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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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12 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 ...
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1answer
26 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
27 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 ...
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46 views

How many polynomials are O(x^4), if each of the coefficients is either 0 or 1? Stuck on this question.

A polynomial of degree $n$ can be written in the form $a_n x^n+a_{n-1} x^{n-1}+⋯+a_2 x^2+a_1 x+a_0$,where $a_0,\dots,a_n$ are constants (i.e. the coefficients) and $x$ is the variable. How many ...
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0answers
12 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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13 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
13 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 ...
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0answers
28 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 ...
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0answers
21 views

Open problems in Cellular Automata field

here there is a link on Wolfram about 20 open problems of CA theory. Has anyone of them been solved or tested? I'm searching for literature.
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0answers
17 views

Computation with infinite Weyl algebra

My question here is very computational. My problem is in mathematical physics, so I want to ask the community what kind of software they use to do the following computation if there is any? Let $$L_{...
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1answer
31 views

What's the mathematical model for this matrix.

I need to find the mathematical model for the relationship between number of columns and all possible number of combinations when including two elements, one is not repeating and the other repeats ...
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0answers
34 views

polynomial interpolation exam paper

Good evening everybody, I have to calculate the interpolator $g(x)$ of degree at most 2 of values ​​$g(x_o) = f_0 g (x_1) = f_1 g (x_2) = f_2$ on the nodes $x_0 = -1, x_1 = 1, x_2 = 2$ (this passage ...
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2answers
116 views

What is the solution of $x^a+(1+x)^b=0 $?

I am not a mathematician, but a theoretical physicist. I am faced with this equation coming from some plasma phenomenon and I am unable to 'recognise' it. Mathematica software cannot solve it. I ...
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0answers
21 views

Minimum path distance from a source

Suppose I have a path-connected subset $I$ of $\mathbb{R}^n$ (not convex, but can be contained in a product of finite-measure closed intervals), and I define a "source point" $a \in \mathbb{R}^n$. ...
3
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1answer
49 views

Find divisors $n_i$ of $a_i$, mutualy coprime, with $\prod n_i=\operatorname{lcm}(a_i)$

We are given $k$ positive integers $a_i$, and want as many positive integers $n_i$ each dividing the respective $a_i$, with the $n_i$ mutually coprime, and the product of the $n_i$ equal to to the ...
3
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1answer
76 views

Primes $p=n^6+1$

Which is the least odd prime $p=n^6+1$ for some $n\in\mathbb N$? I have tested for $n\leq 10,000$ without finding any. Due to a conjecture of Bunyakovsky there are an infinite number of such primes, ...
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2answers
15 views

Theory of Computation (Regular/Non-Regular proof)

Suppose that L0, L1, L2 are languages over the same alphabet and that L0 ⊆ L1 ⊆ L2. Is it true that if L0 and L2 are regular, then L1 must be regular as well? ...
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0answers
21 views

Generating functions and algebricity

I study Generating Functions and i see the example 5.1a p.p 9 in paper below. i can't understand the author how solved it? can you help me? Paper: "SOME APPLICATIONS AND TECHNIQUES FOR GENERATING ...
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1answer
50 views

Prove a recursive function $f(p_i) = (p_i/2)g(p_i) \to \infty$ as $p_i \to \infty$?

Let's say we have a set of numbers $\{5,9,13,17,21,\ldots, 5+4i,\ldots\}$ and each $p_i$ is a member of this set, namely $p_0 = 5, p_1 = 9, p_2 = 13$ , etc. Consider the following functions defined ...
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0answers
17 views

Efficient way to iteratively compute span of vectors.

The problem itself is not difficult, but the naive approach seems very wasteful (computationally speaking). Given an ordered list of $n$-dimensional vectors $(v_1 ,\dots ,v_m)$, that span $\mathbb{R}^...
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16 views

Kernel evaluations of and order approximations of 2nd order Volterra integral equation

The integral equation $u:[a,b]\to \mathbb{R}$ $$u(t) = f(t) + \int\limits_a^t K(t,s)u(s)ds$$ defined on the interval $[a,b]$, with $f:[a,b]\to \mathbb{R}$ and $K: [a,b]^2 \to \mathbb{R}$ some known ...
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0answers
27 views

Why do we not need information of the eigenvalues of $A$ for the CG method, but we do need it for the Chebyshev method?

