1
vote
1answer
74 views

Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
2
votes
1answer
78 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
0
votes
0answers
65 views

some strange sums ramanujan type

i found sum as $$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...
-1
votes
0answers
26 views

product distinct prime factors of prime(n)-1 and prime(n)+1 [on hold]

The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are such primes common? Can a value of ...
1
vote
0answers
61 views

The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
-1
votes
0answers
63 views

Finding the unique Nash equilibrium [on hold]

$$ m^2(1-m)[(1+m)^2 - R] x^3 + [6m^2R + 12mR - 2m(1-m^2)(6+2m) - 4tm^2(1+m)^2] x^2 + [(1-m)(6+2m)^2 + 8tm(1+m)(6+2m)] x - 4t(6+2m)^2=0 $$ where: m ∈ (0, 0.5), R ∈ [0, 0.25], t ∈ [0, 1], x ∈ [0, 2]...
2
votes
0answers
51 views

On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...
-1
votes
0answers
11 views

Markov Chain: Number of communicating classes of a power of the irreducible transition matrix [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P_k$. In terms of $d$ and $k$, how many communicating classes does $P_k$ have, and what is the period ...
6
votes
0answers
115 views

Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
4
votes
0answers
40 views

Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the cofiber doesn't increase?

Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One ...
0
votes
0answers
54 views

Application(s) of complex dynamical system into some other areas of mathematics [on hold]

Complex dynamical system is a very active branch in mathematics. I wondered are there some nice applications of complex dynamical system into the some other areas of mathematics.
1
vote
0answers
63 views

relative chernoff bound

Is the following true? Is there a contradicting example? Let $x_1,\ldots,x_n$ be independent random "bits", with $\forall u:\Pr[x_i=u]\in\{0,\frac{1}{2}\}$. Denote $x=\sum_i x_i$, and assume $\mathrm{...
-1
votes
0answers
71 views

Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...
-5
votes
0answers
21 views

How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2? [on hold]

How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2?
1
vote
0answers
135 views

Computational number theory

Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...
0
votes
0answers
62 views

Boundary conditions for Klein-Gordon equation [on hold]

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...
5
votes
2answers
477 views

Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
-2
votes
0answers
62 views

topology- Compute π1(M3) when M3 be the 3-manifold [on hold]

Let M3 be the 3-manifold obtained by gluing two handlebodies of genus g by the identity map. To be precise: Let H1, H2 be two copies of a “standard” genus g handlebody in S 3 . If we denote Σ1 = ∂H1, ...
3
votes
1answer
212 views

What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?

Let $X$ be a smooth projective surface over a field $k$. Is there a way to compute $H^1(X - \{x\},\mathcal{O}_{X-\{x\}})$ in terms of similar invariants for $X$? Actually I'd like to remove even ...
0
votes
0answers
18 views

Generating artificial time series with gaussian noise [on hold]

I have been using BoolNet package in R. Artificial time series are generated with Gaussian noise (SD) =0.1. I understand that Gaussian noise has a probability density function similar to that of ...
3
votes
0answers
56 views

Non-homotopic cdga maps with same effect on cohomology

Let $f, g : \mathcal{A} \to \mathcal{B}$ be maps of commutative differential graded algebras. An easy way to tell if they are homotopic is by looking at their effect on cohomology. However, if $f$ ...
4
votes
1answer
135 views

Number of walks on integer lattice with self-edge at zero

Consider the graph with vertices $V=\mathbb Z$ and edges $$E=\{(n,n+1):n\in\mathbb Z\}\cup\{(0,0)\},$$ that is, the usual integer lattice with a self-edge at zero. For some fixed parameters $a,b,n\in\...
-3
votes
0answers
72 views

Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [on hold]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...
0
votes
1answer
55 views

the relation between projective and quasi-projective modules

An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$. What are the rings $R$ for which every ...
3
votes
1answer
43 views

Matrix model for “$\beta$-Ginibre” ensembles

A very well known result in random matrix theory is that there exists "nice" (i.e., with independent entries) tridiagonal matrix for the $\beta$-ensembles of random matrix theory $$\propto\prod_{i<...
4
votes
0answers
50 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
-1
votes
0answers
120 views

Stalks of the sheaf

Let $X$ be a scheme. For $m < \infty$, given the surjection ${\cal O}_X^{\oplus m} \twoheadrightarrow {\cal F}$ between sheaves on $X$, where ${\cal O}_X$ is the structural sheaf of $X$. Choose a ...
12
votes
1answer
247 views

Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
2
votes
0answers
79 views

Cohomology of Mumford line bundle on abelian variety

Let $X$ be an abelian variety over a field $k$, and let $L$ be a line bundle on $X$. I would like to calculate the cohomology of the Mumford line bundle $$\Lambda(L)=m^*L\otimes p_1^*L^{-1}\otimes p_2^...
2
votes
0answers
63 views

Approximating formal surfaces by analytic surfaces

Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there ...
1
vote
0answers
46 views

A two variable recurrence relation with conditionals

I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence $$ f(n,m) = \begin{cases} f(n, \...
3
votes
0answers
59 views

Turán's inequalities for Hermite functions

Given $\lambda \in \mathbb{R}$ let $H_{\lambda}(x)$ be the solution of the Hermite differential equation: $$ \frac{d^{2}}{dx^{2}} H_{\lambda}(x)-x\frac{d}{dx}H_{\lambda}(x)+\lambda H_{\lambda}(x)=0, ...
2
votes
0answers
55 views

How to find a closed form of following matrix's determinant [on hold]

I wanna find a closed form of determinant of the following matrix $$A(n) = \begin{pmatrix} B_{1} & B_{2} & \cdots & B_{n} & 1 \\ B_{n} & B_{1} & \cdots & B_{n-1} &...
6
votes
1answer
248 views

A question about composition of functions

Recently, I heard this question: are there two functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is strictly increasing on $\mathbb{R}$ and $g\circ f$ is ...
0
votes
0answers
45 views

Variant of Holder's inequality [migrated]

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
2
votes
0answers
77 views

Does attach-one-cell have a stable homotopy transfer?

Specifically, I am thinking of attaching one ordinary cell to an ordinary space in your favourite convenient category of spaces; so, given a cofiber sequence $$ \mathbb{S}^k \to_c X \to_p X', $$ on ...
5
votes
1answer
314 views

Renorming a Banach space to make projections contractive

Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$. Can the same be done for a family of projections? That is, given finitely many ...
1
vote
0answers
34 views

Combination of certain linear-programming topics new?

Consider the combination of the following topics, aimed at a future book on Linear Programming: Generalization of certain parts of the polyhedron theory and of the Simplex Algorithm to arbitrary ...
-4
votes
0answers
69 views

four consecutive primes ending in 1,3,7, or 9 [on hold]

Examine the last four digits of four consecutive primes to seek 1,3,7,9 in any order. You will find that they occur more than by chance. Do the same for the frequency of two, three, four,.......
10
votes
1answer
237 views

Polylogarithm sheaves

In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...

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