# Calculus

Is there a definition of the Mittag-leffler function for variable-order fractional calculus?
E_q(t) with q a bounded function.
Any set-valued map which not satisfied any properties of subdifferential calculus?

I'm looking for optimizing multivalued  vector valued function.

Michael Patriksson · Chalmers University of Technology

This is a case of "vector optimization" or "multi-objective optimization". Consult, for example, the book Vector Optimization by Jahn or Nonlinear Multiobjective Optimization by Miettinen.

What does it mean to the fractional calculus if the constant is equal to zero in some definitions and not equal to zero in another's definition?
Riemann–Liouville fractional derivative & Caputo fractional derivative & Modified Riemann–Liouville
Is it possible to apply the Leibniz product rule for variable-order fractional derivation?
D_q(t) (f.g) = D_q(t) f .g + f.D_q(t)g
What is integral of (secx)^3 wrt x ?
If limits considered from 0 to (pi/4) and without limits also.

My answer from wolfralpha.com is in the file attachment. Your sincerely; Anna Tomova

Any suggestion to represent "thinking" in mathematical language or logic?

How could we describe the act of "thinking" with mathematical tools? Which paradigm is best suited for? What does "thought" mathematically mean? Is there any alternative to the procedural (linear) conception of neural calculus?

Federico Bugnoli · University of Verona

The signal can definitely be represented as mediated by neurons. Alas, a bijection between subsequent neurons involved in the transmission is as elucidative as "machine code": as it is a low-level programming language (a.k.a. close to the hardware), it does not provide any hint to effectively describe the process [of thinking] above.
After all, the brain is essentially an interpreter: it receives impulses from the outside, then translates them into a more abstract language so that it is possible for it to compare them with others in a more effective way; without abstraction, thinking ability and remembering are pointless. In order to be conscious, a thinking machine should at least recognize the domain[s] (which could actually be microlect[s]) the elements it uses to postulate its propositions belong to: it is not a problem of codomain (you could define, as J.Gruenwald suggested, a set of <<saving options ... for the description>> as codomain), but of domain - since, if you have a unique homogeneous set (it is so because of the low-level syntax you advance), you can not discern effective groups for such a discrimination. For which I mean "effectiveness" as the capability to segregate in order to refer to something in particular.
Your architecture lacks in terms of extrapolation of a law for which these groups are recognizable among all the possible parts of such a set - what does tell you that the progressive activation of a particular chain of neurons implies a one-to-one relation to a said model? Actually, nothing; moreover, there are evidences that such a connection is non-linear (I spoke about this before).

What do you think about the definition of *-differentiation which is given via "Generalized Runge-Kutta method with respect to non-Newtonian calculus?

The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. In July of 1967, they created an infinite family of calculi that includes the classical calculus, the geometric calculus, the harmonic calculus, and the quadratic calculus. In August of 1970, they created infinitely-many other calculi, including the bigeometric calculus, the biharmonic calculus, and the biquadratic calculus. All of the calculi can be described simultaneously within the framework of a general theory. They decided to use the adjective "non-Newtonian" to indicate any of the calculi other than the classical calculus.

M. Grossman sent a letter to R. Katz on 21 July 2014; he said that

"As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that NNC has considerable potential for application in science, engineering, and mathematics. And we were right!! ".

I agree strongly with you Professor.

Anatolij K. Prykarpatski · AGH University of Science and Technology in Kraków

Dear Ugur, thanks - yet I guess you are slightly mislead concerning my  in no sense critical comments (being strongly math backgrounds educational, in particular concerning   things, which are related to general of analytical nature   math problems - as you well, I am sure, understand too - that any  such a math problem is not solvable on the whole, if it does not allow some  either local  or global linearization. - Even more, the math analysis as science would not exist,  if there were no possibility to linearize a mapping in some  vicinity of a domain point! )  - As you can also easily observe,  my  clearly formulated  intentions  were completely aimed at  the pure mathematical essence of these "calculi" activity subject to this fundamental linearity property,  and concerned in no sense their , eventually,  possible applications! Why not? - -nobody knows!

