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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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Find the “region of interest” of an unknown function

Given an unknown function $f:\mathbb{R} \rightarrow \mathbb{R}$, is it possible to find it's region of interest? By that I mean either the range in which $f$ does not converge or diverge, e.g. $f(x)=(...
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A* implementation for 4D tesseract (or higher dimensional structures)

I am an undergrad Computer Science student. I have seen implementations of A* (shortest path algorithm) for a 2D square grid of nodes using the Euclidian distance as a heuristic. I have also seen it ...
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0answers
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A subset of N that is recursively enumerable is the range of a function [on hold]

I was trying to solve this problem but all the ideas that i had are wrong. I have to prove that if a set A is recursively enumerable( i.e $ ∃f $computable and a program P that computes f such that $ ...
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1answer
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Set that is recursively enumerable but is NOT decidable

I was trying to find a set that is recursively enumerable i.e $$ \exists f\; \text{computable and a program $P$ that computes}\; f $$ such that $$ A = \{ x\; :\; P(x)\downarrow \}. $$ But it is ...
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Efficiently computable, nonempty sets with no known element

I'm looking for sets $S$ of natural numbers with the following properties: There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial ...
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What is the formula for pi used in the Python decimal library?

(Don't be alarmed by the title; this is a question about mathematics, not programming.) In the documentation for the decimal module in the Python Standard Library, ...
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1answer
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About qoutients of Lower Exponent$-p$ central Series

Let $G$ be a finite $p-$group of number of generators $d$ and exponent$-p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G) =1$ in the series $$ G=P_0(G) \geq ...\geq P_{i-1}(G)\geq ...
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1answer
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Computing approximation of cos function

i have an assignement in which the whole point was to approximate $\cos$ function using 2 methods : Using series expansion using a more algebric method with a linear system The teacher also defined ...
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+50

How accurately must I compute the twin prime constant to get the twin prime density?

Let $\pi _{2}(x)$ denote the number of primes $p\leq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that $$ {\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}...
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Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. ...
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Time complexity (Big-O notation) of Posterior Probability Calculation

I got a basic idea of Big-O notation from Big-O notation's definition. In my problem, a 2-D surface is divided into uniform M grids. Each grid (m) is assigned with a posterior probability based on A ...
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1answer
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Solving second order ordinary differential equation with variable constants

I'm having trouble solving a differential equation I found: $$ u''(x) + x\int_0^xu(t)dt = f(x) $$ where: $ x\in[0,1], \quad u(0) = 1, \quad u(1) = -1 $, and $f(x)$ any given function. One of my ...
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Rapid evaluation of Daubechies Scaling function

Let the Daubechies 4-tap scaling function $\phi\in C_{0}([0,3])$ be defined by \begin{align} \phi(x) &= \frac{1+\sqrt{3}}{4} \phi(2x) + \frac{3+\sqrt{3}}{4}\phi(2x-1) + \frac{3-\sqrt{3}}{4} \phi(...
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2answers
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Different database queries

We have the table person (pid,pname) We want to find the names of the persons. What is difference between a and d? I know that d is the correct answer but I don't understand why. $$a)\, \{\langle X\...
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1answer
26 views

Find Tangent Points of Circle and Two Lines in First Quadrant

I need to define explicit expressions to find the points (x1,y1) and (x2,y2), which are the two tangent points of a circle with radius r (known) and two lines (equations known). The center of the ...
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The mass matrix and the stiffness matrix in finite element method for heat equation

On the page 99 in the Chapter 5.2 of Prof. Endre Süli's lecture notes on FEM for PDEs (see: https://people.maths.ox.ac.uk/suli/fem.pdf), he derived the mass matrix for forward Euler scheme and the ...
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Reference request about experimental mathematical using computers

I have graduated last year. During my corsus I've done a lot of abstract algebra especially coding theory. I wish to put that "knowledge" in use and I think that the first step is to master a ...
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1answer
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Solving a system of second order tightly coupled nonlinear ODE with six initial conditions in Matlab

I am solving a problem from fluid dynamics; in particular tightly coupled nonlinear ordinary differential equations. The following is a scaled-down version of my actual problem. I have solved ...
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0answers
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Designing a PDA to accept $wa^{n_a(w)}$ where $w$ is over $\{a,b\}$ and does not contain the string $bbb$.

I'm very new to this topic! I see that the PDA must push $a$ on the stack until after the last $b$ when it must pop $a$'s after that. I know that I'm looking to make sure the $a$ stack is empty at the ...
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1answer
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Can I find an infinitely differentiable function of of bounded moments closest to triangle wave?

Based on this question regarding existance of closest function in Schwarz class, where answer was negative. What if we add a new constraint. Not only infinitely differentiable compact support but with ...
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2answers
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Which function in the Schwarz class of functions is “closest” to triangle wave in $L^2$ sense?

Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave? Let us measure closeness as $$<f-g,f-...
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2answers
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Cost of solving systems of simultaneous linear equations

Given $A,$ a $n \times n $ non-singular matrix and $B,$ a $n \times k$ matrix, I am interested in estimating the computational cost of solving $$AX=B$$ for different values of $k.$ Take as a reference ...
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1answer
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If $f = ax - by + 0c$, with non-negative $x$, $y$, $c$, and $x+y+c=n$, all integers, then how many distinct values does $f$ take?

Given a function $f = ax - by + 0c$, with constraints $y\geq 0$, $x\geq 0$, $c\geq 0$, and $x+y+c=n$, where all are integers. I am interested to find the count of all unique values of $f$. I know ...
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1answer
29 views

Approximating eigenvalue and eigenvector pair with Lanczos iteration

Using the Lanczos algorithm for some matrix $A$, I obtained a tridiagonal matrix $T$ and an orthogonal matrix $Q$ such that $$Q^T A Q = T$$ I also approximated some eigenpair of $T, \ (\lambda_T, \...
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0answers
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Different type of BOussinesq equation

I currently study about Boussinesq equation. The boussinesq model was constructed for describe the behavior of soliton wave (permanent wave), which occurred in several physics phenomenon. Since the ...
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0answers
44 views

Stability of numerical algorithm for computing $e$

Let an algorithm compute $e$ by $e=\sum\limits_{k=0}^\infty 1/k!$ (assume that $fl(k)=k$). The last term of this summation will satisfy $1/k! < \epsilon_{mach}$. My attempt at solution: Let $m$ ...
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0answers
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Stability of numerical algorithms with complex numbers

Let $x\in \mathbb{C}$. I want to analyze the stability of the numerical algorithm computing $2x$ by $f(x) = fl(x) \oplus fl(x)$. So here is what I did: $|fl(x)+fl(x)|(1+\epsilon_1)=|fl(a+ib)+fl(a+ib)|...
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1answer
70 views

Sum to n terms of the harmonic series [duplicate]

We know that $\sum_{k=1}^{\infty}\frac{1}{k}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots$ diverges. But for any natural number $n$, $\sum_{k=1}^{n}\frac{1}{k}$ is finite. The question is; how to ...
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0answers
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Self-study - Numerical Optimization

Is there any available online course on Numerical Optimization other than NPTEL courses? I'm also looking for lecture notes or textbooks that are suitable for self-study with lots of examples and ...
2
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1answer
44 views

Detect (catastrophic) cancellation in sums

In finite precision arithmetic, what are ways to detect (catastrophic) cancellation when adding $N\in\mathbb{N}$ numbers? Example ($N=2$): When adding two numbers $a$ and $b$ the result $a+b$ might ...
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46 views

Normal distribution with Z-Score

A school board is considering implementing a new literacy program to help improve their students’ literacy skills. As a first step, they wish to find out how their students’ literacy skills compare to ...
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1answer
31 views

Find explicitely an element of SL$_2(\mathbb Z)$

Let $\Gamma=\text{SL}_2(\mathbb Z)$ act on $\mathbb H=\{z \in \mathbb C: Im(z) > 0\}$ by $\gamma z=\frac{az+b}{cz+d}$, where $\gamma$ is the matrix $\left[\begin{matrix} a & b \\ c & d \end{...
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Why won't my Gamma difference density function run in R?

I'm trying to find the pdf of X where X is the difference of two iid Gamma distributions. the pdf is given in page 341 Theorem 2 of https://www.sciencedirect.com/science/article/pii/S0047259X83710365 ...
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1answer
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Find the general formula of a sequence generate by the order by the distance of each element to the matrix diagonal.

Some examples: Case 1: n*(n-1) $\mathbb {R}^{5\times4}$ $$\begin{bmatrix} {a_{00}}&{a_{01}}&{a_{02}}&{a_{03}}\\ {a_{10}}&{a_{11}}&{a_{12}}&{a_{13}}\\ {a_{20}}&{a_{21}}&{...
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2answers
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Volume of air in the box (packing problem)

Suppose I have a box with dimensions $L \times W \times H$. What is the volume of air in the box, if I pack balls with radii $r$? With increase of radius, does volume of air decrease?
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1answer
163 views

How to efficiently calculate the Manhattan distances between all pairs of points?

Given $N$ pairs of points in the form $(x,y)$. How can we efficiently calculate the Manhattan distance between each pair of point? One way is to simply calculate the distance between each pair of ...
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Dihedral angles of a $k$-simplex

Given a $k$-simplex $(p_0, ..., p_k)$, where $p_i$ are $n$-dimensional points. Define the dihedral angle $\theta_j$ as the angle between the (hyperplanes of the) two $(k-1)$-facets incident to the $(...
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1answer
47 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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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
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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 ...
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2answers
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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}...
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0answers
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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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1answer
67 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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35 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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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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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
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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 ...
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2answers
42 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 ...
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0answers
56 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 ...