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Questions tagged [computational-mathematics]

This tag concerns computational problems central to mathematical and scientific computing. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

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2answers
27 views

What is this recursive function approximating? ($x_i = x_{i-1}^2 / x_{i-2}$)

I came across this recursive function in some code, and it is within a function called "interpolate". Essentially the rule is: $x_i = x_{i-1}^2 / x_{i-2}$ which can also be defined as: $x_i = \...
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0answers
14 views

Is the method of calculation correct for the possible permute orderings of 3x3 matrices of a valid sodoku grid? [on hold]

Screenshot of link Now suppose, we make the list mentioned but in reverse order. map in reverse ...
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0answers
14 views

Fourier transform of a product of two functions

How can I compute the Fourier transform of a function of this form $f(u)\vert u-v \vert^{a}??$ with v is a variable that appears in my computations and lives in $\mathbb{R}$. Any idea how to ...
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0answers
31 views

Workaround for Windows Magma Memory Limit [duplicate]

I found this question which said that Magma for Windows has a strict memory limit of 1.3 GB as a consequence of something weird with long ints. Is there any known workaround for this? Would running ...
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0answers
28 views

Limit of Square root 2

i want compute the limit of this function $\lim_{x \to 2} \sqrt{2-x}$ =? and draw $f(x)= \frac{x^2-x-2}{x+1}$. can you help me?
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0answers
36 views

Variance of sum of correlated variables

I want to compute the variance of this estimator $\hat{\sigma}^2 = \frac{n}{N}\sum_{i=1}^{N}\big(R_{i} - \frac{1}{N}\sum_{j=1}^{N}R_{j}\big)^2$, where $R_{1}, \ldots, R_{N}$ are i.i.d such that: $ R_{...
0
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0answers
21 views

Infinitely many or unique solution of ODE?

Consider the boundary problem $$u''+k^2u=0 \qquad (1)\\ u(0)=u_l, u(1)=u_r \qquad\\ \text{domain } \Omega = (0,1)\qquad$$ When I discretize this problem using the Finite Element Method I get the ...
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0answers
8 views

How to find basis of functions for best representation of function described in several different truncated ON systems?

I have two ON bases where I to the best $L_2$ precision possible represent the same function. Let us say that these have $n_1,n_2$ basis functions respectively for which we know the coefficient ...
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0answers
21 views

Multivariable Taylor's series [matlab]

I'm looking for a program that displays taylor's series graphs from $f$ from 1 to N. Where f is : $f : [-2,2] \times [-2,2]\rightarrow R$ $(x,y) \rightarrow sin(x+y^2)e^{-x^2-y^2} $ I'm looking ...
1
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2answers
45 views

Finite element discretization example clarification

We introduce on the domain $\Omega = (0,1)$ a mesh $0=x_0<x_1<x_2<\dots<x_{n+1}=1$ and let $V_h$ be the space of piecewise linear hat functions $\varphi_i$ such that $$\varphi_i'= \...
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2answers
22 views

Round off the number 8.03567 to four significant digits and compute the percentage error.

I rounded off the number 8.03567 to four significant digits and it'll be 8.036 but, the problem is how can I compute the percentage error because there's not enough data to compute.
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0answers
17 views

Existence and uniqueness of solution of discretized Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
0
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1answer
17 views

If $b_i = a_{i+1} - a_i$ why does $a_j - a_i = b_i + \cdots + b_{j-1}.$ for $1 \leq i \leq n-1$?

Let's say we have an array $a_1,\ldots,a_n$ and a new array $b_i = a_{i+1} - a_i$ for $1 \leq i \leq n-1$. Hence $$ \max_{\substack{i < j \\ a_i < a_j}} |a_j-a_i| = \max_{i<j} (a_j-a_i) = \...
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0answers
21 views

sum up the series $f(x) = \sum_{n\geq 1 } \cos n x /\sqrt{n}$ numerically

For $x$ close to zero, the series converge slowly. Is there a way to accelerate the convergence?
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0answers
39 views

Lambert involution

Let begin our discussion with the following curve $$ x=ye^{-y}.$$ Let choose a parametrization variable $z$ and we parametrize the above curve $$ x(z):=ze^{-z}, y(z)=z $$ We find the ramification ...
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2answers
448 views

Preserving historical information of the Collatz function?

