# All Questions

**-3**

votes

**0**answers

11 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

**0**answers

15 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

**0**answers

15 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

**0**answers

15 views

### Determine number of directed trees and rooted trees obtainable

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

**0**answers

67 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

**2**answers

25 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

**0**answers

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

**0**answers

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

**1**answer

58 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

**0**answers

38 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

**0**answers

23 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

**1**answer

55 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

**0**answers

29 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

**1**answer

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

**0**answers

156 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

**0**answers

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

**1**answer

113 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

**0**answers

43 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

**1**answer

326 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

**0**answers

28 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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

97 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

**0**answers

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

**0**answers

30 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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

45 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

**1**answer

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

**1**answer

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 ...

**0**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

101 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

**0**answers

55 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

**0**answers

23 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

**0**answers

172 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

**2**answers

115 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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

45 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

**0**answers

65 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 $x\in C$ ...

**6**

votes

**1**answer

248 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 ...

**1**

vote

**0**answers

18 views

### Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?

Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?
Where $\omega\subset\subset \Omega$ with $\Omega$ is a $C^2$ ...

**1**

vote

**0**answers

28 views

### Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zeta Zeros?

Mertens function has, by residues, an explicit formula of
$M(n)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$
...