1
vote
0answers
8 views

A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection

$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...
0
votes
0answers
5 views

Intuition behind shrinking and subsampling in gradient boosted regression/classification?

I'm using gradient boosted decision trees from here: http://scikit-learn.org/stable/modules/generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier In ...
3
votes
1answer
33 views

a question about connected open sets in $R^2$

Let $U,V$ be two nonempty connected open sets in $R^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty connected ...
2
votes
0answers
24 views

On the local root number(or local $\epsilon$-factor)

I want to ask some question related to the local root number. Let $E/F$ be a quadratic extension of p-adic local fields and $\psi:E \to \mathbb{C}$ is an additive character of $E$. Let $\phi:WD(E) ...
-2
votes
1answer
46 views

finitely generated subgroups of SO(3)

Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each ...
1
vote
0answers
11 views

Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...
0
votes
0answers
12 views

Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group. I am ...
1
vote
1answer
34 views

n-cube connectivity problem

Given $n\geq 2$, let us consider the n-cube $H_n=(V,E)$, i.e. vertex set $V$ is $\{0,1\}^n$. Here, the edges of $H_n$ are directed, oriented by set inclusion, i.e., $(x,y)\in E$ iff $x\subseteq y$ and ...
1
vote
0answers
55 views

On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems". Proposition 1.4: When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
0
votes
1answer
76 views

Twin primes for polynomials in $\Bbb Z[X]$

The following paper provides result on an analog of twin primes conjecture for $\Bbb F_q[X]$ http://www.math.uga.edu/~pollack/twins.pdf Is there an analog of twin primes conjecture for $\Bbb Z[X]$? ...
3
votes
1answer
59 views

Localizations or quotients of categories?

Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category ...
0
votes
0answers
32 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
0
votes
0answers
38 views

Does the associated Lie algebra determine a group?

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
3
votes
1answer
34 views

Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs. Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
-4
votes
0answers
37 views

Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$ [migrated]

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...
-4
votes
0answers
24 views

Distributions properties [on hold]

Let $\varphi\in\mathcal{D}(\mathbb{R})$ the set of functions $\mathcal{C}^\infty$ with compact support, $\delta_n$ is the Dirac in $n$ and the functionals : $$ T = \sum_{n=0}^{+\infty} e^n ...
1
vote
0answers
84 views

Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
4
votes
0answers
141 views

Generalization of a property of $A_n; n\geq 5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_n; n\geq 5$. Then there exists a maximal subgroup $M$ of $A_n$ such that $H\not\leq M$ and $K\not\leq M$. To see this ...
-1
votes
0answers
32 views

Random Spanning Tree Edge Probability [on hold]

I am working on a problem with a Loop Erased Random Walk used to create random spanning trees. The problem has many parts, but there are two hints to help with the more complicated problems Figure ...
2
votes
0answers
54 views

Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties: For every two points in the plane there exists a unique geodesic joining them. Every geodesic determines exactly two points on the ...
4
votes
1answer
110 views

Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a simple (non-self-intersecting) closed curve of length $L$ on the unit-radius sphere $S$. So if $L=2\pi$, $\gamma$ could be a great circle. I am seeking the most equitably ...
-1
votes
0answers
54 views

Are there any special properties of graph eigenvalues of perfect matchings?

Say if the graph is just a perfect matching of its vertices OR if it is an union of a few perfect matchings OR its an union of a few perfect matchings and a part of another? Anything if one further ...
4
votes
0answers
74 views

Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
1
vote
0answers
39 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ ...
-4
votes
0answers
26 views

Irreflexivity of relations on sets [on hold]

How can I know if the relations: xy >= 1 and x=y+1 or x=y-1 Are irreflexive on Z(all integers)? Thank you!
2
votes
1answer
98 views

Isomorphism problem for two radical extensions [on hold]

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether ( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...
13
votes
0answers
106 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to \mathbb{R}$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
3
votes
1answer
202 views

Does this function have any exponential growth?

Has anyone seen any function of the following type? $$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$ The question is whether for some constant ...
-2
votes
0answers
69 views

What is the characterization of a graph Laplacian? [on hold]

Given a matrix, what properties must it have so that its ensured that there exists a graph whose Laplacian it would be? (...may be you can consider weighted and unweighted cases separately...) And ...
0
votes
0answers
14 views

the ratio between product of two trace functions maximization

Consider the following Optimization [\begin{array}{l} \mathop {\max }\limits_{\bf{X}} \,\frac{{trace\left( {{\bf{XA}}} \right)trace\left( {{\bf{XB}}} \right)}}{{trace\left( {{\bf{XC}}} ...
0
votes
0answers
67 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ ...
0
votes
0answers
33 views

Random Matrix Theory: question on a quantity on page 87 in book of Bai and Silverstein 2010

Anyone else is reading carefully the book of Bai and Silverstein 2010, titled "Spectral Analysis of Large Dimensional Random Matrices"? On page 87 of this book, when they state the final step in the ...
4
votes
2answers
368 views

Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence. I'm a bit ...
0
votes
1answer
56 views

A decomposition of incompressible vector fields

In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved: Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$. Call $w$ its ...
2
votes
1answer
119 views

A question about generalized Dyck words

I am interested in counting the following. How many words using $n-1$ copies of $u$ and ${n \choose 2} - n+1$ copies of $d$ begin with $uu$ and, in general, the $k^{th}$ $u$ is among the first ${k ...
2
votes
1answer
115 views

Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...
0
votes
0answers
112 views

Self homotopy equivalence

Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ...
-1
votes
0answers
89 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant [migrated]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
0
votes
0answers
124 views

Homotopical nilpotency [on hold]

For a connected grouplike space $G$, the homotopical nilpotency of $G$ is the invariant defined by Berstein and Ganea as follows: ${\rm Hnil} (G)$ is then the least integer $n$ such that the ...
7
votes
1answer
185 views

Busy beaver function vs low Turing degrees

Let $BB(n)$ denote busy beaver function. It's well known that $BB(n)$ dominates all computable functions (I'm quite certain it includes partial computable functions too). However, I was wondering if ...
2
votes
1answer
86 views

A (“Rice-like”) conjecture about the decidability of primitive recursive (PR) problems

Question: is the conjecture below true? Consider decision problems in which the instance is (the PR index, definition, or LOOP program of) a primitive recursive function. Denote the PR function (with ...
0
votes
0answers
23 views

proving that a complicated function is concave or strongly uni-modal

I am trying to prove the concave property for a complicated function during my research project (imperfect maintenance modelling for starter) which has the following form: $\eta(t)= \alpha \beta ...
2
votes
0answers
25 views

Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset \mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that this pair share a joint cyclic ...
-1
votes
0answers
39 views

What is the difference between disturbance and noise for dynamic systems? [on hold]

In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) ...
1
vote
1answer
75 views

Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
0
votes
0answers
61 views

Estimating the moments of a random variable

Suppose i wanted to estimate the expectation and variance of a random variable $X$. More over suppose i could write a variable $X$ as a sum of indicator random variables $X=\sum_{i=1}^{k} X_{i}$. Are ...
3
votes
1answer
97 views

Averaging maps of Riemannian manifolds

Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ...
2
votes
1answer
77 views

Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
3
votes
0answers
35 views

Behaviour of Markov type under uniform homeomorphism of spheres

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has $$ ...
2
votes
1answer
46 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...

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