0
votes
0answers
6 views

Generation in finite simple groups of Lie type

Let $S$ be a finite simple group of Lie type and $p$ be a prime such that $|S|_p=p$. I want to get some restrictions on $S$ with the conditions that $S$ is generated by elements of order $p$ and the ...
0
votes
0answers
18 views

Systole of a flat surface

Is the systole (length of the shortest saddle connection) of a flat surface $(X,\omega)$ ($X$ is a Riemann surface and $\omega$ an abelian differential on it with zeros in the points $\Sigma=\{p_1,\...
2
votes
2answers
20 views

Expected distance between points drawn from different distributions

Let $X$ and $Y$ be independent random variables, with $X_1,X_2$ and $Y_1,Y_2$ identical to $X$ and $Y$. Then (is it true that) $2\mathbb{E}|X-Y|\geq\mathbb{E}|X_1-X_2|+\mathbb{E}|Y_1-Y_2|$? This is ...
0
votes
0answers
26 views

Strong convergence on Dual of Reflexive Banach Space

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathfrak{S}_{p}(\mathcal{H})$, $1<p<\infty$, denote the $p$-th Schatten class of compact operators acting on it. Suppose we have a net ...
0
votes
0answers
39 views

congruences: number theory

We have the following Diophantine equation on $l, m, n$ (all belong to natural number) $(4a^2 - y^2)^l + (4ay)^m = (4a^2 + y^2)^n$, where $a$ and $y$ both belong to natural number with $(a, y) = 1$, $...
0
votes
0answers
25 views

Mappings of random processes $\varphi(X(t))$

I am interested in problems of the following type. Let $X(t)$ be a planar random process and $\varphi:\mathbb R^2\to\mathbb R^2$ be a mapping. Then what can we say about the image $Y(t) = \...
-2
votes
0answers
34 views

proof Hadamard's Inequality [on hold]

Theorem 4.2. Hadamard’s Inequality. Suppose A is positive semidefinite of size n. Then |A|≤ [A]11···[A]nn. Proof. Let A be any positive semidefinite matrix of size n. Note that In is a postive ...
0
votes
0answers
35 views

The midpoint geodesic

Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment. Pick three distinct ...
1
vote
1answer
85 views

Does composite number of the form $6k + 1$ has at least three non-totient divisors?

If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call ...
-2
votes
0answers
21 views

How to compare to geometric curve [on hold]

noisy image original image In original image i have a curve representing human contour, and in noisy image with human contour curve some additional noisy curves are there. I want to remove the noisy ...
1
vote
0answers
26 views

Lattices without nontrivial dense elements

This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed. An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\...
1
vote
0answers
25 views

Unique Fixed Point in a Simplex

I have a vector $(X_1,X_2,...,X_n)$ and satisfy the constrain $\sum_iX_i=1$. Then an operator is defined as $X_i=F_i(\textbf{X})\textbf{X}$, so in fact the operator is $T:\Delta^n\rightarrow\Delta^n$. ...
7
votes
0answers
149 views

Has the following problem posed by Deligne in the official description of the Hodge conjecture been solved?

Towards the end of his official description of the Hodge conjecture, Deligne asked the following question: Let $A$ be an abelian variety over the algebraic closure $\mathbb{F}$ of a finite field ...
-1
votes
0answers
25 views

Determinants of block matrices with non-square diagonal and square anti diagonal elements [on hold]

Is there a way to find the determinant of $X$ in terms of its sub-matrices $A,B,C_0$ and $R_0$? $$X = \begin{bmatrix} AC_0 & -I_n\\ 0_{(n-1)} & R_0B \end{bmatrix} \in \mathbb{R}^{(2n-1) \...
2
votes
0answers
22 views

Proof that the length function $\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$ is injective without the $9g-9$ theorem

In Chapter 10 about Teichmüller spaces of Farb and Margalit's "A Primer to Mapping Class Groups", the length function $$\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$$ is described, where $...
1
vote
1answer
50 views

Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$

I know that the MCG (isotopy-classes of orientation preserving homeomorphisms) of 3-torus $(S^1\times S^1 \times S^1)$ is $SL(3,Z)$, since it is an Eilenberg–MacLane space, giving $ MCG(T^3)=Out(\pi_1(...
-4
votes
0answers
22 views

which universties in USA and europe have intersted in delay differentail equations or functional differential equations? [on hold]

I am in master degree now about delay differential equations and I need after master degree get scholarship for Phd so which university is intersest on delay differential equation or functional ...
20
votes
0answers
384 views

What was achieved on IUT summit, RIMS workshop?

