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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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16 views

Modified conjugate gradient methods for densely optimized calculations?

Sometimes when solving very sparse equation systems $$Ax = b$$ with conjugate gradient using computers, if $A$ is a very sparse matrix, it can be difficult to utilize the hardware computational power ...
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0answers
22 views

Question on matrix representation of equations

The question is: In a flock of wild Hoatzins, the females can be classified as being either chicks (up to 1 year old) or adults. Each year, for every 100 adult females, 50 female chicks are born. ...
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0answers
30 views

Verify that an eigenvector is orthogonal to 2 other eigenvectors

Verify that the eigenvector v3=(4,-2,1) corresponding to the eigenvalue e2=16 is orthogonal to the eigenvectors v1=(1/2,1,0) and v2=(-1/4,0,1) (both) corresponding to eigenvalue e1=-5 All I can ...
2
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2answers
39 views

Infinite summation question and i need to make an algorithm for finding the summation?

$$\sum_{n=1}^{\infty} \frac {x^{2n -1}} {2n!}$$ for the algorithm i use == $d=\frac {a_{n}} {a_n - 1}$ And other hint that I have is the for $\sum_{n=1}^{\infty} \frac {x^{2n}} {n!}$ ; $d = \frac{x}...
1
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1answer
33 views

Estimating probabilities of switching between two states

I wasn't sure on what Stack Exchange forum to ask this. I'm trying to computationally model a situation where I have an object that has two possible states and can switch between them. I know that ...
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0answers
21 views

calculate hessian matrix in markov random field

i try to learn markov random field parameters , for this i want to calculate the hessian of the probability function, i calculate the hessian in 2 way and got two different answer and i cant ...
1
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1answer
47 views

Can we mesh a rectangle using only heptagons?

I know that according to Euler's formula in graph theory the average number of edges or vertices in a planar connected graph cannot exceed six. So it seems according to Euler's formula its not ...
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0answers
29 views

Computational method 2nd order differential equations : 2 conditions at two boundaries

I have the following system of second order differential equation : $\left\{ \begin{split} &\frac{dy}{dt}=F(t,x,y) \\ &\frac{dx}{dt}=y \end{split} \right.$ with $y(t)$ and $x(t)$ defined ...
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0answers
30 views

Consider the expression $1-\cos x$. [closed]

Consider the expression $1-\cos x$. For what values of $x$ does this expression suffer from loss of significance? Construct an example which illustrates the loss of significance. Derive an alternative ...
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0answers
19 views

How to solve this problem by induction method, and derive neumann formula?

Some explaination would be helpful if$\quad$ $a^n = a(a - h)....[a - (n - 1)h ]$ $\quad$ and $\quad$ $a^0=1$ $\quad$ then $\qquad$ prove $\quad$ $(a + b)^n = \sum_{m=0}^n C_n^m a^{n - m} b^{m}$,...
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0answers
15 views

Tensor fields or operators of order $\geq 2$ being diffused by tensor fields of order 2?

Background: I am aware that in for example physics tensor fields can be used to describe things like properties of materia. Like heat conduction in macroscopic media (imagine a thermos, heat can flow ...
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1answer
25 views

Optimizing Polygon Search

I split de world in X random polygons. polygons on map Then I am given a coordinate C1, for instance (-21.45, 7.10), and I want to attribute the right polygon to this coordinate. The first solution ...
2
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2answers
36 views

How many bits to represent these numbers precisely?

Consider the following numbers: $$19=10011_b, 12.75=1100.11_b, 7.125=111.001_b$$ What is the minimum number of bits necessary to represent the above three numbers precisely? A system like the IEEE ...
2
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0answers
53 views

Instability in 2D steady state heat equation with variable thermal diffusivity

Thank you in advance for your time and consideration with this issue. Any suggestions would be greatly appreciated! I'm trying to numerically solve the steady state heat equation in 2D (x,y) with ...
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1answer
47 views

How many ways to pick up 1 or 3 chocolates from $n$ chocolates?

There are $n$ chocolates. We can pick either $1$ or $3$ chocolates at once. In how many ways can we pick all the chocolates? The variable $n$ can have any value, $30,100,500$ etc. What is the ...
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1answer
19 views

Number of ways of not placing k same color balls together ?? [closed]

there are n red balls and m white balls .How to find number of ways to place the balls such that at max k balls of same color are together.How to understand and solve these kind of problems .I have ...
6
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0answers
35 views

Identifying ODE types for solving by hand and when to use computers instead

So this questions relates to my specific ODE but also ODEs in general. I am a big fan of solving ODEs by hand, but I also know when to give up and use, say, Mathematica to solve it for me. Having ...
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0answers
34 views

Compute Machin's function by hand

I use Machin's function \begin{equation} \pi = 16 \cdot \left(\frac{1}{5} - \frac{1}{3\cdot 5^{3}}+ \dotsm\right)-4 \cdot\left(\frac{1}{239} - \frac{1}{3\cdot 239^{3}} +\dotsm\right) \end{equation} ...
3
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1answer
63 views

Fast Fourier Frames (FFF), do they exist, and if so, how to calculate them?

