4
votes
0answers
64 views

Understanding homotopy t-structure

The following question came up while reading Hoyois' From algebraic cobordism to motivic cohomology. Let $S$ be a Noetherian scheme of finite Krull dimension and let $SH(S)$ denote the homotopy ...
0
votes
3answers
156 views

Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which $$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...
0
votes
1answer
187 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
0
votes
1answer
50 views

Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$: $\require{AMScd}$ \begin{CD} a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...
3
votes
1answer
56 views

Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...
0
votes
0answers
96 views

Ramanujan graphs and stable real polynomials

I have a problem with the lemma 6.5 and the theorems 5.1, 6.6 in the paper of Adam Marcus, Daniel Spielman and Nikhil Srivastava. http://arxiv.org/pdf/1304.4132.pdf I do not understand how they use ...
1
vote
0answers
46 views

Can a semigroup be defined on a Banach algebra? [on hold]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...
0
votes
0answers
25 views

Why do we say a level 1 Menger Sponge has 5 holes? [on hold]

I've heard that a level 1 Menger Sponge has 5 holes, but what is the justification for this? I can understand starting with a hole down the center, and making 4 more to meet it from the sides, but ...
1
vote
0answers
32 views

Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix ...
0
votes
1answer
75 views

In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made: Let $u_m$ converges weakly to $u$ in ...
0
votes
0answers
22 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
12
votes
1answer
404 views

Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.
1
vote
1answer
147 views

Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...
2
votes
0answers
33 views

Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...
0
votes
0answers
35 views

Is preimage of closure equal to closure of preimage under continuous topological maps? [on hold]

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$ Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
9
votes
1answer
191 views

The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says: The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$. The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the ...
4
votes
1answer
77 views

Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...
0
votes
0answers
57 views

Prove in GL that no statement can be proven consistent with PA unless PA is inconsistent [on hold]

Edit: I moved this question to math.stackexchange.com here as sugested. I'm trying to do a exersie on page 16 of this paper. It says: Exercise. Show, using the rules of Godel-Lob modal logic ...
0
votes
1answer
620 views

Hard maths on viXra? [on hold]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...
1
vote
0answers
35 views

How to calculate the derivative of logarithm of a matrix? [migrated]

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(1-M)^k}{k}$$ The derivative of ...
12
votes
1answer
160 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...
2
votes
0answers
49 views

Zauner's conjecture [migrated]

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 ...
-4
votes
0answers
43 views

Distance Between Sets- Jaccard Coefficient [on hold]

How do I estimate whether a given distance between two sets obeys triangle law of inequality. lets say d(x,y)= |A-B|+|B-A| or d(x,y)=(|A-B|+|B-A|)/|A U B|
1
vote
0answers
96 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
1
vote
1answer
76 views

Vectors which average to zero over any graph neighborhood

Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum_{j \in N(i)} x_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set ...
1
vote
1answer
97 views

The space of loops as a Banach space [on hold]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...
0
votes
0answers
28 views

Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space. I'm intrested in informative examples and applications of such systems. I know about ...
1
vote
2answers
51 views

Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...
2
votes
1answer
71 views

Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
3
votes
2answers
85 views

References about the matrix generators of the finite subgroups of the orthogonal group O(4)

"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
3
votes
0answers
77 views

henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme? For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme. Let $Y_{Z}^{h}$ the ...
7
votes
0answers
52 views

Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
0
votes
0answers
18 views

Characterization of Dedekind complete Riesz spaces by strictly positive functionals

I was browsing throughout the literature and I found the following fact: Every $\sigma$-Dedekind complete Riesz space $E$ which admits a strictly positive functional $f$ is Dedekind complete. I ...
0
votes
0answers
27 views

Minimum size set of partition-of-unity smooth functions with bounded support [on hold]

Let D be the unit disk in $R^2$, and take fixed $\epsilon <1$. I am looking for smallest possible set ${\left\{f_i\ \right\vert{}\ {0\!\leq\!f}_i\!<\!1\}}_{i=1}^n\,\,$ with the following ...
1
vote
0answers
71 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
3
votes
1answer
68 views

Is the variation of two BV functions the same in the set in which they coincide?

Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set $A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$ is it true that ...
8
votes
2answers
283 views

What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
2
votes
1answer
138 views

Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$

Is the following limit result correct: $$\lim\limits_{x \to 0^{+}}\sum\limits_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle ...
2
votes
0answers
57 views

Asymptotic behavior of a function [migrated]

Let $n\geq 2$, and define $$\phi(t) = \int_{S^{n-1}} \cos(t \omega_1) d\omega,$$ where $S^{n-1}$ is the unit sphere in $n-$dimensional euclidean space, and $d\omega$ denote the surface area on ...
4
votes
2answers
119 views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
1
vote
1answer
258 views

Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$ A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise} \end{cases} $$ How does the ...
-3
votes
0answers
26 views

How to obtain a closed form expression for a function of type floor(min{a, b}) and ceil(min{a, b})? [on hold]

I am working on finding average distance in mesh based structures, and I need to find a closed form expressions for functions that involve floor, ceil and min operators. An example expression has been ...
4
votes
0answers
75 views

Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...
1
vote
1answer
72 views

p-summable sequence

Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...
2
votes
0answers
60 views

Any bi-invariant distance on a group is inverse-invariant? [migrated]

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
0
votes
0answers
41 views

Hilbert's “The Foundation of Geomtry” theorem 4, 7 [on hold]

http://www.gutenberg.org/files/17384/17384-pdf.pdf?session_id=7f857052706d896691753ea34b68d334536fb7c7 Can you give prooves to the theorem 4 and 7?
4
votes
1answer
135 views

Examples of NIP fields of characteristic $p$

Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the ...
2
votes
1answer
113 views

The source-side-opposite of the arrow category

Given a category $C$, is there a name for the following category: $\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$ $D((x, y, f), (x', y', f')) = ...
5
votes
1answer
159 views

Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series $$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$ This series definitely converges when all the arguments are small enough. My goal is to ...
0
votes
0answers
42 views

How to check numerically iterated logarithm law ? (How to choose cutOff lim_n sup_{m: n<= m<= CutOff} ) ?

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then $$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...

15 30 50 per page