0
votes
0answers
2 views

equivalence of Lie group and Lie algebra intertwiner

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
0
votes
0answers
25 views

Prove a function is polynomial?

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
0
votes
0answers
12 views

Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group. Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
2
votes
2answers
61 views

What is a Floretion?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
0
votes
0answers
12 views

An obstacle problem

Let $f:[0,T]\to \mathbb{R}$ be an increasing function with $f(0)=0$. We want to maximize $f(T)$ with the following constraints: $|f^\prime(t)|\le M,\quad \forall t\in[0,T]$ $f(t)\le g(t),\quad ...
2
votes
0answers
28 views

isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
-1
votes
0answers
27 views

Can I calculate another model? [on hold]

Let's assume that I have following linear model: y = b + c*x can I just do following to get linear regression model for x? It will be a good model? y - b = c*x x = -b/c + 1/c*y What if I had more ...
1
vote
1answer
25 views

Differences of consecutive ordered fractional parts

Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...
0
votes
0answers
25 views

Automorphism group of the gamma factor of a certain type of L-function

Let $F$ be an element of the Selberg class with polynomial Euler product, $\gamma_F$ its gamma factor appearing in the functional equation of $F$, which is defined up to a multiplicative factor. Is ...
4
votes
0answers
48 views

What is the most transparent, rigorous definition of the Univalence Axiom?

I've been studying homotopy type theory and trying to grasp the Univalence Axiom. I have yet to find a concise, accessible, rigorous definition of Univalence. I have several excellent survey papers ...
8
votes
1answer
150 views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
8
votes
2answers
291 views

Is Every Holomorphic Near an Entire?

Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded. Let $U$ be the interior of $K$. Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction ...
1
vote
0answers
34 views

Number of unitary representations of a Kazhdan group

It was proved by de la Harpe, Robertson, and Valette that for a discrete group $\Gamma$ with Kazhdan's property (T), there is a constant $c$ so that the number of irreducible unitary representations ...
0
votes
1answer
40 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow. In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma ...
0
votes
2answers
23 views

Generate Bernoulli vector with given covariance matrix

I am from different background, so please forgive me if the answer is so well known. Let $C=(c_{ij})$ be a given $n\times n$ matrix. Do we have a way to generate samples of random Bernoulli vectors ...
0
votes
0answers
23 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
-3
votes
0answers
37 views

which graph data structure generalize tree which generalize finite group [on hold]

i use tree to generalize finite group such as left small, right big for generalize the ordering and number of finite group into tree, if want to further generalize this kind of tree, which data ...
5
votes
2answers
105 views

PSL(2,p) as quotient of triangle groups

As a by-product of some Magma computations, I've observed that, for each prime $p$ such that ${\rm PSL}(2,\mathbb{F}_p)$ can be a quotient of the $(3,3,4)$-triangle group (i.e. $p \equiv \pm 1 ...
2
votes
0answers
24 views

Complements of unknotted tori (higher dimensions)

It is weil-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori. It is also known that an unknotted 3-torus in ...
0
votes
0answers
11 views

Solving the difference equation $h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$

I am trying to solve the following difference equation: $$h(x_1,x_2,x_3,x_4) \cdot A(x_1,x_2,x_3,x_4)=A(x_1-1,x_2,x_3,x_4)+A(x_1,x_2-1,x_3,x_4)+A(x_1,x_2,x_3-1,x_4)+A(x_1,x_2,x_3,x_4-1)$$ Or more ...
4
votes
1answer
78 views

Hyperbolic knot complement groups and relative dimension

Let $G$ be a the fundamental group of a hyperbolic knot complement. Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$. The knot complement has a $2$-dimensional spine ...
2
votes
0answers
87 views

Convex polyhedron and its Gauß-curvature

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here. What I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. ...
0
votes
0answers
39 views

Matrix representation

Let $\mathbf{c}\in \mathbb{R}^n$ and $\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...
3
votes
0answers
37 views

Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence: $$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$ and this ...
0
votes
0answers
45 views

A graph constructed using non edge intersecting cliques

Suppose I have a graph $G=(V,E)$ which is the union of non edge intersecting cliques $K^{i_{1}},K^{i_{2}},.....,K^{i_{m}}$ where $1 \leq i_{j} \leq n$ for $j \in \{1,...,m\}$. I'm interested to know ...
1
vote
0answers
49 views

Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$? I know, that $\frac{\partial ...
8
votes
0answers
84 views

When is alternating sum $\sum_{i}f(a_i)-\sum_{i<j}f(a_i+a_j)+\ldots+(-1)^{n-1}f(a_1+\ldots+a_n)$ always positive?

