All Questions

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115 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
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19 views

Represent this differential equation as a set of n+1 equations w n+1 unknowns

Given the following differential equation: $$s''w'' + 2s'w''' + sw'''' = q$$ We use these approximations: $$w''''(x_i) \approx \frac { { w }_{ i+2 }-4{ w }_{ i+1 }+6{ w }_{ i }-4{ w }_{ i-1 }+{ w ...
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11 views

A name for the number system used in versioning software

Software often uses a numbering system where one "digit" increments independently of the others. For instance, the next version of Software 2.9 might be Software 3.0 or Software 2.10 or Software ...
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34 views

The meaning of correlation coefficient and p-value

I found the following quote in a study, but I'm not sure exactly what it means: Using linear regression, the immunization schedules of these 34 nations were examined and a correlation ...
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23 views

Sphere degenerates to point in discrete space?

Is it - or can it be - correct to say that a sphere degenerates into a point in discrete space? When I say "degenerate", I mean in the same way a torus can degenerate into a sphere. I specify a ...
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31 views

Volume of a rotated regular polygon

I want to calculate the volume of the shape which is created when you rotate a regular $n$-sided polygon around the $y$ axis with a major radius $r$. (like a torus, but with a polygon as rotated ...
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16 views

Phrases for uniform boundedness and uniform convergence

I have some doubts about using prepositions. I. Let $f_a : \mathbb{R} \to \mathbb{R}$, $f : \mathbb{R} \to \mathbb{R}$. Assume that $f_a (x)$ converges uniformly to $ f (x)$, $x \in [0;1]$, as $a ...
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59 views

Strange sum divergent

Could you Find /Check the strange sum i have calculate in a unformal way it is like a analitycal continuation $$\sum _{k=1}^{\infty } (-1)^k k \log \left(\frac{k+3}{k+2}\right)=\frac{1}{6} (-36 \log ...
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26 views

Domain of boundedness of a power series

In several texts about several complex variables, like Krantz for instance, domain of convergence and domain of boundedness of a given power series are defined, and the easy result that the former is ...
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31 views

Does $\sum_{k = 1}^{\infty}\frac{1}{z - a_{k}}$ have a pole at each $a_{k}$?

Suppose $f(z) = \sum_{k = 1}^{\infty}\frac{1}{z - a_{k}}$ with $|a_{k}| = 1$ for $z \in \mathbb{D}$ is analytic. Suppose $f$ can be meromorphically continued to a domain larger than $\mathbb{D}$ which ...
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25 views

Codimension of $\ker $ $\alpha $

Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $? Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to ...
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31 views

Sum of function applied to parts not equal to function of total

The general goal is to determine the effectiveness of the test pill's ability to keep the test subjects from getting sick using the following data. | Test Subjects | Took Test Pill | ...
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0answers
43 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
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0answers
15 views

Weighted probabilities assigned to a two stock portfolio

Does anyone know how to input a two variable portfolio with uneven weighted probabilties into an HP10bii+? By uneven I mean the weightings are not in increments of 10% and thus you can't simply equate ...
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20 views

Find if the given size of the rectangle fits on the rectangle

Given a rectangle of size row = 3 and col = 2 which are occupied by other rectangles as in figure. The shaded region in green are occupied. Now what I want to find is, if a rectangle of size (p = 2 ...
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29 views

Optimizing choice of data points with known model

my question is fairly simple to explain but I'm not quite sure how to solve it. Basically lets say I am measuring some value at 8 time points. I get to choose these 8 time points. I also know the ...
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16 views

How to standardise with respect to two variables?

I'm programming an app and I have a resizeable zone and a map of police incidents and where they've occurred. I've trying to get a result between $0$ and $1$ (or $0$ and $100$) that is calculated with ...
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39 views

A question on two sequences of real functions

Define $$S\left( \alpha ,C \right)=\left\{ f:\mathbb{R}\to {{\mathbb{R}}^{+}}:\int_{\mathbb{R}}{f\left( u \right)du}=1,\,\,\int_{\mathbb{R}}{{{\left| {{f}^{\operatorname{ft}}}\left( t \right) ...
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0answers
46 views

A non-unital commutative ring with infinite elements such that each element $a$ satisfies $ab =0$ for infinitely many $b$'s

Beside the usual rules for non-unital commutative ring (that is, a ring without multiplicative identity) $R$, I want $R$ to satisfy the following: $ab = 0$ for each element $a$ and there are ...
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0answers
79 views

Transformation of variables

Let variables $U$ and $V$ be uniformly distributed on $[-\pi, \pi]$, and independent. Let: $$(x,y) = (\cos(U+V),\sin(U-V))$$ What is the probability distribution function of $f_{x,y}(x,y)$ My ...
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20 views

Fint $N$ games played by minimal number of players

I got a following problem: There is a database which looks like this: $$\operatorname{GameId}||\operatorname{PlayerId}$$ $$1||1$$ $$1||2$$ $$\dots||\dots$$ where every game was played by 10 ...
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22 views

Usage of the phrase “constant parameter”

In general, constants are globally fixed, while parameters are a bit more free. But suppose I wish to use the word "constant" as an adjective to emphasize that a parameter is fixed w.r.t. other ...
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39 views

Is a set of some $m \times n$ matrices a relation?

