0
votes
0answers
17 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(...
-2
votes
0answers
13 views

which universties in USA and europe have intersted in delay differentail equations or functional differential equations?

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 ...
11
votes
0answers
140 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
27 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.
0
votes
0answers
40 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)...
0
votes
0answers
11 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 ...
6
votes
0answers
56 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 ...
2
votes
0answers
54 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/...
1
vote
0answers
25 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 $\...
0
votes
0answers
33 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
31 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 ...
3
votes
0answers
46 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
26 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
32 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
39 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
58 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
33 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 ...
0
votes
1answer
70 views

Example 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
30 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
95 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 ...
1
vote
0answers
66 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
44 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
97 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
33 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
54 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
63 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
39 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
56 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
45 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
81 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
50 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
58 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_{...
7
votes
3answers
139 views

For which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras?

Let $G$ be a finite group. Then the rational group algebra $\mathbb{Q}[G]$ has a wedderburn decomposition of the form $\prod_i M_{n_i}(D_i)$ where each $D_i$ is a division algebra. My question is: ...
-2
votes
0answers
47 views

Applications of group [on hold]

I have heard that Group ( i.e a non empty set defined by a binary operation satisfying closure,associative,identity,inverse axioms ) are applied in the usage of credit cards. How is that used exactly ?...
2
votes
1answer
73 views

Examples of canonical bases

Let $A=(a_{ij})$ be a generalized Cartan matrix of order $n$ and $D=diag(d_1,\ldots,d_n)$ the diagonal matrix such that $DA$ is symmetric. Let $$E_{ij}=\sum_{r+s=1-a_{ij}} (-1)^r E_i^{(r)} E_j E_i^{(s)...
2
votes
0answers
41 views

Uniform mean-square-error estimates

Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...
-5
votes
0answers
40 views

How to flip a graph over the x-axis but retain the original equation [on hold]

I know this question seems really very basic for this forum, but after about an hour of trying to work it out for myself, I decided to look here for help. The premise I'm trying to create the graph ...
0
votes
0answers
40 views

Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $c$ such that $c\geq0$, and $\rho\geq0$. We want to prove ...
4
votes
1answer
91 views

When is a finite $R$-algebra isomorphic to $R$?

Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there ...
2
votes
1answer
77 views

Intuitional feeling of harmonic measure on one-third Cantor set

It is known that the harmonic measure on classical one-third Cantor set has Hausdorff dimension strictly less than $\frac{\log 2}{\log 3}$. Even harmonic measure has a close relation with brownian ...
6
votes
0answers
54 views

Decomposition of an induced representation of $GL(2, q)$

Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\...
2
votes
0answers
57 views

A (second-order) axiomatic characterization of the integers which rules out surreal/hyperreal versions

I've seen it stated, for example here, that the integers are the unique commutative ordered ring with identity whose positive elements are well-ordered. I understand why the integers are the ...
3
votes
2answers
129 views

Reference for Using Group Cohomology to calculate Etale Cohomology

I'm looking for a reference for the following statement: Let $X$ be a variety (over an algebraically closed field $k$), and let $F$ be a locally constant etale sheaf. Let $x \in X(k)$. Then $ \mathrm{...
-4
votes
0answers
49 views

I need to know the most active research topic which depend on Real analysis and functional analysis? [on hold]

I need to know the most current topic in pure math with depend mainly on real analysis and functional analysis and not need a good knowledge in algebra and geomtry ? is delay differntial equations is ...
1
vote
1answer
70 views

Unique Stationary Distribution of A Markov Chain

I have a Markov Chain like $Y_i=\sum_n\pi_{n,i}(Y)Y_n$, i=1,2,3...N. So the Markov chain has N states and the transition matrix depends on the vector $\textbf{Y}$. Moreover, $Y_i$ is continuous and ...
2
votes
0answers
79 views

Randomly put $k$ balls in $2n$ circular boxes, pick $n$ consecutive boxes such that the number of balls is minimum!

You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing ...
0
votes
1answer
54 views

Random variable of random variable [on hold]

This is confusing and difficut, but I hope it makes a sence. I am interested in kind of like Random variable of Random variable. This issue might've been mentioned below before. The Probability ...

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