**1**

vote

**0**answers

37 views

### maximal elementary abelian p-subgroups of finite groups of Lie type

Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...

**2**

votes

**1**answer

72 views

### If two knots in $S^3$ are invertible cobordant (from both ends), are they equivalent?

Let $K_1,K_2$ be two knots in $S^3$ and assume that there exists a cobordism $(W;K_1,K_2)$ which is invertible from both ends. Does this imply that $K_1, K_2$ are equivalent? In the paper by D.W. ...

**3**

votes

**0**answers

96 views

### Comparing spectra of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators ...

**0**

votes

**0**answers

58 views

### Does every odd prime divide a Mersenne number? [migrated]

For a given odd prime number $p$, is there an integer $n>0$ such that $p$ divides $2^n-1$?

**-4**

votes

**0**answers

32 views

### Is there any closed form for the following nested series? [on hold]

I am wondering if there is any closed form of the following summation?
$\sum \limits_{i=0}^{\infty} (q^i \sum \limits_{j=0}^i \dfrac{a^j}{j!})$ where |q|<1
I know that $\sum \limits_{i=0}^{\infty} ...

**2**

votes

**1**answer

97 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a ...

**12**

votes

**2**answers

193 views

### Can $L$ be thin?

I have recently been wondering if the following is consistent with ZFC:
For every infinite ordinal $\alpha$: $|V_\alpha\cap L|=|\alpha|$.
Intuitively, this states that for $L$ is very "thin", in ...

**3**

votes

**0**answers

51 views

### Equality of codimension under Lusztig-Spaltenstein induction

Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G$ with the $\tilde{\Delta}=\Delta\cup\{\alpha_0\}$ being the simple roots plus the negative highest root. Any ...

**1**

vote

**0**answers

22 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

**11**

votes

**1**answer

161 views

### Is there an integrable complex structure on SU(3)?

Is there a complex manifold diffeomorphic to SU(3)?
This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
...

**-2**

votes

**0**answers

42 views

### Explain this (PDE) [on hold]

I started studying PDE from REF : advanced engineering mathematics and in the beginning of the chapter there was something I didn't get it
"A solution of a PDE in some region R of the space of the ...

**4**

votes

**1**answer

141 views

### Historical developement of analysis and partial differential equations (especially in the 20th century)

What are some comprehensive surveys or monographs that describe (in
enough technical detail) the historical development of the various
subareas of analysis and partial differential equations?
...

**1**

vote

**1**answer

135 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...

**-2**

votes

**0**answers

17 views

### Individual contribution to a global change in a ratio [on hold]

I have about four thousand records, each consisting of an entity ID and four variables that represent two data points from two different years. x2010 and x2013 are counts of all cases for each entity. ...

**2**

votes

**1**answer

64 views

### Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...

**1**

vote

**0**answers

56 views

### The Irreducible Representations of the Sekine Quantum Groups

Here Y. Sekine introduces a one-parameter family a finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$ (I have a feeling this should actually be $e^{\pi i/n}$ - ...

**4**

votes

**0**answers

34 views

### Is there a name for a bifurcation where a stable fixed point bifurcates into three fixed points: stable, unstable and saddle?

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by
$$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & ...

**-3**

votes

**0**answers

64 views

### Examples of ill functions in physics [on hold]

I am asked to find some ill functions in physics, but after having some research I could not find an example that was understandable for me.help me with some examples which are well explained please.

**5**

votes

**1**answer

128 views

### Is there a way to simplify the following trace expression?

I'd like to simplify the following expression:
$$\text{tr}\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}_{M\times M} & \mathbf{0}_{M\times N} \\ \mathbf{0}_{N\times M} & ...

**1**

vote

**0**answers

74 views

### Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = ...

**0**

votes

**0**answers

34 views

### Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary ...

**6**

votes

**1**answer

73 views

### Weighted Permutation Sum

I am trying to find out a closed-form formula (or a generating function at least) for the number of permutations $\sigma$ that satisfy $$ S = \sum_{i = 1}^{n} i\sigma(i)$$ for a given value of $S$. We ...

**2**

votes

**0**answers

33 views

### On the relationship between the factorizations of an operator $T$ and its second adjoint $T^{**}$

Let $T$ be an operator from a space $X$ into a space $Y$ and let $1\leq p<\infty$. If $T^{**}$ has a factorization $T^{**}=RS:X^{**}\xrightarrow{S} l_{p}\xrightarrow{R}Y^{**}$, where $S$ is compact ...

