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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50 views

Find binomial coefficient by its value

Given any positive integer $~m~$ there always exist pair of positive integers $~(n,k)~$ such that $~\binom{n}{k} = m~$. At least we can take $~n = m~$ and $~k = 1~$. How can we efficiently find all ...
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48 views

Good books on Algorithms for a math major without any programming experience?

I couldn't find this question anywhere else so it may not be apt. I am an undergraduate mathematics major and during my discrete math class I really enjoyed the study of algorithms and recursive ...
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1answer
25 views

Expected number of rows of the full rank matrix

Let A be a m by n random matrix over finite fields F_q. Suppose the rank of A is n. How much does expected number of m? I think m maybe qlogq by bins and balls property But I do not know exactly ...
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17 views

Matrix reduction in Computational Topology: An Introduction

I'm working on learning Persistent Homology from "Computational Topology: AN Introduction" by Herbert Edelsbrunner and John L. Harer. In section VII.1, Persistant Homology, they start with a ...
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462 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = ...
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16 views

Bezout coefficients with least absolute sum

Let $a$ and $b$ be some integers and $d$ is their gcd. By Bezout's identity there exist such $x$ and $y$ that $ax+by=d$. I wonder when sum of absolute values of $x$ and $y$ is minimal? I'm ...
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34 views

How to Solve an Integral Equation for an Unknown Integrand numericlaly?

I am working on an astrophysical research in which we relate the cumulative number of Damped Lyman Alpha HI clouds/galaxies, namely their number densities, $\frac{dN_{DLA}}{dz}(>M, z=0),$ to the ...
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2answers
251 views

Playing with Fermat's little theorem

Fermat's little theorem states that if $~p~$ is a prime number then for any integer $~a~$ the number $~a^p - a~$ is divisible by $~p~$. What if one fixes the exponent $~n~$ and tries to find all ...
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1answer
42 views

Proper code for $p\to q$ in GAP

I’d like to know if gap> IsBool(not(p) or q)=true; is the only code for checking the trueness of a conditional statement ...
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30 views

Using only postage stamps of value 64 and 55, how can I work out the way to get closest to a high parcel value?

Searching has shown many questions like this for values of 4 and 7 cents, but nothing for higher values. For British postage, first class stamps are £0.64 and second class are £0.55. Low value stamps ...
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18 views

In the Miller–Rabin primality condition why do we know the odd power case is congruent to 1?

A prime number n satisfies $a^{d} \equiv 1\pmod{n}$ or $a^{2^r\cdot d} \equiv -1\pmod{n}$ where $n - 1 = 2^s·d$, d is odd and $r = 0, 1, ..., s-1$ Why does the second condition being false imply ...
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164 views

What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
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1answer
33 views

Harshad numbers with given sum

By definition Harshad number for base $~10~$ is any number divisible by sum of its decimal digits. Wikipedia gives some information on such numbers but i still have some questions and unforunately i ...
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23 views

A Maths Budgeting Puzzle

A maths puzzle is as follows: Bonger have a printing budget of $119.40. Bonger have 5 children. Each children may do some printing, subject to the printing limit that their father impose, at any ...
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11 views

Determining eigenvalues of a differential or integral operator in Matlab?

Say I have a differential operator such as $L[\phi] = \frac{\partial \phi}{\partial x}$, or $L = \Delta \phi$, or an integral operator such as $L[\phi](x) = \int_{\partial D} \log(x - y) \phi(y) ...
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46 views

Can this integral be computed? $\int_E^{\infty} \frac{1}{4\pi t} e^{\omega^2 t - \frac{|x - n - y|^2}{4t}} dt$

I am working with periodic Green's functions for a scattering problem and the form of the Green's function given is as follows - $$G(x, y) = -\sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha} ...
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30 views

Product of first values of totient function

Let $~p~$ be prime and $~n~$ some positive integer below $~10^9$. Is there an efficient way to compute product $~ \phi(1) \cdots \phi(n) \mod p~$? It is known that $~p > \sqrt{n}~$ (i don't know if ...
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1answer
48 views

Can this integral be evaluated numerically?

