Questions tagged [discrete-calculus]
Discrete calculus is an analog of the continuous version where the 'shift parameter' $h$ remains a non-zero positive number instead of being passed to a limit.
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Is it possible to reverse construct a first-order homogeneous (no forcing term) difference equation from its general solution?
The solution to $y' = a(t)y$ is $y = Ce^{\int a(t)dt}$. If I start with a general solution to this ODE such as $y = Ct$, then I can use algebra to find that $a(t) = \frac{1}{t}$, and reverse ...
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How to evaluate the exact value of the sum $\sum_{x\in \Lambda_N}\frac{1}{\|x\|_2^3}$ in 2d?
Let $\Lambda_N=\{1,2,\dots,N-1,N\}^2\subseteq\mathbb{Z}^2$ be a set of lattice points in 2d. Could we evaluate the exact value of the sum below?
$$(*)\quad\sum_{x\in \Lambda_N}\frac{1}{\|x\|_2^3},\...
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Differentiating Entropy with respect to Convolution Parameters
First, some formula reminders for the sake of completion:
$H(X) = -\sum_{i} p(x_i) \log p(x_i)$ is the entropy of a sequence $x_i$, where $p(x)$ is the discrete probability of x.
A discrete ...
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Gauss Hypergeometric Function Simplification
Recently I've been developing a strategy to handle a large class of Feynman integrals that uses some series expansions to simplify the angular integrals. This always leads to multiple sums over ...
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Problem of difference quotient : $\displaystyle\sum_{B\subseteq A} (-1)^{\mathrm{card}(B)} \cdot {{100+s(B)}\choose {\mathrm{card}(A)}}$
For any finite set $B⊂\mathbb N$, we note $\mathrm{card}(B)$ its cardinality as well as $s(B)$ the sum of its elements, and $A=\{1,2,\ldots,10\}$. Calculate :
$\displaystyle\sum_{B\subseteq A} (-1)^{\...
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Proving the discrete Laplace operator is an approximation of the continuous one
Given the lattice $\mathbb{Z}^d,$ one defines the discrete Laplace operator to be $$Hu(n) = - \sum_{\|m-n\|_1=1}(u(m)-u(n)), u:\mathbb{Z}^d\rightarrow \mathbb{C},$$ whereby the sum is running over all ...
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what is the general form of $T(n)=nT(n-1)+1$ [closed]
$T(n)=nT(n-1)+1$ where $T(0)=0$ there is hints $nCr ~ nPr ~T(n-1) = (n-1) T(n-2) + 1 n * ((n-1) * T(n-2) + 1) + 1 n * (n-1) * T(n-2) + n + 1 T(n) = n * (n-1) * (n-2) * ... * 1 * T(0) + n * (n-1) * (n-...
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Discretizing and solve $u''=\lambda u $ with Matrix scheme
as a part of a computer project, I need to solve
$u''=\lambda u\Longleftrightarrow (T-\lambda) u, $
with $T$ the discretized central difference second derivative (stencil/kernel: 1,-2,1).
Details are ...
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Discrete Gamma function
I am trying to find a closed expression for summations of the following form:
$\bar\Gamma_{n}(s)=\sum_{t=0}^{\infty} 2^{-ts}t^n$
I have managed to find closed expressions for $\bar\Gamma_{\{0,1,2\}}$ ...
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$\sin x \cos x$ defined as a Cauchy Product of their Taylor Series.
The goal of this project is to show that the Cauchy Product of two Taylor Series, $\sin x$ and $\cos x$, is equal to the Maclaurin Series of $\sin x\cos x$ . I am having trouble simplifying the ...
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Calculating the area between two plots
I have a series of $(x,y)$ data points where all the Y values are in ascending order. Since these are data points, it is possible to fit a linear, exponential or a polynomial trend line. Given a ...
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About a discrete Fourier property
Let $F(G)$ denote the space of functions from the abelian group $G$ to $\mathbb{C}$ and let $f\in F(G)$. Define the discrete furier transform of $f$ as $\hat{f}:\widehat{G}\to \mathbb{C}$ (where $\...
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Discrete convolution and supports: can we have $\operatorname{supp}(f * g) \subsetneq \operatorname{supp}(f) + \operatorname{supp}(g)$?
Let $G$ be a finite abelian group. For $f,g:G\rightarrow \mathbb{C}$ functions. Define their convolution in a point $x\in G$ as
$$(f*g)(x)=\sum_{y\in G}f(x-y)g(y)$$
From the definition of convolution ...
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Going from discrete Poisson equation to (discrete) divergence calculation
Just to give some background: I am currently working on a fluid simulation and am trying to clear any divergence from my discretized velocity field (i.e. it's split up into grids).
To eliminate such ...
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Lagrange multiplier method in a discretized way
I succeeded in using the Lagrange multiplier method to solve continuous case, but I failed to solve discretized case.
Assume $\min f(x,y) = x^2 + y^2$
constraints $s.t. g(x,y) = x + y -1 = 0$
use ...
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