This tag concerns computational problems central to mathematical and scientific computating. 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

Literature Reference for transformations through vector spaces

I am trying to understand the transformations through vector spaces: Problem 1. Let's say we have orthonormal basis $B=\{v_1, v_2, \ldots, v_n\}$ spanning the vector space $V$ and basis $B_1=\{u_1, ...
12
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1answer
319 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
5
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1answer
440 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
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0answers
35 views

Algorithm for finding zero of an odd function from n-sphere -> R^n

There is a well-known Borsuk-Ulam theorem stating that each continuous mapping $f : S^n \rightarrow \mathbb{R}^n$ that is odd in sence of $f(v) = -f(-v)$ for each $v \in S^n$ (where $-v$ denotes the ...
7
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1answer
156 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
0
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1answer
15 views

Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
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2answers
293 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
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0answers
52 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
8
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1answer
83 views

How to find this number, which is probably a very big prime or a product of big primes?

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Which is the smallest natural number $n>0\;$ such that: $\mathcal N(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot n)−2\cdot 3\cdot 5\cdot 7\cdot ...
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0answers
18 views

Why steepest descent gives a wrong direction search?

I have to minimize the function $ƒ(x_1,x_2)=(x_1-1)^2+x_2^3-x_1x_2$. The initial point is $[1,1]^T$. The gradient of this function is $∇ƒ(x_1,x_2)=[2(x_1-1)-x_2,3x_2^2-x1]$. This gradient evaluated ...
3
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1answer
160 views

$\pi$, disjunctive numbers, and finite sequences of given length

It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion of $\pi$. Of course, every sequence ...
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0answers
28 views

Mathematical equivalent to curve fit between polynomials

I am adapting a calculation done in an Excel workbook to code. Right now, we are predicting a variable based on three other variables, say $x,y,z$. We are creating six functions of $x$ and $y$ at ...
7
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1answer
82 views

Carmichael numbers of form $m^3+1$ and Ramanujan's $1729$

While researching for a post on tetranacci pseudoprimes I came across a list of Carmichael numbers, $$C_n = 561,\, 1105,\, 1729,\, 2465,\, 2821,\dots$$ Of course, Ramanujan's taxicab number $1729 = ...
0
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1answer
30 views

Consider the recursively defined language, L2

Consider the recursively defined language, $L_2$ i) $x \cap L_2$ and $y \in L_2$ ii) if $w \in L_2$, then so is $wxw \in L_2$ Find all strings in L_2 with length less than $7$ ...
1
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1answer
10 views

Logistic Regression - malty classification

I want to understand why the probability of P(D|p) is presented as a product of mentioned probabilities. I read a lot of texts, but everywhere the explanations are full of terminologies to confuse ...
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2answers
902 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...
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0answers
6 views

Modulo 3 operation on days/seconds

Simple question.. I want to do a modulo 3 operation on the number of days in a month (28/30/31). and based on that i want to put my user into 3 different groups.. i am also willing to use seconds ...
4
votes
1answer
452 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
2
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3answers
30 views

Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$.

I need help proving the following: Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. I know that the smallest $n$ is 8 by ...
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1answer
23 views

Entropy Calculation and derivation of logarithm

I have probabilities as $$p_1 = 0.4,\ p_2 = 0.3,\ p_3=0.2,\ p_4=0.1$$ hence entropy is given by: $$H(x) = -\big(0.4\cdot \log_2(0.4) + 0.3\cdot \log_2(0.3) + 0.2\cdot \log_2(0.2) + 0.1\cdot ...
4
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1answer
108 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
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0answers
53 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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0answers
22 views

Maximize polynomials

Hi guys I need some help. I am reading a paper and I cannot understand something simple. The author has 4 polynomials with a constrain and is trying to find the optimal solution to the problem. ...
12
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1answer
165 views

On $1^2+2^2+\dots+24^2 = 70^2$, and $15^3+16^3+\dots+34^3 = 70^3$

It is quite well-known that, $$1^2+2^2+\dots+24^2 = 70^2$$ Not so well-known is, $$15^3+16^3+\dots+34^3 = 70^3$$ The formula for the sum of $m$ consecutive squares starting with $a^2$ is, ...
0
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0answers
36 views

If a computer can check 1 million colorings per second, about how long would it take to check all possible three-colorings on 100 vertices?

