• James F Peters added an answer:
    How can situation calculus and event calculus be integrated?

    Situation calculus and event calculus.

    James F Peters · University of Manitoba

    In addition to the incisive observations by @Abedallah M Rababah, the relationship between a discrete event calculus and the situation calculus is investigated in

    E.T. Mueller, Discrete event calculus with branching time, 2007:


    See Section 5, starting on page 6, where the equivalence of a branching discrete event calculus (BDEC) and the situation calculus is covered.   Perhaps you will find Section 7 (Actual situations and events), starting page 17, interesting.    In that section,  the BDEC is extended and used to distinguish between hypothetical and actual situations.    Also, Section 8 (Related work), starting on page 18, is very detailed.

  • İ. Onur Kıymaz added an answer:
    How can I prove that the Caputo fractional derivative of a polynomial of degree N is a polynomial of degree at most N?

    The left-sided Caputo fractional derivative of u(x) is defined as
    ^C_a D^{\beta}_x u(x) = \frac{1}{\Gamma(1-\beta)} \int_{a}^{x}

    \frac{u'(t)}{(x-t)^{\beta}} dt,  where  0<\beta <1.

    Under what condition the Caputo fractional derivative of a polynomial of degree N is a polynomial of degree at most N?

    İ. Onur Kıymaz · Ahi Evran Üniversitesi

    Yes x^{n -\beta } , for 0<\beta<1, is not a polynomial but one can change it to a polynomial by replacing x by u^p when n-\beta=\frac{p}{q} where p,q \in N.

    But my answer is still wrong. I wrote "the Caputo fractional derivative of a polynomial of degree n is a polynomial of degree at most n when 0<\beta <1" but it must be "at least". Because after the change of variable to u, the degree of new polynomial is p where p<qn.  

  • Feng Qi added an answer:
    What is the power series expansion at zero of the secant to the power of three?

    It is well known that the secant $\sec z$ may be expanded at $z=0$ into the power series
    \sec z=\sum_{n=0}^\infty(-1)^nE_{2n}\frac{z^{2n}}{(2n)!}
    for $|z|<\frac\pi2$, where $E_n$ for $n\ge0$ stand for the Euler numbers which are integers and may be defined by
    \frac{2e^z}{e^{2z}+1}=\sum_{n=0}^\infty\frac{E_n}{n!}z^n =\sum_{n=0}^\infty E_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\pi.
    What is the power series expansion at $0$ of the secant to the power of $3$? In other words, what are coefficients in the following power series?
    \sec^3z=\sum_{n=0}^\infty A_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\frac\pi2.
    It is clear that the secant to the third power $\sec^3z$ is even on the interval $\bigl(-\frac\pi2,\frac\pi2\bigr)$.

    Feng Qi · Tianjin Polytechnic University

    Dear All, How are you going? I have a good news to tell you: Several days ago I found a method to give a closed form for coefficients of MacLaurin series of the function $(sec z)a$, where $a$ may be any given number. When I complete themanuscript, I would announce here.

  • Philippe Jacquet added an answer:
    What is the physical significance of poles in Laurent's series?
    In Laurent's series expansion of complex functions we encounter a term pole.
    Philippe Jacquet · Alcatel Lucent

    The poles of a Laurent series can give a lot of information about the asymptotics of the coefficients of the series via the integral Cauchy formula. This is the fundation of Analytic combinatorics.

  • Jacques Labelle added an answer:
    How does one describe two integration contours as a set?

    We knew that an integration contour can be described as a set of points. How one can describe the two integration contours as sets? Two figures describing sets are enclosed, Which one of these figures describes the two integration contours.

    Jacques Labelle · Université du Québec à Montréal

    I find the question not clear enough. Most answerers try to formulate the question (all differently) and then answer it. After all a curve is a mathematical object and two curves form a set with two elements, the first curve and the second curve. If one thinks of a curve as a set of points in the plane then the union is a new set of point; but neither a simple nor a single curve. A curve is mostly consider as a function from [0, 1] to C then sometimes two curves can be combined (like loops) to give another curve.

  • Mariano Pierantozzi added an answer:
    What is the best Calculus book for students?

    Dear all, I studied Calculus 1, 2 and so on, in some italian and Russian (e.g. Nikolskij) books. They are good books, but very theoretical with few examples or applications.

    When I graduated, I began to read american books, but I don't know other resources. For example I love Calculus 1 of Apostol or Anton because this books are clear, and explain with simple words difficult concepts. 

    So, what is the best Calculus 1 book for students?

