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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1answer
19 views

What are the solutions of the following exponential inequality for $n \in \mathbb{N}$?

What are the solutions of the following exponential inequality for $n \in \mathbb{N}$? $$n^3 < 4^{{(2n - 1)}^8}$$ I tried using WolframAlpha, but only an Inequality plot is returned, which I do ...
2
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0answers
28 views

Good method for finding roots that *usually* fall within an interval?

I've been using Brent's method to find the roots of a monotonic, nonlinear, non-differentiable function. The roots often fall within a known interval, but Brent's method fails if they occasionally ...
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0answers
42 views

How to Solve this Implicit Equation

Let $Y$ to be a uniformly distributed random variable. Consider function $z(\gamma)$ defined by the following equation. $$ \int_{\left\{y\in(0,1), -\frac{\log ...
3
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1answer
60 views

Simple factorials

I've been doing some work with factorials and the normal way of calculating them is simply not working so well. When the numbers get really big, doing iterative multiplications is not viable and gets ...
1
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1answer
16 views

Formula for maximal usage

I'm a programmer with a way to easy question for this site, please correct me if I'm wrong. I have the following given: ...
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0answers
19 views

after hajek, can we guarantee that annealing has reached a solution in the best 1%?

hajek showed in http://web.mit.edu/6.435/www/Hajek88.pdf that there are conditions under which an annealing process is guaranteed to find the global minimum. these constraints are pretty tight, but ...
1
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1answer
51 views

Stopping criterion for approximating exponential series

Given $e^{x}=1+x+\frac{x^2}{2!}+\frac{x^{3}}{3!}+\cdots $. Summing in the natural order, what stopping criterion should you use? Can you rearrange the series or regroup the terms in any way to get ...
1
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2answers
83 views

Floating-point arithmetic error

Suppose you need to generate $n + 1$ equally spaced points on the interval $[a, b]$, with spacing $h = \frac{b-a}{n}$. In floating-point arithmetic, which of the following methods: $x_0=a$, $x_k = ...
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1answer
55 views

Min exponent range in normalized floating-point system

In a floating-point system with precision $t = 6$ decimal digits, let $x = 1.23456$ and $y = 1.23579$. (a) If the floating-point system is normalized, what is the minimum exponent range for which ...
2
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1answer
69 views

Using inequalities to find vertices of a polytope

Consider a set $C$ of vectors of integers $x\in\mathbb N^d$ satisfying $$ \begin{align} \forall\ i=1..d & \ \ [0 \leq \ell_i \leq x_i \leq u_i]\\ \forall\ i=1..d-1 & \ \ [x_{i+1} \leq x_i] ...
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5answers
85 views

A equation of Trigonometry

hello dear friends please help me to solve this problem.Thanks very much. How much are $a$ and $b$ in the problem below? $$1-2\cos 3a +2\cos 3b=0$$$$1-2\cos 5a +2\cos 5b=0$$
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1answer
61 views

Generalized eigenvalue problem for symmetric, low rank matrix

I'd like to solve a generalized eigenvalue problem of the form: $$\mathrm{A}x = \lambda \mathrm{B}x$$ $$s.t. x_i^T\mathrm{B}x_i=1.$$ Where $\mathrm{A}$ and $\mathrm{B}$ are symmetric but low-rank ...
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0answers
42 views

Embedding an Implicit Runge-Kutta Algorithm

I am interested in implementing a fourth-order RK method with variable step-size. However, in order to test for accuracy, the solver needs to periodically utilize a higher-order method to see if the ...
1
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0answers
56 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 ...
0
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1answer
27 views

Interpreting high p value and low correlation value

I am trying to run regression on financial data in R. I am new to regression analysis so I am finding it to difficult to interpret certain scenarios. I have the code as follows: Regression analysis ...
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0answers
37 views

Finite Difference Approach for the 1D Conservative Advection Equation with Spacially Varying Velocity

I am attempting to numerically solve the following conservative advection equation in 1D, using a finite difference method. $\frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial ...
5
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0answers
76 views

A summation involving the ceiling function

I'm trying to find a better method of calculating the sum $$\sum_{k=1}^N\lceil ak\rceil^2$$ where $a$ is an irrational number. So far, my only idea is to somehow use a best rational approximation. ...
1
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0answers
59 views

Is there any efficient progam or software to calculate the fractional chromatic number?

The fractional chromatic number $\chi_f(G)$ is a generation of the chromatic number of a graph $G$. It can be formulated as a linear programming question: Let $\mathcal{I}(G)$ be the set of all ...
5
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0answers
175 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 $\{ ...
4
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2answers
92 views

How to solve this least square problem effectively?

I want to solve the least square problem, $\min\|Ax-b\|_2$, but the condition number of $A'*A$ is very large, How can I solve this problem effectively?
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0answers
33 views

How to calculate p-value using an algorithm on R assuming that distribution is unknown.

