# All Questions

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### Reference on Infinite Dimensional Manifold

I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold. Now I have two questions or ...
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### a question about $SL(2n, \mathbb Z)$ and symplectic group $Sp(2n, \mathbb Z)$

My question: what are the generators of $SL(2n, \mathbb Z)$? And what are the generators of $Sp(2n, \mathbb Z)$? Thank you !
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### combinatorial and linear duality

Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual ...
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### How to rank a separate population using elo points/system [on hold]

Background: I have a website where students vote on the attractiveness of their peers: they are presented with two images, and they must pick one (the "winner")- then the elo score for each is ...
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### Looking for Soviet Math. Dokl

I'm a undergraduate student and I'm trying to study research level papers for the first time. I'm studying William Weiss's "Countably compact spaces and Martin's axiom" and he gives "Soviet Math. ...
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### Pan geometry trigonometry unification [on hold]

This is a very important and interesting question, the answer to which is perhaps known only partially. Reference to Robert Bonola " Non-Euclidean geometry ", "Pan- geometry" . Circular functions ...
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### What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?

If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
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### Finding the peak of a bivariate Gaussian distribution via a guess-and-check protocol with noise

I have a bivariate Gaussian distribution $f(x,y)$ with mean $\mu$, covariance matrix $M$, and square roots of the eigenvalues of the covariance matrix $(\sigma_1,\sigma_2)$. We now play a kind of ...
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### Solving a difficult equation for a variable?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
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### A definition of ($\infty$-)cat(egorie)s [on hold]

Yesterday I uploaded an updated version of my paper on "Cats". It will be available on the arXiv as of monday; until then, you can download a copy here. In that paper, I take Simpson's category ...
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### Contour deformation and inversion

I am working on a research problem and would like to know peoples views about what is contour deformation and its inversion. It will be appreciable if one can explain it with the help of an example.
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### Counting function for sums of three squares [migrated]

Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any ...
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### a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
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### Do you know their definitions? [on hold]

contragredient representation extremal weight fine K-type fine representation generalized infinitesimal character good roots Harish-Chandra map Harish-Chandra module Hochschild-Serre spectral ...
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### Characterizing Inf and Sup sets

Suppose $X$ is a set. Let $\Xi$ be the set of all pairs $(\mathcal A,\mathcal B)$ such that $$\mathcal A, \mathcal B \subseteq \mathcal P(X)$$ and for some partial order $\le$ on $X$, $\mathcal A$ is ...
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### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
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### In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?

What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?
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### Comparison between operators

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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### Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic? Update: Which (different) methods can be used to ...
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### Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for the shape that encloses three given volumes in $\mathbb{R}^3$ and has minimal surface area? I know the planar triple bubble conjecture was ...
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### unique types and decidability

Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is ...
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### Existence of local frames in Hilbert manifolds

Thank you for looking at my queation. I'm studying about Hilbert manifolds. I would like to ask you if there exists some concept like a local frame for every Hilbert manifold. I'm refering to the ...
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### In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
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### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
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### Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
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### Prove the functor $[n]\to [n]\star [n]$ preserves inner anodyne

Let $f:\Delta\to \Delta$ be the functor given by $[n]\mapsto [n]\star[n]=[2n+1]$. We can extend $f$ cocontinuously to a functor $$f_!: SSet\to SSet$$ (that is, the left adjoint of the functor $f^*$. ...
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### A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
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### upper bound for an incomplete exponential sum

Let $q$ be a large prime and $\delta\in(0,1)$. Let $k$ be an integer which is not a multiple of $q$. Define $e(x)=e^{-2\pi ix}$. Can we get any non-trivial upper bounds for ...
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### Inducing a comodule structure on Hom

If $C$ is an $R$-coalgebra and $M$ is an $R$-module... then is it possible to endow $Hom_{_RMod}(C,M)$ or $Hom_{_RMod}(M,C)$ with the structure of a $C$-comodule?
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### cohomology of a space from a map to affine plane

Suppose $X$ is an affine variety,complete intersection in $\mathbb{C}^{2n}$, but with a high dimension of singularities. I also have a surjective finite algebraic map $f:X\rightarrow \mathbb{C}^{d}$. ...
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### Generators of Ideals in Integers in a Number Field [migrated]

Let $R$ be the ring of integers in a number field $K$. It is known that each ideal of $R$ can be generated by two elements. In fact if $I$ is an ideal of $R$ and $a\in I$ is a nonzero element, then ...
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### Cut locus, conjugate points and smoothness of distance function

I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance ...
Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
### Finite Subgroups Of $GL(2,\mathbb{R})$ [migrated]
I have the following question: Is it true that every finite subgroup of odd order in $GL(2,\mathbb{R})$ is cyclic? Thanks!