# All Questions

**0**

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**0**answers

5 views

### Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence

Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let
$$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$
be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...

**0**

votes

**1**answer

31 views

### Hyperplane generic to a given arrangement

At the moment, I am reading the paper "on the connectivity of the realization spaces of line arrangements" of Nazir and Yoshinaga.
I would like to extend their Lemma 3.2 to higher dimension. However, ...

**2**

votes

**1**answer

18 views

### A follow up question to: Number of walks on integer lattice with self-edge at zero

Let $a(n)$ be the number of lattice paths in ${\mathbb{Z}^2}$ of length $n$ which start at the origin $(0,0)$ and have only up-steps $U:(i,j) \to (i + 1,j + 1)$, down-steps $D:(i,j) \to (i + 1,j - 1)$...

**3**

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**0**answers

37 views

### approximating smooth simplicial sheaves by manifolds?

Bit of background
Let $C$ denote the category with objects given by $\mathbb{R}^n$, $n\in \mathbb{N}$ and morphisms given by smooth maps $f:\mathbb{R}^n\to \mathbb{R}^m$. Then we can form the ...

**2**

votes

**0**answers

49 views

### Fourier Mukai transform for non-quasi coherent sheaves

Let $A$ be an abelian variety and $\hat A$ be the dual abelian variety. If $P$ is the (normalized) Poincare line bundle, then Mukai defines $R\hat S:D(A)\to D (\hat A)$ via $R\hat S(?)=Rp_{\hat A,*}(...

**0**

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**0**answers

44 views

### Can we use GAP to find index of subgroup of an infinite group?

Can we use GAP to find index of subgroup of an infinite group? If yes, please tell how, I tried kgmag package of GAP but could not find. From various questions here, I guessed that in MAGMA, one can ...

**3**

votes

**1**answer

47 views

### Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?

Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\...

**2**

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**0**answers

15 views

### A-priori $C^0$-estimates for a semi-linear vector Poisson equation

Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which is bounded and satisfyies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) =...

**1**

vote

**0**answers

14 views

### Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution ...

**2**

votes

**0**answers

30 views

### How do spectrums interact with bi-Lipschitz maps?

If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...

**0**

votes

**0**answers

39 views

### generic divisibility equation for two natural numbers [on hold]

Given two natural numbers N and B, so that N > B, is there a generic equation that contains only multiplications of N and B which can tell whether N is divisible by B?
Basically, something like a ...

**3**

votes

**2**answers

141 views

### Combinatorial interpretation for coefficients of reciprocal of power series

I've seen a number of combinatorial interpretations for the coefficients of the compositional inverse (aka reversion) of a power series. Is there a known combinatorial interpretation for the ...

**-4**

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**0**answers

30 views

**3**

votes

**1**answer

132 views

### Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...

**1**

vote

**0**answers

26 views

### Integral of Daubechies wavelets [on hold]

For Daubechies wavelets according to this paper (above eq 19) this relation holds
$$
\int_{-\infty}^{-\infty} \phi(2x-i)\phi(2x-j)dx = \frac{1}{2} \int_{-\infty}^{-\infty} \phi(x-i)\phi(x-j)dx
$$
...

**0**

votes

**0**answers

73 views

### Number Theory Characterization Problem

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...

**5**

votes

**0**answers

53 views

### Homology of inverse limits over inverse systems more complicated than towers

Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...

**0**

votes

**0**answers

35 views

### Questions related to a summation of fraction equation

I am struggling the following problems. It is not ensured to solve completely because these problems are generated by myself. Particularly, I guess the second problem is very hard if we try to solve ...

**4**

votes

**0**answers

55 views

### Is there a name for groups of the form $Sp(1)^n$?

A (compact) torus is a Lie group isomorphic to the product of finitely many circles: $T^n = S^1 \times \cdots \times S^1$. Such groups are extremely important in Lie theory, Differential Geometry, ...

**1**

vote

**1**answer

88 views

### estimating binomial coefficients

There is a beautiful paper on the arXiv by Andrew Suk containing an asymptotic result about the Erdös-Szekeres convex polygon problem. I am struggling with one of the estimates he makes on page 4. He ...

**-3**

votes

**0**answers

31 views

### How many vertical or horizontal asymptotes does this equation (x^2 - 1)y = x^2 - 4 have? [on hold]

How many vertical or horizontal asymptotes does this equation (x^2 - 1)y = x^2 - 4 have?
Actual question is this(Qs11)
Answer is given as C but I think there are two horizontal asymptotes

**5**

votes

**2**answers

155 views

### Constructively, are all fibrations cloven?

A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice".
Firstly, I'm a bit ...