I would like to know what was achieved in the workshop towards the verification of abc conjecture's proof and the advance of understanding of IUT in general. A comment from a participant: C ...
-4
votes
0answers
29 views

Differentiation with composite, product and quotient rule [on hold]

This is a simple question but I hope someone can give a detailed explanation of how to solve the question. Differentiate y=xtan√x.
1
vote
0answers
64 views

How did the $\operatorname{div}_x$ “disappear”?

Given $E(x,t) := \nabla_x (\frac 1{|x|} * \rho)$ where $\rho : \mathbb R^3 \times [0,\infty) \to \mathbb R$ is defined by $\rho(x,t):=\int_{\mathbb R^3} f(x,v,t) dv$, I have understood that $$\rho(x,t)...
1
vote
0answers
23 views

Quantifying how much a vector gets turned toward the expanding direction of an $\mathrm{SL}(2,\mathbb R)$ matrix

Consider a matrix $B\in \mathrm{SL}(2,\mathbb R)$. Let $s$ be a vector that is pointing in the most contracted direction of $B$, and let $u$ be the image under $B$ of a unit vector pointing in the ...
8
votes
1answer
122 views

How to compute modular forms of weight one on Shimura curves?

Classical modular forms of weight $k\geq 1$ can be computed explicitly by exhibiting their $q$-expansion. When the weight is at least two, the most standard method uses modular symbols. When the ...
3
votes
0answers
80 views

How do sutured TQFT fit into the larger TQFT picture?

In https://arxiv.org/abs/0807.2431, Honda--Kazez--Matic introduce a definition of (1+1 dimensional) sutured TQFT; see also e.g. Mathews http://arxiv.org/abs/1006.5433 and Fink https://arxiv.org/abs/...
2
votes
0answers
41 views

Special values of real analytic Eisenstein series

Given $\tau$ in the upper half plane, define the normalized real-analytic Eisenstein series by $$ E(\tau, s) = \frac{1}{2} \sum_{(m,n)}' \frac{y^s}{|m\tau + n|^{2s}} $$ It is initially defined for $\...
1
vote
0answers
42 views

Reference for $q^{\eta\sum_{i}t_{\lambda_{i}}\otimes t_{\mu_{i}}}$ part of an expression for universal $R$-matrix of a quantum group algebra

The Wikipedia entry for quantum groups states that a quantum group $U_{q}\left(\mathfrak{g}\right)$ has an infinite formal sum that plays the role of an $R$ matrix, which is the product of two factors,...
0
votes
0answers
40 views

Homomorphism from integral module generated by roots of unity to cyclic group?

Let $S$ be the set of all roots of unity. Consider the $\mathbb{Z}$-module, $\mathbb{Z} S$, as an additive abelian group (that is, $\mathbb{Z} S$ is the subset of complex numbers that can be ...
4
votes
0answers
56 views

Traces in finite extensions of integrally closed domains

$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$I'm looking for a reference for the following commutative algebra fact. Let $A$ be an integrally closed integral domain, with field of fractions $K$. ...
2
votes
0answers
29 views

Torsionless not separable abelian groups

A torsionless abelian group $A$ is one for which any element $a\neq 0$ can be sent to a nonzero element of $Z$ by some homomorphism $A\rightarrow Z$ (integers). Equivalently, $A$ can be embedded as a ...
-1
votes
0answers
34 views

Deriving global probabilities from local dynamics

I am interested in growth dynamics and, in particular, how to derive difference/differential/stochastic equations from local behavior of a system. I tried posting first on math.stackexchange (https://...
0
votes
0answers
48 views

conjugacy classes of cyclic subgroups of order $k$ in $ {\rm GL}_n(\mathbb{Z}/p^m\mathbb{Z}) $

Let $p$ a prime numbers and $k$ be positive integer such that $(k, p) = 1$. And $m$ be the order of $p$ in $\mathbb{Z}/k\mathbb{Z}^×$. How many conjugacy classes of cyclic subgroups of order $k$ does ...
2
votes
1answer
64 views

Coefficients for Powers of the Mittag-Leffler Function

Considering the one parameter Mittag-Leffler function, $$E_{\alpha}(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$ Considering then the generating function for $E_\alpha(z^...
0
votes
0answers
35 views

Why is $R_{\Omega}R_{\Omega}^{*}=R_{\Omega}$ in matrix completion?

