1
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0answers
49 views

Understanding a proof by R.C Lyndon and J.L Ullman.

Here in this article I have difficulties understanding the theorem on page 162. Theorem. Let $A, B$ and $C=AB$ be an elements of group $GL_2(\mathbb{Z})$, all with real fixed points. Suppose that ...
1
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0answers
114 views

Continuity of $\mathfrak{a}$-adic topologies on completion ring and modules

Let $A$ be a commutative ring with identity and $\mathfrak{a}$ and ideal of $A$. Then $A$ has a topological structure which is defined by the following chain of ideals \begin{equation*} A \supseteq ...
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0answers
45 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
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0answers
99 views

Formula for monthly payment of mortgage

What is the formula for monthly payment of mortgage including Term, Interest Rate, Cost of Home Down, Payment Insurance, Property Tax, HOA Fee. I'm a programmer and want to add this functionality to ...
1
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0answers
62 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
1
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0answers
121 views

Maximum Modulus Principle used on boundaries of domains in \Bbb C

Let $D$ be a domain in $\Bbb C$, and $u$ be the real part of a function $f$ which is analytic on $D$. Assume $u$ is constant on the boundary of $D$. Show that $f$ is constant on $D$. Well, I know ...
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0answers
143 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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0answers
42 views

Meaning behind variable transformation

At times we change variables to ease computation for example while solving indeterminate forms of limits which have repeating pattern on applying lopital's rule. For ...
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0answers
40 views

Counting roots of sums of sigmoids

Let $f(x)=\sum_i a_i\tanh(b_ix+c_i)+d$ be the class of sums of $n$ sigmoids parameterized by $a,b,c,$ and $d$, with all values being real. I suspect, but can't prove, that the number of roots of $f$ ...
1
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0answers
53 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
1
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0answers
100 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
1
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0answers
171 views

Problem on convergence of Series (Rudin)

This is problem $3.11$ of Rudin's real Analysis. Suppose $a_n>0$ , $s_n := a_1+a_2+...+a_n$ and that $\sum a_n$ diverges. Prove that $\sum \dfrac{a_n}{1+a_n}:=\sum b_n$ diverges. I came across ...
1
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0answers
112 views

Prove that the set N of normal numbers has negligible complement.

I'm taking a graduate course called Real Variables I, without having taken the prerequisite of Real Analysis II, and having taken only Real Analysis I. Therefore, I'm brand new to measure theory and ...
1
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0answers
67 views

Can a condition for a global maximum (of some specific function) be given?

Suppose we have a twice continuously differentiable function $h(x) := \frac{g(x)}{1 - \delta + \delta F(x)}$, $0<\delta<1$, defined on the interval $[0, a]$ (where $a$ may be infinite). The ...
1
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0answers
61 views

Linear Complementarity Problem - multiple solutions, which one will it find?

If I have a inequality constrained system: w = Mz + q <= 0, z<=0, z^T w = 0 that for some given properties M and ...
1
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0answers
33 views

Length of the voronoi diagram

Does there exist an algorithm for computing the length of the voronoi diagram of a set of points or just gives the intersection points of the voronoi diagram?
1
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0answers
58 views

Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

Charles Weibel writes in his survey of homological algebra Riemann de fined a surface $S$ to be $(n + 1)$-fold connected if there exists a family $C$ of $n$ closed curves $C_j$ on $S$ such that ...
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0answers
25 views

(Reference Request) Canonical forms for Real and Complex binary forms of low degree.

I am asking for a reference for Canonical forms for Real (and Complex) binary forms of low degree with respect to the natural action of the Real (and Complex) special linear group $SL_{n}(\mathbb{R})$ ...
1
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0answers
73 views

Isomorphism between quotients of linear groups

Suppose that $n$ is even. Is it true that $$\mathrm{SL}_n (\mathbb{R})/\{ \pm I \} \times \{\pm 1\} \cong \mathrm{GL}_n(\mathbb{R})/{\sim}$$ where $A\sim B$ if and only if $A=aB$ for some $a\in ...
1
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0answers
92 views