I am working with periodic Green's functions for a scattering problem and the form of the Green's function given is as follows - $$G(x, y) = -\sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha} \int_0^E ...
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1answer
98 views

Largest prime gap under $2^{64}$

Thanks to Tomás Oliveira e Silva's extensive calculations, it is known that the largest prime gap less than $4\cdot10^{18}\approx2^{61.8}$ is 1476. I'd like an upper bound for the largest prime gap ...
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54 views

Instability of Kuramoto solutions with Runge Kutta 4th order method

I am currently trying to solve the Kuramoto model: $\ddot{\theta_i} = P_i - \alpha\dot{\theta_i} + K \underset{i \neq j}{\sum}\sin(\theta_i - \theta_j)$ I split this second order differential ...
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1answer
24 views

Help with Legendre Plot Matlab

I've written a code to change a Chebyshev into a Legendre Polynomial, however it will not plot the graph after and I'm not sure why the graph will not plot? The code i have is: function ...
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3answers
104 views

How to calculate $10^{0.4}$ without using calculator

How to calculate $10^{0.4}$ without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?
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2answers
34 views

Computing midpoint of an interval overflow

For computing the midpoint m of an interval $[a, b]$, which of the following two formulas is preferable in floating-point arithmetic? Why? When? (Hint: Devise examples for which the "midpoint" given ...
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14 views

Equivalence and interoperability of computation systems using a calculus

I am trying to prove that two computational systems are interoperable and both can be converted into another parent system . So computational system A,B can be mapped into computational system C. I ...
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1answer
44 views

Cannot understand solution (Turing Machine & Reduction)

Photo of my problem that I don't understand About question above in photo, I just can't understand its solution provided. We know the complement of Atm = {...
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0answers
18 views

Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
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22 views

Different methodology for maximizing entropy in continuous random variable case

Suppose we want to maximize the well-known Shannon entropy $S=-∫_{0}^{x_{max}}f(x)lnf(x)dx$ subject to the following constraints $∫_{0}^{x_{max}}f(x)dx=1$, $∫_{0}^{x_{max}}xf(x)dx=x ̅$ and so on ...
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12 views

Transitivity Proof on graphs: computational or by hand?

I am stuck to transitivity consideration with graphs where $x^a,x^b$ are boolean monomials such as $x_1x_3$ and $x_5$, the cut sets contains $C=\{x^{C_i}=0\mid C_i\in C\}$, ...
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1answer
69 views

Algorithm for isomorphic groups?

As I understand There can't be a general algorithm to decide if two finite groups are isomorphic, Wikipedia. But are there efficient algorithms for all subgroups of $S_n$ for say $n=10$ or so? ...
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50 views

Problem with 4-body Matlab code

I'm trying to model the 4 body problem to see how Jupiter, Earth and Mercury orbit the Sun. I found a two body script and adapted it as accordingly to modify my problem, but for some reason the ...
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1answer
40 views

How to simplify this equation regarding pronic numbers for integer solutions

A pronic number is a number that can be expressed as the product of two consecutive positive integers. For instance, $42 = 6 \cdot 7$ is a pronic number. I've become interested in solving for the ...
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1answer
45 views

Converting Integers from One Base to Another Digit by Digit

So I’ve done some hands-on work with converting integers from one base to another using the well-known method of division and taking the remainder. The most generic algorithm involves dividing the ...
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1answer
31 views

Compute Christoffel symbols & Riemann tesors in Maple 17

I invented a metric tensor g and now I'm trying to compute my first Christoffel symbol but an error message is popping up "Error, bad index into matrix" Is there a way for maple to compute ...
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1answer
25 views

Method to study obvious properties

Most of the time studying mathematics we come across various properties like associative, commutative,...etc. These properties are obvious and sometimes I feel why at all they are given in the text. ...
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28 views

