# All Questions

**11**

votes

**0**answers

255 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 ...

**3**

votes

**0**answers

81 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

**0**answers

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 ...

**1**

vote

**1**answer

109 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

**1**answer

61 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: ...

**3**

votes

**1**answer

79 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, $\beta_r(F)$, ...

**10**

votes

**1**answer

97 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

**0**answers

32 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

**1**answer

97 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

**0**answers

97 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 ...

**12**

votes

**2**answers

298 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

**0**answers

54 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

**1**answer

70 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
$$
...

**-6**

votes

**0**answers

127 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

**1**answer

91 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

**0**answers

44 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

**0**answers

62 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
$$\int\frac{x^2}{\exp(ax^2)+\exp(bx^2)}dx$$
...

**4**

votes

**0**answers

191 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

**1**answer

56 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

**0**answers

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

**3**answers

69 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

**0**answers

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

**0**answers

36 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

**0**answers

110 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

**0**answers

49 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

**0**answers

76 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

**0**answers

26 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: ...

**1**

vote

**0**answers

37 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

**1**answer

119 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

**0**answers

45 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 ...

**3**

votes

**0**answers

63 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

**0**answers

131 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

**1**answer

107 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

**0**answers

89 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 ...

**-2**

votes

**0**answers

32 views

### Hypothesis Testing Diagram [on hold]

Consider the null hypothesis H0 : μ = μ0 vs the alternative hypothesis H1 :μ>μ0. Supposethetruemeanisμ∗ >μ0.
(a) Draw a diagram representing the sampling distribution of the mean number of nails in a ...

**0**

votes

**0**answers

61 views

### Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them are idealizations and valuation domains. But the first non-noetherian ring we are thinking about ...

**1**

vote

**0**answers

74 views

### Why is cellularization the fiber of nullification for slice cells?

I'm a bit confused about the nullification functors that come up when constructing the slice tower in HHR.
Let $\mathcal{A}$ be a set of compact objects in the $G$-equivariant stable homotopy ...

**6**

votes

**0**answers

77 views

### Reconciling the affine grassmannian and the based loop group

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...

**1**

vote

**0**answers

49 views

### Nonlocal Stefan problems

Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality?
A search on Google and MathSciNet give me only a handful of results which greatly ...

**2**

votes

**0**answers

163 views

### Why care about Fourier-Mukai partners?

Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool ...

**4**

votes

**0**answers

68 views

### Triangle inequality for $L^1$-norm with respect to a state

It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity.
To wit, let $\varphi$ be a normal ...

**-2**

votes

**0**answers

31 views

### Verification of Chain Rule for Covariant Derivatives [on hold]

I am stuck on one particular step. I'm sure this is a simple property but I'm not sure what I'm looking for.
Exercise 10.12, pg 261, Gravitation, Misner et al.
VERIFICATION OF CHAIN RULE:
Let ...

**4**

votes

**1**answer

79 views

### Positivity of Ehrhart polynomial coefficients

Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?
What methods are available for proving such a property for some family ...

**15**

votes

**0**answers

186 views

### Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps
$$
B G \longrightarrow B \mathrm{GL}_1(A)
$$
for $A$ an $E_\infty$-ring carrying an oriented ...

**0**

votes

**0**answers

41 views

### Do knot Floer homology detects genus of knot in rational homology sphere

My question is the following:
Do knot Floer homology detects genus of knot in rational homology sphere ?
If the answer is yes, I would like to have a reference for the statement of the result and the ...

**2**

votes

**0**answers

44 views

### Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...

**-3**

votes

**0**answers

42 views

### Help with determining onto (surjective) [on hold]

The question is to determine if the following function T(x,y,z) = (ysin x,zcos y,xy) is onto.
So far I have only learned of creating a coefficient matrix and checking if the determinant is 0 to figure ...

**3**

votes

**2**answers

262 views

### A Question related to the Formula Hierarchy

Let large Latin symbols as $X$ and $Y$ denote sets of natural numbers and small symbols as $n$ and $n´$ denote natural numbers and small Greek letters stand for formulas.
Suppose $\alpha$ is ...

**2**

votes

**0**answers

13 views

### Central-Slice-Theorem Analogue for Wavelet Transforms?

The 2D Radon transform and the 2D Fourier transform are related by the so-called Central Slice Theorem (cf e.g. http://en.wikipedia.org/wiki/Projection-slice_theorem) and I would like to know, whether ...

**-3**

votes

**0**answers

59 views

### Computability Theory [on hold]

If $A\equiv_T B$ then $A^{\omega}\equiv_1 B^{\omega}$. Odifredi in his book says hint $A\leq_T B$ implies that $A\leq_1 B'$, but don't know how this helps.