In some sense this two equations are the same, namely $f_2$ preserves the historical information of $f_1^n$, where the exponent is function composition, but I am not sure how to show this rigorously. $...
0
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0answers
44 views

what is the mapping reduction of $A_{TM}$ to $\overline{CF_{TM}}$

first post here :) I am trying to find a reduction from $A_{TM}$ to $\overline{CF_{TM}}$. definitions: $CF_{TM}\:=\:\left\{<M>| M\:is\:a\:TM\:and\:L\left(M\right)\:is\:a\:context-free\:...
3
votes
1answer
137 views

Why wolfram alpha claimed that this $\sum_{n=1}^{\infty}\sin (\frac{n}{\sqrt{n!}}) $ is converge by test and in the same time is diverge?

I'm confused why wolfram alpha claimed that this sum $$\sum_{n=1}^{\infty}\sin \left(\frac{n}{\sqrt{n!}}\right) $$ is convergent by test criterion, and in the same time is divergent in result below ...
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0answers
17 views

$\mathcal{R}\{\bullet\}$ for matrix-vector product

I am studying about Hessian free optimization methods and I came across a method to convert Hessian into Gauss Newton approximation: $H \approx J^THJ = G$, here G is the Gauss newton approximation. ...
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0answers
11 views

Discretizing a parabolic PDE with finite volume method

I want to discretize the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$ Given ...
1
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3answers
61 views

Inverse of SPD matrix + identity

I didn't know exactly where to post this question. I feel like it falls between Computer Science and Mathematics, so I'm sorry if it doesn't fit here. I need to calculate $(A+\alpha I)^{-1}$, given $\...
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0answers
32 views

Solving linear system of equations by block

I want to solve the following large system of linear equations: \begin{align} \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2}...
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0answers
16 views

Discretization Dirichlet boundary condition for Elliptic PDE with finite volume method

I want to discretize the following equation using FMV: $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ To this end, let $V_i \subset\Omega$, $i=1,\dots,N$...
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0answers
24 views

Finding MSE for cauchy distribution estimators in R. Is this method and findings correct?

Hi I'm required to find the MSE of a few estimators of the cauchy distribution for location $\theta$. I am using the following R code but my MSE values are coming out incredibly worrying. I am ...
0
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0answers
11 views

Proof Using Strong Reducibility

I read the following proof so many times and I could not, for the life of me, figure out how the professor went from (1) to (2). I know to prove $\overline{K} \le A$ , where $\overline{K}=\{x:\...
0
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0answers
20 views

Angle between ray and triangle mesh

Are there any algorithms to quickly compute the angle between a ray and a large triangle mesh? (Or, at the minimum, whether the angle is less than some fixed threshold.) By angle from a ray I mean ...
0
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1answer
128 views

Integer solutions to the equation $a^3+b^3+c^3={\color{red}{6}}$ [closed]

I once thought that this equation $x^3+y^3+z^3=6$ has only a few smaller integer solutions. Until somebody told me this $6=192722201207819^3+162765491944499^3+(-225522344776678)^3$ existed. I don't ...
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0answers
22 views

How do I determine significant “moments of change” in a forecast?

I have a function returning a trendline and a forecast for that line. I need to identify "significant" moments of change (ie, in slope) in the forecast region. Each timestep in the series is 15 ...
0
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0answers
6 views

Find CDF of $Z=\{X_1+X_2, X_1<X_3\}$ using computational software

Let $X_1, X_2$ and $X_3$ be three independent exponential random variables. The PDF and CDF of $X_i$ with parameter $\beta_i\geq 0$ are $$ f_{X_i}(x_i)=\beta_i e^{-\beta_i x_i} \text{ for } x_i\geq ...
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0answers
24 views

Byzantine Fault Tolerance Threshold of Bitcoin: 1/2 or 1/3?

Although, this question is related to the Bitcoin network; however, its calculation is more relative to the Mathematics. So, let's bring it up here: According to this answer: https://bitcoin....
3
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1answer
36 views

Adding artificial dissipation to continuity equation

I'm trying to solve the system of equations that relates to blood flows in arteries i.e. flow in elastic tubes. The system looks as follows $$\frac{\partial A}{\partial t}+\frac{\partial\left(Au\...
0
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0answers
17 views

Why is there noise for higher bandwidth intensity distributions?

