# All Questions

**1**

vote

**0**answers

89 views

### Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable.
Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...

**0**

votes

**0**answers

14 views

### Reducing the degrees of freedom in unitary columns

Let $U = diag([U_1, U_2, ..., U_N])$ be a block-diagonal $NM \times NM$ unitary matrix, where each $U_j$ is a unitary $M \times M$ matrix.
Furthermore, let $Q_e = kron(I_M, Q)$ be the Kronecker ...

**3**

votes

**1**answer

102 views

### A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

**7**

votes

**1**answer

140 views

### Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square.
To be exact, the group $G$ has a special element $1$, a unary ...

**8**

votes

**0**answers

147 views

### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...

**1**

vote

**0**answers

89 views

### probability of a quadratic form being nonnegative at a random point

I am looking for good and explicit lower and upper bounds on the probability that $x^TGx\ge 0$, where $G$ is a symmetric matrix with zero trace and $x$ is a vector whose components are independent ...

**17**

votes

**3**answers

791 views

### Is there any pattern to the continued fraction of $\sqrt[3]{2}$?

Is there any pattern to the continued fraction of $\sqrt[3]{2}$ ? Wolfram Alpha returns for cube root of 2:
$\sqrt[3]{2}=$ [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...

**7**

votes

**1**answer

129 views

### Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...

**0**

votes

**0**answers

34 views

### Error in the paper “Oscillating-Decaying Solutions, Runge Approximation and its Applications to Inverse Problems”?

I am going to use simpler notation then the paper. The potential problem is not very technical.
Theorem 4.2. in the paper states the following:
a) If $t < t_0$, then $I(t,\tau) \to 0$ as $\tau ...

**1**

vote

**1**answer

106 views

### Sum of Eigenvectors Entries of an Adjacency Matrix

I have a question regarding the sums $\sum_{i=1}^{n}v_{j}\left(i\right)$ where $v_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length.
Ordering the eigenvectors by ...

**6**

votes

**1**answer

136 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 ...

**1**

vote

**0**answers

113 views

### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an ...

**2**

votes

**0**answers

58 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**4**

votes

**1**answer

209 views

### Unit in cyclotomic field

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...

**4**

votes

**2**answers

373 views

### What does it mean for a differential equation “to be integrable”? [duplicate]

What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...

**3**

votes

**0**answers

32 views

### Cartan integral formula for a p-adic group?

Let $G$ denote a reductive group over a local field $F$. Suppose that $G$ is split over $F$ and fix a maximal (split) torus $A$. Let $A^+$ denote a Weyl chamber in $A$ and let $K$ be a suitable ...

**1**

vote

**0**answers

42 views

### Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade

Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has:
and in Theorem 16 of Chapter 5 proves that:
$p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...

**5**

votes

**0**answers

152 views

### Theories introduced by a class of forcing notions

The following notion is introduced by Mohammad Golshani. Let $V$ be a model of set theory and let $\mathcal{P}$ be a class consisting of non-trivial forcing notions in $V$. Let
$$Th(V, ...

**1**

vote

**0**answers

83 views

+50

### A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...

**-1**

votes

**0**answers

47 views

### Is this one of equivalent formula to compute riemann gamma, or something “new”? [on hold]

Rimeann-Siegel formula is the next:
$$\zeta(s)=\sum_{n=1}^{N}\frac{1}{n^{s}}+\gamma(1-s)\sum_{n=1}^{M}\frac{1}{n^{1-s}}+R(s)$$
Where
$\theta(t)=arg(\Gamma(\frac{2it+1}{4}))-\frac{log(\pi)}{2}t$
...

**5**

votes

**0**answers

86 views

### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here.
Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...

**2**

votes

**1**answer

76 views

### Prescribing an induced metric

We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial ...

**4**

votes

**0**answers

38 views

### $G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$.
My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

**1**

vote

**0**answers

30 views

### Algebraic invariants of linear ODE's with constant coefficients

Let consider linear ODE with constant coefficients:
$$y^{(n)}(x) + A_{n-1} y^{(n-1)}(x) + ... + A_1 y'(x) + A_0 y(x)= 0.$$
It admits some equivalence point transformations, that preserves its ...

**3**

votes

**0**answers

69 views

### Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.
If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...

**2**

votes

**0**answers

73 views

### $\mathbb{C}[x_1, \dots, x_n]$ is a free $\mathbb{C}[x_1, \dots, x_n]^{S_n}$-module with certain generators [duplicate]

Let the symmetric group $S_n$ act on $\mathbb{R}^n$ by permutation of coordinates. This makes $S_n$ a subgroup of $\text{GL}_n(\mathbb{R}$ and the algebra $\mathbb{C}[x_1, \dots, x_n]^{S_n}$ is the ...

**1**

vote

**0**answers

75 views

### Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated?
A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or ...

**-5**

votes

**0**answers

105 views

### New way to solve Quadratic Equations [on hold]

A couple months ago, I was challenged to discover a new way to solve quadratic equations with one rule: I must think of it myself. At first I was hopeless, trying various sorts of random things, but I ...

