**1**

vote

**0**answers

18 views

### 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?
...

**4**

votes

**0**answers

29 views

### 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 ...

**3**

votes

**2**answers

105 views

### 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], ...

**4**

votes

**2**answers

50 views

### 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 ...

**5**

votes

**0**answers

43 views

### 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 ...

**7**

votes

**2**answers

71 views

### 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 ...

**5**

votes

**0**answers

52 views

### Improving Baumgartner's result?

Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...

**5**

votes

**1**answer

213 views

### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow.
Besides, I know that there ...

**0**

votes

**1**answer

56 views

### Time Hierarchy Theorem and P vs NP

One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...

**5**

votes

**0**answers

93 views

### Grothendieck's paper on principal bundles, reduction to a torus step

In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ...

**13**

votes

**1**answer

255 views

### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...

**8**

votes

**2**answers

493 views

### A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$.
I am a prospective undergraduate mathematics student in Zimbabwe ...

**3**

votes

**0**answers

75 views

### What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...

**2**

votes

**0**answers

55 views

### Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)$ imply $A\cong B$? [duplicate]

Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)\implies A\cong B$ hold for arbitrary sets $A, B$?
Notation:
We write $\mathfrak{P}(M)$ to denote the power set of a set $M$.
$M_1\cong M_2$ is just an ...

**0**

votes

**0**answers

11 views

### F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...

**-2**

votes

**0**answers

92 views

### Linear forms that avoid numbers with lot of factors

Is following true?
For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log ...

**3**

votes

**1**answer

52 views

### 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. ...

**3**

votes

**1**answer

204 views

### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...

**2**

votes

**1**answer

41 views

### Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ ...

**0**

votes

**0**answers

15 views

### 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 ...

**4**

votes

**1**answer

39 views

### 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 ...

**0**

votes

**0**answers

14 views

### 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 ...

**6**

votes

**2**answers

91 views

### 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 ...

**4**

votes

**0**answers

47 views

### 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 ...

**5**

votes

**3**answers

204 views

### 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 ...

**3**

votes

**1**answer

119 views

### 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 ...

**5**

votes

**0**answers

35 views

### 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 ...

**18**

votes

**2**answers

294 views

### 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 ...

**3**

votes

**0**answers

91 views

### 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 ...

**0**

votes

**1**answer

37 views

### 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 ...

**13**

votes

**3**answers

812 views

### 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 ...

**3**

votes

**0**answers

56 views

### 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 ...

**6**

votes

**1**answer

103 views

### 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 ...

**3**

votes

**0**answers

62 views

### 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 ...

**0**

votes

**0**answers

57 views

### 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 ...

**4**

votes

**2**answers

252 views

### A property of uncountable almost disjoint families

Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge ...

**4**

votes

**0**answers

56 views

+100

### Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...

**10**

votes

**1**answer

156 views

### equivariant stable class of quaternionic Hopf fibration in RO(G)-degrees of ADE-type

Does the quaternionic Hopf fibration possibly represent a non-torsion element in the $G$-equivariant stable homotopy groups of spheres, for $G$ a finite subgroup of $SO(3)$ and in RO(G)-degree being ...

**3**

votes

**1**answer

98 views

### Progress on isospectral plane domains

Has there been any progress on the $\textit{smooth}$ isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to ...

**6**

votes

**0**answers

181 views

+150

### A characterization of quadratics similar to an inverse sieve problem

Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties:
$|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and
the discriminant $\prod_{\alpha \neq ...

**8**

votes

**1**answer

493 views

+100

### 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 ...

**3**

votes

**0**answers

239 views

+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 ...

**5**

votes

**1**answer

75 views

### 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 $\text{char}\,k > 0$?

**1**

vote

**1**answer

188 views

### 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 ...

**11**

votes

**2**answers

315 views

### 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, ...

**3**

votes

**0**answers

76 views

+50

### 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 ...

**6**

votes

**1**answer

215 views

+50

### 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 ...

**1**

vote

**0**answers

40 views

### 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 ...

**9**

votes

**2**answers

201 views

### Evolution operator for a linear parabolic equation

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 ...

**71**

votes

**25**answers

27k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...