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1
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2answers
36 views

Maximizing the sum of the squares of numbers whose sum is constant

I wonder how one goes about to find the maximum of $\sum v_i^2$, the $v_i$'s being positive integers whose sum $\sum_i v_i$ is fixed.
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0answers
11 views

Discrete Approximation to Dynamic Lagrangians

Suppose I have the following dynamic optimization problem, where I want to maximize the function $u(c,h)$ over time that's differentiable in both $c$ and $h$. I'm going to assume that the function $u$ ...
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0answers
15 views

Estimate for the discrete Green's function associated with the discrete Laplace operator $\Delta$

Consider the discrete lattice $\mathbb{Z}^{d}$ with $d\geq 2$. Let $x\in\mathbb{Z}^{d}$. We define the discrete Laplace operator $\Delta$ as follows: $$-\Delta u(x) := \sum_{\vert x-y\vert=1}\left(u(...
0
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0answers
31 views

Double Integral handling to calculate gradient directions

I would like to calculate the gradient direction of a simple 2d grayscale image for different areas/regions of interest by the given formula: $$\tan2\alpha=\frac{2 J_{xy}}{J_{yy}-J_{xx}}$$ the J - ...
1
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0answers
10 views

Sufficient condition for the classical differentiability of a function defined via an integral over $\mathbb{Z}^{d}$

Let $$a:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to\mathbb{R}:(x,y)\mapsto a(x,y)$$ such that ($0<\alpha\le\beta<\infty$): $$a(x,y)=\begin{cases} 0 &\text{if }\Vert x-y\Vert\neq 1\\ a(y,x)\in[\...
3
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1answer
69 views

Extended Kalman filter for the model x_dot=f(x,u,w)

There is a lot of info about EKF out there but everything I find explains it for the simplified model of the form x_dot = f(x,u) + w; i.e. the process noise is a ...
0
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0answers
15 views

Discrete functional notation

My question has two parts: 1) Is $ H: f \rightarrow \sum\limits_{Y \in \mathbb{R^+}^3}^{} f \cdot \operatorname{Log}_2(f) $ a valid discrete functional ? (H is the entropy defined in here and $f$ ...
2
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2answers
50 views

Discrete Calculus — Summing $\sum _{k=1}^{n+1} \frac{k}{(n+1)^k (-k+n+1)!}$

$$\sum _{k=1}^{n+1} \frac{k}{(n+1)^k (-k+n+1)!}$$ Mathematica gives that the answer is quite surprisingly the reciprocal of $n!$. The first way I thought of trying to prove this was by induction, ...
0
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0answers
12 views

Discrete logarithm in specific case

Suppose we have $a^{(x_1-x_2)z}=-b (modp)$ Is it posible to recover $log_{a}b$ knowing above statement to hold true?
0
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1answer
39 views

How would you solve the equation $-4450(1.05)^{n}+240n+4800=0$?

I've formed a difference equation to calculate the rate of change in a population of weasels, which I then solved to find tan expression for the population at time $n$. The expression I got was $y(n)=-...
0
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0answers
47 views

About the Mueller's (summation) formula?

Let $f$ be a real function with $\lim_{n\rightarrow\infty}f(x)=0$. (1) What is the necessary conditions for the Mueller's formula: $$ \sum_{x}f(x)=\sum_{n=1}^\infty(f(n)-f(n+x))+C ? $$ (2) What is $...
7
votes
1answer
85 views

Is there a non-constant function $f$ such that $f'(x) = f(x - 1)$?

In discrete calculus, where the difference operator $\Delta f = f(x + 1) - f(x)$ takes the place of $\frac{d}{dx}$, Fibonacci sequences are given by the functions satisfying: $$ \Delta f(x) = f(x - 1)...
9
votes
3answers
126 views

What are the $\sin$ and $\cos$ of discrete calculus?

I'm getting acquainted with Discrete Calculus, and I really like taking functions that arise in traditional calculus and finding what their counterparts in discrete-land are. For example, if we ...
1
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0answers
24 views

Are there efficient algorithm to split a discrete function into piecewise convex functions?

Hi guys let's say we have a function $f(x)$ where $x$ belongs to a discrete range of values (but finite). Are there algorithms to split $f$ as piecewise convex functions? If yes what about the ...
2
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4answers
97 views

Explanation of a method to compute $\sum_{k \le n} k^2$

I was searching for methods to compute $\sum_{k\le n} k^2$. I stumbled across this (which is an answer provided by Gareth Rees to this question). "..Represent $k^2$ in terms of falling powers (...
2
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0answers
83 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
0
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0answers
44 views

how to show a concave function on discrete domain increases in x?

Define $g(x)=\frac{f(x)/x}{f(x)-f(x-1)}$ where x$\in$ $\mathbb{Z}$. Known that $f(x)$ has the concave extension in every consecutive $x$, i.e: $f(x+1)+f(x-1)-2f(x)<0$ holds $\forall x$. My question ...
0
votes
1answer
27 views

Finding the decomposition of a function on a cubic spline basis of functions

In a computational project, I need to solve a partial differential equation. Standard procedure is to consider the weak formulation of the problem which maps it onto an algebraic problem. With cubic ...
0
votes
2answers
43 views

Given this operator what is inverse operator?