For a given sample x, where x is distributed with a normal distribution mean known but variance unknown. I am testing the hypothesis that variance is equal to one or greater than 1. The question is to ...
3
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0answers
38 views

Tool for construction of graph with specified properties

Is there a tool (class of algorithms for graph generation) that can construct graph with specified properties. E.g. construct graph who is homeomorphic with both to K5 and K3,3. Construct planar graph ...
1
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1answer
76 views

Write two (or more) numbers as sum of multiples of other numbers (one, two or more)

I have the following problem: Numbers 32, 35 and 57 can be written as sum of multiples of 7 and 9: 32 = (7*2) + (9*2) 35 = (7*5) + (9*0) 57 = (7*3) + (9*4) Is ...
1
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1answer
50 views

How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
0
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1answer
156 views

Probability of breaking the enigma cipher

I assume that most of you are already familiar with how the ENIGMA machine works, that the germans used during WWII. We now that the enigma machine has 3 scramblers with each 26 setting each. That ...
0
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1answer
27 views

Designing a Pushdown Automation to accept a language

Im a novice trying to understand the theory of computation.Im trtying to learn about PDA.I understand that it is a machine counterpart of CFG.Im basically referring to Introduction to Automata Theory ...
0
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1answer
41 views

Numeric calculation of partial derivative: proper sequence of operations?

I am calculating a second order mixed derivative by the following formula $$\frac{\partial^2 f(x, y)}{\partial x \partial y} \approx \frac{f(x + h, y + h) - f(x - h, y + h) - f(x + h, y - h) + f(x - ...
1
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1answer
68 views

Can 2 items be added/taken away from a stack in push down automata at once?

Here is a language and 2 ways (I hope) of representing it with a PDA. Can I use the notation (b,a $\to$ ee) or anything of the like, to take away 2 items from the top of a list at once? Such as I ...
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0answers
64 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...
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0answers
147 views

Help with proving a language is regular (Sipser problem 1.49a)

I am working through Sipser's Introduction to the theory of computation on my own, so I don't have access to a teacher. Hopefully you can help me! Giving hints is highly appreciated. This question is ...
2
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1answer
19 views

Intersection symbols

I am writing a scientific paper. I need to express the intersection of two space, e.g. A and B where A and B can be a line, plane or a 3-D space. What is the appropriate symbol to state this concept. ...
3
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1answer
54 views

Hypercomputation & Higher Dimensional Variants of Conway's Game of Life

Conway's Game of Life is a simple and important mathematical game with some rules of evolution in a two dimensional space. It appears in many subjects in mathematics, artificial intelligence and ...
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0answers
98 views

Solution of equations involving determinant and matrix inverse

$x$ and $y$ are two scalar unknowns. The two equations are $$|\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2|=R$$ and ...
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0answers
84 views
-2
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1answer
51 views

how to do the opposite of mod in this equation

if $X=((A*Y)+C)\mod m$ how does one calculate $Y$? If you have all other variables except Y? I have already tried everything I can think of just don't know how to do the exact opposite of mod, I can ...
0
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1answer
29 views

How to take the integral of a derivative to obtain desired result?

I am aiming for the form of derivative below computed over time that causes its differentiated variable V to go from an initial -.001 and increase to reach 10. I will explain my current calcs below ...
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0answers
18 views

Mix of two bivariate distributions (two correlations hidden in data)

We have two sample vectors $X$ and $Y$ which are realizations (observations) of metric (continuous) random variables, and are interested in a sample correlation between them. Actually, a correlation ...
0
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1answer
47 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
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0answers
68 views

Is there any direct method for Lagrange multiplier based domain decomposed problem?

In elastic problem, we often solve K * u = f, where K is the stiffness matrix, f the external force vector and u the displacement vector. I'm trying decompose the mesh to domains, using Lagrange ...
1
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1answer
38 views

Polynomial representation of intersection of polynomials

How to minimally represent intersection of two degree $d$ polynomials intersecting at $d^2$ points as a single polynomial?
3
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0answers
57 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
0
votes
1answer
145 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
3
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3answers
117 views

Formal notion of computational content

In constructive mathematics we often hear expressions such as "extracting computational content from proofs", "the constructivity of mathematics lies in its computational content", "realizability ...
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0answers
30 views

Efficient way to compute the strong convexity modulus of a function?

I have a strongly convex function $f:X\to\mathbb{R}$, where $X\subseteq \mathbb{R}^n$, with strong convexity parameter $\sigma>0$. By definition $f$ satisfies, for all $x,y\in X$ and $t\in[0,1]$, ...
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0answers
115 views

Double integral involving Dirac delta

I came across a problem in which I have to computationally evaluate an expression like this: $\underset{\vec{r}}\int{F(\vec{r}) \delta(\zeta)\,dx\,dy}$ where $\delta(\zeta) = 1$ over a curve ...
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0answers
42 views

How to estimate the local error and the global error for Runge-Kutta method

How to estimate the local error and the global error for Runge-Kutta method in practice? I have no idea. I recieved a nice answer on the question at other site
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0answers
19 views

Is there a software package to solve large (128 x 128) non-positive-definite quadratic programming problems?

I am trying to solve a quadratic program of the form maximize $\sum_{i=1}^n\sum_{j=1}^nA_{i,j}w_iw_j$ subject to $\forall i:w_i\ge 0$ and $\sum_{i=1}^nw_i=1$ for a 128$\times$128 matrix $A$. ...
2
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2answers
85 views

How does mathematics fit into fractal generation for computer graphics?

I have to do a research paper on any mathematical concept. The mathematical concept must be complex, so I thought fractals would be a good choice (I was told it was a complex idea). I have been ...
0
votes
1answer
40 views

Pumping Lemma for a regular language

I've done pumping lemma proofs in the past but I'm honestly not even sure where to start on this problem. Using the Pumping Lemma for Regular Languages show that the language $$L = \{a^i b^j c^k ...
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0answers
27 views

Which boundary condition dominates in elliptic boundary value problem?

I am working on a solution to a boundary value problem (which is too complicated for me to reproduce here) but have a question about the boundary. In many dimensions, my function is infinite along ...