0
votes
0answers
2 views

Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...
0
votes
0answers
4 views

A question for the inverse orbit in the construction of conformal measure

Recently, I read a theorem of existence of conformal measure for the rational map. I did not understand two places in the proof. The author claims that there exists an open set $V\subset ...
-3
votes
0answers
17 views

Calculate minimal number of nodes? [on hold]

Calculate minimal number of nodes? in a loopless simple undirected pi-partite graph. that has exacatly 144 nodes
-3
votes
0answers
18 views

Directed and undriected trees [on hold]

How many different directed trees can be obtained if we assign all possible orientation to the edges of an undirected tree having exactly 7 nodes? how many of them will be rooted(directed) trees?
-3
votes
0answers
16 views

Can you give me an example of signed distance function [on hold]

Can you give me an example of signed distance function? Thank you!
0
votes
0answers
17 views

Determine number of directed trees and rooted trees obtainable [on hold]

I've been doing some exercices about graph theory and I find myself stuck on this one with no idea of to proceed. Here's the question : how many different directed trees can be obtained if we assign ...
0
votes
0answers
70 views

About the proof of the Morse lemma

In the Chang's book "Infinite dimensional Morse theory and multiple solution problems" the Morse lemma is a special case of the spliting lemma but i dont understand in the proof why ...
1
vote
2answers
27 views

Linear Programm with matrix

Is there a name for problems like this min norm(Cx) Ax = b where C is a matrix and norm is the maximum norm. This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
0
votes
0answers
28 views

Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
-5
votes
0answers
27 views

How can I compute the singular homotopy? [on hold]

Let D2 be a 2-dimensional disc and M be the Mobius strip. Note that the boundary of both D2 and of M is homeomorphic to the circle S1. (a) Consider the space X := (D2 ∪D2) /~ where ~ is the ...
6
votes
1answer
68 views

Questions about Prikry forcing and Cohen forcing

I have two unrelated questions. The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...
0
votes
0answers
41 views

How to define the distributional Hessian for a convex function on a $C^0$ Riemannian manifold?

M is a $C^1$ manifold with a $C^0$ Riemannian metric, f is a convex function on M. How to define a functional on M which can represent $Hessf$? For example: for $\Delta f$ we can define the ...
0
votes
0answers
24 views

Hom-Lie algebras induced by derivations

Let $(\mathfrak{g},[,])$ be a Lie algebra, and $D\colon \mathfrak{g} \rightarrow \mathfrak{g}$ be a linear vector space map satisfying the Hom-Jacobi identity $$ ...
2
votes
1answer
58 views

Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$). His main technique of ...
0
votes
0answers
30 views

Is there such a thing as cyclic Hasse diagram for posets?

If so can you name one ? If not how to prove that there is none? Thanks !
0
votes
1answer
52 views

State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 ...
4
votes
0answers
161 views

Computing $\Pi_p(\frac{p^2-1}{p^2+1})$ without the zeta function?

We see that $\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\Pi_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\Pi_p(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)})=\Pi_p(\frac{p^2-1}{p^2+1})$ ...
-1
votes
0answers
32 views

Context Free Grammar [on hold]

does anyone know how to find the Context Free Grammar for this language? L = {anbm | n > m}
1
vote
1answer
117 views

Blow-ups in Motivic Homotopy Theory

I have what I hope is an easy question in motivic homotopy theory: Let $X$ be a smooth scheme over a field $k$, and let $Z\subset X$ be a closed sub-scheme of codimension $>1$. Let $Bl_Z(X)$ ...
5
votes
0answers
46 views

Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
13
votes
1answer
335 views

Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...
2
votes
0answers
29 views

Almost-Monotone Kernels - Examples and/or Covering Theorems

I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation. Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures ...
0
votes
0answers
23 views

adjacent matrix directed or undirected [on hold]

I'm having trouble seeing how you can determine if a graph is directed or directed based off of the adjacent matrix. Can someone explain to me how to determine ths? Thanks!
0
votes
0answers
7 views

