1. G

    Generalized Fourier Integral and Steepest descent path, Saddle point near the endpoin

    I am looking forward to solving the integration in the following equation with the assumption that $ka$ is very large \begin{align} H = 2jka\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{jka[\cos{\varphi}+\cos{(\varphi-\phi)}]}\ d\varphi \end{align} I used the steepest descent path method to...
  2. H

    Generalized Pythagorean theorem

    Hi, let S be bounded piece of a plane in the space E3 and let's note Si an orthogonal projection of S into xy, xz and yz planes respectively. Then it can be proved that (1) area(S)^2=area(S1)^2+area(S2)^2+area(S3)^2. But there is also a general theorem, that in a vector space with dot product...
  3. H

    Generalized Turing machine (TM)

    So called "Church-turing thesis" says that Turing machine (TM) is the "strongest" computation device. (There exists of course other devices with equivalent strength as TM - even if it seems at first sight that these are stronger than TM.) But what about following definition of "generalized"...
  4. L

    Generalized Fermat equation, N+1 terms, power N

    (a1)^N+(a2)^N+...+(aN)^N=(a0)^N Is there at least one solution to this equation for every natural, nonzero N?
  5. D

    Riesz potential in generalized grand Lebesgue spaces

    I would like to draw your attention to the mathematical research in the fields of functional spaces and operator theory, which is carried out at Academy of Sciences of Chechen Republic in cooperation with University of Algarve and Southern Federal University. The paper is in Russian, so if there...
  6. M

    proving primality of generalized Fermat numbers

    let $q=a^{m}+1$ with $a,m \in \mathbb{N}^{+}$ and $a>1$, then $q$ can only be prime if $a$ is even and $m$ is a power of 2. This leads to the form of generalized Fermat numbers: $\large q=a^{2^{n}}+1$ with $a,n \in \mathbb{N}^{+}$ and even $a$. Given two generalized Fermat numbers...
  7. C

    Generalized Tijdeman problem

    To show once again how intriguing is my complicate modulus algebra method I prophose here how to work on the Generalized Tijdeman problem 1) With: n,m = Even >=2 B^m = A^n + K As I've shown: A^n = A^{2p} = \sum_{x=1}^{A^{(n/2)}} (2x-1) and: B^m = B^{2q} =...
  8. N

    Application of Cauchy generalized integral theorem

    I am asked to find $\displaystyle\oint_\gamma \frac{z^2 + e^z} {(z-i\pi)^2}dz$. Using Cauchy's Generalized Integral Formula. I am getting $\boxed{ - 1+2\pi i}$. Is this correct? (I don't have an answer key) Also I need to find $\displaystyle\int_0^{2\pi} e^{\alpha\cos \theta}...
  9. M

    Generalized Holder Inequality

    Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ Then show the following inequality by assuming that there are for every $i = 1, ... ,k$ one $N \in...
  10. R

    generalized gelfond-schneide theorem

    Can anyone explain Baker's generalisation of Gelfond-Schneider theorem? In his book, I just dont get lemma 2.
  11. M

    generalized formula of an infinite hyperbolic sine and hyper

    Recently i found a generalized formula of an infinite hyperbolic sine and hyperbolic cosine ceries in analytic continuation. You can download the .pdf document containing 4 pages here: ... ine-series I'm not sure if this formula is known. Please report...
  12. S

    Generalized algorithm for Collatz-like "hailstone" sequences

    Considering the recent flurry of posts here on the Collatz Conjecture, this may be of interest to some of you. So thinking about multiplicative groups, I realized that under certain constraints they too could be used to generate similar "hailstone" sequences. Moreover, it *should* be fairly easy...
  13. mathbalarka

    Generalized Taxicab Number Problem

    The conjecture is about whether there exists a number that can be written as sum of 5-th powers in two ways. How much currently is known about this. What approaches has been taken?
  14. B

    Generalized integral

    I tried to solve these two integrals but did not succeed. How to study the convergence? -) Integral from 0 to +inf 1 / (sqrt(x^3(abs(cosx-sinx)))) dx -) Integral from 0 to +inf (x^3 / ((1+x^2)^2))) *cosx dx
  15. A

    Generalized Product Rule?

    Suppose f is a function of a and that a occurs in several arguments of f. For example, f(a x_{1},...,a x_{n}) . Apparently the derivative of this function w.r.t. a is \frac{\partial f}{\partial a} ~=~ \sum_{i=1}^{n} \left [ \frac{\partial f}{\partial x_i} \cdot x_i \right ] . The product...
  16. S

    proof using the generalized triangle inequality

    I need to prove |d(x,y) - d(z,w)| <= d(x,z) + d(y,w). I know that there are two cases, when d(x,y) - d(z,w) < 0, and d(x,y) - d(z,w) > 0. I am just not sure how to apply the generalized triangle inequality to get the result I want. Can anyone help?
  17. M

    Generalized Rayleigh Quotient help

    Hello, I am in a college multivariable calculus class, and right now we have a homework assignment where we need to prove certain things about a Generalized Rayleigh Quotient. Q(x) = \frac{Ax\cdot x}{Bx\cdot x} (A) First, we need to prove that Q has at least one critical point. (B) Next, we...
  18. M

    Formula for generalized derivative

    Hello, does anybody know where this formula comes from? \frac{1}{2}(D^+g(t)+D^-g(t))=\lim_{h\rightarrow 0}\frac{3}{2h^3}\int_{0}^{h}{r(g(t+r)-g(t-r))dr} Thanks a lot.
  19. The Chaz

    Academic Guidance Generalized derivative

    Has anyone here studied this? In light of the various methods (forms?) of integration out there - R, R-S, L, etc I was curious if there were an analogue in differential calculus and came across this topic. Sadly, I have never encountered a...
  20. A

    Efficient Solution to Generalized Traveling Salesman Problem

    What's the most efficient method to solve the Generalized Traveling Salesman (Traveling Politician) Problem? (E-GTSP)... An approximation method is fine also. I just need an efficient method because the number of nodes in my problem might go as high as 3000-4000. Is there any code shared...