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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Creation of "customisable" structures in Magma - Computer Algebra System

Good evening! I am trying to implement some structures in the Computer Algebra System "Magma". My goal is to be able to specify a set and a binary operation on that set. I want to create a ...
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need to show working of how to obtain the largest choicest given a preference relation

Consider the relationship x ≼ y if and only if x^2 ≥ y. What is the largest choice set (a subset of the real numbers ] − ∞, ∞[) such that ≼ defines a preference?
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Understanding this Inclusive Notation

I am learning about splines in my computational analysis class, and this question came up: I have not seen this notation before (that is "$(-1 \leq x \leq 0.5)$"), and it is confusing me ...
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Example of matrices where computing the inverse is the most efficient method

I know there exists matrices where for example LU-factorization is not the most efficient way of solving the linear system of equations: $$Ax=b \tag{1}$$ Examples of such matrices are triangular or ...
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Consider the non-homogeneous linear recurrence relations $a_n = 2a_{n−1} + 2^n$ find all solutions.

I can show that $a_n^{(h)}$characteristic equation $r-2=0 \to a_n^{(h)}=\alpha2^n$ But I'm stuck on $a_n^{(p)}$ characteristic equation $A2^n = 2A2^{n-1} + 2^n$ Simplifies to $-A = 2$ $A = 2A + 2$
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Solving the matrix equation $X^T A X = B$ where X is the unknown matrix.

I have seen people ask about solving this equation where A is the unknown matrix but I have never seen people talking about X being the unknown. The matrices are square matrices. In my situation X ...
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1answer
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Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit

In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is: How can I reproduce Schwartz's result ...
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Euclidean division algorithm for (quadratic) number rings

I implemented in the computer Euclidean division for Gaussian integers. I’d like to implement it for other Euclidean number rings, or at least for the few norm-Euclidean quadratic rings. Any reference?...
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Determine programmatically whether an expression is always defined

I would like to determine programmatically whether a given expression is always defined, or if it could have an undefined result. For instance, take the following computation: ...
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1answer
76 views

How to integrate PDEs back to the original function, e.g. $x e^{-x^2-y^2}$? (Generic or numeric approaches are welcome.)

I have to solve tasks computationally, where integrations increasingly play a role. This practically addressed question aims later at a general approach. However, I would first like to understand the ...
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Need to check the logic of this theorem related approximation ratio among A, B,C

I am a student outside the mathematics major. I need a help to check the logic and formulation of this theorem. Let A= 2B =2C is best-case, where A and B are Known values. C is unknown value. Given ...
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1answer
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Efficiency of a root finding algorithm

I'm writing some code regarding polynomials and I have a few questions about a root-finding algorithm I've come up with based on my current knowledge. Moreover, Given a polynomial, I would like to ...
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Fast mod 29 on 28 bit number

I have a 64-bit number ($x$) that I want to programmatically display as base 58. I have currently found two solutions, the first one is the double dabble and the second one is mod-and-divide. I'm ...
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Can we compute all digits of the Euler-Mascheroni constant?

Let $\gamma$ be the Euler-Mascheroni constant from calculus. Is there an algorithm that computes the $n$-th digit of the decimal expansion of $\gamma$ given $n$ as input? For all we know $\gamma$ ...
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Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation?

I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ...
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Convex Optimisation - How to convert from the LP Formulation into Interior Point Format

I am currently learning about convex optimisation and now looking at implementing some methods with code. I am starting from the Linear Programming formulation for the L1 norm, which I have ...
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clustering coefficient and average distance in Watt-Strogatz networks

we want to prove that clustering coefficient in Watts–Strogatz Graphs when p=0 (probability of having random edges) can be obtained using the following formula: C(0) = 3(c-1)/2(2c-1) where 2c= mean k(...
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1answer
59 views

Finding appropriate step size for ODE system numerically solved without ODE solver in Matlab

This is a post piggy backing off a post made earlier yesterday, I have made considerable progress towards completion but I'm having difficulty with this part, and I'm sure it's last hurdle I need to ...
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Determine L(M), the language recognized by M.