Short background to question: Maybe one of the most famous mathematical transforms is the Fourier Transform, which has countless applications across all possible sciences and engineering branches. One ...
0
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0answers
12 views

Compute the hull of nonnegative linear combinations of a finite set, and the extreme points of the intersection of two polyhedra

Let $\mathbb{R}^d$ be $d$-dimensional Euclidean space Let $\Delta=\{x\in\mathbb{R}^d_+:\sum_{i=1}^dx^i\leq1\}$ ($x^i$ is the i-th coordinate of $x$) (Equivalently, $\Delta$ is the convex hull of $\{(0,...
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2answers
97 views

A dollar amount $\${-}67.9{-}$ is divisible by 72. What is this number?

A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $\${-}67.9{-}$. Find a way to make the price of each turkey round at the second digit ...
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0answers
22 views

Definition of a “Column Saxpy”

In GVL's book Matrix Computations, Problem 1.2.3 says to "Give a column saxpy algorithm for the $n$-by-$n$ matrix multiplication problem $C=C+AB$ where $A$ is upper triangular and $B$ is lower ...
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0answers
17 views

Energy minimization functionals for Bayesian statistics?

The latest years I have spent some time learning to do numerical minimization of energy functionals for.. ehm various applications, however I sometimes feel my background in probability theory is ...
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0answers
13 views

Algorithm to keep scrolling lyrics' current line vertically-centered as much as possible?

I'm working on a web app that allows users to generate lyrics videos. You can see an example of how it works so far here. As you can see, it currently always shows the current line of lyrics in the ...
1
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0answers
60 views

Efficient way to check a prime of Prime digits

I was trying to solve a question on number theory. It says, given a range (start, end $ \le 10^{15}$ and end - start $ \le 10^9$), how many prime digit prime numbers exist? Generating prime digits ...
1
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1answer
36 views

Making an expression stable for small values

The following expression shows significant numerical differences in a program when I compile in x86 (32 bit) versus x64 (64 bit), when $a$ is small: $$ \left( \dfrac{1}{a} - b \right) \left( 1- \exp(-...
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0answers
43 views

Faster Fourier - Bessel Coefficient Calculations on Python

So I'm trying to get the fourier-bessel coefficients of a very large array of numbers that are around 1 million points in size, but I'm coming across some speed issues with calculating $J_{o}(x)$ for ...
0
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0answers
25 views

Compute specific elements of the inverse of a matrix of the form $M=X^\mathrm{T} (Q+D)X$, with symmetric $Q$ of low rank and diagonal $D$

I have a square matrix $M\in\mathbb R^{m\times m}$ of the form: $$M=X^\mathrm{T} (Q+D)X$$ where $X\in\mathbb R^{n\times m}$, $Q,D\in\mathbb R^{n\times n}$. I want to invert $M$, and I know that $Q$ ...
1
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0answers
28 views

How do we compute homology from the Čech complex $\text{Čech}(r)$ with $B^{\infty}$ balls covering the Hilbert manifold $M$?

Suppose a Hilbert manifold $M$ is covered by the union of $\infty$-balls (in the sense of Baire spaces), namely $M=\bigcup_{\alpha\in A}B^{\infty}$, without knowledge of intersections. The only ...
1
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1answer
67 views

There are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$

Conjecture: Given $a,b\in\mathbb Z^+$ there are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$. I would like help with a proof or a counter-example. Tested for millions of pseudo random numbers.
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1answer
78 views

Compare Calculation-time for two methods

Method $1$ (Machin): $$ \pi = 16 \left(\frac{1}{5} - \frac{1}{3\cdot 5^3} + \frac{1}{5\cdot 5^5} - \frac{1}{7\cdot 5^7} + \ldots\right) -4\left(\frac{1}{239} - \frac{1}{3\cdot 239^3} + \frac{1}{5\...
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0answers
56 views

What does it mean that if the cover of $k$-fold intersections is not contractible it takes the form of a spectral sequence of cohomology?

I asked the following question in a previous post: Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only ...
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0answers
38 views

Formula for this point distrubution on a sphere?