**4**

votes

**1**answer

113 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**0**

votes

**0**answers

10 views

### Gaussian upper bounds for heat kernels

I am studying heat kernel estimates.
Let $U$ be an open subset of $\mathbb{R}^{d}$, endowed with the Lebesgue measure $dx$. Define a bilinear form
$$
\mathcal{E}(f,g)=\int_{U} \left\{ ...

**-4**

votes

**0**answers

127 views

### Power equivalence of two positive real numbers [on hold]

I posted the same question on Math.Stackexchange but I didn't get any precise answer until now. Thus I asked it here.
Assume $a,b>0$ are two real numbers. Define the sequences $a_n, b_n$ as ...

**2**

votes

**1**answer

59 views

### Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes:
The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a
parallel subbundle $Q$ which is locally spanned by $3$
...

**1**

vote

**0**answers

43 views

### Colimits in n-fold categories

An $n$-fold category is an internal category in the category of $(n-1)$-fold categories (and a $0$-fold category is just a Set).
General results about internal categories assure that the category of ...

**1**

vote

**0**answers

82 views

### Family $(X_y,D_y)$ with trivial canonical bundles

Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...

**2**

votes

**0**answers

49 views

### holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...

**13**

votes

**3**answers

1k views

### How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...

**5**

votes

**0**answers

74 views

### Free generators for the fat commutator subgroup

There is a homomorphism $\langle x,y\rangle\to\langle x\rangle$ of free groups, sending $y$ to $1$. We can combine this with the other obvious homomorphism to get a surjective homomorphism
$$ ...

**-1**

votes

**0**answers

26 views

### Why is a weakly dominated strategy never played in a mixed strategy equilibrium? [on hold]

I'm having trouble proving that a weakly dominated strategy can never be played in a mixed strategy equilibrium. I have an idea that it can be proven by contradiction, using the definition of best ...

**2**

votes

**1**answer

52 views

### Post composition of integral

Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...

**1**

vote

**1**answer

37 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**4**

votes

**3**answers

209 views

### Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...

**1**

vote

**0**answers

75 views

### Additive combinatorics and a Diophantine equation

Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
...

**1**

vote

**1**answer

65 views

### Existence of real solutions for a system of linear and quadratic equations

Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not ...

**-2**

votes

**0**answers

28 views

### Real time application of Generalized closed set in point set topology [on hold]

Hai i am doing reserach in point set topology.i like to know what are the real time application for a new set in point set topology

**0**

votes

**0**answers

59 views

### Improvement on $\phi\sigma$ bound [migrated]

We have:
$$\dfrac{6}{\pi^2}\lt\dfrac{\phi(n)\sigma(n)}{n^2}\le1$$
with equality iff $n=1$.
Wikipedia
Are there any known improvements on these bounds?

**4**

votes

**3**answers

419 views

### Topological structure of SO(n) as a product

I’m interested in the question for which $n$ the special orthogonal group can be written as a product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] ...

**3**

votes

**0**answers

58 views

### Power's Theorem for irreducible representations

Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...

**1**

vote

**0**answers

27 views

### Cubulating non-compact hyperbolic manifolds

Let $X$ be a hyperbolic manifold of arbitrary dimension. When does $X$ admit a cell structure of a CAT(-1) cube complex? of a hyperbolic CAT(0) cube complex?
I suspect that the question is widely ...

**2**

votes

**0**answers

23 views

### Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f(x)$ satisfying
$$ \int_0^\infty f(x)x^{s-1}dx=e^{-p\psi(s)} $$
where $\psi(s)$ is the usual digamma function and $p>0$. The inverse Mellin formula is
$$ ...

**5**

votes

**1**answer

63 views

### Convex embedding with a positivity condition

We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...

**1**

vote

**0**answers

50 views

### associativity of the extension of finie groups

Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then ...

**4**

votes

**0**answers

29 views

### Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action

Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation ...

**4**

votes

**1**answer

100 views

### Closeness graph of a topological space

Let $(X,\tau)$ be a topological space. We say that $x, y \in X$ are close if for every neighborhood $U$ of $x$ and $V$ of $y$ we have $U\cap V \neq \emptyset$. Let $E$ be the set of $\{x,y\}$ where ...

**0**

votes

**0**answers

30 views

### Constructing an additive set function from on a non-additive one

repost from math.se.
I was trying to generalize some results from measure theory to functions that are "almost" measures but not additive. Then, I thought it could be interesting to do this in a ...

**-1**

votes

**0**answers

33 views

### If the total number of divisors of the square of a number is 7 less than the number, what is the total number of divisors of cube of that number [on hold]

if the total number of divisors of the square of a natural number N is 7 less than N, then what is the total number of divisors of cube of N