# All Questions

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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 $f$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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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### (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 $SL_{n}(\mathbb{R})$ ...
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### 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?
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### 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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### Probability and correlation function, interpretation of a result

My question is originated from the paper ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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 ...