Given operator $$\Delta_{sym}[f(x)]=\frac{f(x+\varepsilon)-f(x-\varepsilon)}{2\varepsilon}$$ what is inverse operator in terms of summations? For instance, given operator $$\Delta_{full}[f(x)]=\...
1
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1answer
26 views

Finding a $f(n)$ such that $\sum_{k=1}^{n}f(k)$ equals the natural logarithm of a sinusoid

Let $g(n) = A\sin(\omega n + \phi)$ be a generic sinusoid, where $n$ is a positive integer. Suppose that we want to find $f(n)$ such that $f(1) + f(2) + \cdots + f(n) = \ln[g(n)]$, or equivalently ...
1
vote
1answer
44 views

Unifying Perspectives on Discrete/Continuous Differentiation

I was prompted by some recent readings, and also by this question, to try to rectify the fact that my notions of discrete and continuous differentiation have slipped away from one another. I'm most ...
0
votes
1answer
34 views

Establishing a stable state withing a discrete dynamic model

Given the following discrete dynamic model: $P_0=360$ and $R_0=80$ and $t[0,200]$ $P_t=1,18*P_{t-1}-0,002*R_{t-1}*P_{t-1}$and $R_t=0,86*R_{t-1}+0,0004*P_{t-1}*R_{t-1}$ One can rewrite the ...
1
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0answers
126 views

Mean value of discrete periodic signals

It is clear that a continuous periodic signal always takes at some point $x_m$ the absolute mean value. Then we could define the absolute mean value of the signal as the value that it takes at $x_m$. ...
1
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0answers
41 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ D_{j}f(x):=...
0
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0answers
37 views

Integration matrix

I want to do integration(summation) of a signal(x) using matrix multiplication. I am looking for a transformation matrix, I corresponding to integration such that F = I * x , where x is the signal ...
1
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0answers
150 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
1
vote
1answer
70 views

Closed form for estimated sum with different asymptotic bounds?

I found asymptotic lower and upper bounds for a summation as follows: $$ 1 - O\left(\frac{\log_2^2 n}{n}\right) \le \sum_n f(n) \le 1 + O\left(\frac{1}{n}\right).$$ If you want to write it in a ...
1
vote
1answer
627 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi (...
1
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0answers
61 views

Discretization of an Euler-Bernoulli

Given the following Euler-Bernoulli equation: $$ (s(x) w(x)'')''= q(x),\ \ x \in [0,1]$$ Could someone explain why the following discretization scheme may not be a good idea? \begin{align*} (sw'')''...
0
votes
1answer
37 views

Can any analytic function be represented as a sum of a Newton series and a periodic function?

Suppose $f(x)$ is a real analytic function whose Newton series converges. Conjecture. $f(x)$ can be always represented as a sum of its Newton expansion and an 1-periodic function $g(x)$: $$f(x)=g(...
4
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0answers
63 views

If two functions are equal to their Newton series, is their composition also equal to its Newton series?

Suppose we have two real functions $f(x)$ and $g(x)$, both equal to their Newton series expansion (let's call such function Newton-analytic): $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\...
1
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2answers
101 views

About Recurrence Relations.

I need help in order to solve the following question, Here RR is for Recurrence Relations.
4
votes
2answers
109 views

Purely “discrete” PDEs?

Usually, one formulates a system of continuous PDEs and then discretizes it in order to approximately solve it. Is there a view point that instead formulates a system of "discrete" PDEs, which ...
3
votes
1answer
104 views

Continuous differentiation for polynomials?

Has this concept been explored & if so what name does it go by? Taking a simple polynomial & its derivatives: $$y = x^3 + x ^ 2 + x + 1$$ $$\frac{dy}{dx} = 3x^2 + 2x + 1$$ $$\frac{d^2y}{dx^2}...
0
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2answers
239 views

Differencing -vs- Differentiating

When you $\tt{diff}$erence discrete observations of a function $f$ you lose one observation each time you apply $\tt{diff}$. When you $\partial$ifferentiate a $\mathcal{C}^n \ni$ function $f: \mathbb{...
1
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0answers
58 views

Need help to prove this by using natural deduction.

i m concerned to prove these by using Natural Deduction. And i am also concerned to prove it for both sides. $$\exists x(P (x) \implies A) \equiv \forall xP (x) \implies A$$ I have some difficulties ...
0
votes
1answer
201 views

How to prove this just by using Natural Deduction?

I need your help to prove this by using Natural Deduction: $$(\exists x)(p(x) \implies q) \dashv\vdash (\forall x)(p(x) \implies q).$$ I want to show the proof for both sides. It is a bit easy for ...
4
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0answers
103 views

Using discrete calculus to study convergence of series and sequences

From some personal investigation, I've noticed that all convergence tests for infinite series (at least, the real kind) can be rephrased in terms of the discrete derivative $∆f(x)$ of a function $f(x)$...