Let $a_1,a_2,\ldots,a_n\geq 1$, and let $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$. Consider the sum ...
3
votes
2answers
107 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
0
votes
1answer
96 views

Smooth morphism to homogeneous spaces and fibers

Let $f:X \to Y$ be a smooth morphism between projective varieties. Suppose $Y$ is a homogeneous space. Under what additional condition on $f$, can we conclude that every fibers of $f$ are isomorphic?
0
votes
0answers
47 views

Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic. Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$. Is there some ...
1
vote
0answers
73 views

Artin's criterion for étale, quasi-separated algebraic spaces

it is known from Knutson's work that an algebraic space which is separated and étale over a scheme is a scheme. Let $S$ be a locally noetherian scheme. I am looking for a reference giving an Artin's ...
0
votes
1answer
84 views

Picard group of a quotient of a group by its maximal parabolic subgroup

Let $G$ be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero and $P$ be the maximal parabolic subgroup. We know that the quotient $Z=G/P$ is a ...
2
votes
1answer
47 views

Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of ...
0
votes
0answers
13 views

Is there any description for 1-Yoneda-Extension of $A$-moduls in the spirits of factor systems?

The 1-Yoneda-Extension group of two abelian groups ($\mathbb{Z}$-modules) in the category of abelian groups can be described explicitly by symmetric factor systems(modulo coboundaries). Now my ...
0
votes
0answers
27 views

solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...
5
votes
0answers
92 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
0
votes
0answers
149 views

Non-university jobs suited for pure mathematician turned computational neuroscientists, with coding experience [on hold]

I also asked this question on academia stack exchange, http://academia.stackexchange.com/questions/48057/type-of-non-university-research-jobs-suitable-for-a-mathematician-turned-comput but asking ...
1
vote
1answer
35 views

An inequality with critical Sobolev exponent

Let $\Omega\subset \mathbb{R}^n, n\geq 3$ be a nice bounded domain and $2^*=2n/(n-2)$ the critical Sobolev exponent. One may expect that $\forall \epsilon>0$, $\exists C_\epsilon<\infty$ such ...
0
votes
1answer
57 views

free action on product of two spaces [on hold]

Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group ...
1
vote
0answers
27 views

Regarding a result of I.Vekua on integral equations of first kind

Suppose $k\in L^2([0,1]\times[0,1])$ is a symmetric kernel. Presumably the following is true(under the assumption that $k$ is weakly singular): Either $$\displaystyle\int_0^1f(y)k(x,y)dy=0$$or ...
-4
votes
0answers
80 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements [on hold]

I am preparing to my abstract algebra exam and I try to find an example of such ring. Does it even exist? Thank you in advance.
23
votes
2answers
1k views

What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the ...
0
votes
0answers
35 views

Solve complex exponencial equation [on hold]

I need to solve an expression of this kind(solve for x): e^(pi*i*x) -e^(-pi*i*x) = y*2i Both x and y are real numbers, y is given. I have no clue on how to solve it analytically. All I know is that ...
0
votes
0answers
54 views

Functor of order $n$ in Mumford's abelian variety

Let $T$ be a contravariant functor on the category of complete varieties into the Category $\underline{\mathrm{Ab}}$ of abelian groups. Let $X_0,\ldots,X_n$ be any system of complete varieties, ...
4
votes
1answer
84 views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ...
0
votes
0answers
29 views

Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
0
votes
0answers
51 views

tessellation of an arbitrary shape

Is there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate by some shapes?
3
votes
1answer
123 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
-4
votes
0answers
48 views

Cayley graph of dihedral group is isomorphic to which kind of graphs? [on hold]

Let D_{2n}= be dihedral group of order 2n. Also let D_{2n}= in which 1\ notin S=S^{-1}. In this case Cay(D_{2n}, S) is isomorphic to which kind of graphs? This is my conjecture that this graph is ...
-3
votes
0answers
55 views

Find the joint density function?

Assume that $X_t$ is the OU process , i.e, $dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$. Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$. I want ...

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