    Mariano Pierantozzi · Università Politecnica delle Marche

    Dear Daniela,

    I agree with you.

    Thanks a lot.

  • Is there some general FFT approach optimizing the calculus of an integral by reducing the number of calls?

    Is there any conventional approach to calculate the following integral:

    (x,y) = int [ A (u,v) * (x,y,u,v) * exp [ i (x*u + y*v) ] dudv

    The part A(u,v)*exp[ i (x*u + y*v)] represents a conventional FFT. But I also have a set of functions Hx,y(u,v) which have their own distributions in u,v - plane.

    One direct way is to perform N calculations substituting Hi,j function one by another. The question is if there is some general FFT approach optimizing the calculus by reducing the number of calls?

    Georgy Sergeevich Kalenkov · Moscow Institute of Physics and Technology

    Dear Uche,

    Of course this is an integral of discrete functions. And of course it can be represented by a number of summations. There are different approaches, which depend on the exact type of the function Hx,y(u,v).  The question was how to get use of the FFT algorithm to reduce the number of calls, and NOT breaking it into individual summations.

  • Harish Kumar Kotapally added an answer:
    What is integral of (secx)^3 wrt x ?
    If limits considered from 0 to (pi/4) and without limits also.
    Harish Kumar Kotapally · Indian Institute of Technology Indore

    Sir, take sec(x)=t and change the variable to t from x, Now you can solve the equation easily :)

  • James F Peters added an answer:
    What are the applications of Ostrogradsky equation?

    While teaching calculus of variations, I came across the Ostrogradsky equation for extremizing functionals involving functions with many independent variables. What are the other applications of Ostrogradsky equation? Thanks. - Sundar

    James F Peters · University of Manitoba

    For a recent study of Ostrogradsky and higher order Lagrangians, see

    K.I. Nawafeh, Canonical quantization of higher-order Lagrangians, Journal of Applied Math. 375838, 2011, 11 pages:


    Ostrogradski's approach in formulating higher-order Lagrangians is presented in Section 3, starting on page 3.

    For an interesting application, see

    M. Chaichian,  M. Oksanen, A. Tureanu, Arnowitt-Deser-Misner representation and Hamiltonian analylsis of covariant renormalizable gravity, arXiv, 2011:


  • Malay K. Pakhira added an answer:
    Can the steepest descent method be success if the initial assumption for the solution is poor?

    In other words which method is not sensitive method for the initial assumption for the input ? Secondly can I use it to minimize function such as F=(x,p,u1,u2) with two inputs to minimize u1,u2 at the same time (by using steepest descent method)

    Malay K. Pakhira · Kalyani Government Engineering College

    There is no relation between selection of a poor or good initial solution, and finding a poor or good solution.  What matters is that what path your logic is following. In case of steepest descent method, we cannot guarantee about the result quality. You need to use some reliable method that can guarantee to some extent.

  • Andrzej Hanyga added an answer:
    Does the Mittag Leffler Function follow a semigroup property?

    Semi group property means : f(x).f(y)=f(x+y),

    Mittag Leffler Function is in the following link & beta=1;

    My question is do they follow semi-group property? I tried to search some papers in the past 2 days, In some papers they said that statement is true & in some they said that statement is False. {Example in one of the paper which was published in 2009 they said statement is correct & in the same journal another paper 2010 they proved that Statement is wrong, In 2012 in some other journal they said that statement is correct & In 2014 paper they are saying that Statement is Wrong ? }  Any suggestions? Thanks in advance.

    Andrzej Hanyga · Polish Academy of Sciences

    Any function with the semigroup  property is an exponential. 


    Set y = 0 => f(0) = 1.

    f'(x) f(y) = f'(x+y). Set x = 0 => f'(y) = f(y) => d ln f(y)/dy  = 1 => f(y) = exp(y) exp(C),

    f(0) = 1 => C = 0.


    Corollary. E_{a,b} (a, b >_ 0) has the semigroup property iff a  = b = 1.


    E_{a,b}(z)= exp(z) for a,b >_ 0 iff 

    Gamma(a n + b) = Gamma(n + 1), n =0,1,2,... (1)

    Hence Gamma(a + b) = 1 = Gamma(b).

    Gamma(z), z >_ 0,  has a unique minimum at z_0 = 1.46..., with G(z_0) < 1.

    Hence either a = 0 or b < z_0. The first case implies that Gamma(n) = Gamma(b) for n = 0,1, 2,... (impossible).