I'm currently reading up on information about the Chebyshev method and the Conjugate Gradient method for finding the solution of a system $$A\mathbf{u}=\mathbf{f}$$ where $A$ is an SPD matrix, but I ...
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1answer
53 views

Proving error bound for Simpson's rule

The Simpson's rule can be stated as follows: $$\int\limits_{x_0}^{x_2}f(x)dx\approx \frac{h}3\left[f(x_0)+4f(x_1)+f(x_2)\right]$$ The way I'm trying to find the error bound for the Simpson's rule is ...
2
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1answer
26 views

Efficient computation of incremental standard deviation (removing first value)

I found this link about incremental standard deviation where it computes the standard deviation every time a new element was added to the dataset. Is there a similar method when adding a new value ...
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2answers
42 views

Numerical differentiation and the product rule

It is well-known that for two functions $p$ and $q$, $$[p(x)q(x)]' = p'(x)q(x)+p(x)q'(x)$$ But if one uses numerical approximation, say the centred difference method $$f'(x) = \frac{f(x+h)-f(x-h)}{...
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0answers
14 views

Alternative Parabola Formula for Fortune's Algorithm using Strange Subscripts $f$ and $d$.

I am trying to figure out how to create a voronoi diagram using Fortune's algorithm. I have found a pretty good explanation of the algorithm here. However, I don't understand his style of math ...
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0answers
35 views

Are there any other internet-based computation engines that plot special functions?

I am looking for alternatives to Wolframalpha, specifically which can plot many rare functions like the polygamma function, the dilogarithm, the cosine integral, etc. However, it doesn't always do ...
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0answers
10 views

Deriving quadratic interpolating function

I want to derive the piecewise interpolating function on the interval $[x_i, x_{i+2}]$, $i\in \{0,\dots, n\}$: $$p_2(x) = \frac{(x-x_{i+1})(x-x_{i+2})}{(x_i-x_{i+1})(x_i-x_{i+2})}f(x_i) + \frac{(x-x_{...
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1answer
64 views

Is Theory of Computation by Sipser a bad math reference?

So, Professor Sipser's bio is unreal, I mean no disrespect to the man, nor do I think I could be half the mathematician he is. My issue is--and maybe this is me just not looking at these questions ...
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0answers
29 views

Negative order of accuracy

Suppose we analyze the order of accuracy of a finite difference approximation of a derivative, $$f'(x)=\frac{1}{2h} \left[f(x-2h) -4f(x-h) +3f(x)\right]$$ and we conclude that the order of accuracy ...
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2answers
37 views

Maths notation for iteration steps

A function $$F(n)=f(F(n+1))$$ is called $n$ times, from $n=b$ to $n=a$, where $b>a$, with the purpose of acquiring $F(a)$. Is there an elegant, mathematical way of depicting this? To illustrate ...
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3answers
37 views

result of multiplication having zeroes after the decimal

Given the multiplication $3.25 \times 0.4$, the primary school students learn that we multiply the digit 4 to the number 325 which result in 1300 we count and then add the number of decimal place ...
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1answer
31 views

How can $O(fg)=fO(g)$

So it is known that $\mathcal{O}(fg)=f\cdot \mathcal{O}(g)$. But what if $f$ is always negative, say a constant? Then $h=\mathcal{O}(fg)$ implies there exist $C>0, x_0$ such that $|h(x)|\le Cg(x)f(...
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0answers
14 views

Order of product of functions is product of one function and order of other function

I want to show that if $f$ and $g$ are functions then $$f\mathcal{O}(g) = \mathcal{O}(fg)$$ As we know, $\mathcal{O}(f)$ formally means that there exist a constant $C$ and $x_0$ such that $|f(x)|\le C|...
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1answer
26 views

