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10 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 ...
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11 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 ...
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2answers
38 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 ...
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1answer
25 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 ...
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1answer
35 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 ...
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1answer
29 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 ...
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0answers
54 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 $x_m$. ...
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28 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 $$ ...
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22 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 ...
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89 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 ...
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1answer
59 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 ...
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1answer
58 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 ...
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1answer
49 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*} ...
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1answer
34 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 $g(x)$: ...
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58 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 ...
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2answers
66 views

About Recurrence Relations.

I need help in order to solve the following question, Here RR is for Recurrence Relations.
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2answers
96 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 ...
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1answer
61 views

Continuous differentiation for polynomials?

Has this concept been explored & if so what name does it go by? Taking a simple polynomial & its derivatives: $$y = x^3 + x ^ 2 + x + 1$$ $$\frac{dy}{dx} = 3x^2 + 2x + 1$$ ...
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2answers
157 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: ...
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0answers
57 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 ...
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1answer
147 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 ...
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0answers
91 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 ...