# Tagged Questions

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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### 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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### 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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### 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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### Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering now:...
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### How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
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### Formula for MLM like system

I'm trying to figure out a formula for a system similar to a MLM system such that all members will receive 50/50 of the shares. So for example, X recieves 50% and A recieves 50%. When A recruits B and ...
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### Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...
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### Minimizing computations for evaluating two polynomial simultaneously

I want to evaluate two polynomials $f$ and $g$ simultaneously, on the same input (in a computer program). These polynomial have only coefficients $0, 1, a , b$ and their degree is less than 700. I ...
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### Finite element method books

I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples. The topics I would ...
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### For what values of $k$ does $(1+x)^{500+k}(1-x)^{500-k}$ exceed $10^9$?

Pretty simple question, for what values of $0\leq k \leq 500$ do we have $\max\{(1+x)^{500+k}(1-x)^{500-k}|x\in[0,1]\} \geq 10^9$ ? Some trivial observations: The problem is equivalent to finding ...
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### More efficient method of computing the square root of $-1 \mod p$

I am currently doing collecting some preliminary data about elliptic curves over finite fields of order $p$ where $p$ is a prime congruent to 1 mod 4. Part of the data collection process requires me ...
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### c(x) = k for all positive k is primitive recursive [closed]

How can I show this function is whether primitive recursive or not? Do I need to use Godel number?
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### Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
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### Squeezing primes

Any positive odd number $n$ can be coded one binary digit smaller by the rule $\frac{n-1}{2}$ and that's obviously the best squeeze: a bijection from $\mathbb N$ such that $f(n)\geq n$. I'm looking ...
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### Digits of $\pi$ using Integer Arithmetic

How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...
559 views

### A valid floor function trick?

Given $x\in\mathbb R_+$ and $m,n\in\mathbb Z_+$, is it true that $$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$$ Thanks for at least three ...
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### Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
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### Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
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### Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
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### How to obtain fair competition between two teams

Consider a class with $4$ students having min goals as $\big\{ 1, 3, 4, 5 \big\}$ and max goals as $\big\{ 2, 5, 8, 6 \big\}$. Find the best way to divide the class in such a way that the match is ...
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### Finite element boundary conditions

I have a boundary condition given by $\mathbf{n}\cdot \nabla m=\phi$, where $n$ is a vector normal to a surface, $m$ is a physical quantity (say mass) and $\phi$ is a constant. The boundary condition ...
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### How can I implement Newton-Raphson's method with a function of one vector and one matrix?

I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite). I somehow successfully computed ...
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### Converting Java Code to Mathematic formula

I have algorithm , but I don't know how to convert it to mathematic formula. ...
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### bio heat equation modification

I have the bio heat equation as described .here And the solution to it is, But to this I am trying to include the effect from exercise intensity as well. So the modified bio heat equation is ...
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### Given a set of integers $S$, what is the maximum integer that is a product of one or more integers from $S$ not exceeding $X$?

Given a set of integers $S$, which will contain no more than $100$ integers. Now, what would be the fastest approach to find $M$ which is a product of one or more integers from $S$ (and multiple usage ...
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### summation of even and odd numbers

In my video processing algoritm, I do some processing even and odd frames seperately. F = E(x) + O(x) where F is the video, E and O contains its even and odd ...
$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ ...