# All Questions

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### Prove that the positive existential theory is undecidable

Could you tell me how we could prove that the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ is undecidable? ...
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### If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here. I am curious as to whether or not we can, in a similar way, show that if $k$ is an algebraically closed field (of any characteristic) then the ...
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### Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [[2, 0, 1, 7], [3, 4, 5, 6]] [[4, 0, 6, 7], [2, 3, 1, 5]] [[0, 7], [4, 6], ...
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### What characterizations of relative information are known?

Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as relative information, relative entropy, information gain or Kullback–Leibler divergence is defined ...
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### all subsets borel

Assume Martin's axiom plus not CH. It is well know, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space each of whose subsets is ...
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### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$? For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...
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### Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. ...
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### Fractal Dimension Estimate from Hurst Paramater using Higuchi's Algorithm [on hold]

I am using R Stats package to calculate the Hurst parameter of my Center of Pressure Data set. I am using Higuchi's Algorithm to calculate it. I went and read his paper and saw that "index H is ...
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### Proving moduli of uniform continuity in RCA_0

Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ...
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### regularization and conversion of sgn(x) to difference of convex [on hold]

sgn(x) or sign function has discontinuity in 0 which make it nonconvex function. however i have tried to represet sgn(x) as a limit for a sequence of converging function which are smooth and ...
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### Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let ...
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### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...
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### family of polynomials with square discriminant

The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of ...
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### Does the nearby cycle functor commute with the Verdier duality?

I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
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### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...
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### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...
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### Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers. My question is on moduli space of varieties of ...
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### expressing in terms of sum of (double) schubert polynomial

It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,x_3,...]$. I am interested in knowing how to express a particular polynomial into sum of Schubert ...
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### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...
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### Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$

I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
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### Eigenvalue inequality for regular graphs

I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim ...
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### Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting. Let us consider a smooth projective algebraic variety ...
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### Modifying tensor to be positive definite everywhere [on hold]

Consider a (0,2)-tensor. It is known that it is positive definite somewhere and it is negative definite otherwise. Is there a theory how to "make" that tensor positive definite everywhere, while ...
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### Hilbert schemes and moduli of ideal sheaves

Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to ...
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+50

### Properties of the solution of Heat equation

Note 1: the following question has been post on Math Stackexcahgne here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
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### Center of $k$-algebra with two generators and sole defining relation $yx - xy = 1$ when $\text{char}\,k > 0$

Let $A(k)$ be a $k$-algebra with two generators, $x$, $y$, and one defining relation: $yx - xy = 1$. What is the center of the algebra $A(k)$ in the case $\text{char}\,k > 0$?
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### What is the cofinality of the positive measure sets of reals?

What is the minimal cardinality of a family of sets of real numbers, each of positive Lebesgue measure, such that every set of real numbers of positive Lebesgue measure contains some member of the ...
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### Quotient rule, differential operator on a localization is well-defined, underlying geometry?

Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, ...
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### Eigenvalues and eigenvectors of the q-Bernstein operator

The Bernstein operator maps $f\in C[0,1]$ to its Bernstein polynomial $B_n f.$ The eigenvalues and eigenfunctions of the Bernstein operator on $C[0,1]$ have been described in [1]. Similar description ...
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### Etale fundamental of a parahoric group scheme

Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$ i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by ...
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### capacity of a channel with random phase rotation

Consider a wireless channel $h=e^{j\theta}$, where $\theta$ is a uniform random variable in $[0,2\pi]$ independent of the input messages and the independent of the noise. The channel randomly rotates ...
Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...