A relation between sets $A_i, i = 1, \dots, n$ is defined as a subset of $\prod_i A_i$. Given $m, n \in \mathbb N$, is a set of (some or all) $m \times n$ matrices over $\mathbb R$ considered a ...
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26 views

Bounding the operator norm of some operators

Let $T:X \to Y$ be an operator between Banach spaces with $Tx:= A(Bx+Cx)$ where $A$, $B$ and $C$ are operators too. Is it possible to find a upper bound of the form $$\lVert{(I+T)^{-1}} \rVert \leq ...
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34 views

Finding Optimal Threshold Values For Ensemble Predictor

I have a range of eight models (each providing a p-value as measure of significance of a certain property of an instance), which, for a final prediction (binary, with 1 being a positive hit) I combine ...
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32 views

traingular matrix

I am not much in to the matrices. so I am sorry if I can not put forward the question properly. Assume $G_{n\times n}$ is a rank $n$ matrix (Indeed $G$ is the generator matrix of a lattice). I need ...
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30 views

Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...
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36 views

just a question

This is kinda silly question. But just asking to be sure of some fact. Someone just said today that $Hom(\mathbb{R}^{n},\mathbb{R}^{m})$ is isomorphic to $M_{n\times m} (\mathbb{R})$. But all $f \in ...
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46 views

A question about a Markov Chain

I encountered a question about Markov Chains which looks interesting. Given a homogeneous, irreducible, non cyclic Markov Chain with $K$ possible states and a transition matrix $Q$. We define $T_i$ ...
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18 views

Can I calculate the overall % each drug is prescribed if the list of drugs shown to each respondent varies?

I have a survey where the Q's are: Q1: Which of these drugs are you aware of - list of 10 drugs shown Q2: What % of your patients receive each drug (only drugs identified at Q1 shown). As the drugs ...
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20 views

Attracting basin is simply connected

How to prove that the immediate attracting basin of a (finite) attracting periodic point is simply connected? It's a question from Devaney's An Introduction to Chaotic Dynamic System and a hint is ...
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0answers
18 views

$LU$ decomposition of a matrix

My question is: Is it okay to switch rows to find the $LU$ decomposition (Lower, Upper Triangle) of let's say matrix $A$. Let's say we found matrix $U$ easily without having to switch rows. Now, can ...
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28 views

Tree-width of a graph

What is the tree width of the graph? Here are the relevant definitions from my textbook: We define the width of an induced graph to be the number of nodes in the largest clique in the graph ...
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37 views

Definite integral with dx=d

I'm currently reading a paper (Baye & Morgan & Scholten, 2006. "Information, Search, and Price Dispersion") where the following integral is computed (p.14) : $\int^{r}_{\underline{p}}p^2 ...
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43 views

Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
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0answers
26 views

How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
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0answers
11 views

Vertex Cover Approximation

Is there any Vertex Cover approximation algorithm that gives the optimal solution for some graphs but otherwise near-optimal solutions for other graphs? Would an algorithm like that be useful? From ...
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0answers
40 views

Estimate coefficients in an analytic expansion

Suppose I have an analytic function $f(z)$ in the open unit disc $$f(z)=\sum_{n=0}^\infty a_nz^n$$ Assume that $|f(z)|$ is less than $M$ for all $z$ in the open unit disc. Show that $M|a_1| \le ...
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0answers
44 views

Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
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51 views

Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
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0answers
44 views

What can we say about shape of intersection area of $N$ disks on a plane?

Intersection area of two disks can be bounded by at most two arcs. Intersection area of three disks can be bounded by at most four arcs. It looks like (I'm not sure) that four disks can have common ...
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0answers
14 views

network design: why can't an almost satisfied proper function violated by all given sets?

I'm reading a book about (survivable) network design and i have a problem understanding a lemma. Given an undirected graph G and $V(G)$ its nodes and $E(G)$ its edges. The book defines a proper ...
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0answers
57 views

Integrating by parts with exponential and power-law functions

I have a question about integrating by parts for $$\int_{L}^{U}\left[x^{a} \cdot e^{-bx}\right]\,dx$$ for positive reals $L,U$ with $L<U$ ($L, U \in [0, +\infty) $). I'm interested in cases with ...
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33 views

Homotopic maps of a compact polyhedron

My friend and I are trying to solve the following exercise. Problem: Let $X \subset \mathbb{R}^n$ be a compact polyhedron. Show that there exists $\alpha > 0$ such that for any pair of maps $f, g ...
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14 views

$w \in \operatorname{Span}(T) \leftrightarrow [w]_b\in \operatorname{Span}[T]_b$?

$$w \in \operatorname{Span}(T) $$ lets apply coordinate function on both sides $$[w]_b=\alpha_1t_1+\cdots+\alpha_nt_n=[t_1,\ldots,t_n]_b$$ $$\operatorname{Span}(T)=\sum \beta_it_i=\left[\sum ...
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0answers
51 views

Can a Computer Algebra System 'experiment' with expressions?

I have recently been reading about software for symbolic manipulation, and I can see its use as a tool for performing large calculations that would be unfeasible otherwise. Given that these systems ...
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0answers
52 views

Diophantine Equations involving cubes

I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations): $$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters) $$z^3 = x^2 + ...
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24 views

Find the reordering of a matrix rows based on vector.

Let $A\in\mathbb{R}^{m\times n}$ and $\mathbf{u}=(u_1,\ldots,u_n)^T\in\mathbb{R}^n$. Is there any map $f\colon\mathbb{R}^{m\times n}\times\mathbb{R}^n\to\mathbb{R}^{m\times n}$, $$ f(A,\mathbf{u})=B, ...
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23 views

Change of Variables in an Asymptotic Big-Oh Situation

I'm looking at the function $cos(x)^n$ as $n$ varies. It appears to be gaussian. The book says it's easy to verify that it is Gaussian: set $x=\omega/\sqrt n$, and then a local expansion yields: ...
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0answers
13 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...

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