Below is provided a nondeterministic Turing machine. Could anyone explain to me how can I determine the language that is recognized by it? Thanks a lot!
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Computing Complex[CReal] roots of a polynomial

I have a type CReal = (precision: Int) -> (approximation: BigInt). I have a Polynomial[CReal] as well. I want to get the ...
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Reproduce this figure using matlab for system of ODE's solved numerically

So I believe that I am working with the Becker-Doring equations or the spatially uniform mass-action equations, either way, this was taken from a research paper and I'm trying to reproduce the results ...
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Efficiently solving two system of equations from block-matrix

I know that the solution of the system $(X^\top X)\beta = X^\top y$ is simply $\hat{\beta} = (X^\top X)^{-1}X^\top y$. Suppose that $X$ and $y$ are a block-matrix and block-vector respectively. Then ...
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Normalizing floats dataset to (0f-1f) range while minimizing variance

I have software that will output a fixed amount of floats every iteration. I've got no idea about the magnitude of the outputs, but I need to remap them to stay in range (0f, 1f). I could use a simple ...
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1answer
60 views

Model Theory Question for Predicate Logic in van Dalen's Logic and Structure

Let L be a language without identity and with at least one constant. Let σ = ∃x1 ··· xnϕ(x1,...,xn), where σ is a formula in predicate logic. Let Σ = {ϕ(t1,...,tn)|tᵢ closed in L}, where ϕ is ...
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Bilinear form of a matrix [duplicate]

Let V be the vector space Mn×n(R), and let B: V×V→R be the bilinear form B(X, Y) = tr(XY). Calculate the signature of B. Here, tr means the trace of the matrix. Edits I'm stuck here as I was only able ...
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1answer
40 views

Need help to write a proof regarding first order formula

I'm trying to write a proof for the following question: For any formula φ, given two assignments g and g′ which differ only in what they assign to variables not in φ, and any model M (of appropriate ...
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2answers
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Palindrome Subtraction

A palindrome number is a number that is inverted, but the number remains the same. A four-digit palindrome number is 6226. Suppose, you subtract another four-digit palindrome number from such a four-...
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unexpected high precision of the Simpson's rule

I tried to use the Simpson's rule to calculate the following integral $$ \pi = 4 \int_0^1 \frac{dx}{1+x^2} .$$ The Matlab code is as follows. I expected a scaling law $error \propto h^4 $ ...
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1answer
40 views

To find a solution of non-linear PDE

Consider the following non-linear PDE: $u_x^2+u_y^2-u_{xx}-u_{yy}=2a+bz$ where $x,y,z$ are independent variables with $u=u(x,y)$ and $a,b$ are constants. I am trying to find at least one examples of $...
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Grouping/Sorting using matrices. Large number of Items and unknown number of groups.

I have a program that reads user entries and needs to appropriately group them. The user entries are NOT exact each time. Entries are strings(text). I have a function that compares 2 strings and ...
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1answer
35 views

numerical error after matrix multiplication

I have these two 3D vectors m_v and f_v (the numerical values are given below in matlab ...
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1answer
30 views

How to determine parameters that satisfy a given equation (using python)

I am working for a mathematical model in which I intend to fit the model itself to statistical data to determine the values that my parameters must have. I am dealing with equations of the sort: $$0....
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2answers
64 views

Why 8-bit signed binary numbers go from -127 to 128

I'm studying floating point representation using "Numerical Mathematics and Computing 6ed." By Cheney and Kincaid. This paragraphs, which can be found in page 46, puzzle me: I know that $m$ ...
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How to calculate probability based on multiple variables?

So let say I want to calculate how probable it is for you to win an Olympic gold medal if you are an athlete competing in Olympics as a part of team USA.To do that I might compare the number of gold ...
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What is Iteration Number in modeling?