In the matrix completion literature, a restriction or sampling operator is defined as follows. Let $X$ be the unknown matrix to be recovered and $\{w_{\alpha}\}_{\alpha=1}^{n^2}$ be some orthonormal ...
1
vote
1answer
141 views

Identity from Lectures on the Theory of Elliptic Functions by Harris Hancock

I'm working on Example $4$, page $262$, of Harris Hancock's book Lectures on the Theory of Elliptic Functions which reads: Prove that $\dfrac{1}{\operatorname{sn}(iu,k)^2} + \dfrac{1}{\...
-1
votes
0answers
33 views

Trigonometry from two graphs [on hold]

My problem is that i have an image or rectangle which has line inside it which is rigid it will not move. I also have a graph with the same line but may be slightly bigger and may have a slightly ...
1
vote
0answers
113 views

Fibration when central fibre is a Calabi-Yau variety

Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that X and Y are projective varieties and central fibre $X_0$ is Calabi-Yau variety and Y is also Calabi-Yau variety, then can ...
3
votes
0answers
76 views

Twisted derived Morita theory of schemes

It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a ...
0
votes
0answers
18 views

Rank of the Matrix under the following Constraints? [on hold]

Case 1: An nXm Matrix of Non-Negative Integers, and the scalars are allowed to have only binary values (i.e. 0 or 1)? Case 2: The calculation of the Binary Matrix in Gf(2) is a standard algorithm....
3
votes
0answers
53 views

Bott-type projections in $C^*$-algebras

Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element $$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$ is an ...
5
votes
1answer
107 views

Can a projective solvable group be transitive?

Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup. Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
-3
votes
0answers
36 views

Estimation of Uncertainty of parameters defined from Lognormal Particle Distribution [on hold]

I think I previously posted too simplified math question (OTL), so I would like to ask again with more specific examples and problems that I currently have for my cloud radar research. Let us assume ...
-3
votes
0answers
57 views

Group theory application [on hold]

I have heard that group theory is applied in using credit card. How is a group theory applied in using the credit card? What fact is used there ?
1
vote
0answers
69 views

Number of conjugacy classes of cyclic subgroups of order $pq$ of $GL_n(\mathbb{Z}/p\mathbb{Z}) \times GL_m(\mathbb{Z}/q\mathbb{Z})$

Let $p$ and $q$ be distinct prime numbers, and let $m$ and $n$ be positive integers. How many conjugacy classes of cyclic subgroups of order $pq$ does the group $$ {\rm GL}_n(\mathbb{Z}/p\mathbb{Z})...
3
votes
0answers
41 views

Injectivity of the Chern character in $K$-homology

Let $(\pi,H,F)$ be a Fredholm module: here $\pi:A \to B(H)$ is a representation of an algebra on the Hilbert space $H$ and $F$ is a self adjoint operator with square one such that for each $a \in A$ ...
-1
votes
0answers
27 views

show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$ [on hold]

For $n\geq 3 $.Let $u\in C^2(R^n), \Delta u\leq 0,u>0$ in $R^n$, show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$. I was reading the article <Liouville-type theorems and Harnack-type ...
9
votes
0answers
66 views

For which finite groups $G$ is every character a virtual permutation character?

Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the ...
-1
votes
0answers
48 views

Singular locus of codimension 1 for a hypersurface [on hold]

If $V$ is a hypersurface and it is reducible, then I know that $\dim Sing(V)= \dim V-1$. Is the contrary true? I.e., if $\dim Sing(V)= \dim V-1$, then $V$ is reducible? I am only interested in ...
5
votes
1answer
102 views

$\kappa$-homogeneous topological spaces

Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if $|X| \geq \kappa$, and whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\...
0
votes
0answers
10 views

Hypergraph clustering conductance Formula

Consider the Hypergraph $H=(V,E)$, with $V$ being the vertices and $E$ being the hyperedges. What is the formula of conductance $\Phi(S)$ for hypergraphs, with $S$ being a set of vertices (cluster ...
2
votes
1answer
54 views

On the notion of conelike stratified (cs-) space

The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
2
votes
1answer
63 views

Universal bundle of grassmannian of planes and projective bundle over grassmannian of lines

Let $p:Y=\mathbb P(\mathcal E_3^{\vee})\rightarrow G(3,n+1)$ be the universal family of hyperplanes (i.e. lines) of the planes of $\mathbb P^{n}$. The following isomorphism seems natural $$\mathcal O_{...

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