Linearity of discretized ODE

I'm currently reading a section of a book on an implicit Ruge-Kutta method, and the following is written: We start, say, with a diagonally implicit Runge-Kutta method $$k_i = hf\left(y_0+\sum\...
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0answers
16 views

Optimization of nonlinear $f(x)$ where $x$ is a vector of binary variables

I'd like to find a solution (potentially approximate) to the problem $$ \max_{x_{i,j}} \sum_{k=0}^K\left[ 1 - \prod_{i=1}^{I} \left(1 - \prod_{j=1}^{J}(1-b_{k,i,j} \, x_{i,j}) \right)\right] $$ ...
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0answers
54 views

Qualified majorities consensus algorithms and Block chain algorithms

I have a question below. Could you please help me to find the answers or reference of document about it? Let $A = f(a_1 ;... ; a_n)$ be a set and suppose that $t$ elements of $A$ are colored red; the ...
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2answers
38 views

Program or computing system [closed]

What programs do you suggest to me such that when introducing a set of numbers, it will answer possible functions that generate them? Different wolfram, neither OEIS. For example, if I input 2, 3, ...
2
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1answer
103 views

A conjecture about irreducible polynomials with integer coefficients

Let $f\in\mathbb Z[X]$, define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments. Theorem: If $f\in\mathbb Z[X]$ is non constant and reducible ...
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1answer
20 views

Number of lattices inside mixed-integer polyhedron

Given a mixed-integer polyhedron $P = \{(x;z) \in \mathbb{R}^n \times \mathbb{Z}^d \mid A x + B z \leq c \}$ with $A \in \mathbb{Q}^{m \times n}$, $B \in \mathbb{Q}^{m \times d}$ and $c \in \mathbb{...
2
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1answer
89 views

Split the matrix 8-ways PUZZLE

I have a problem that has been bugging me for the last month, there is a matrix with 8x8 squares, so 64 squares, and with 8 balls placed randomly each in a square. I need to find the solution of how ...
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0answers
32 views

Two's complement addition issue

In a two's complement system, 16-bit binary numbers $0010010010010010$ and $1111110011111100$ can be represented in the decimal system as $9362$ and $-772$, correspondingly. Now, from what I've read ...
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1answer
37 views

Determining a variational formulation of $u^{(4)} = f$ with $3$-rd order BC.

Below is a problem from a recent exam and I have some questions about it. Given the boundary value problem:$$\dfrac{\partial^4 u}{\partial x^4} = f,\quad f\in L^2(0,1)$$ $$u(0) = u''(0) = u'(1) = u''...
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0answers
19 views

Numerical integration of $\int_0^1 \int_0^1 d x dy |x-y| f(x,y)$

Here the function $f$ is smooth and bounded in $x$ and $y$. The problem comes from the factor $|x-y |$, which has a ridge along $x= y $. Is it possible to develop a scheme like the Gauss quadrature ...
0
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1answer
26 views

46- and 64-bit integers

Some Cray supercomputers used to support 46-bit and 64-bit integer data types. What are the maximum and minimum values that we could express in a 46-bit integer? in a 64-bit integer? Is my ...
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0answers
22 views

23-bit mantissa and 9-bit exponent range and precision

I have the following problem which I would like to make sure that I understand correctly. So I would appreciate your help in this matter. Some computers (such as IBM mainframes) used to implement ...
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0answers
54 views

Most efficient root finding algorithm for a monotonic function

This is my first time asking a question here, so I may not be asking this in the right place. I am trying to find the roots of a monotonic function with as few function evaluations as possible. I ...
4
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2answers
82 views

Best programming language for solving numerical PDEs

I have been using matlab to write my finite difference, finite volume and level set codes when solving PDEs. The codes become even complicated hence taking longer to run when I go to 3D cases. This ...
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0answers
17 views

What do I need to do and know to find an extremum of a function in a parallelogram?

I think in a very short time on my work, I will be face issue of a finding extremum points of a restricted functions. What math topic it is? I didn't find any similar problems in multivariable ...
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1answer
24 views

Unique combinations with elements from a set

I am not a mathematician per say, so please forgive me if I am not using the correct terminology. I have the following question: I have a set of 6 elements, i.e ...