All the best and regards!

Should we teach limit or derivative first?
Should we teach limit as a tool for derivation or should we teach in depth as a subject before derivative concept?
Vitaly Voloshin · Troy University

Everything depends on the audience. Compare it with driving.   Now everybody drives a car.   But not everybody needs to know why it runs.  Users and developers need different approaches in teaching the "same" things..

What is your opinion that students find difficult in learning introductory calculus?
Michael Livshits · mathfoolery

@Andrew Messing, it looks easy until you try, and when you try you will understand how difficult it is, especially in our real world, not in the ideal world that you may be imagining while discussing this or that "improved" approach. And if you tell me how "dumping the Riemann integral" or "rigorous formulation of infinitesimals" would make introductory calculus more understandable, I will be all ears.

What is difference between greatest value and maximum value of a real valued function? When do they coincide?
Real valued functions
George Stoica · University of New Brunswick

Dear Suruchi,

Max is the largest number within a set, whereas sup bounds the set from above. Sup may or may not be part of the set itself. If sup is part of the set, it is also the max. Examples are straightforward.

Sincerely,

George

When is the power series of an even function alternating?

This is motivated by a calculation using Fourier transforms of filters / mollifiers. Since it is such a classical question, I suspect there is a classical answer out there and would appreciate help satisfying my curiosity.

Roman Sznajder · Bowie State University

Dear Bill:

If I understand it correctly, you are expanding a function f in Taylor series at a=0, which is natural, as the function is even. As such, all odd-order derivatives of f at 0 must be zero. Thus, we deal only with f^(2k)(0) and this sequence should alternate. That's it. It seems not to exist any reasonable classification of functions with this property.

All the best,

Roman

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 @Professor Afag Ahmad, your contribution is highly appreciated. I want to know in particular who did first define orthogonal polynomials.

What do you think about the non-Newtonian calculi which have considerable potential for use as alternatives to the classical calculus?

"The non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore anyone may be surprised to learn that there is a uniform relationship between the corresponding operators of the non-Newtonian calculus and the classical calculus" (Grossman and Katz_1978)

R. C. Mittal · Indian Institute of Technology Roorkee

Dear Dr. Kadak, thanks for giving me this link. It is a good paper and some new ideas for me.

Can a multivariate function f(x,y,..) be converted to a lower dimension function?

I am trying to find if there is a mathematical formula that can, for example, convert a  function f(x,y) to f(r) such that f(x0,y0) = f(r0) and f(x+a,y+a) = f(r+a).

Thanks.

Rogier Brussee · Hogeschool van Utrecht

Well you can do

Map(X_1 \times X_2 \times X_3 , Y)

\iso Map(X_1, Map(X_2\times X_3, Y))

\iso Map(X_1, Map( X_2, Map(X_3, Y)))

I think in some functional programming languages (Haskell IIRC) they even do this in practice, so  they write the "Curried" form

f: int -> string -> bool -> double

which means f: int -> (string -> (bool -> double)))

for what in most computer languages you would write as

f(int, string, bool) -> double

or C/Java  style

double f(int, string, bool)

How can we construct surfaces of revolution in Cartan-Vranceanu Space?

How can we construct surfaces of revolution in Cartan-Vranceanu Space? Frankly, we should define a rotation matrix, and its determinat must be zero. But, I have difficulties to find that matrix. Thank you.

James F Peters · University of Manitoba

This is a good question.   Earlier I gave an answer to this question but for some reason RG did not record my answer.

A start on answering this question can be found in

A. Balmus, Differential Geometry--Dynamical Systems, 2009:

http://www.mathem.pub.ro/dgds/mono/D-10-Bal.pdf

There are 20 places in this monograph where surfaces are revolution, starting with Example 3.17 (Biharmonic curves on a surface of revolution), p. 49.   See Example 3.18 (Biharmonic curves of Cartan-Vranceanu spaces), starting on page 50.