I have been trying to computationally evaluate the following E-field computationally: It is basically the sum of waves of a given wavelength, weighted by the square root of a gaussian distribution ...
1
vote
1answer
30 views

Creating a connected undirected graph with connectivity k

I am working on a computing mini-project and I am stuck because of the following problem: "Given a number of nodes N and a connectivity k such that $N > k + 1$, is it always possible to construct ...
0
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1answer
11 views

Could use some help in probability question

Two distinct numbers are selected simultaneously and at random from the set {1, 2, 3, 4, 5}. What is the probability that the smaller one divides the larger one? Express your answer as a common ...
1
vote
1answer
41 views

Cartesian Product over a list of objects in MAGMA

I'm currently trying to create the Cartesian Product of certain objects (namely: Character Tables of different finite groups). However, it seems like I don't really understand the car<$...$> ...
1
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0answers
30 views

Any evaluatory cheap polynomial regression alternatives for bounded input?

Background: I was trying to approximate $\ f(x) =1/2^x,(f(x) <10^{-38}):127>x>0 \\$. The polynomial had to be of relatively low degree such that when evaluated using horner's method to be ...
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0answers
24 views

Trying to understand formula $T(N) = |Ops| \times \frac{Time}{Op} $ for the time an algorithm takes

I am not sure how to interpret the formula given in my professor's slides. We assume (for simplicity) that each operation takes the same amount of time to complete. Time an algorithm takes on ...
2
votes
1answer
136 views

How do I solve this combinatorics problem?

I have eight sets, $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$ and I know that (note that I mean the cardinality of these sets, not the elements!) $|A_i| = |B_j| = 4$ for all $i, j$ $|B_i \...
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0answers
9 views

Prove the languages |L<M>| = 2 and |L<M>| $\not=$ 2 to be non-Turing recognizable or non-recursively enumerable

I am trying to prove the non-recursively enumerable property of two languages. L = {$\langle M \rangle$: |L$\langle M \rangle$| = 2}. and L = {$\langle M \rangle$, |L$\langle M \rangle$| $\not=$ 2}. ...
2
votes
1answer
99 views

Modelling congestion games in python without tons of for loop

In the bidirected triangle network as shown below, 4 agents $\{s1,s2,s3,s4\}$ have their own destination $\{t1,t2,t3,t4\}$. I am trying to model this problem with a python script without any game ...
0
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0answers
39 views

Maximum principle for discretized ODE

I discretized the following ODE using central finite differences for 1st and 2nd derivatives: $$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$ The ...
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0answers
15 views

Compare Geometrical Shape and Subset (Transformation Invariant comparission)

I am working on CAD Automation projects, which requires comparison of two geometrical shape (Transformation invariant comparison), the expected outcome could be exact match or subset match. Please do ...
0
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0answers
27 views

Quick way to compute $\int_{0}^{\infty} f(x) g(x+y)\, dx$ for $y \in [0,\infty)$

When we want to compute: $$ h(y) = \int_0^y f(x) g(y-x)\, dx, $$ for $y \in [0,\infty)$ we can make use of a Fast Fourrier Transform to quickly compute these integrals. I have a similar problem where ...
0
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0answers
39 views

Finding the Proportions of Topological Disks

I am currently in the process of writing an internal software package that will be used for computational geometry research. I am interested in being able to programatically generate isotoxal ...
0
votes
1answer
32 views

Proving that product of general matrices has small spectral radius

In a Jacobi type of iteration for finding solution to a linear system $Ax=b$, one writes $$x_i^{(k+1)} = Gx_i^{(k)}+c,$$ where $x_i$ is the $i$-th component of vector $x$ and $G=D^{-1}N$, $c = D^{-1}...
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1answer
51 views

Discretization matrix for 3D Poisson equation

It is known that the 2D Poisson equation defined on a domain $\Omega$ (let's say $\Omega := (0,1)^2$) with Dirichlet boundary conditions $u(x,y)_{|\partial \Omega}=g(x,y)$, $$u_{xx} + u_{yy}=f$$ can ...
2
votes
0answers
14 views

How do we plot a domain wall between two strings?

Intuitively, an angular vector field $\theta(x)$ such as the following would describe a domain wall between two vortices/strings of opposite winding number. I would like to draw this on a computer ...
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0answers
17 views

Connectionism and AGI

Marvin Minsky and Papert prove that the XOR function($\newcommand{\xor}{\oplus} x \oplus y$) cannot be implemented by a single layer perceptron and ended connectionism research for some time. My ...
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0answers
17 views

Doubt on plotting bivariate Gaussian contours

So I'm trying to understand the implementation of a function that plots the bivariate normal contours of two variables. Thought I'd post here since the question is primarily mathematical in nature and ...
0
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0answers
31 views

Is there a program able to compute index of quotient sets and the size of the orbits?

I am interested to compute Hurwitz-Kronecker class numbers and this asks to compute a sum over representatives of a quotient set. Do you know any implemented program that does the job for any quotient ...