Is a constant such as 8 considered an expression? [migrated]

The question asked was "Which of the following expressions are considered polynomials?" 8 was one of the answers, and though it is clearly a monomial, it was part of the answer and I'm confused as to ...
0
votes
0answers
19 views

Finding a particular solution to the non-homogenous system [on hold]

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
3
votes
1answer
98 views

Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$. Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real? What I knew is that if $N=1$ and ...
4
votes
0answers
102 views

NP Problems with unique solution [migrated]

Is there any class of NP problems that have one unique solution? I'm asking that, because when I was studying cryptography I read about the knapsack and I found very interesting the idea.
0
votes
0answers
31 views

Extensions on Higher-dimensional local fields

I have the following question: Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...
1
vote
0answers
29 views

Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it: Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...
0
votes
0answers
43 views

Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...
3
votes
0answers
164 views

Can we prove an open affine subscheme of a noetherian scheme is noetherian without Axiom of Choice?

I'm interested in proving basic results of algebraic geometry without Axiom of Choice. As for why I think this is interesting, please see Pete L. Clark's answer to this question. To state my problem, ...
4
votes
2answers
94 views

On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...
1
vote
0answers
38 views

Collecting terms with nested sums and combinatorics

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
3
votes
0answers
46 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
3
votes
1answer
156 views

Can we sometimes define the parity of a set?

Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if $k=2^\alpha-1$ and $n=2k$.) In this case, can we select ${n\choose k}/2$ sets of size ...
3
votes
1answer
34 views

variation of the Lieb concavity theorem

A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B: $$ (A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \} $$ for $s \in ...
-1
votes
1answer
90 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
5
votes
1answer
104 views

Areas of Triangles in (Non-Riemannian) Metric spaces?

I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this ...
0
votes
0answers
72 views

Time derivative of an integral on a moving surface? [on hold]

I need to take the time derivative inside the surface integral, $$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}t}\left(\oint_{\partial B} \left(\mathbf{x} \times ( \mathbf{n} \times \mathbf{u} ) ...
1
vote
0answers
103 views

Quadratic - Ternary Forms

Hi I have the following problems concerning quadratic and ternary forms. Any help would be greatly appreciated. $3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{x^2+xy+7y^2}=3\displaystyle\sum_{x, ...
1
vote
0answers
56 views

Are semigroups with finite-to-one right multiplication “moving”?

A semigroup $S$ is moving if $S$ is infinite, and for all finite $F\subseteq S$ and infinite $A\subseteq S$, there are $a_{1},\dots,a_{k}\in A$ such that, for all but finitely many $s\in S$, $$ ...
0
votes
0answers
24 views

How can I decode efficiently a triple-error-correcting binary BCH code?

In a given BCH(N,K) T=3 code over GF(2^m), there are ways to find the error locations in a given N-bit codeword directly from the syndromes without going through the normal Berlekamp-Massey and Chien ...
11
votes
0answers
174 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
6
votes
2answers
116 views

Random walk in a convex body or convex polytope

Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in $\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside ...
4
votes
2answers
157 views

Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
4
votes
1answer
174 views

Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
0
votes
0answers
26 views

Injectivity of a linear logistic transform

The motivation for this question has to do with neural networks, but it is essentially a purely mathematical question. Suppose you have a perceptron with one hidden layer, a bias, and a logistic ...
-1
votes
0answers
46 views

Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized Kinetic Energy'. On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...
0
votes
0answers
66 views

Fixed point of a function on the circle

Consider a circle $C$ with radius of $r$, we place $m$ balls(treated as point) randomly on it, and each ball $i$ has the mass $m_i$. We define a function $\varphi:C\rightarrow C$ which maps $x\in C$ ...
7
votes
1answer
253 views

Prime races à la Mertens

I have just read the nice survey by Granville and Martin about prime races. I wonder what happens if one changes the rules for the prime races as follows. Fix $q$ a modulus (an integer $>1$). For ...

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