    In the second case b = 1. Hence  a + b > z_0 and a + b = 2 +> a = b = 1, q.e.d.

  • P. K. Mohanty asked a question:
    If we go back 400yrs and evolve again, should we discover calculus ?

    This is just a opinion generating question; not that i want an  answer.

    How  unique is a  "mathematical description  of a natural phenomena"  ?

    Classical mechanics  (special relativity) has a range of validity; 

    there  Quantum  mechanics (general relativity)  can  provide an alternative

    description (though complicated and unnecessary). 

    The simplest  description could be unique, but  how do I  know that

    the one I am using is simplest ?

    Wigner's article :


  • Theo K. Dijkstra added an answer:
    How do I calculate statistical sampling error and confidence limits as my sample approaches the entire population?

    My understanding of the statistical sampling error is derived from the binomial distribution where the variance, which is the square of the standard deviation, is simply N, the size of the sample. I believe that I know, or once knew, that the confidence limit is calculated from this distribution using an error function integral, Erf(x). In my physics career, we always worked with samples that were small compared with the whole population, and one got used to calculating the statistical component of the total error from sigma = SQRT(N). Estimating systematic errors was where most of the error analysis effort was spent.

    Now I am doing social science research where one has access, on some occasions, to entire populations, for example, the number of women graduating with Electrical Engineering degrees from U.S. institutions year-by-year. When I propagate the sampling errors through a calculation of the ratio of women to women and men graduates, I get an enormous error, larger than and comparable to the ratio itself.

    Clearly, I am doing something wrong. For example:

    1.) Elementary calculus or algebraic errors (see attached)
    2.) Violating assumption(s) of the simple propagation of errors scheme:
     a.) the component errors are small compared with the measured quantities.
     b.) the component errors are uncorrelated with one another.
    3.) Applying sigma = SQRT(N) when I have the entire sample
    4.) <strike> Something else I remain blind to. </strike> My main mistake was adding the central value to the error before graphing the main value with error bars. Merely graphing the main value with error bars works as expected. (Edit made 27 August 2014.)

    For what it is worth, another physicist kindly checked (1.) and (2.) for me and said they were correct. 

    It seems to me that it should be possible to calculate the sampling error as the sample approaches 100% of the population and the confidence limit approaches 100%. In this case, the sampling error should approach zero, smoothly, I would imagine.

    If the sampling error is simply zero when one has the entire sample, this formula should tell me. If I had this formula, and understand how to derive it, it seems that the sampling error being zero when one has the entire sample would be easier to accept.

    It seems to me that there would still be systematic errors. But these are hard to estimate, particularly where self-reported data is aggregated nationwide.

    It also seems to me that there may still be errors related to the population size, but I have no understanding or intuition for that, other than the fact that the data look "naked" to me without their error bars and that to the naked eye, smaller populations (e.g. astronomy) appear to have more year-to-year statistical fluctuation that larger populations (e.g. biology)

    If one of you could get me back on the statistical path, I would greatly appreciate that.


    Mark Frautschi

  • Peter T Breuer added an answer:
    Calculus of the number of ideals class group.

    What is the best method, in economic terms of explicit calculation of ideals class group and its order of a number field. Case cubic or quartic field.

    Peter T Breuer · Birmingham City University

    Please alter your question so it is comprehensible .. I GUESS you are talking about the polynomial ring over the integers, and its quotient by the principal ideal formed from an irreducible cubic or quartic polynomial?

    If so - and that was only a guess based on desperate reaching - please re-ask the question. I think you are asking "how many elements does the resulting field have", or something like that, but it's impossible to tell (see below).

    Here are some ideas:

    1. remove the topics "Cases", "Calculations", "Economics", "Calculations", "Numbers"

    2. Add topics "FInite Fields", "Computational number theory".

    3. Ask "what is a suitable representation of the elements on a computer and how do I efficiently compute addition and multiplication in this field".

    If that IS what you meant, please try again.

  • Lucas Oliveira added an answer:
    What is a good contemporary book (notation-wise) on calculus of variations?

    I'm re-reading Gelfand and Fomin, which is a great classical treatise on the subject, but is there anything contemporary that is recommendable?

    Lucas Oliveira · Universidade Federal do Rio Grande do Sul

    Try this one: "Moser, Jürgen Selected chapters in the calculus of variations." It is short, but have good ideas and explanations.

  • Ismat Beg added an answer:
    Learning of Mathematics for biology/life sciences

    What are the good study material for learning the application of mathematics for biology?

