# All Questions

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $M|a_1| \le ... 0answers 44 views ### Proof Strategy: Induction Summation of Series Let$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: $$P(1):1 + r = ... 0answers 51 views ### Plotting parametric form of a gradient This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ... 0answers 44 views ### What can we say about shape of intersection area of N disks on a plane? Intersection area of two disks can be bounded by at most two arcs. Intersection area of three disks can be bounded by at most four arcs. It looks like (I'm not sure) that four disks can have common ... 0answers 14 views ### network design: why can't an almost satisfied proper function violated by all given sets? I'm reading a book about (survivable) network design and i have a problem understanding a lemma. Given an undirected graph G and V(G) its nodes and E(G) its edges. The book defines a proper ... 0answers 57 views ### Integrating by parts with exponential and power-law functions I have a question about integrating by parts for$$\int_{L}^{U}\left[x^{a} \cdot e^{-bx}\right]\,dx$$for positive reals L,U with L<U (L, U \in [0, +\infty) ). I'm interested in cases with ... 0answers 33 views ### Homotopic maps of a compact polyhedron My friend and I are trying to solve the following exercise. Problem: Let X \subset \mathbb{R}^n be a compact polyhedron. Show that there exists \alpha > 0 such that for any pair of maps f, g ... 0answers 14 views ### w \in \operatorname{Span}(T) \leftrightarrow [w]_b\in \operatorname{Span}[T]_b?$$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 ... 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 ... 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) $$z^3 = x^2 + ... 0answers 24 views ### Find the reordering of a matrix rows based on vector. Let A\in\mathbb{R}^{m\times n} and \mathbf{u}=(u_1,\ldots,u_n)^T\in\mathbb{R}^n. Is there any map f\colon\mathbb{R}^{m\times n}\times\mathbb{R}^n\to\mathbb{R}^{m\times n},$$ f(A,\mathbf{u})=B, ... 0answers 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: ... 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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