a binary operation on Classi cation of binary operations by their properties Associative and Commutative Laws DEFINITION 2. A binary operation on Ais associative if. m and Complete Residue Systems 43 17 Addition and Multiplication in Z 21 Probabilistic Primality Tests 55 22 Representations in Other Bases 57 23 Computation of aN mod m 59 24 Public Key Cryptosystems 63 A Proof by Induction 67 B Axioms for Z 69 C Some Properties of R Chapter 1 Divisibility In this book, all numbers are integers, unless. Proof Marks, Arsenal & Inspector Marks. In addition to arsenal marks, you will find other marks or stampings. These include the date, serial number and property marks as well as various acceptance and proof marks. I have not been able to locate an authoritative resource for identifying the acceptance and proof marks as it appears, many. Me×d ≡Me×d (mod φ(n)) ≡M (mod n) •The result shown above, which follows directly from Euler’s theorem, requires that M and n be coprime. However, as will be shown in Sectionwhen n is a product of two primes p and q, this result applies to all M, 0 ≤M. m!n!(m+ n). is an integer. 2 IMO /3 A Show that the coe cients of a binomial expansion (a+ b)n where nis a positive integer, are all odd, if and only if nis of the form 2k 1 for some positive integer k. A Prove that the expression gcd(m;n) n n m is an integer for all pairs of positive integers (m;n) with n m 1. Putnam A. In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions-or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems .The system x a k modm k, k 1, 2,n has a unique solution modulo M m 1 m 2 m n. Proof. First we prove that the system has a solution. Proceeding as in the above example, we define solution to each equation as y k a k modm k y k 0 mod m j, k j Combining them together yeilds the equation V. Adamchik 9.notion of ordering mod m. In R you know x2 = y2)x= y, but it is false that a2 b2 mod m)a bmod m in general. Consider 42 12 mod 15 with 4 6 1 mod In R you are used to x3 = y3)x= y. But 23 13 mod 7 and 2 6 1 mod 7. When we add and multiply modulo m, we are carrying out modular arithmetic. That addition and multiplication can be carried out on.

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What is the history of geometry? The term “geometry” is derived from the ancient Greek word “geometria,” which means measurement (-metria) of earth or land (geo), but this branch of mathematics covers much more than mapping. Geometry explains the relationship between shape and size, and also the nature of mathematics and numbers. Find out more about the invention and history of geometry, as well as the pioneers of geometry and the groundbreaking discoveries they made that have shaped the field. Discovery and Origin of Geometry Ancient civilizations in places like the Indus Valley and Babylonia roughly 3000 BC are credited with laying the groundwork for geometry. Geometry first appeared in the ancient world as a set of rules and formulas suitable for planning, constructing, astronomy, and solving mathematical problems. These principles included length, area, angle, and volume. Cubic and spherical Indus weights and measures were crafted from chert, jasper, and agate. Beginning around the 6th century BC, the Greeks expanded this knowledge and, using it, developed the conceptual field currently recognized as “geometry.” Greek philosophers such as Thales (624-545 BC), Pythagoras (570-490 BC), and Plato (428-347 BC) realized the fundamental relationship between the nature of space and geometry and reinforced geometry as an important field of study belonging to mathematics. Euclid (325-265 BC), who was probably Plato’s student and worked as a teacher in Alexandria, summed up the early Greek geometry in his magnificent work, “Elements,” written in 300 BC, and created scientific principles for geometric models using a handful of simple rules and axioms. The Elements became a standard geometry textbook for over 2000 years. Let no one ignorant of geometry enter.Plato, Greek philosopher, and mathematician The Turning Point in the History of Geometry Throughout the Middle Ages, mathematicians and philosophers from different cultures continued to use geometry to create a model of the universe. But the next major milestone came with the work of the French mathematician and philosopher René Descartes (1596-1650), who lived in the 17th century. Descartes developed coordinate systems to define the positions of the points in two-dimensional and three-dimensional space led to the birth of the field of analytical geometry, a new tool of mathematical algebra to solve and define geometry problems. Descartes’ work also led to the emergence of far more exotic forms of geometry. Mathematicians had long known that there were regions, such as the surface of a sphere, where the axioms of Euclidean geometry did not apply. The discovery of non-Euclidean geometry helped clarify many more fundamental principles that combined numbers and geometry. In 1899, German mathematician David Hilbert (1862-1943) developed new and more generalized axioms. Throughout the 20th and 21st centuries, these axioms were applied to a wide variety of mathematical cases. Timeline of the History of Geometry The timeline of geometry begins with the birth of practical geometry and concludes with fractal geometry. 3000 BC – Practical Geometry The history of geometry first arose in the Indus Valley and Babylonian civilizations from the need to solve problems such as calculating the volume of material required to build a pyramid. The level of sophistication of some of these early concepts is so high that a contemporary mathematician can struggle to deduce them without resorting to calculus. 