How can one show that the left limit is smaller than the right limit?

In the figure lines are on the real axis.

What I am interested in to prove that the left limit of the red interval is to the left of the left blue one and correspondingly for the right end.

I was thinking that we can prove it by the definition of the left limit and right limit but I stuck with in it.

Any suggestions to prove it mathematically will be highly appreciable.

Peter T Breuer · Birmingham City University

Somehow, I don't think we need to! (see Apostol).

f(x) <= supx f(x)

is good enough for me!

Can anyone suggest a method to solve the following constrained functional minimization problem?

When a functional of one function is linear in the derivative of the function, I=int(f(y) + A(y)y'), the Euler-Lagrange equation leads to an algebraic equation as the terms involving y' cancel each other. But if the functional depends on more than one function this doesn´t happen:
I=int(f(y,z) + A(y,z)y' + B(y,z)z'), the E-L equations are:

df/dy + dB/dy*z' - dA/dz*z'=0
df/dz + dA/dz*y' - dB/dy*y'=0

These are first-order differential equations. Can this problem be solved, considering that there are at least two boundary conditions (imposed or natural)?

Actually my problem comes from a functional I=int(f(y,z)), subject to a constraint linear in the derivatives, so the E-L equation of the augmented functional using the linear constraint lead to first order differential equation. BUT, if I re-write the linear constraint:

A(y,z)y' + B(y,z)z' + C=0

as

C/(A(y,z)y' + B(y,z)z') + 1=0

and I construct the augmented functional with this constraint, the augmented functional is no longer linear in the derivatives and the E-L equation is a second order differential equation.
Does it make sense? Can the same problem lead to differential equations of different orders? What happens then with the boundary conditions?

Ezequiel Soule · Universidad Nacional de Mar del Plata

Ha, it was much easier (and more obvious) than I thought... Thank you very much Vladimir!

What is the physical significance of poles in Laurent's series?
In Laurent's series expansion of complex functions we encounter a term pole.
Luiz C. L. Botelho · Universidade Federal Fluminense

Remember that a Laurent series is a Taylor series on the inverse variable .That is whole useful when understand one complex variable on the Rieman sphere , where analytycal functions around the infinity (essential singularities around the origem -or any other point by translation )must be analytic complex analysis understood

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

Luiz C. L. Botelho · Universidade Federal Fluminense

There is an important fourth order lagrangian in string theory :that relative to theextrinsic string .I think that imposing APROPRIATE AND CORRECT  boundary conditions on higher order elliptic problems is somewhat non trivial issue (Path integrals for higher order Lagrangians may be an interesting issue on the subject!)see

1-J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, John Wiley & Sons, New York, NY, USA, 1976. View at Zentralblatt MATH

:2-ISRN High Energy Physics
Volume 2012 (2012), Article ID 674985, 25 pages
http://dx.doi.org/10.5402/2012/674985
Research Article
Basics Polyakov’s Quantum Surface Theory on the Formalism of Functional Integrals and Applications
Luiz C. L. Botelho

How do you show that the integral is convergent or divergent?

I have an integral and wanted to see whether it converges or diverges. I know there are many methods that can be used to show it.

Note that -l(ell) to l(ell) belong to real numbers.

I have the comparison method in my mind, but I am stuck with it. Any other suggestions will be highly appreciated.

Luisiana Cundin · Die Wand : leben heißt kampf

Peter is right... But, complex analysis is becoming somewhat of a dinosaur these days of computer assisted knowledge. Anyway, a very good book for complex analysis and analytic expansion is Zeev Nehari's "Conformal Mapping", but it is out of print, although there are other books to get on the subject.

The reason I suggested Widder's book is that he provides general theorems that either prove an integral either convergent or not, based on very loose and wide constraints.

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:

http://alumni.media.mit.edu/~mueller/papers/ecsc.pdf

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.

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.

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
\label{secant-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

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?

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.

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.

• 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.

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.

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.

Proof.

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.

End

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

Proof.

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.

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.

Thanks,

Mark Frautschi