Keeping a constant resolution scale of an image

Intro I'm writing a program that zooms in and out on an image and the only way to do it with the data I have is to change the limits of my graph. The aspect ratio (scale of x to y) always remains 1:1 ...
1
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0answers
27 views

Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess). First of all some definitions: Definition 1 (möbius-transformation): Let $A \in \mathbb{C}^{2 ...
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0answers
103 views

Sufficient conditions for the convergence of countably infinite products of matrices

I'm interested in countably infinite products of matrices of the form $$T_k=\begin{pmatrix}1&0&\cdots &0&0&0\\ 0 &\ddots&0&\cdots&0&0\\ a_{i1}& \cdots& ...
1
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0answers
162 views

If a sequence of smooth functions converge in the Sobolev norm, what can one say about the limiting function?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of $k$ times weakly differentiable functions on $\mathbb{R}^n$, having finite Sobolev $p$ norm. Suppose $\{f_i\}$ is a Cauchy sequence in ...
1
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0answers
39 views

Laurent Series Expansion Coefficient.

For the function $f(z)={1\over z}$ where $z_0=1.$ I'm not exactly sure how they get $c_n=(-1)^n$ using the formula for finding the coefficient. Any help is appreciated, thanks.
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0answers
79 views

Speeding up solution of a binary integer program

To solve the problem of making a "good" schedule for a tournament between N teams, using memories from my (long gone) student days, I expressed it as a binary integer program. With the current set of ...
1
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0answers
44 views

Binet's differential equation yields a circular path

Binet's equation is $$\frac{d^2u}{d\theta^2}+(1-\frac{\lambda}{a^2v^2})u=0$$ I need to show that for $\lambda=a^2v^2$ path will be a circle. For $\lambda=a^2v^2$ I have $$\frac{d^2u}{d\theta^2}=0$$ ...
1
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0answers
148 views

Help with spinor indices

Let's have $$ \varepsilon^{\alpha \beta} = \varepsilon^{\dot {\alpha }\dot {\beta }} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}^{\alpha \beta}, \quad \varepsilon^{\alpha \beta} = ...
1
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0answers
42 views

Simple finite supergroups and sporadic supergroups

Is there an analogue list with "finite simple supergroups" similar to the finite simple group classification? Are there sporadic "finite simple" supergroups?
1
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0answers
110 views

Transversal and complete intersection of hypersurfaces in $\mathbb{P}^{n}$

(a) Let $k<n$ and $F_{1},\dots,F_{k}$ be homogeneous polynomials of degrees $d_{1},\dots,d_{k}$ of $n+1$ variables in generic case. Prove that the corresponding hypersurfaces in ...
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0answers
59 views

homework probability question involving indicators with geometic variable

I am solving a question and came across something new to me, first - the question - in a box there are $2$ regular coins, $p=q=0.5$ and $4$ "fake coins" $p=1/4$ define $X = \begin{cases} 1, & ...
1
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0answers
143 views

Strong continuity in time vs uniform continuity in time

I have a problem with understanding the definition of strong continuity and uniform continuity for the families of operators, e.g. semigroups. Let $(X_t)_{t \geq 0}$ be a family of bounded linear ...
1
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0answers
170 views

Proving discontinuity of a Dirichlet like function

Let $ f(x)=\left\{\begin{matrix} x &if &x\in \mathbb{Q} & \\ 0& if&x\not\in \mathbb{Q} \end{matrix}\right. $ Prove that $f(x)$ is continuous only at $0$. Here is what ...
1
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0answers
118 views

Dimension of the root spaces of a semisimple complex Lie algebra

I have problems in understanding the proof that the root spaces of a semisimple Lie algebra are all 1-dimensional and that the only multiples of a root $\alpha \in \Phi$ which occur in $\Phi$ are $\pm ...
1
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0answers
136 views

Probability and correlation function, interpretation of a result

My question is originated from the paper ...
1
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0answers
63 views

decimal digit grouping delimiters

I feel a bit silly having to ask this but I just can't seem to find any resources that give an answer to this. When dealing with decimal values that have a large number of digits to the right of the ...
1
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0answers
51 views

Question about partitions and primes.