300 BC – Spherical Geometry The Greek astronomer Theodosius of Bithynia (169-100 BC) compiled “spherics” in a book that consolidated the earlier work by Euclid (325-265 BC) and Autolycus of Pitane (360-290 BC) on spherical astronomy. In his Elements book, which was regarded as authoritative all the way up to the early 19th century in the history of geometry, Euclid provided the foundation for geometry. The spherical geometry allowed calculating areas and angles on spherical surfaces, such as star or planet positions in the imaginary sky sphere used by astronomers, or the locations of points on a map. However, this system does not follow Euclidean rules. In spherical geometry, the sum of a triangle’s angles is more than 180 degrees, and lines that run parallel to each other eventually meet. Euclid is considered the “father of geometry.” 500 BC – Pythagoras The Greek philosopher named the “Pythagoras theorem”, which calculates the hypotenuse (the longest side) of a right-angled triangle from the lengths of the other edges. The theory that a triangle’s angles would sum to 180 degrees, or two right angles, is attributed to Pythagoras of Samos. He said that the sum of the squares of the other two sides of a right triangle equals the square of the hypotenuse (the side opposite the right angle). Whenever the ancient engineers knew the lengths of two sides of a right triangle and needed to figure out the third, they used this theorem to do so, just like we do today. The idea of similar triangles, whereby two triangles are similar if they have the same form but need not have the same size, was also created by Pythagoras (570-490 BC). Pythagoras is remembered as a pivotal figure in the development of geometry. 4th Century BC – Geometric Tools Since the geometric tools have been around for thousands of years in the history of geometry, their precise historical beginnings are difficult to ascertain. However, the use of geometric instruments to measure, sketch, and build geometric forms and constructions may be traced back to at least the ancient Egyptians and Greeks. The Egyptians employed geometric tools in the building of their pyramids. The Greeks founded the science of geometry and wrote extensively on the use of geometric tools. They have undergone significant changes throughout the time. The Greek philosopher Plato (428-347 BC) stated that the tools of a true geometer should be limited to a straightedge and a compass, thereby establishing geometry as a science rather than a practical mastery. Euclid (325-265 BC), the “father of geometry,” and subsequent geometers defined the method of creating geometrical forms using certain tools. The ancient Greeks were the first to present building challenges using just a straightedge and compass. Some examples are building a line that is twice as long as another line or a line that divides an angle into two equal halves. 360 BC – Platonic Solids Plato introduced the concept of the Platonic solids first in his dialogue “Timaeus.” The Platonic solids are known as the five regular convex polyhedra (polygonal bodies), but Plato also combined them with his ideas about the structure of matter in 360 BC. The Platonic solids include five shapes that can be formed by joining similar faces along the edges. The tetrahedron has four faces, the cube has six, the octahedron has eight, the dodecahedron has twelve, and the icosahedron has twenty faces. In this “Timaeus,” Plato equated the five Platonic solids with the classical elements (earth, air, fire, and water) and the fifth element of the universe, which he named the “quintessence.” In the history of geometry, mathematicians and philosophers have always had a healthy respect for and fascination with the Platonic solids. They have also found applications in the fields of art, design, and architecture, and have served as inspiration for the development of several additional three-dimensional forms. The Platonic solids are still widely studied and admired for their symmetry and elegance, making them an integral element of modern geometry. 240 BC – Archimedean Solids The Greek mathematician Pappus of Alexandria (who flourished in 320 AD) described 13 convex polyhedrons, which are uniform polygons with similar edges and corners. In all, there are 14 Archimedean solids, 13 of which are attributed to Archimedes. In his 240 BC book, “Measurement of a Circle,” he detailed his study on these solids. The value of pi was approximated in this book by the Greek mathematician using the Archimedean solids. Archimedean solids are a class of polyhedra. To qualify as one of these solids, each of the faces of an object must be a congruent regular polygon, and there must be vertices at which two or more polygons intersect. Archimedean solids differ from Platonic solids in that their vertices do not always have the same number of faces meeting. 9th Century – Islamic Geometry Mathematicians and astronomers of the Islamic world explored the possibilities of spherical geometry. The geometric models used in Islamic decoration during this period are similar to modern fractal geometry. Geometric principles and the recurrence of geometric patterns are central to Islamic geometry, which is known for its ornate and aesthetically pleasing forms. Stars, polygons, and both regular and irregular tessellations are common components of these patterns. During the Islamic Golden Age (about the 8th to 13th centuries), Islamic geometry emerged as its own culture. Structures, tiles, textiles, and other ornamental arts all made use of these patterns and designs. 