Questions tagged [discrete-calculus]

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15 views

Discretizing a function with 0 indexing

I have already asked this question and got it answered : Discretizing a mathematical equation. Now i want to adjust the indexes, i.e originally it was asked for natural numbers correspondence i.e $\{x,...
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1answer
11 views

Number of equal pairs from a set of size n in range 1-nlog

Let A be a set of n DIFFERENT natural numbers in range [1, 2,...,n*log n] The number of pairs from this set is n choose 2 (number of different ways to choose 2 objects out of n where repettion is not ...
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0answers
18 views

What is the characteristic polynomial exactly and why can we use it to solve linear ODEs and recurrence relations?

Disclaimer, from now on when I say ODE or recurrence relation I am referring specifically to the linear and homogeneous kind. I know how to solve linear ODEs and recurrence relations. I know when I ...
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0answers
10 views

Discrete equivalent of Sobolev norms and numerical experiment

I am solving a boundary value problem (BVP) that involves a system of equations (similar to the Euler or Navier-Stokes equations) for which, at this moment, there exists no sufficient theory to define ...
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0answers
33 views

How to solve linear difference equations with non-integer steps?

While reading Theory of Probability by Jaynes (p. 281), I came across this difference equation: $$ M(n,G) = \sum_{j = 1}^{m} M(n-1, G - g_j) $$ where $g_1, \dots, g_m \in \mathbb{R}$. The first ...
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21 views

Solution to 2D discrete laplacian on a rectangle

I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)\in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary ...
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0answers
9 views

extension of discrete time process to the unit interval

I am reading a proof and i don't understand some lines of cumputations. We have a Euler Scheme SDE given by $X_{n}(\frac{k+1}{n}) =X_{n}(\frac{k}{n}) +b(X_{n}(\frac{k}{n}))/n + \sigma(X_{n}(\frac{...
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2answers
44 views

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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0answers
57 views

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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1answer
108 views

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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0answers
25 views

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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0answers
16 views

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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0answers
18 views

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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1answer
44 views

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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2answers
296 views

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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0answers
15 views

Discretization and grids approach

In a text about multigrid methods, there is the following in the introductory section. $L_h=-\Delta_h\equiv \text{standard five-point } O(h^2) \text{ approximation of the PDE operator }L\text{ as ...
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2answers
71 views

Finding a general formula for a summation with discrete chain rule

I'm trying to find a general formula for the summation $$\sum_{k=1}^{n} \frac{k+1}{2^k}$$ I think I can use the discrete chain rule to do this, but I'm only able to figure out some of the steps. Here'...
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0answers
33 views

Show that a simple graph $G$ with $n$ vertices and $m$ components has at most $(n - 1)(n - m + 1) / 2$ edges. [closed]

Show that a simple graph $G$ with $n$ vertices and $m$ components has at most $(n - 1)(n - m + 1) / 2$ edges. How can we prove this using the graph teory? Can any one help me is it by qudratic ...
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0answers
18 views

Testing for Mean Differences for Two Discrete Random Variables

Two discrete random variables come from two different distributions. If you know the mean and variance of each discrete distribution, is there a test to show that the two means are statistically the ...
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1answer
64 views

Using summation by parts on a combination

I am trying to expand the series $\sum_{k=1}^{n}\binom{n}{k}$ when $n$ is a integer greater then zero, by using summation by parts. I am using the following definition of summation by parts.\begin{...
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1answer
32 views

fundamental matrix solution for difference equation

I need your help please, we have the system X(t+1)= \begin{bmatrix} -1 & \frac{2+(-1)^{t}}{2} \\ \frac{2+(-1)^{t}}{2} & -1 \end{bmatrix} X(t) By using this formula $\phi(t)=A(t-1)...A(1)A(0)$...
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3answers
181 views

Prove by Induction on k. using Fibonacci Numbers

$$F_{k-2}+F_{k-4}+...+F_{k\,mod\,2+2}=F_{k-1}-1, \quad \quad if\: k\geq2.$$ This equation is to prove by induction on $k;$ the left-hand side is zero when $k$ is $2$ or $3$. Therefore $k_{1}$ is the ...
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0answers
42 views

Help debugging a $2 = 1$ proof and clearing further confusions [duplicate]

I found this proof of $2 = 1$ online and as usual tried to debug it. Consider the following true statement: $x^2 = x + x + x + ... + x$ ($x$ times) If we differentiate both sides, we get: $$2x =...
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2answers
53 views

Is there a reason why this function does not exist/can't be found?

