# Questions tagged [discrete-calculus]

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93 questions
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### How to prove periodicity Modulo b

We have the sequence $$x_n=a^n \mod{b},$$ where $a$ and $b$ are positive integers. How to show that it's periodic? It is intuitively clear but I have no clue how to prove it rigorously from first ...
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### Discretization of continuous model with white noise to use Kalman filter later

I have this system which describes dynamics of a car in 2D space. The dynamics are governed by Newton's law g(t) = ma(t). The final task is to use Kalman filter on discretized system to estimate it's ...
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### Find the sum of $\sin^2(n)$

I have no clue how to solve this a detailed solution would be great$$\sum_{n=1}^N \sin^2(n)=?$$
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### Necessary and sufficient conditions for the existence of the Newton Series of a function $f: \mathbb{N} \longrightarrow R$

I’m wondering if a function $f: \mathbb{N} \longrightarrow R$ can be represented as a Newton series given that all its forward differences exist. The first thing I searched up was a result in complex ...
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### Deriving the discrete-time lowpass filter from the Laplace equation

I was trying to demonstrate the discrete-time expression of a lowpass filter: $y_i = \alpha x_i + (1-\alpha) y_{i-1}$ However my result is completely off the target so I am wondering which of my ...
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### Nonlinear Least-Squares Deconvolution Help

Starting with the following equation $p_m(t) = p_0 - \int_{0}^{t} q_m(\tau) g(t-\tau) d\tau$ ....... (1) Where $p_m$ and $q_m$ are measured data points, $p_o$ is the initial value and $g$ is ...
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### Derivative of a sum w.r.t its limits

As an analogue of $$\frac{d}{\,dx} \int_a^x f(y) \,dy = f(x)$$ Could we define something for an expression of the derivatives of partial sums; something like $$\frac{d}{\,dn} \sum_{k=a}^n f(k)$$?
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### Integration with discrete data - Matlab

I need to integrate function $\int_0^1 pur\mathrm{d}r$, where I only have discrete values for $p$,$u$ and $r$. So, if I multiply these values, would it be correct to integrate only that final value ...
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### Quantifier Order Matters?

Does the quantifier order matter in this problem? I don't think it does. (∃y in R)(∀x in R)(x+y=x) For some real number y and for all real numbers x, x plus y equals x. I assert this to be true. (I ...