1619 – Kepler’s Polyhedron German mathematician Johannes Kepler (1571-1630) discovered a new class of polyhedra, known as the star polyhedron. In his 1619 treatise “Harmonices Mundi,” Kepler described a set of four polyhedra. These “Kepler-Poinsot polyhedra” are the small stellated dodecahedron, the great stellated dodecahedron, the great dodecahedron, and the great icosahedron. The Platonic and Archimedean solids, which are likewise polyhedra but have regular polygons for faces, are closely linked to Kepler’s polyhedra. Kepler’s polyhedra differ from these solids in that not all of their vertices are shared by the same number of faces. In Kepler’s mind, these geometric patterns reflected the underlying structure of the cosmos and hence had a cosmic meaning. 1637 – Analytical Geometry “La Geométrie,” a fascinating book by the French mathematician and philosopher René Descartes, explains how points in space can be measured by coordinate systems and geometric structures can be described by equations. This is called “analytical geometry,” and it is a field of study. The study of geometric forms and their attributes is the domain of analytical geometry, often referred to as coordinate geometry or Cartesian geometry. Many consider René Descartes to be the “father” of analytical geometry because of his work in this area. Prior to the advent of analytical geometry, the measurement and qualities of physical objects and forms were at the center of the geometry field. Descartes’ ideas made possible the abstract analysis and description of geometric forms and their attributes using algebraic techniques. In addition, he proposed the use of equations to define geometric forms and curves, which allowed for a more accurate and rigorous investigation of these objects and their attributes. Today, many disciplines rely on the tools and techniques developed in analytical geometry, making it an indispensable branch of mathematics in the history of geometry. 1858 – Topology During the 19th century, mathematicians began to be fascinated by topology, or geometric edges and surfaces rather than specific shapes. The visualized Möbius strip above is an object with a single surface and a single continuous edge. Topology is the mathematical study of the features of geometrical objects and spaces that remain unchanged while they are continuously deformed. It can be deformed by stretching, bending, and twisting, but not by ripping or gluing. Leonard Euler, a Swiss mathematician who flourished in the 18th century, is considered the father of modern topology. Using what would become known as graph theory, Euler investigated the topology of polyhedra. Graph theory is the study of networks made up of points and lines. The notion of the Möbius strip, a surface with just one side and one border, was established by August Ferdinand Möbius and Johann Benedict Listing, who contributed to the advancement of topology in the 19th century. To explore various mathematical objects and structures such as manifolds, knots, and topological spaces, topology evolved into a more abstract and generic discipline in the 20th century. 1882 – The Discovery of the Klein Bottle German scientist Felix Klein (1849-1925) discovered a shape that has a one-sided surface without any surface borders, which proves to be a geometry with more than three dimensions. Felix Klein initially characterized it in the 19th century as a tool for investigating the characteristics of non-orientable surfaces. The Klein bottle is not characterized by its geometric form but rather by its attributes and connections to other things, making it a topological entity. It is a mathematical description of a surface with no obvious boundaries, and it may be thought of as a loop that has been twisted and linked to itself. One of the fascinating features of the Klein bottle is that it can’t be immersed in three-dimensional space without crossing over into itself. Because of this feature, the Klein bottle is notoriously hard to draw to scale, prompting the creation of a number of computer techniques for studying and understanding its features. 20th Century – Fractal Geometry Our ability to use computers has led to the discovery of fractals, which are equations of detailed models that repeat each other at different scales and produce shapes like the well-known Mandelbrot set and display them in a graphical form. Mathematically speaking, fractal geometry is the study of the characteristics of geometric objects with both self-similarity and a non-integer dimension. Many definitions of fractals focus on the fact that they are geometric forms with “fractional dimension,” or a dimension that is between a whole number and a fraction. The French mathematician Henri Poincaré used the word “fractal” to characterize things having a “fractional dimension” in the early 20th century. However, the German mathematician Georg Cantor researched the idea of self-similarity, a central aspect of fractals, far earlier, in the 19th century. Benoit Mandelbrot, a mathematician born in Poland, is widely recognized as the pioneer of fractal geometry. He made significant contributions to the area in the 1970s. Mandelbrot used the term “fractal geometry” to characterize the emerging area of study he established by using computer graphics to show and analyze the characteristics of fractals. Fractal geometry has relevance in many modern disciplines, such as physics, biology, and computer science. It’s also used in the development of computer graphics as well as the research of chaotic systems. The computer’s power allowed us to solve