    Ismat Beg · Lahore School of Economics

    Two good books to start are;

    1. Essential mathematical biology - ‎Britton
    2. Introduction to mathematical biology - Rubinow

  • Mahmoud Moussa asked a question:

    I have a data of four variables and I don't know whch program do generate me the equation of those variables?

  • Michael Patriksson added an answer:
    Can mathematics conferences help to promote the researchers?
    In the whole world, different conferences are going on. Different subjects are they given. Are they any inspiration to new researchers?
    Michael Patriksson · Chalmers University of Technology
    I can say for sure that when I was a PhD student meeting the masters in my fields at conferences, workshops and PhD courses was a real treat and was very uplifting. Most of whom you meet have an open mind, and do not at all mind talking with you on your subject(s); you will find them supportive, and you get inspiration from hearing them give talks. I would not hesitate to go, and in fact what we do in our research group is to let a PhD student go to at least one conference as a "listener" before he/she gives his/her first conference presentation, so that he/she can enjoy the conference without that pressure. What is probably the best is to focus first on tutorial talks - they will be easier to understand for a fresh PhD student in comparison to technical talks.
  • Patrick Solé added an answer:
    Why could the product of two divergent series not be divergent?
    I need a counter example or proof to confirm this statement: The product of two divergent series may not be divergent.

    The product here is Cauchy Product but not pairwise product (in that case is easy to say harmonic series).
    Patrick Solé · Institut Mines-Télécom
    @Peter: thanks your example is simpler and more convincing than mine. Its a language problem: in french series NEVER mean sequence. Always the partial sum sequence...
  • Kirill Tsiberkin added an answer:
    Is there any way to effectively calculate the confluent Heun function?
    I need to make a relatively quick calculation of confluent Heun function (It is HeunC in Maplesoft Maple notation) values with different parameters in different points. I understand that it is possible to solve a Heun equation numerically, but that way seems not to be effective.
    Of course, it is possible to use, e.g. Maple, for numerical procedure realization I need this in my study, but applying some series expansion or hypergeometric functions is more preferable.
    So, I would appreciate simple (because I'm not a pure mathematician) information about series expansions of the Heun confluent function, its expression by hypergeometrics or any other known functions which are available to realize the numerical calculation.
    Kirill Tsiberkin · Perm State University
    Thank you, Dr. Homeier

    I think the reference and its' references will be useful.

    Actually, the main problem for me is really slow calculation Maple HeunC function because it uses numerical integration outside the series expansion convergence region (i.e., where |z| > 1). Probably, it is impossible and there is no coordinate mapping can turn the outside area into the inside region due to function and Heun equation singularities. I continue the search.
  • Kevin Larkin added an answer:
    What is a good way to teach the linear equation to non mathematics students?
    Suppose we want to teach the Linear Equation to other student who have don't know the mathematics. They do not have mathematics background.
    Kevin Larkin · Griffith University
    What I do with the students I teach (who are training to be primary school mathematics teachers) is to get them to create and build patterns based on information provided. For instance input 1 and output 3; input 2 and output 5; input 3 and output 7 etc. They then have to build the next few steps - e.g. input 4 output ? or output 13 input ?. They then determine the output for any input (x) and an input for a given output (y) and write in words the relationship between input and output. In this instance the relationship is multiply the step number by 2 and add one to determine the output. They can then write this in symbols as y (output) = input (x) time 2 add one. It is then a small step to understand y = mx+ b as output = input of x times m add the value of b.

    Once I have done this a few times I then use the Hands On Equations materials suggested by Helia. By this stage they can see that the linear equation describes a relationship between input and output governed by the function. My experience has been that for many students, this is their first experience of understanding that algebra isn't about the symbols, but rather it uses symbols to express relationships. Only later do I start to graph the relationship to strengthen the understanding.
  • Michael Wendl added an answer:
    Testing a new pedagogical tool
    As a case study for a planned research on the pedagogical advantage of a new mathematical definition, I'd like to request a reference to what's considered a good practice for this kind of research.

