0
votes
0answers
3 views

Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space: $\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; \forall ...
0
votes
0answers
3 views

Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
0
votes
0answers
15 views

Laplace equation in polar coordinates [migrated]

Hy, I'm new in here but i have a question please help me. I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = ...
0
votes
0answers
11 views

Is there a unique saddle value for a convex/concave optimization?

I asked this question previously in math.stackexchange and couldnt receive any answer. The question is fundamental, though I couldn't find a reference or a proof somewhere. Here is the question: ...
4
votes
0answers
64 views

Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as $$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...
1
vote
0answers
18 views

Is the associated order of a minimal $T_0$ space always total?

Let's call a space $(X,\tau)$ minimal $T_0$ if it is $T_0$ and for every topology $\sigma\subseteq \tau$ with $\sigma \neq \tau$ we have that $(X,\sigma)$ is not $T_0$ any more. We say $x\leq y$ in a ...
0
votes
0answers
9 views

Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$. Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...
-1
votes
0answers
41 views

Simple algebras [on hold]

I like to prove the following statement. Proposition Let $A$ and $B$ two $K$-algebras such that $ A \otimes_{K} B$ is simple. Then $A$ and $B$ are simple. Proof We only proof this proposition for ...
1
vote
0answers
21 views

Infinite non-splittable graphs

Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is splittable if there are $S,T\subseteq V$ with $S\cap T=\emptyset$ and $S\cup T = V$ such that for all ...
2
votes
0answers
27 views

What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question. Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...
3
votes
1answer
43 views

Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?

Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$. Clearly, $\kappa$ is a cardinal. Question: Is it consistent that $\kappa = ...
2
votes
0answers
27 views

Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let ...
2
votes
1answer
63 views

When two Dedekind sums are equal

The (classical) Dedekind sum $s(h,k)$ is defined as $$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\bigg(\frac{hr}{k}-\Big[\frac{hr}{k}\Big]-\frac{1}{2}\bigg)$$ for $\gcd(h,k)=1$. A natural question is, when ...
1
vote
1answer
22 views

What is the smallest partition lattice PART(M) containing the lattice P(N) of subsets of a finite set of N elements

It's been known for 35 years that every finite lattice can be embedded in a finite partition lattice ( Pudlak and Tuma algebra universalis 1980, Volume 10, Issue 1, pp 74-95). I don't follow the ...
1
vote
0answers
39 views

rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...
-5
votes
0answers
24 views

Prize allocation of scratch codes to ensure correct number of prizes to give away [on hold]

We would like to give away 100 prizes. We have 13.5 million codes, divided into 20 categories. To win a person must collect one code in each category. If 15 of those categories have 877,195 codes ...
0
votes
0answers
45 views

Finiteness of geometric valuations

I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference: Let $X$ be a normal variety, and $Z \subseteq X$ be a ...
15
votes
1answer
653 views

Mathematicians wearing hats on arbitrary total orders

I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$: Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat ...
1
vote
0answers
47 views

How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...
-2
votes
1answer
54 views

Show that among any 6 non-negative integers one can find 2 integers so that their difference is divisible by 5 [on hold]

I have the following question in homework I have been assigned in Discrete Mathematics relating the pigeonhole principle: "Show that among any 6 non-negative integers one can find 2 integers so that ...
1
vote
1answer
45 views

A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...
1
vote
2answers
70 views

Partial Sum of the Binomial Theorem [duplicate]

The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation} \sum_{k=0}^mC_n^kr^k, \quad m<n \end{equation} for fixed $n$ and $r$, and both $m$ and ...
1
vote
0answers
23 views

Classes of knots that have known Bridge spectra

Bridge spectra is a knot invariant first defined by Doll, who established some basic properties. Tomova has shown that high distance knots have bridge spectra $(n,n-1,\ldots,2,1,0)$. Zupan has ...
3
votes
0answers
37 views

Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...
2
votes
0answers
27 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
0
votes
0answers
52 views

Behaviour of first $l^2$-Betti number under quotienting

Let $G$ be a finitely generated group, and let $H = G / N$ be a quotient of it. We have two observations: 1) In general, it is $\textbf{not}$ true that $\beta_1^{(2)}(G) = 0 \Rightarrow ...
0
votes
0answers
75 views

Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies. Do we have the following ...
2
votes
0answers
46 views

$t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...
6
votes
1answer
79 views

On a permutation module for GL(n,q)

Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ...
8
votes
3answers
530 views

“Epicycles” (Ptolemy style) in math theory?

By analogy: The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell ...
3
votes
1answer
185 views

When is the category of small (pre)sheaves a(n elementary) topos?

When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos. When $C$ is ...
3
votes
1answer
35 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
5
votes
0answers
37 views

A “lower-central” filtration of Steenrod algebra?

$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...
5
votes
1answer
80 views

Singular/Smooth locus of Schubert variety of the affine grassmannian

Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat ...
4
votes
2answers
115 views

Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ ...
2
votes
2answers
150 views

Applications of Szemeredi's Theorem

Szemeredi's Theorem is a famous theorem in Additive Combinatorics, Ergodic Theory and maybe some other parts of Mathemtaitcs: (Szemeredi's Theorem) Let $\Lambda \in \mathbb{Z}$ be a subset of ...
0
votes
0answers
64 views

Reformulation of a theorem for special case [on hold]

This question is about some theorems in the book "Analysis Now" from Pedersen. In particular Proposition $5.3.2$ and Theorem $5.3.3$. Here one has a essential $*$-isomorphism of some algebra $L(X)$ ...
1
vote
1answer
94 views

Dieudonné modules -reference request

I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of Reference for Dieudonné modules, ...
0
votes
0answers
24 views

Finite element method p2 2d [on hold]

I'm preparing my graduate project and i really need some help to implement FEM 2D with quadratic element triangles, so i have done everything so far and i think the problem is in the assembling of the ...
9
votes
2answers
426 views

Consecutive numbers with mutually distinct exponents in their canonical prime factorization

Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers ...
1
vote
1answer
38 views

Decomposing large symmetric banded sparse matrices

I'm investigating 3D image deblurring and one of the approaches I'm interested in is applying spatial regularisation. To do this I have generated a matrix $A$ which encodes the 6-connectivity of each ...
5
votes
0answers
64 views

Max min of functionals

I have an interesting question which I believe was probably already studied, but I could not find anything. Let $n, m \geq 1$ be fixed. Suppose that $|| \cdot ||$ is a norm in $\mathbb{R}^n$ and $f_1, ...
4
votes
2answers
68 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
2
votes
0answers
96 views

Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...
0
votes
0answers
34 views

How can I calculate the global log canonical threshold?

Let $X$ be a normal variety and let $D$ be a $\mathbb{Q}$-divisor. The log canonical threshold of a pair $(X,D)$ is $lct(X,D)=sup\{c|(X,cD)$ is log canonical$\}$. If $X$ is $\mathbb{Q}$-Fano ...
4
votes
1answer
134 views

A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied: $\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$ where $a\in\mathbb C^*$ and ...
3
votes
1answer
104 views

Is there a maximal connected Hausdorff space?

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. Is there a maximal ...
1
vote
1answer
73 views

Maximal connected topologies

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. If $(X,\tau)$ is ...
0
votes
1answer
35 views

Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
-5
votes
0answers
19 views

LyX preamble to auto add space before and after math (CTRL+M) [migrated]

I tried everything, but nothing works. How can I make auto space bofore and after math formulas in the same line? Thank you.

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