If we imagine a graph G with 100 vertices, how would we find all possible colorings for G if G(v) = 100? I think that to solve this problem we would start with vertex 1 with 99 edges for the first ...
3
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1answer
206 views

What is the sixth Martin quadruple $\sqrt[n]{x_1^k+x_2^k+x_3^k+x_4^k} =\text{Integer}$ for $k=1,2,3$?

Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system, $a+b+c+d = x^2$ $a^2+b^2+c^2+d^2 = y^2$ $a^3+b^3+c^3+d^3 = z^3$ It can be shown that there are an infinite ...
0
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1answer
39 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
2
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1answer
52 views

Prove that sets don't intersect

I am trying to solve a computer algorithm problem that boils down to solving the following. I would appreciate some mathematician assistance on the proof. So here goes: Having: Set $S$ - rational ...
1
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1answer
24 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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0answers
26 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
1
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1answer
24 views

Pollard Rho intuition

I have been reading about pollard rho factorization, however their is something I don't understand if we don't use a polynomial that is pick two random numbers and see the gcd(a-b,n) > 1 if it is ...
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0answers
52 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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1answer
461 views

Calculating eigenvectors and eigenvalues of a 2x2 complex matrix

I've previously asked elsewhere, http://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a ...
4
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2answers
500 views

“Opposite” of idempotent operation?

What is the adjective given to a mathematical operation/expression on a variable whose new value can only be described in terms of that variable's existing value? Sequential operation? Example: i = ...
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0answers
172 views

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
1
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1answer
21 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
2
votes
2answers
59 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...
46
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4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
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1answer
671 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
2
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1answer
34 views

How do I find the sum of first N numbers common to 2 APs?

Here is the question - Certain numbers appear in both arithmetic progressions 17, 21, 25, ... and 16, 21, 26, ... . Find the sum of first 100 numbers appearing in both progressions. The ...
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0answers
23 views

numerically solving linear integral equations

I want to solve a 3*3 linear equation system but the equations are integral equations and he coefficients of solutions are to be extracted NUMERICALLY from some other integrals.I do not know how. I ...
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0answers
14 views

Norm (modulus) of the derivative of complex function and Newton Method

I am implementing a function that approximates a root of a complex function, say $f(z)$. As we know, at iteration $i$ we ave $$z_i = z_{i-1} - \frac{f(z_{i-1})}{f'(z_{i-1})}$$ The division of ...
1
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1answer
45 views

is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Given the equation $$\frac{635x+326}{637+x} = y$$ where $$x>0$$ Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number. While I ...
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2answers
54 views

If negative of negative (--) = positive then why not positive of positive(++)= negative

As per my understanding positive and negative are just indicative of direction of number axes with zero at the center. If that is the case we should apply same laws to both positive and negative ...
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1answer
80 views

(x,y) coordinates from gluing together a sequence of right triangles with arbitrary angles [duplicate]

I have been scratching my head all day over this question for one of my assignments. I haven't made any progress and I'm at the point of giving up. Here's what I need help with. Start by gluing ...
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0answers
20 views

Simpson's rule is not good enough for the best approximation in L2 problem

The problem came from my computation methods (practice) class. It was to write a program which does the following: Original problem statement: We have a [0; 1] segment. Let us divide it into $2^n$ ...
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0answers
8 views

Fourier Analysis of a p2 continous Galerkin Scheme for the Laplace & Poisson Equation

Background: I am obtaining residual calculations for the 3D Laplace and Poisson Equation using finite element continuous galerkin scheme with lagrange polynomial basis functions for p1, p2, p3 and ...
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0answers
36 views

does any polyhedral partition admit a convex piecewise quadratic surface defined over?

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. ...
0
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1answer
18 views

Golden Section Search termination condition

From textbooks I found that the tolerance for Golden Section Search method should be set to $\sqrt{\epsilon}$, where $\epsilon$ - is the machine epsilon. This can be derived from Taylor series. So, in ...
3
votes
3answers
292 views

A bug with the WolframAlpha computational search engine?

I think I may have discovered a bug with WolframAlpha. So I was trying to determine all $x$ such that $$\sum_{i=0}^{5}{x^{-i}} < \frac{13}{12}.$$ WolframAlpha spit out $x > 13$ (see this ...