All Questions

-2
votes
0answers
10 views

Legendre symbol problem

Let $p,q$ different odd prime numbers and $a$ a positive odd number, Prove that : $$p \equiv - q\ \ \ \ (mod \ \ 4a )\Longrightarrow \Bigg(\frac{a}{p}\Bigg) =\Bigg(\frac{a}{q}\Bigg) $$ Where ...
0
votes
0answers
13 views

Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum: $$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$ as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...
0
votes
0answers
13 views

Canonical divisor for conic bundles

Is there a general formula for the canonical divisor $K_X$ of a smooth conic bundle $X$? Motivation: for smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $K_X = \mathcal{O}_X(d-n-1)$. But ...
-2
votes
0answers
19 views

How to show this Legendre Symbol Problem [on hold]

Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$, Prove that: $\exists x, y \in \mathbb{Z};\,\, \gcd(x,y) = 1$ such as $x^{2} +ny^{2} \equiv 0\, (\mod p) \iff ...
0
votes
0answers
13 views

Variance of sums of correlated variables when sampling without replacement?

Background Let $X_t$ be an alternating renewal process, a stochastic process on the state space {0, 1} where 0 = 'Broken' and 1 = 'Working'. Let $U$ be the cumulative sojourn working during an ...
5
votes
0answers
42 views

Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$: $n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
-1
votes
1answer
26 views

extension of a continuous function

Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$? ...
5
votes
1answer
48 views

Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...
-2
votes
0answers
15 views

Formula for unequal share distribution [on hold]

What formula would I use to distribute $M$ shares among $N$ shareholders, such that shareholder $X_i$ has 3/2 as many shares as shareholder $X_{i+1}$? P.S. I apologize if the tag isn't relevant. I ...
-5
votes
0answers
46 views

Exactly 2 Girls - Conditional Probability [on hold]

This is very confusing to me. I am really new with this stuff. A couple wants to have 3 or 4 children, including exactly 2 girls. Is it more likely that they will get their wish with 3 children or ...
-5
votes
0answers
36 views

Integral of (2-x)/(x-1) Really stumped [on hold]

So I tried doing this: I have integral (2-x)/(x-1) I used a substitution ; u = x-1 x= u+1 du = dx So then (2-u-1)/u du then : 1/u - 1 Then I integrate and get ln u - u But when I plug ...
4
votes
0answers
68 views

A question about three forcings

Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 ...
1
vote
0answers
37 views

The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number. Now ...
2
votes
1answer
56 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
-2
votes
0answers
52 views

Combinatorical configuration. Proof [on hold]

Given integers $k$ and v with $1 < k < v$ show that there exists a $$(v, \binom v k ,\binom {v-1} {k-1}, k, \binom {v-2}{k-2} ) $$ design. Please give me a hint. For: $(a, b, c, d,e )$ ...
4
votes
0answers
93 views

The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property: The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$. ...
7
votes
0answers
191 views

Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...
2
votes
0answers
45 views

Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
0
votes
0answers
29 views

Saddle point method for asymptotic expansion [migrated]

Any idea on how to derive an asymptotic expansion of the following integral for $x$ around zero (using Saddle point method): $$\int_{0}^{x}\left(u+\alpha\right)^{-a - \left(b-a\right)e^{-\lambda ...
0
votes
0answers
54 views

Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way: Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$. Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...
0
votes
0answers
38 views

Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of Dynamical Systems. The motivation for this question can be found in: ...
2
votes
0answers
44 views

Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$): $$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$ where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...
9
votes
1answer
86 views

Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...
2
votes
0answers
27 views

Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...
2
votes
1answer
80 views

Lower regularity version of Moser's theorem on volume elements

A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965), shows that if a $C^\infty$ compact manifold $M$ has two ...
6
votes
0answers
90 views

Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space: Let $X_1,X_2,\cdots$, be ...
10
votes
1answer
187 views

How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...
-3
votes
0answers
51 views

Mathematically compute - species diversity [on hold]

Using a simple model, and having these as paramters numberofspecies <- 100 meaninitialpopulationsize <- 50 sdloginitialpopulationsize <- 1 #determines variation in initial population ...
4
votes
1answer
64 views