A series of points are plotted on a unit radius sphere, see image above. Each point represents a vector coming from the origin. First, points on XY and XZ planes are created using a certain equal ...
6
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1answer
198 views

How do we compute homology from the nerve $\text{Nrv}(\Sigma)$ for $\Sigma=\{S^n,\dots,S^n\}$ covering the CW complex $M$?

Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only requirement is that the union covers $M$. Let $\Sigma=\{...
0
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1answer
64 views

What is the exact value of this : $5\int_{0}^{\infty}\exp(-x^2 \text{erf}(x))x^{\sin x+\frac12}dx$?

My curiousity is to get more integrals about the constant $\pi$ using special functions. I have used some special functions as shown below in the integral: $$5\int_{0}^{\infty}\exp\left(-x^2 \text{...
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1answer
48 views

Factorisation of algebric expression

If $a+b+c+d+e=0$ than factorise $a^3+b^3 +c^3+ d^3 +e^3$. . I know that if $a+b+c+d=0$ than $a^3+b^3 +c^3+ d^3 = 3(a+b)(a+c)(a+d)$. Similarly can we say something about $a^3+b^3 +c^3+ d^3 +e^3$. It ...
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1answer
40 views

How can I calculate these using a program or WolframAlpha?

I have this expression: $ S_{n-1} = \frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)} r^{n-1} $ I have this sequence: $ S_{n} $ $ \frac{1}{2}(S_{n}) + S_{n-1} $ $ \frac{1}{2^2}(S_{n}) + \...
1
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1answer
32 views

Golub and Van Loan 3rd Ed Basic Algos Getting Started

I am trying to get my brain back into matrix algebra, and I am starting right from the beginning of the standard Matrix Computations, by Golub and Van Loan ( 3rd Edition ). A couple of the practice ...
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1answer
45 views

Heuristic to integrate (or find all extrema of) a function

I have a function $f:[0, T] \to [0, 1]$ which is expensive to evaluate and cannot be differentiated numerically. I am interested in computing $$\int_0^Tdt \min\left(\frac{Df}{Dt}(t), 0\right) \,.$$ ...
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1answer
31 views

Can we combine convolution and higher powers for locally maximising a function?

Can we somehow find local maximum function value (for strictly positive functions) using a convolution? My idea is based on the result that $$ \lim_{p\to \infty}\left[\frac{1}{N}\sum_{k=1}^N {(a_k)} ^...
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0answers
16 views

Recalculating or estimating percentile statistics for new data points

After reading The Math Form: Re-Calculating the Standard Deviation I'm curious what sort of statistics can be calculated or estimated in a similar manner. The post details how—given just the mean, ...
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2answers
48 views

Defining a Neighborhood Around a Point Whose Elements Have Different Units

Suppose I have some n-dimensional point $ \mathbf{x} = [x_1,...,x_n] $ where some of the $x_i$ have different units (for example, $x_1$ in Ohms and $x_5$ in seconds). Now if all the elements had the ...
0
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0answers
23 views

Minimal number of clues required to uniquely determine a Latin Square

Let's suppose I want to fill a n*n array with numbers occuring from 1 to n. One number is allowed to appear only once in every column/row. The solution becomes unique when there is only one possible ...
4
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3answers
381 views

How do I convert a simple algorithm to a mathematical notation?

I am trying to write a simple algorithm given below in mathematical notation. I wrote the formula up to a certain point, but I have no idea of restrictions. For example, I used the NULL statement even ...
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2answers
158 views

Why are there researchers in PDE theory when we can instead build numerical solvers?

I have a bit of a naive question: Why are there researchers in PDE theory, e.g. people who work on analysis and PDEs, when instead one can spend their research days building numerical solvers to get ...
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0answers
36 views

How to estimate the order $\max{(\alpha, 1-\alpha)}$

The data sequence is generated from simulations using one algorithm. It should have the order $O(n^{\max{(\alpha, 1-\alpha)}})$. I know it can be approximately modeled as $a(n) = b_1n^{\alpha}+ b_2n^{...
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0answers
13 views

Space-Efficient Calculations using Symmetric Kronecker Product: $(A\otimes_sB)^{-1}x$

Introduction: The Kronecker product of two matrices is defined as: $$\mathbf{A} \otimes \mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots &...
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1answer
46 views

An algorithm to reduce a factorisation problem into an easier one?

Question I recently wondered about my own factorisation method (see method) to generate a smaller number to factorise than the original one. What are some "good methods" to choose $\lambda$ and $\...
0
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0answers
16 views

Approximate bound algorithm

I was reading an article, an I saw the following reduction: If we can solve a problem $P$, then we can find an algorithm that approximates the decision version of the shortest vector problem (SVP) to ...