Let $A_1\cup A_2\cup\cdots\cup A_n = P$ , where $P$ stands for the set of odd primes $<\sqrt{x}$ and $A_i$ is nonempty. Also $\#A_k\gg \# A_l$ iff $k>l$ ($\#$ is cardinality ). In fact we ...
1
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0answers
30 views

Complexity of Earlist Avaible Due Date for Scheduling Problem 1|ri, pi=1|Lmax

Let us consider the scheduling problem 1|ri,pi=1|Lmax (basically, this means there is one machine on which we have to schedule n jobs (all with identical procssing time 1) in such a way that the ...
1
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0answers
79 views

Prove Hom$_{\mathbb{Z}}(\mathbb{Z}/n \mathbb{Z},A) \cong A_n$

Let $A$ be a $\mathbb{Z}$-module, let $a$ be an element of $A$ and let $n$ be a positive integer. Prove that the map $\phi_a:\mathbb{Z}/n \mathbb{Z} \rightarrow A$ given by $\phi(\bar{k})=ka$ is a ...
1
vote
0answers
54 views

Notion of Distribution

I have some difficulties in understanding the notion of distribution. As I understand, distribution, is some source of data, where each element of the data have some probability to occur, for example ...
1
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0answers
146 views

Dimension analysis and explaining the $\varepsilon$

Reference of this post (page no 6 from equation 39) The time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} ...
1
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0answers
185 views

Convergent series involving logarithms

So I have a test next week and I found something which I do not understand. I probably have a wrong conclusion but I do not know where it is. I was told that the series $ \sum_{n=1}^\infty ...
1
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0answers
274 views

Find CFG for Lisp-like expressions

All Lisp-like expressions. A Lisp expression may be an unsigned integer or a list. A list, enclosed by left and right parentheses, consists of one operator ($+$, $-$, $*$, $/$, $\max$, $\min$) ...
1
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0answers
41 views

General form of the relation between three Hypergeometric functions

I'm reading this article, where the main result which relates the tree Hypergeometric functions ${_2F_1}(a_1+\alpha_i,a_2+\beta_i;a_3+\gamma_i;z)$, $i=1,2,3$ is given in theorem 3, page 297. In tables ...
1
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0answers
167 views

Intuition about absolute Continuity/Singularity of measures

Let $\mu$, $\nu$ be two measures, $X= d\nu/d\mu$ their Radon-Nikodym derivative (in general a random variable). I want to gain an intuition about the following statements: $$\int X\; d\mu = 1 ...
1
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0answers
76 views

partial differential equation problem

I have the pde $$\ \dfrac {\partial u}{\partial t} + \ 3\dfrac {\partial u}{\partial x}=0.$$ $$ x>0, t>0$$ $$u(x,0)=e^{-x^2}, x>0,$$ $$ u(0,t) = e^{-t^2}, t>0 $$ The characteristics ...
1
vote
0answers
133 views

Inverse Laplace transform of functions with jump discontinuities

Given a function $F(s)$, suppose we define its inverse Laplace transform as: \begin{equation} f(t) = \lim_{k \to \infty} \frac{(-1)^{k}}{k!}\left(\frac{k}{t}\right)^{k+1}F^{(k)}\left( \frac{k}{t} ...
1
vote
0answers
344 views

Trace theorem, Sobolev space

Let $\Omega$ be a domain of $\mathbb{R}^d,$ and let $\Omega_1$ and $\Omega_2$ be a partition of $\Omega$, and let $\Gamma=\partial \Omega_1 \cap \partial \Omega_2 \subset \Omega$. Let $u_i=u/\Omega_i, ...
1
vote
0answers
37 views

Complexification of the inclusion $\text{U}_n\subset \text{GL}_n(\mathbb{C})$

What is the map $\text{GL}_n(\mathbb{C}) \to \text{GL}_n(\mathbb{C}) \times \text{GL}_n(\mathbb{C})$ named in the title? I guess it has something to do with the polar decomposition, but I can't manage ...
1
vote
0answers
136 views

Grid Function Norm

I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the ...
1
vote
0answers
62 views

Error in Lang's definition of weak topology?

On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims Let $Y$ be a topological space and let $\mathscr{F}$ be a family of mappings $f \colon X \to Y$ of $X$ into $Y$. Let ...

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