5
votes
0answers
62 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 ...
-3
votes
0answers
23 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
26 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
9 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 ...
4
votes
0answers
37 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
47 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
24 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
28 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
26 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
44 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
52 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
31 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
59 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
29 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
90 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
63 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
43 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 ...
4
votes
1answer
91 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 ...
-2
votes
0answers
51 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
55 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
44 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 ...
3
votes
1answer
75 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
48 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
57 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
46 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
39 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
39 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
76 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
123 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
48 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
53 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 ...
0
votes
0answers
41 views

Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. http://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
4
votes
1answer
86 views

Importance of $E_n$-algebras over ring structures on $\pi_*(E)$

Hopefully this question is not too vague to be closed. I am looking for examples of when a construction/theorem that involves $E$-(co)homology or even simply the ring $E_*$ requires an understanding ...

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