2
votes
0answers
93 views

Can such categorical notion of action be formalized?

I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...
3
votes
2answers
187 views

Example of measure of non-compactness

I can't understand the following example of measure of non-compactness, which was given in this article. Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...
7
votes
3answers
579 views

Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
3
votes
2answers
184 views

bar construction and loop space cohomology

Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that ...
5
votes
1answer
48 views

Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints? $x_1+x_2+...+x_n = 1$ $a_1 \le x_1 \le b_1$ $a_2 \le x_2 \le b_2$ $...$ $a_n \le ...
6
votes
1answer
421 views

Segal's 1999 Stanford lecture notes on TQFT, where to find them?

I am trying to track down a copy of Graeme Segal's 1999 lecture notes on topological field theory. These are sometimes referred to as the "Stanford lectures" or something similar. For many years ...
-4
votes
0answers
26 views

What is the upper bound for training Linear separable set with Perceptron, Rosenblatt rule? [on hold]

I have the following neural networks problem and couldnt find any answer on the web. Any hints would help. I am not looking for a complete solution, just some pointing in the right direction. ...
4
votes
1answer
128 views

Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
4
votes
1answer
85 views

unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ ...
4
votes
0answers
80 views

Are all transversely oriented foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...
-2
votes
0answers
86 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...
3
votes
2answers
182 views

Connectedness of moduli of vector bundles

Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...
3
votes
0answers
149 views

Probability that an integer contains no $1\bmod 4$ prime factor

$n$ represents integer variable. What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime ...
0
votes
1answer
33 views

floating point representation via the perspective of TTE/computable analysis

Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...
-4
votes
0answers
105 views

Are polls good approximations [on hold]

Let $X$ be a finite set and $A\subseteq X$ and $m$ be a natural number satisfying $m\le |X|$ and $\epsilon$ be a small positive number. I'm interested to know if one selects a random $Y\subseteq X$ ...
-1
votes
0answers
33 views

Obtaining z-transform of a multivariate nonlinear difference equation [on hold]

I need to obtain the z-transform of difference equations that are as follows: My problem however is multivariate and looks like this: x[k+1]=ay[k]+ ((x[k])^2)(y[k]) ...
0
votes
1answer
71 views

Counting number of primorial factors

Denote $$P(n)=\prod_{p\in\mathsf{Primes}\leq n}p$$ signifying $n^{\mbox{th}}$ primorial. We know that $P(n)$ has approximately $n/\log2$ bits ...
-3
votes
0answers
17 views

segment intersecting a tetrahedron coordinates [on hold]

I am trying to write C++ code to find the intersection points of a segment intersecting a tetrahedron. I reduced the problem like this: For each face of the tetrahedron (a triangle), find the ...
-3
votes
0answers
49 views

Solution manual of real mathematical analysis PUGH [on hold]

Is there anybody know that where I can find the solution manual of real mathematical analysis PUGH ? Thanks a lot.
5
votes
1answer
412 views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
5
votes
0answers
101 views

Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody's answering this question so I'll try it here. This is really a reference request: Has a certain kind of proof ever been used? A series $\displaystyle\sum_n a_n$ converges absolutely if ...
7
votes
1answer
184 views

Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer. Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have ...
1
vote
1answer
47 views

approximate diameter of polytopes in high dimensions

I just came across the following problem: Let us consider the unit corner of the n-cube $$ \Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...
3
votes
0answers
172 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
2
votes
0answers
118 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
2
votes
1answer
119 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
2
votes
0answers
85 views

Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
1
vote
0answers
79 views

Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
2
votes
0answers
54 views

Explanation that Twistor Space of $S^4$ is $\mathbb{C}P^3$?

I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example: Let $S_-$ be the $SU(2)$-bundle of anti-self dual ...
1
vote
1answer
97 views

Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...
1
vote
1answer
63 views

The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the ...
4
votes
1answer
138 views

Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...
4
votes
1answer
113 views

Parametrized Atiyah-Singer index theorem

Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D_s:E\to F,s\in S$ be a continuous family ...
0
votes
0answers
46 views

Conditions for Mellin inversion

Under which conditions is the function $$ g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R} $$ the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...
0
votes
0answers
23 views

Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds ...
1
vote
0answers
57 views

Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$ [migrated]

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...
3
votes
1answer
64 views

Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$? It is a standard fact that Krull domains, and thus noetherian ...
1
vote
0answers
100 views

Is this permutation-sum problem NP-complete? [migrated]

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In ...
2
votes
1answer
79 views

Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
4
votes
2answers
146 views

Relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? $\pi_*(X, A)$ and $\pi_*(Y \cup_f X, Y)$?

Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let $f: A \to Y$ be a cellular map. What is the relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? Is there a similar ...
3
votes
2answers
249 views

Is there a nonabelian free group inside a group of positive rank gradient?

Let $G$ be a finitely generated residually finite group with positive rank gradient, and let $F_2$ be the free group on $2$ elements. Must there be an embedding $i \colon F_2 \to G$ ? A group ...
-1
votes
0answers
37 views

A question on spectrum [migrated]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
4
votes
0answers
56 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
0
votes
1answer
82 views

Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...
4
votes
1answer
105 views

Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors? In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...
5
votes
2answers
48 views

convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...
-3
votes
1answer
36 views

upper bound for a convex fractional function [on hold]

Consider the following convex fractional function $$f\left( {\bf{x}} \right) = \frac{1}{{1- {\bf{x}}}}$$ where ${1- {\bf{x}}} > 0$. Is it possible to obtain a linear or quadratic upper bound ...
-1
votes
0answers
112 views

Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It is clear that ...
12
votes
2answers
340 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
7
votes
1answer
404 views

Sums of unique squares

Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } ...

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