Estimate on sum of $J_n^4$

If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$ What is known about $$ ...
-5
votes
0answers
111 views

Mathematics Research and The Internet [on hold]

I reformulate here a question about Mathematics and The Internet. My questions are: What was the vital role of Mathematics research in the foundation of the Intranet ($\rightarrow{Internet}$) and, do ...
1
vote
1answer
81 views

Upper bound of sum of binomial coefficients

I am looking for an upper bound - up to constant factor - for: $\sum_{k=t}^{t+l} {n \choose k} \cdot 2^{-n}$ where: The values of $t$ are between: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. ...
3
votes
0answers
43 views

Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and ...
-1
votes
0answers
61 views

Integration of the reciprocal of sum exponential [migrated]

Any one know the method to do the integration as $$\int\frac{x^2\cdot \exp (-ax^2) \exp(-bx^2)}{\exp(-ax^2)+\exp(-bx^2)}dx$$ It can be simplified as $$\int\frac{x^2}{\exp(ax^2)+\exp(bx^2)}dx$$ ...
3
votes
0answers
164 views

“Partition” of a smooth function in $\mathbb R^2$

This is a question asking for reference. I have a proof of the following. Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions ...
0
votes
1answer
51 views

Uniform approximation of increasing function in $C^{\infty}$

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function ...
-4
votes
0answers
36 views

fluid flow through an orfice [on hold]

Forgive me for my ignorance. What would be the method to determine the pressure a non compressible fluid creates when forced though an orifice? Keep in mind this orifice does not have a constant ...
0
votes
3answers
63 views

Maximizing/Minimizing the Operator norms of 0-1 matrices subject to a constraint

Fix $n$ and let $B, C$ be two $n \times n$ 0-1 matrices of full rank such that $\sum_{i,j} b_{i,j}^2 = \sum_{i,j} c_{i,j}^2$, in other words they have the same number of $0$ entries and the same ...
-1
votes
0answers
65 views

complete compact open topology [migrated]

Let $X$ denotes a path-connected and compact manifold and $PX$ its path-space (the set of continuous maps $\gamma: [0,1] \longrightarrow X$) topologized with the compact open topology. It is true that ...
0
votes
0answers
35 views

How to obtain a solution of polynomial recurrence relation? [on hold]

How to obtain a solution of 2-termed recurrence relation? I want to have a generalized form of solution, applied to every 'n' value.
4
votes
0answers
102 views

What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$? The $p$-Laplacian ...
-5
votes
0answers
44 views

basic : modulo and division [on hold]

how can one prove that a mod b < a/2? I understand why is that happening: if a mod b > a/2 that means that ...
1
vote
0answers
71 views

Clebsch Gordan coefficients of compact quantum groups

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
0
votes
0answers
20 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: ...
0
votes
0answers
23 views

Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation $$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$ can ...
2
votes
1answer
106 views

The definition of computational complexity or complexity measure of computing reals [on hold]

A real $r$ is computable,if for any $i\in \mathbb{N}$,the $i$ bits can be outputed by a Turing Machine or an algorithm. So, what is computational complexity or complexity measure of computing the ...
1
vote
0answers
42 views

clustering in a graph on boolean functions

Fix some n and k. Consider the following directed graph: vertices are all functions $2^n\rightarrow 2^n$ and a vertex f has an edge to a vertex g for every h such that $f=h\circ g$ and h depends only ...
2
votes
0answers
56 views

Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories. How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...
0
votes
0answers
122 views

Canonical identification between 3-manifolds cohomology and group cohomology [on hold]

I am trying to understand why this 3-manifold cohomology is equal to this group-cohomology. $$ H_\ast (\mathbb{H}^3/PGL_2(\mathbb{Z})) \simeq H_\ast (PGL_2(\mathbb{Z}))$$ In both cases, use the base ...
4
votes
1answer
103 views

Counting Boolean Normal Matrices of size $2n \times 2n$

Fix $n$ a natural number. Consider the set of all $2n \times 2n$ matrices with entries from {0,1}. This is clearly a finite set. I would like to count the number of such normal matrices for fixed ...
1
vote
0answers
86 views

Are convolution algebras ever “topologically noetherian”?

For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means ...

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