I'm looking at a function $f\colon \mathbb N \rightarrow \mathbb R$, defined such that $(\Delta f)(x) = 1/x$. However, I know such a function does not exist or has not been found yet. I'm interested ...
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0answers
43 views

Solving $div(A)=B$ for unkown $A$

I'm mechanical PhD student and I'm trying to solve the following problem discretized on the domain $(x,y)\in[0,1]^2$. Find $D$ such that $$\frac{\partial}{\partial x}\left(\rho D\frac{\partial c}{\...
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0answers
48 views

Can a recursive, continuous integral be approximated with Gauss-Legendre or similar?

Maybe I need to reformulate(?). Suppose there is this simple function: $$f(x)=\int_a^b{x \text{d}x}$$ If it were to be discretized, there would be losses due to sampling. In order for the errors to ...
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1answer
113 views

Closed forms of sums of rational or real powers

I am curious about whether a closed form expression of $$\sum_{k=1}^{n}k^\alpha $$ for $\alpha \in \mathbb{R}$ exists in terms of special functions. Clearly for the natural number case we have the ...
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0answers
51 views

Help identifying mistake during sum manipulation

This is a problem from the book Concrete Mathematics. It is about finding a closed formula for a sum. I came up with one but when checking with values of the sum at different points the formula fails. ...
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1answer
38 views

Stacking Uncertainty Over Time

I am currently working on a probabilistic model of a known linear system. It is desired to estimate the value of a system process at discrete time steps. Each time step could be described as follows:...
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0answers
85 views

Moser invariant curves in discrete dynamical systems and how they give stability

I have seen a theorem which says that under certain hypothesis, given an elliptic fixed point $P$ for particular discrete dynamical systems ($X_{k+1}=S(X_k)$ with $S$ a conservative diffeomorphism not ...
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0answers
73 views

Can we derive the analytical expression for this discrete fractional differential operator?

Experimenting about on this question, I some days ago investigated the polynomial equation: $$p(x) = x^6 - d =0, p'(x) = 6x^5$$ I started with ${\bf X}_0{\bf = I}$ Then the iteration $$\begin{...
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1answer
105 views

Does this Newton series describe an interesting function, if any?

I was reading about how the harmonic numbers are analogues to the logarithm in that $\displaystyle \log(x) = \int \frac{1}{x}dx$ and $\displaystyle H_x = \sum \frac{1}{1+x} \delta x$ Where indefinite ...
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2answers
195 views

“Discrete” $\sin(x)$, $\cos(x)$ power series derivation

The forward difference operator (discrete derivative) $\Delta$ is defined as $\Delta f(x) = f(x+1) - f(x)$. The "discrete $e^x$" / eigenfunction of $\Delta$ is $2^x$. Since $2^{x+1} - 2^x = (2-1)2^x ...
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0answers
36 views

Can we find a solution for this indefinite sum using finite calculus?

I was recently introduced to Finite Calculus through Graham and Knuth. I stumbled upon this sum earlier today in a problem posted in this site, although I cannot link to it right now because I'm on ...
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0answers
185 views

Solving a difference equation with two variables

Consider the following difference equation: $$ {c_1}{a_{n + 3}} + {c_2}{a_{n + 2}} + {c_3}{b_{n + 2}} + {c_4}{p_{n + 2}} + {c_5}{p_{n + 1}} + {c_6}{p_{n + 3}} + {c_7}{p_{n + 3/2}} + {c_8}{p_{n + 5/2}} ...
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1answer
81 views

