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### Discretization of an EXP function

I have a function in the form of $V = a[1-exp(t-t_0)]$ and $V_0=0$. I'm using this formula in discrete system and I need to discretize this formula and solve it every T seconds and get the $V_{k+1}$ ...
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### Approximation of “smooth” discrete functions

Assume $f : \{1,\ldots,n\} \to \mathbb{C}$ satisfies $|D^\ell f(i)| \leq C$ for all $i \in \{1,\ldots,n-\ell\}$ and all $\ell \in \{0, \ldots, k\}$ for some $k \in \mathbb{N}$. Here, $D$ denotes ...
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### 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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### 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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### 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 ...
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### 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 $f$ ...
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### 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, ...
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### 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?
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### 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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### 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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### 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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### 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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### 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'')''...
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### About Recurrence Relations.

I need help in order to solve the following question, Here RR is for Recurrence Relations.
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### 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 ...
Has this concept been explored & if so what name does it go by? Taking a simple polynomial & its derivatives: $$y = x^3 + x ^ 2 + x + 1$$ $$\frac{dy}{dx} = 3x^2 + 2x + 1$$ $$\frac{d^2y}{dx^2}... 2answers 246 views ### Differencing -vs- Differentiating When you \tt{diff}erence discrete observations of a function f you lose one observation each time you apply \tt{diff}. When you \partialifferentiate a \mathcal{C}^n \ni function f: \mathbb{... 0answers 58 views ### Need help to prove this by using natural deduction. i m concerned to prove these by using Natural Deduction. And i am also concerned to prove it for both sides.$$\exists x(P (x) \implies A) \equiv \forall xP (x) \implies A$$I have some difficulties ... 1answer 202 views ### How to prove this just by using Natural Deduction? I need your help to prove this by using Natural Deduction:$$(\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 ...
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 $f(x)$...