problems such as the four-color theorem, which distinguishes the various regions within a complex map using nothing but four colors in such a way that no two adjacent regions on the map have the same color. Francis Guthrie introduced the four-color theorem in 1852, then in 1976, Kenneth Appel and Wolfgang Haken used computer systems to verify it. Extensive computer computations were used to prove the four-color theorem, which states that every map can be colored using just four colors. Useful in many disciplines, including cartography and computer science, the four-color theorem is worth studying. It’s used to make comprehensible maps, and it’s spawned new computational and mathematical methods for investigating and comprehending the characteristics of maps and other spatial systems. Types of Geometry Euclidean geometry is analyzing shapes in two dimensions and three dimensions according to the rules established by Euclid, an ancient Greek mathematician. One defining feature of Euclidean geometry is that its definitions and theorems are grounded in axioms, or universally accepted facts. Euclidean geometry, named after the ancient Greek mathematician who developed it, is one of the most widespread kinds of geometry. Axioms, or fundamental principles, form the foundation of Euclidean geometry. For example, it is an axiom that any two straight lines will meet at a single point. This branch of geometry is very applicable since it explains the behavior of solid forms and objects in the physical world. This branch of geometry is distinguished from Euclidean geometry by its use of various axioms. Two examples of non-Euclidean geometries are hyperbolic geometry, which uses a different set of axioms for parallel lines, and elliptic geometry, which uses a different set of axioms for the sum of the angles in a triangle. Some of the assumptions of Euclidean geometry are rejected in non-Euclidean geometry. Hyperbolic geometry, one of the most well-known types of non-Euclidean geometry, is predicated on the premise that two lines can meet in more than one place. This branch of geometry is often used in the investigation of cosmological structure and space-time characteristics. The focus here is on geometric shapes and transformations that keep length-to-width ratios (the ratios of distances) the same. Perspective drawings and other forms of graphics can be analyzed using projective geometry. Projective geometry is the study of the characteristics of figures and forms when they are projected onto a two-dimensional plane, making it the third type of geometry. Creating perspective and the appearance of depth on a two-dimensional surface is made possible using this form of geometry, making it useful in art, architecture, and photography. The fourth type of geometry is topological geometry, which investigates the features of geometric objects that remain the same under continuous deformations like stretching or bending. This geometry is often used in the investigation of sub-atomic structures and cosmological characteristics. Preserved features of geometric forms during continuous deformations like stretching or bending are the focus of this field of research. Deformable forms have interesting qualities that can be investigated using topology. Differential geometry is a calculus-based research field in geometry. Curves and surfaces in three dimensions can be analyzed with the help of differential geometry. Numerous branches of science and technology rely heavily on the concepts and methods of differential geometry, such as physics, engineering, and computing. It is used in the investigation of spatial phenomena and the properties of dynamic physical systems. It is a tool used in physics to learn more about the nature of space-time and how particles and fields behave. Manifolds are high-dimensional spaces used to simulate physical processes and investigate their characteristics. This type of geometry has many applications in engineering, including the study of surface qualities and the design and analysis of complex systems like airplanes and cars. In computer science, it is used to study data structures and algorithms and what makes them work the way they do. In algebraic geometry, we apply algebraic concepts and methods to the study of geometry. This field of study focuses on the qualities of shapes whose characteristics can be expressed in terms of equations. This type of geometry studies Algebraic curves, Algebraic surfaces, Algebraic varieties, and also manifolds. These are curves, surfaces, objects, and high-dimensional spaces that can be defined using algebraic equations such as parabolas, planes, ellipsoids, lines, circles or the curvature of space-time. In general, geometry is a fascinating and intricate mathematical field with several practical applications. Geometry is essential to our knowledge of the world around us, whether we’re trying to figure out how the cosmos formed or conjure up some impressive optical illusions. - Ray C. Jurgensen, Alfred J. Donnelly, and Mary P. Dolciani. Editorial Advisors Andrew M. Gleason, Albert E. Meder, Jr. Modern School Mathematics: Geometry. Houghton Mifflin Company, ISBN 0-395-13102-2. - Frits Staal, 1999, “Greek and Vedic Geometry”, Journal of Indian Philosophy, doi:10.1023/A:1004364417713, S2CID 170894641 - Nineteenth Century Geometry – Stanford Encyclopedia of Philosophy - Rosenfeld, B. A., 1988. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, New York: Springer. - Euclides, Elementa, I. L. Heiberg (ed.), Leipzig: B. G. Teubner, 5 volumes., 1883–88. - Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322.