**0**

votes

**0**answers

18 views

### Permutation Invariant Color Class

$G$ is a $d$ regular graph, it has $n$ vertices.
$S_n$ acts on $n$ vertices of graph $G$.
Question: Does there exist a coloring algorithm for which color classes is invariant under all ...

**2**

votes

**0**answers

40 views

### Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...

**8**

votes

**2**answers

270 views

### What is the intuitive meaning of the coskeleton of a simplicial set?

Consider the inclusion $i_n: \varDelta_{\leq n} \hookrightarrow \varDelta$.
This induces a truncation functor $tr_n:\mathbf{sSet}\to\mathbf{sSet}_{\leq n}$, which has a left and a right adjoint, $tr_n^...

**1**

vote

**0**answers

9 views

### The jump set of $SBV$ function for different value of parameter in image denoising problem

The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...

**2**

votes

**0**answers

30 views

### Negative-curvature behaviour of higher-rank symmetric spaces

Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...

**3**

votes

**2**answers

72 views

### A question on invariant measures

Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures.
I do not know is there some nice functional characterization of the following set
$\{...

**1**

vote

**0**answers

35 views

### Comodules of the $B,C$ and $D$ series quantum groups

In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...

**6**

votes

**1**answer

100 views

### Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...

**4**

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**0**answers

38 views

### Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$

I am struggling with figuring out the details of Proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor.
Setting is as follows.
Let $\Omega$ be a noncompact Riemann ...

**3**

votes

**3**answers

391 views

### sum of four squares with some coefficients

Is there an ordered 4-tuple of rational numbers $(a,b,c,d)$ such that $(b,d)\ne(0,0)$ and $2a^2+3b^2+30c^2+45d^2=2$?
The former (deleted) question was just about cases $(a,b,c,d)\ne(1,0,0,0)$ but it ...

**-3**

votes

**0**answers

65 views

### Is a surjective mapping of R2 to itself with full rank derivative everywhere injective? [on hold]

Let $f:\mathbb R^2 \rightarrow \mathbb R^2$, and $rank(\frac{df}{dx}) = 2$ everywhere. If $f$ is surjective $f$ necessarily injective?
Also, what if $f$ maps $\mathbb R^{2+}$ (i.e. $\{x_1>0, x_2&...

**3**

votes

**0**answers

85 views

### On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [on hold]

(Note: This question has been cross-posted to MSE.)
Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$.
A number $M$ is called almost perfect if $\sigma(M) = 2M -...

**0**

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**0**answers

63 views

### Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...

**10**

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**0**answers

183 views

### Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...

**0**

votes

**0**answers

42 views

### Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$.
Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is
$M[i,j]={(...

**0**

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**0**answers

61 views

### Action is determined by a braiding

Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by
\begin{align}
& \Psi: V \otimes W \to W \otimes V, \\
&...

**2**

votes

**0**answers

25 views

### Realisation of the preprojective algebras as $F(\Delta)/T$ over some quasi-hereditary algebra

Let $A$ be the Auslander algebra of $K[x]/(x^n)$ for some $n \geq 2$, which is quasi-hereditary with some characteristic tilting module $T$.
Dlab and Ringel showed in their paper "The Module ...

**2**

votes

**1**answer

61 views

### Generalized height of elements in abelian groups

In the book Infinite abelian groups Vol. I by L. Fuchs, on page 154, the notion of the generalized $p$-height of an element in an abelian group is defined, as follows:
Let $A$ be an abelian group ...

**2**

votes

**1**answer

60 views

### An inequality on partitions into distinct bounded parts

Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$.
After some numerical experiments it appears
$...

**0**

votes

**0**answers

97 views

### An ideal and its annihilator

Let $I$ be an arbitrary ideal of commutative ring $R$ with identity.
If $ann (I)=AB$, where $A$ and $B$ are comaximal, how can we fine two ideal $I_1$ and $I_2$ with $I=I_1+I_2$ such that $I_1$ ...

**2**

votes

**1**answer

56 views

### Bounds on Matrix Exponential

Suppose $A$ and $B$ are (non-commuting) hermitian $n\times n$ matrices and $k$ is a large positive number. Suppose we write the product of matrix exponentials as
$e^{kA + B} e^{-kA} = e^{C(k)}$
for ...

**5**

votes

**2**answers

622 views

### Euclid vs Eratosthenes

Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes ...

**5**

votes

**0**answers

77 views

### Has a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ a non-scattered fiber?

Question. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-...

**0**

votes

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50 views

### Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor.
For ...