1
vote
0answers
12 views

L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$ \|q\|_1 \leq O(n) \|p\|_1 $$ where we define $\|f\|_p := (\int_{-1}^1 |f(x)|^pdx)^{1/p}$? ...
-4
votes
0answers
34 views

Prove relations between hypotenuse and cathetus in geometry problem [on hold]

Given a triangle ABC with cathetus 'a' and 'b' and hypotenuse 'c', prove that for every odd 'a' >= 3 (3, 5, 7...), there is an integer 'b' and integer 'c' -> c - b = 1
0
votes
0answers
28 views

Showing that $|F(x,p)|+\left|\frac{\partial F}{\partial x} \right|+\left|\frac{\partial F}{\partial p} \right| < C \exp[-(|x-pt|+|p|)]$

Given: $F(x,p,t)=F_0(X(x,p,t),P(x,p,t))$, $$|F_0(x,p)|+\left|\frac{\partial F_0}{\partial x} \right|+\left|\frac{\partial F_0}{\partial p} \right| < C \exp[-(|x|+|p|)],$$ and $|P-p|<C$, $|X-(x-...
3
votes
0answers
44 views

On the cardinal arithmetic of accessible categories

If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and $$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$ Here $P_\lambda(X)...
0
votes
0answers
13 views

Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$

Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008) observed that if $(x_{n})_{n}$ is a sequence in $L_{p}$ that is ...
1
vote
0answers
39 views

Bounding the union of conjugates of a maximal subgroup of the Symplectic group over a finite field

Let $g \geq 1$ be a positive integer, and let $p$ be a prime. Consider the symplectic group $G := \operatorname{Sp}_{2g}(\mathbb{F}_p)$ of symplectic matrices with entries in $\mathbb{F}_p$. Let $M \...
3
votes
0answers
114 views

Corollaries of the Yoneda Lemma in Analysis?

This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: http://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis. If this question isn't ...
2
votes
0answers
69 views

Semisimplicity of Frobenius on *integral* Tate module

Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
0
votes
0answers
10 views

Constrained optimization with an integral

I am trying to maximize the parameters $\alpha$ and $\beta$ of the following equation $ max_{\alpha, \beta} \sum_{n=1}^N ln (q(\lambda, \beta)) + \sum_{n=1}^N \sum_{k=1}^K \alpha_k \psi_k(\lambda)$ ...
0
votes
0answers
22 views

Infinite dimensional version of a simple fact on certain singular matrices

We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds; Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with ...
0
votes
0answers
28 views

What's the name of the following distribution?

What's the name of the following distribution? $f(\lambda, \beta)=\frac{\beta}{2\alpha\Gamma(1/\beta)}e^{-\frac{(x-\mu)}{\beta}}$ I copied the formula from wikipedia a while ago but I cannot recall ...
0
votes
0answers
30 views

Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...
-4
votes
0answers
46 views
0
votes
1answer
35 views

Time-Variant Markov Chain, Contraction Mapping

I wanted to find some references on showing contraction of a time-variant Markov chain. Specifically, I have a Markov chain whose transition matrix depends on current states, and I want to show it ...
1
vote
0answers
62 views

A universal operator between separable Banach spaces

The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is ...
-5
votes
0answers
66 views

How to proof that this function is 1-1? [on hold]

This is the function: function . I've no idea how to proof that it is 1-1. Any ideas?
0
votes
0answers
9 views

Cubic B-Spline Step function transformation [migrated]

Let $B = (x - x_i)^2(x_{i+2} - x) + (x - x_i)(x_{i+3} - x)(x - x_{i+1}) + (x_{i+4} - x)(x - x_{i+1})^2$ over the interval $x_{i+1} \leq x < x_{i+2}$ Suppose $i \equiv i - 2$, and set $h = x_{i+1}...
0
votes
0answers
43 views

Properties of a subring of a 'completion' of k(X_1, X_2, …, X_n)

I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents". I don't even know the name of this ...
8
votes
1answer
182 views

Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
5
votes
1answer
123 views

Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate. Throughout we work in ZF+AD. My question is: If $A$ is an uncountable OD set of reals,...
1
vote
1answer
84 views

Divisor on variety determined by its restriction to curves

Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?
0
votes
1answer
52 views

Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
9
votes
1answer
224 views

Question on a paper by Benjamini/Kozma/Wormald about a “well known fact”

In "The mixing time of the giant component of a random graph" by the aforementioned authors, in the last proof on page 19 it says something along the lines of "It is well known and easy to verify ...
3
votes
0answers
92 views

'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
3
votes
1answer
70 views

Coherence theorem for symmetric lax monoidal functors

Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements: 1) ...
-5
votes
0answers
42 views

sqrt((i)^2) different from sqrt((-i)^2)? [on hold]