Algorithm for numerically calculating $\log x$

It was while fiddling yesterday that I came up with this rather pretty approximation: $$\log x = \frac{1}{2\epsilon}(x^{\epsilon}-x^{-\epsilon})+\mathcal{O}(\epsilon^2)$$ To be more precise, ...
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14 views

Finding All Combinations of a Hierarchical list Where Conditions Are Involved

I want to find all possible combinations of a list that looks like this. a) Option 1 Sub Option 1 b) Option 2 Sub Option 2 c) Option 3 The catch is that there are some simple and some ...
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2answers
561 views

What does “Mathematics of Computation” mean?

I visited this link: http://www.ams.org/journals/mcom/1950-04-030/S0025-5718-50-99474-9/ And I a bit confused by its title "Mathematics of Computation". I am not a native English speaker. Could ...
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3answers
42 views

upper bound of a function $n^{1/\log(n)}$

I have the following expression $n^{1/\log(n)}, \quad where \quad n \in [1, 10,000]$. When I solve this numericall, I get the resultant value 2.718282 for all $n \in [2, 10,000]$. On this basis, I can ...
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0answers
26 views

proving euclid's algorithm stops after 2k iterations [duplicate]

Apparently, when initialized on $k$-bit integers $A$, $B$, Euclid's algorithm terminates after at most $2k$ iterations. This result is not immediately intuitive to me. I would appreciate help in ...
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38 views

How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
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33 views

Find values of x where error term small between rounded and not calculation in non-linear functions f_{i}(x)

Here's my situation. I have a set theory background so I'm out of my league in applied, computational methods so I'd appreciate a hand-up. I have a set of five functions, $f_{i}(x) \ $ where $\ 0 ...
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3answers
64 views

n(n+1)/2 combinatorial proof (details in description)

Find the number of $2$-lists $(𝑎, 𝑏)$ we can form using the numbers $0,1,2,...,𝑛$ with $𝑎 < 𝑏$. a. Show that the number is $𝑛(𝑛 + 1)/2$ by considering the number of $2$-lists $(𝑎, 𝑏)$ in ...
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1answer
69 views

About Gröbner Bases

Recently I have come across a book of Gröbner Bases written by Adams & Loustaunau. The book is excellent and I have become interested in Gröbner bases after reading the book. I want to read more ...
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2answers
44 views

How to define numbers in a way that a number 'n' is equivalent to the function plus 'n'?

In lambda calculus, is it possible to define (or disprove the existence of) a number system alternative to church numerals such that each number is a function which on application, adds itself to it's ...
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2answers
41 views

Linear Algebra matrices question.

Let $A,B$ be 2 square matrices of the same size. And the following holds true $AB=A+B$ How do I prove that $(I-B)$ and $(I-A)$ are invertible
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1answer
37 views

Explicit piecewise linear approximation of a function of 4 variables

I have a table of numbers for fixed values of 4 parameters $x, y, z, t$, at this $x$ belongs to finite set of natural numbers, $y\in\{1;2\}$, $z\in\{5;10;15;20;25\}$ and $t\in\{1,2,3\}$. Is there a ...
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2answers
49 views

Big O notation: ratio of two $O(\cdot)$'s is $O(\cdot)$ of the ratio?

Is it true that if $f_1=O(g_1)$ and $f_2=O(g_2)$ then $$\frac{f_1}{f_2}=\frac{O(g_{1})}{O(g_{2})}=O\!\left(\frac{g_1}{g_2}\right)$$ ?
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29 views

Distance Geometry Problem (DGP) Programming Language Recommendation

We have been studying DGPs in clinic recently and I was hoping I might be able to get recommendations for computing languages in the processing of large network solutions. Specific computations ...
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1answer
40 views

How to write new algorithm of root finding by combining 2 or 3 standard algorithms(bisection, fixed, etc)

I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 ...
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1answer
36 views

Constructing a specific Rank-One Matrix

Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and ...