Variational Methods In Statistics Mathematics In Science Engineering PDF EPUB Download Variational Methods In Statistics Mathematics In Science Engineering also available in docx and mobi. Read Variational Methods In Statistics Mathematics In Science Engineering online, read in mobile or Kindle. There is a resurgence of applications in which the calculus of variations has direct relevance. In addition to application to solid mechanics and dynamics, it is now being applied in a variety of numerical methods, numerical grid generation, modern physics, various optimization settings and fluid dynamics. Many applications, such as nonlinear optimal control theory applied to continuous systems, have only recently become tractable computationally, with the advent of advanced algorithms and large computer systems. This book reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. The mathematical fundamentals of calculus of variations (at least those necessary to pursue applications) is rather compact and is contained in a single chapter of the book. The majority of the text consists of applications of variational calculus for a variety of fields. Distributions, Hilbert Space Operators, and Variational Methods Author: Philippe Blanchard Publisher: Springer Science & Business Media Physics has long been regarded as a wellspring of mathematical problems. Mathematical Methods in Physics is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines. Variational Methods in Image Processing presents the principles, techniques, and applications of variational image processing. The text focuses on variational models, their corresponding Euler–Lagrange equations, and numerical implementations for image processing. It balances traditional computational models with more modern techniques that solve the latest challenges introduced by new image acquisition devices. The book addresses the most important problems in image processing along with other related problems and applications. Each chapter presents the problem, discusses its mathematical formulation as a minimization problem, analyzes its mathematical well-posedness, derives the associated Euler–Lagrange equations, describes the numerical approximations and algorithms, explains several numerical results, and includes a list of exercises. MATLAB® codes are available online. Filled with tables, illustrations, and algorithms, this self-contained textbook is primarily for advanced undergraduate and graduate students in applied mathematics, scientific computing, medical imaging, computer vision, computer science, and engineering. It also offers a detailed overview of the relevant variational models for engineers, professionals from academia, and those in the image processing industry. Introduction to Variational Methods in Control Engineering focuses on the design of automatic controls. The monograph first discusses the application of classical calculus of variations, including a generalization of the Euler-Lagrange equations, limitation of classical variational calculus, and solution of the control problem. The book also describes dynamic programming. Topics include the limitations of dynamic programming; general formulation of dynamic programming; and application to linear multivariable digital control systems. The text also underscores the continuous form of dynamic programming; Pontryagin's principle; and the two-point boundary problem. The book also touches on inaccessible state variables. Topics include the optimum realizable control law; observed data and vector spaces; design of the optimum estimator; and extension to the continuous systems. The book also presents a summary of potential applications, including complex control systems and on-line computer control. The text is recommended to readers and students wanting to explore the design of automatic controls. This book contains the proceedings ofthe meeting on "Applied Mathematics in the Aerospace Field," held in Erice, Sicily, Italy from September 3 to September 10, 1991. The occasion of the meeting was the 12th Course of the School of Mathematics "Guido Stampacchia," directed by Professor Franco Giannessi of the University of Pisa. The school is affiliated with the International Center for Scientific Culture "Ettore Majorana," which is directed by Professor Antonino Zichichi of the University of Bologna. The objective of the course was to give a perspective on the state-of the-art and research trends concerning the application of mathematics to aerospace science and engineering. The course was structured with invited lectures and seminars concerning fundamental aspects of differential equa tions, mathematical programming, optimal control, numerical methods, per turbation methods, and variational methods occurring in flight mechanics, astrodynamics, guidance, control, aircraft design, fluid mechanics, rarefied gas dynamics, and solid mechanics. The book includes 20 chapters by 23 contributors from the United States, Germany, and Italy and is intended to be an important reference work on the application of mathematics to the aerospace field. It reflects the belief of the course directors that strong interaction between mathematics and engineering is beneficial, indeed essential, to progresses in both areas. The book includes lectures given by the plenary and key speakers at the 9th International ISAAC Congress held 2013 in Krakow, Poland. The contributions treat recent developments in analysis and surrounding areas, concerning topics from the theory of partial differential equations, function spaces, scattering, probability theory, and others, as well as applications to biomathematics, queueing models, fractured porous media and geomechanics.