I have a simple paradox that I can't explain. $$ i^2 = e^{i \pi + 4 k \pi} $$ $$ (-i)^2 = e^{i (-\pi + 4 k' \pi)} $$ $$ \sqrt{i^2}= e^{i \pi/2 + 2 k \pi} = i$$ $$ \sqrt{(-i)^2}= e^{i (-\pi/2 + 2 k' ...
0
votes
0answers
51 views

Spectrum on an unbounded operator

Consider the operator $T_{c}=-\frac{d}{dx^{2}}+ c x^{2}$ with $c\in C^{*}$, $Re(c)>0$ defined on its domain $D_{c}=\{u\in L^{2}; T_{c}(u)\in L^{2}\}$. Put $c=a+ib$ avec $a>0$ et $b\in R$. ...
2
votes
1answer
524 views

Wrong Tits-Index of E7 from Springer book

In the his book Linear algebraic groups, by T.A. Springer, there is a list of possible Tits-Indexes. For the $E_7$ case, there is an index shown, such that vertex $1$ and $7$ are circled (Bourbaki ...
5
votes
1answer
123 views

Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5: "Describe how the correspondence in this paper behaves under the action of Hecke operators." Since ...
0
votes
0answers
21 views

Existence of a proper entropy scaling for a discrete measure

In this (quite elementary) paper, the scaled entropy of an unbounded measure $\mu$ on $\mathbb{N}$ is defined by $$ h_c(\mu) := \lim_{\epsilon \to 0} \limsup_{n \to \infty} \dfrac{H^\epsilon(X_n)}{c(...
6
votes
1answer
171 views

Is a polarization on an abelian scheme an open condition?

Let $A/S$ be an abelian scheme such that the dual abelian scheme $A^{\vee}/S$ exists and let $\lambda : A \to A^{\vee}$ be a morphism of abelian schemes. Is the locus of points in $S$ where $\lambda$ ...
4
votes
2answers
1k views

Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. To ...
0
votes
0answers
112 views

Covering map of classifying space [on hold]

We know that for any $m \in \mathbb{N}$ the map $p_m: S^1 \to S^1$ is an $m$-sheeted covering of $S^1$. Suppose that $BG$ is the classifying space of an arbitrary group $G$. Does there exist such a ...
1
vote
1answer
52 views

Question on a random vector

This relate to that paper: http://www.stat.purdue.edu/docs/research/tech-reports/1982/tr82-17.pdf Let $U_1,...,Un$ be iid uniform on (0,1). Set $L_n=\max_{i\leq n} U_i$. Also $S(n)= \inf\{i\leq n|...
1
vote
0answers
32 views

Ideals of finite codimension in $L^1(G)$

Let $G$ be a non-abelian, locally compact group. Is there any characterization of the two-sided ideals of $L^1(G)$ which are of finite codimension?
1
vote
2answers
189 views

A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
-4
votes
0answers
72 views

how to prove the Hom is not 0 [on hold]

$\mathrm Hom_\mathbb{Z}(\prod{Z_n},Q)\neq0$ how to find the map ,I reckon the map is from $(1,1,,\dotsc,1)$ to 1 what is the answer?
6
votes
0answers
93 views

Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$

Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$. ...
1
vote
1answer
54 views

null infinite product in the p-adic setting

Let $p$ be a prime, $\mathbb C_p$ be the completion of a algebraic closure of $\mathbb Q_p$ and $(u_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ converging towards $0$. Suppose that for all $n\...
3
votes
1answer
95 views

Methods for defining/calculating homotopy limits of quasicategories

I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the ...
7
votes
1answer
277 views

Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
2
votes
1answer
60 views

Meaningful formalization of a continuum of Bernoulli random variables [on hold]

I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...
1
vote
0answers
39 views

A Global Restriction Estimate from Local Estimate

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose ...
0
votes
0answers
38 views

Does the Ruzsa-Szemeredi Theorem also capture graphs decomposable into *nearly* induced matchings?

The well-known Ruzsa-Szemeredi Theorem states that a graph whose edges can be partitioned into $n$ induced matchings has at most $\frac{n^2}{RS(n)}$ edges, for some slow-growing function $RS(n)$. Now,...
2
votes
0answers
97 views

When two non-equivalent binary forms primitively represent the same infinite subset of the integers

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ and $y$ ...
0
votes
0answers
37 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
-3
votes
0answers
42 views

How to figure a complentary set of a Diophantine equation [on hold]

Given a subset of the real numbers defined as 2xy-x-y+1 for x >1, y > 1 how can I determine the complementary set?
3
votes
1answer
104 views

Subspaces of $L_{p}(2<p<\infty)$

Let $p>2$ and $X$ a subspace of $L_{p}$. Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. Question: if $X$ is ...
2
votes
0answers
68 views

a continuous analogue of a graph theory question

I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is ...
0
votes
0answers
60 views

Does A Simple Arrangements of Chords in a Circle Have Hamiltonian Circuits? [on hold]

An arrangement of $s$ chords are drawn over a circle so that no three chords intersect at a common point and no two chords are parallel. Denote the arrangement by $\mathcal{H}_{s}$. Does $\...

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