**-1**

votes

**0**answers

33 views

### weak convergence related problem [on hold]

I am studying for an exam and this was one of the hard problems in my textbook.
Let $f_n(x) = 1 −\cos(2\pi nx)$ for n $\in$ N and x $\in$ $[0,1]$. Verify that $f_n$ is the density of a probability ...

**0**

votes

**0**answers

38 views

### CLT related problems [on hold]

Let $S_n$ = $X_1+···+X_n$ be a sum of i.i.d. mean zero random variables with a finite, positive variance. How can I show that $lim_{n\rightarrow\infty}$ $S_n/\sqrt n$ = $\infty$ a.s. also there is no ...

**0**

votes

**0**answers

27 views

### Regularity of the heat equation: Neumann boundary conditions

I am looking for references to regularity estimates for the solution of a heat equation with homogeneous Neumann boundary conditions on $[0,T]\times D$ for some smooth domain $D\subseteq \mathbb{R}^3$ ...

**1**

vote

**0**answers

52 views

### Chain Rule for Maximal Correlation

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**-6**

votes

**0**answers

46 views

**2**

votes

**1**answer

120 views

### Gradual monotonic morphing between two natural numbers

Let $a < b$ be two natural numbers. I will use these as an example:
\begin{align*}
a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\
b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608
\end{align*}
I seek to ...

**11**

votes

**1**answer

230 views

### Practical advantages of univalent foundations

I'm interested in the machine translation of mathematics from informal to formal (a la Ganesalingam in The Language of Mathematics). As a first step, I am designing a computer language for expressing ...

**3**

votes

**1**answer

104 views

### Algorithm to count the number of perfect matchings in non planar graph

I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing ...

**0**

votes

**0**answers

26 views

### How can I efficiently check a point lie in 4 circles on sphere? [migrated]

Given coordinate of 2 points $A,B$ (Cartisian or longitude-latitude coordinate) on sphere of radius $R_1$. Point $O$ is middle of $AB$, 2 points $E$ and $F$ is derivation from $O$ by a distance ...

**6**

votes

**2**answers

292 views

### Is this almost-cosimplicial object familiar?

I have a sequence $X_0,X_1,\ldots$ of Abelian groups, along with face maps $d_0,\ldots, d_n: X_{n-1}\to X_n$ which satisfy the standard cosimplicial identities EXCEPT at the very bottom, where in ...

**1**

vote

**0**answers

33 views

### An inequality for Maximal Correlation over a Markov Chain

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**1**

vote

**0**answers

59 views

### Understanding the Segal-Sugawara construction

I am trying to understand the Segal-Sugawara construction from the book "vertex algebras and algebraic curves" by Frenkel, Ben-Zvi in 3.4.8. As an absolute layman in the area of mathematical physics ...

**0**

votes

**1**answer

52 views

### Complement of a finite union of convex sets

Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...

**1**

vote

**0**answers

51 views

### On low rank $0/1$ real matrices with one connected component

Let $M$ be an $n\times n$ $0-1$ matrix of real rank $r$.
Moreover assume every row of $M$ is distinct and every column of $M$ is distinct. Such a matrix cannot be bigger than $2^r\times 2^r$.
As in ...

**7**

votes

**1**answer

139 views

### $p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...

**6**

votes

**1**answer

260 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

**0**

votes

**0**answers

91 views

### from finite to $\sigma$-finite measure space [migrated]

This might be rather elementary. I have put it at MSE for a while without getting any answers.
Here is the question:
In the proof of the following theorem, would anyone explain how the general case ...

**0**

votes

**0**answers

30 views

### Voronoi-type summation formula for coefficients of symmetric square $L$-functions

given a primitive form $f$ for the full modular group $SL_2(Z)$ and $\lambda_f(n)$ be the $n$th Hecke eigenvalue. Various Voronoi-type formulas are fulfilled by these coefficients and there are ...

**0**

votes

**0**answers

7 views

### An upper bound for the trace of the product of the (rank-deficient) sample covariance and its “regularized inverse”

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_p)$. Let $\hat S = \sum_{i=1}^n x_i x_i^T$ be the sample covariance.
When $p>n$, $\hat S$ is not invertible. However, by adding a positive-definite diagonal ...

**3**

votes

**0**answers

45 views

### Laplace Equation with Tangential Derivative Prescribed on the Boundary [on hold]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi ...

**-2**

votes

**0**answers

24 views

### $M = P_1U_1$ and $N = P_2U_2$ ,$M $ and $N$ are unitarily equivalent $ \Rightarrow $ $P_1$ and $P_2$ are unitarily similar [on hold]

Let $M, N ∈ M_n$. Let $M = P_1U_1$ and $N = P_2U_2$ be polar decompositions. ($P_i$ is positive semidefinite and $U_i$ is unitary matrix).
Suppose $M $ and $N$ are unitarily equivalent.
Why are ...

**14**

votes

**3**answers

838 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 ...