    For instance, few researchers argue that the number τ (Tau) has a pedagogical advantage; I wonder if and how anyone has ever tested that hypothesis (or similar ones) rigorously.
    Michael Wendl · Washington University in St. Louis
    I'm familiar with Palais' Math. Int. piece and it's never been clear to me whether or not it is tongue-in-cheek. There was a joking follow-up comment in Math. Int. that suggests Palais' piece was, in fact, tongue-in-cheek. There was one other follow-up pointing out that it could (should) equally have been pi/2, rather than 2*pi (MS Klamkin, Math. Int. 24(2) pp3). You should be aware that Palais' paper has never been referenced in the research or other educational mathematical literature in the almost 15 years since its publication. Putting it bluntly, this is not an idea that has (or is likely to have) any traction whatsoever. I would respectfully suggest not devoting any more of your valuable time to it. Best wishes.
  • Saeed Veradi added an answer:
    What is the minimum value of $h(r)=2[\sqrt{a+r}-\sqrt{a}]-4[(a+r)^{1/4}-r^{1/4}]-\frac{r^2}{4(r+2a)^{3/2}},$ where $a>=1$ is fixed and $r>0$?
    Saeed Veradi · Sahand University of Technology
    Minimum is zero in r=0 and a>=1.
    In the graph x means r and y means a.
  • Abedallah M Rababah added an answer:
    Who first defined orthogonal polynomials?
    It is known that Legendre and Chebyshev have contributions in the field.
    Abedallah M Rababah · Jordan University of Science and Technology
    Dear @James, Thank you very much for the information and the article
    Weisstein, Eric W. "Orthogonal Polynomials." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrthogonalPolynomials.html
    It contains some of the properties of the classical orthogonal polynomials.
  • Haridas Kumar Das added an answer:
    Suppose that f(x) is any real valued function, what kinds of relations exist between |f(x)| and f^2(x)?
    Mainly, is it possible to say, the function f^2(x) is increasing function on a set B, if |f(x)| increasing function on a set B?
    Haridas Kumar Das · Concordia University Montreal
    Good suggestions.
  • Kaveh Sadeghi added an answer:
    Do you know where the green theorem in 2 dimension is?
    Green theorem
    Kaveh Sadeghi · University of Kurdistan
    This therom in attachment
  • Andrew Messing added an answer:
    What is your opinion that students find difficult in learning introductory calculus?
    Traditional teaching has shadowed the current learning of calculus.
    Andrew Messing · Harvard University
    What is so hard? We could, for example, stop teaching 19th century integration that is both unintuitive and provides quite little in the way of conceptual connections to any serious use of integral theory. We could recognize that despite the lack of rigor infinitesimals provided prior to the epsilon-delta limit formulation of Weierstraß "most of the results in Calculus were already discovered through infinitesimals" (http://arxiv.org/pdf/1108.4657.pdf), and therefore not require students to grasp that which they have neither the exposure to logic to understand (so long as we have already abandoned rigor anyway, at least until students are sufficiently advanced such that we can teach them what we did not before and taught instead methods and concepts both outdated and needlessly complicated). We could provide the necessary exposure to formal logic as several texts (such as Spivak's do) through formal languages students are already familiar with. We could trim the fat from bloated texts that contain much material useless to the vast majority of students taking calculus. We could stop relying on marketing ploys encouraged by publishing companies such that a "Dummies/Idiot's Guide to Calculus" ceases to be at least as useful as popular university calculus textbooks. We could stop teaching pre-college mathematics as a if math were a series of procedures so that students taking their first course in university calculus (whether they have already been exposed or not) understand something of actual mathematics rather than a series of seemingly pointless ways to implement rules that have no recognizable use and in many cases are indeed useless other than to teach an approximation of actual calculus concepts. And we could stop using excuses like "it's easy to bitch about calculus, but hard to do anything about it" while ignoring the fact that 10 minutes with google could reveal is wrong and possible solutions. There are in fact organizations of mathematicians who have not only stated that we should do something about it but proposed methods and shown solutions. But perhaps you are already familiar with the rigorous formulation of infinitesimals ~40 years ago, the "Dump the Riemann Integral Project", the research in everything from the cognitive sciences to mathematical education journals, and so on. In which case, it would enlighten me at least to have your rejoinder to the several thousand pages of criticisms and solutions to the current approach that warrants your evaluation. In particular, it would prove useful to future attempts of my own, as these attempts to teach the subject to students who lack the background they could have obtained to deal with the best textbooks (even if I could get the department to purchase these) and therefore require my own tests and trials to make out of these poor excuses for calculus textbooks something worthwhile using whatever supplementary material I can find.
  • Adil AL-Rammahi added an answer:
    Is there any geometrical relation between fractional calculus and fractal ?
    Of course there are some concepts of fractional calculus that are co-related to fractal. But how are they geometrically related ?
    Adil AL-Rammahi · University Of Kufa
    I prefer to read the fractional calculus in physics problems where the order of derivative be fractional number.

About Calculus

The theory and application of differentiation, integration and limits.

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