Solving discrete version of Poisson's equation

I have an image, which is a rectangular array of pixel values. I have the Laplacian of the image, which is computed as $\Delta I = I_{xx} + I_{yy}$ where $I_{xx},I_{yy}$ refer to the second ...
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2answers
270 views

discrete Laplacian on nonuniform rectangle grid

I wonder if there is a way to extend the finite difference discretization of the Laplacian on a uniform grid to a nonuniform grid. More specifically, I am not sure that the finite differences ...
2
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1answer
69 views

Simply Connected Complexes and a Combinatorial Poincaré Lemma

On the wikipedia page for conservative vector fields there is a proof sketch given for the following result: Wikipedia: Let $U$ be open subset of $\mathbb{R}^2$. Then all closed differential $1$-...
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1answer
96 views

Discretizing a formula for simulation

I have some molecules in different positions of a plane. Each molecule has a number which is shown by $\hat{f}$. I want to find an continues field is written as: $\int |\nabla \cdot f|^2 d^2r$. ...
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0answers
164 views

How to use explicit Runge-Kutta methods for a PDE with different terms?

Recently, I have been trying to discretise a PDE, however, I realized I may be lacking some fundamental knowledge regarding numerical methods. I have the following PDE to discretise: $$ \frac{\...
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1answer
38 views

how do i open show $(A_1 \cup A_2) -(B_1 \cap B_2)=(A_1-B_1)\cup(A_1-B_2)\cup(A_2-B_1)\cup(A_2-B_2)$?

I'm having problem with proving this following relation: $(A_1 \cup A_2)-(B_1 \cap B_2)=(A_1-B_1)\cup(A_1-B_2)\cup(A_2-B_1)\cup(A_2-B_2)$ i understand the logic behind it, but when i to open $(A_1 \...
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0answers
44 views

Order of difference quotient

I am very new to numerical computation and want to check if it is better to implement $$\partial_x\left(\frac{1}{x}\cdot \partial_x(x\cdot y(x)\right)$$ or if it is faster/more accurate to do $$\...
2
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1answer
474 views

What is in practice an anticausal filter?

Could someone enlighten me about anticausal filters? Actually, I'm trying to understand why the function $\dfrac{1}{1-az}$ is the transfer function of a first order anticausal filter. I think I ...
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1answer
50 views

Inconsistency between a continuous and a discrete version of a stochastic process?

Consider an Ornstein-Uhlenbeck stochastic process $(r_t)_{t\geq 0}$ given through the SDE: $$dr_t=\kappa \left( \mu-r_t\right)dt+\sigma dW_t,$$ where $r_0, \kappa, \mu$ and $\sigma$ are given positive ...
0
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1answer
35 views

How to evaluate the operation of a differential form on a vector?

While studying some Discrete Calculus following Grady and Polimeni (2010), I've found the following problem. I would like to know how the author found the result w(v) = -8. I've found that the vector ...
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2answers
79 views

Can a two-variable discrete linear combination be transformed into a one-variable discrete monotonic sequence?

Consider, for example, the discrete linear combination $$F(m,n) = Amn + Bm + Cn + D$$ where $A,B,C,D$ are non-zero positive integers and constants, and $m,n$ are non-negative integers and variables. ...
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0answers
133 views

How to discretize a certain integral

I would like to discretize the following integral operator: $$\frac{1}{s^2}\sum_{j=1}^N\mu_j\int d\mathbf{x}d\mathbf{x}'f(\mathbf{x})f(\mathbf{x'}) \left(x_j + x'_j - 2\mu_j\right)\hat{a}^\dagger(\...
1
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1answer
345 views

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 ...
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1answer
232 views

Free and Dummy Variables

I have a few questions regarding free and bounded (dummy) variables. Let me lay out what I know. (Please tell me if this is incorrect) and then I'll show you what is confusing me. Let P(x,y) mean ...
1
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1answer
104 views

Fractional derivative for unequal time steps

I would like to implement fractional derivative in my simulation model. Most of the literature I find, discuss techniques of evaluating the fractional derivative using constant time-steps. Are there ...