I am a biologist. Referring to mathematical modeling for biological data in the literature. Could someone explain what is the "Iteration Number" of modeling and the meaning of the up and ...
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2answers
105 views

Why does Alan Turing proof of the halting problem is considered a proof that mathematics is undecidable?

Whenever I read about the Entsheidungsproblem or the halting problem, I read that this proof proves that there is no general algorithm to solve this problem for any given algorithm with a given input. ...
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1answer
38 views

What's Wrong With this Derivative Approximation?

The following problem was given in my math book: I can't see what's wrong with the analysis for the life of me. I suspect it has something to do with $f'''(\xi_1)-f'''(\xi_2)$ being greater than or ...
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Estimating and correcting for clock drift in experimental data

I have 4 instruments, call them $i,j,k$ and $l$, with known coordinates $(x_i, y_i, z_i), (x_k, y_k, z_k), \ldots$. Each instrument carries a radio receiver, and a ...
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21 views

Exclude points from a set of points which prevent them from approximating a function

I am currently writing a small program in python for some calculations and to also learn python. I don't have a problem with python itself at the moment but I lack any idea how I could solve the ...
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Proving Inverse Monotonicity of a Difference Operator

Following is a numerical scheme that uses the 5-point difference approximation for the Laplacian of an elliptic problem: \begin{equation} \mathscr{L}_hU_{l,m}=\begin{cases} \mathcal{L}_hU_{l,m}, ...
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How do I find the maximum value of $e^{-x^2+x}\cos(x^3+4)\cdot(x^4)$?

I've put it through symbolab to derivate and obtained: $$ \frac{d}{dx}=e^{-x^2+x}(-2x+1)\cos(x^3+4)x^4+[-3x^6\sin(x^3+4)+4x^3\cos(x^3+4)]e^{-x^2+x}=0 $$ and then simplified to: $$ \frac{-2x^2+x+4}{3^3}...
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How do you quantize an expression? And would a computer program be considered a math proof in its limits? [closed]

0 is an Expression, 1 is an Expression, Infinity is an expression. If I want to express that during my Educational days or as a student I was disinterested not in the information that I was getting ...
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Local Partitioning of Space when solving PDE's

Not so sure where to put this, since it is very much Applied Math. I am playing around with spectral methods for solving PDE's. As a test example I want to solve the following $\Delta u(x) = \lambda u(...
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1answer
57 views

Weighted mean or average including negative numbers

Let's say I have have multiple values: [12, 8, -6, 0, -1] with weights [20%, 20%, 30%, 15%, 15%], how could I calculate its ...
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Can program be used to simplify algebraic complex expressions?

I have several expressions of this kind $$ \frac{p_1(1+|p_1|^2)}{(1-|p_1|^2)^2}-\frac{(1-|p_1|^2)p_2(1+p_2\overline{p_1})}{(1-\overline{p_1}p_2)|1-p_1\overline{p_2}|^2}\ .$$ Does there exists a ...
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41 views

Reconstructing a cubic form

I am given a number of points $n_1,\ldots,n_\ell$ from an integer binary cubic form $ax^3+bx^2y+cxy^2+ey^3$ (without any information on what $x,y$ generated them). Can I reconstruct the form $(a,b,c,e)...
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73 views

How to quickly compute the initial average half-life of multiple substances?

For example, imagine an object that consists to equal parts of two substances with the following half-lifes: $t_{1, 1/2} = 1d$ $t_{2, 1/2} = 1000d$ I would now like to find out the time after the ...
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33 views

A special generalized eigenvalue problem

I have to solve a generalized eigenvalue problem, $A x = \lambda B x $. Here $A,B$ are both $n\times n $ symmetric and in particular, $B$ is positive definite. I just need the smallest generalized ...
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Diophantine Approximations of high dimensions

I am struggling with this question [enter image description here][1] [1]: https://i.stack.imgur.com/EIvo5.pngstrong text Where Vol_m or Vol_n is the lesbegue measure of the unit ball I would realy ...

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