# All Questions

**5**

votes

**1**answer

104 views

### Symmetry type of non-cohomological automorphic forms

By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...

**21**

votes

**5**answers

805 views

### On an example of an eventually oscillating function

For $x\in(0,1)$, put
$$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$
This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

**0**

votes

**0**answers

62 views

### Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...

**2**

votes

**0**answers

165 views

### When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...

**-1**

votes

**0**answers

10 views

### How to test the significance of covariance [on hold]

I'm using the Mutual Informacion covariance in RNA sequences and I want to know if there exits a way to test if some covariance is significant, let's say, an associated p-value.
Thanks to all for ...

**-1**

votes

**0**answers

15 views

### Combining the output of two functions smoothly for a droplet effect [on hold]

I'm trying to write a function which generates this droplet effect implicitly.
I've got a function which generates both of the shapes and I'm looking for a way to somehow combine these two in such a ...

**-2**

votes

**1**answer

141 views

### Degree of a rational function [on hold]

I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...

**4**

votes

**2**answers

391 views

### How close to an integer can a polynomial root be?

Suppose I have a polynomial $p(x) = a_n x^n + ... + a_0$ where $a_n, \dots, a_0$ are integers. I would like to show that any root of this polynomial is either an integer or is far from an integer. ...

**2**

votes

**1**answer

48 views

### Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...

**1**

vote

**1**answer

151 views

### Why can we not always take a Kähler class to be in rational cohomology?

Given a Kähler manifold $(X,\omega)$ we know that its Kähler class lies in an open cone of $H^{1,1}(X) \cap H^2 (X,\mathbb{R})$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ we should be able to find a ...

**-4**

votes

**0**answers

33 views

### Partially ordered set [on hold]

Show that a subset $C$ of a preordered space $(X, ≤)$ is a chain if and only if
$C × C ⊂ A ∪ A^{−1}$, where $A := \{(x, y) : x ≤ y\}$, $A^{−1} := \{(x, y) : (y, x) ∈ A\}$.

**0**

votes

**0**answers

29 views

### Upper and Down Bound,directed,cofinal [on hold]

I'm learning a partially ordered set.Can you give me some example of each these definition:
1.Upper and Down Bound :
...

**-6**

votes

**0**answers

73 views

### Can you give me some example of each these definition [on hold]

I'm learning a partially ordered set.Can you give me some example of each these definition:
1.Upper and Down Bound :
...

**-4**

votes

**0**answers

70 views

### every(ultra)filter on set I is principle if and only if I is finite [on hold]

1)the filter generated by{a,b} is not ultra filter?
2)the filters generated by singleton are precisely the principle ultrafilters.
3)every(ultra)filter on set I is principle if and only if I is ...

**2**

votes

**1**answer

93 views

### Base of a cone in a vector space: can one always choose a convex base?

Let $C$ be a pointed convex cone in a vector space $V$. This means that $C$ satisfies the three following axioms:
$C + C \subset C$,
$\mathbb{R}_+ \cdot C \subset C$, and
$C \cap (-C) = \{ 0 \}$. ...

**-4**

votes

**0**answers

50 views

### izomorphism of finite abelian group [on hold]

Please help me with rezolving this problem from Romanian "Gazeta Matematica" : "an finite abelian group G have |End G | and |Aut G | coprime numbers. Show that |G| is square free.
Thank you!

**0**

votes

**1**answer

53 views

### Decomposition of semi simple local systems

I found A question similar to this, but the answer wasn't clear to me and I'm not supposed to ask for further clarification in the answer section.
Let $L$ a semi simple local system defined over an ...

**1**

vote

**1**answer

64 views

### Optimal covering

Let consider a problem of optimal covering of Hamming space.
So we have Hamming space $\{0,1\}^n$ and some integer $r$. We want to find a set $A \subseteq \{0,1\}^n$ such that any point from ...

**-6**

votes

**0**answers

36 views

### Summation of Geometric Series [on hold]

Im really desperate please help!!!
how can you show that
a. the sum oscillates between the two values a and b
for the summation of geometric series {a*r^(n-1)}`
provided that this is divergent? ...

**3**

votes

**1**answer

70 views

### Finiteness properties for graph of groups decompositions

My curiosity was raised by the following question
and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting.
Let $G$ be a finitely ...

**0**

votes

**0**answers

22 views

### Interchange summation and differentiation [migrated]

I asked this question already on math.stackexchange, but did not receive any answers
see here
Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$
Now assume we have that ...

**3**

votes

**1**answer

94 views

### Covering finite groups by kernels

Let $G$ be a finite group. When does there exist a finite group $H$ such that every $h\in H$ is in the kernel of some epimorphism $H\to G$?
This is well-known to be true for $G$ abelian, for example ...

**-5**

votes

**0**answers

42 views

### Summation of geometric series divergence [on hold]

The summation of some geometric series a*r^(n-1) is divergent. But what i don't understand is this:
If the summation of a geometric series is divergent, then one of its sum is:
a. the sum oscillates ...

**2**

votes

**0**answers

124 views

### Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...

**1**

vote

**0**answers

70 views

### Asymptotic expansion square root matrix

I am looking for an asymptotic expansion for $\underline\gamma$ which is the "square root" matrix of a symmetric $p\times p$ matrix $\gamma$. Here $\underline\gamma$ is assumed to be symmetric, e.g. ...

**0**

votes

**0**answers

65 views

### Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...

**2**

votes

**1**answer

180 views

### Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?
Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...

**3**

votes

**2**answers

131 views

### Complete sets of incompatible totally ordered down-set in a partially ordered set

Let $(P,\leq)$ be a partially ordered set. A down-set is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ...

**-1**

votes

**0**answers

48 views

### Calculate the intersection numbers by a plane section [on hold]

This question is from the chapter A of Reid's note: Chapters on algebraic surfaces
Let X = X$_d$ $\subset$ P$^3$ be a nonsingular surface of degree d and
suppose that X has a plane section P ...

**6**

votes

**0**answers

212 views

### A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...

**6**

votes

**1**answer

250 views

### Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized)

Let $k > 1$ and $n$ be positive integers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Let $D$ be a digraph which has exactly $k$ vertices $v_0$, $v_1$, ..., $v_{k-1}$ and exactly $k$ arcs $v_0 ...

**-1**

votes

**0**answers

14 views

### probability distribution [on hold]

X is a continous random variable of normal distribution for the length of the rulers produced in a factory. Given X has mode of 15 cm and standard deviation of 1 cm. A ruler is randomly selected from ...

**-1**

votes

**0**answers

34 views

### How to show Well Founded Induction false? [on hold]

The abstract reduction system ({a,b,c,d},→) where the → is defined as:
http://i.stack.imgur.com/TS0Ud.png
Let Q be a monadic predicate on {a,b,c,d} such that Q(a) = Q(b) = false and Q(c) = Q(d) = ...

**3**

votes

**2**answers

231 views

### If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...

**-2**

votes

**0**answers

30 views

### Underlying Set in Model Theory [migrated]

In model theory a structure has an underlying set.
In addition to the interpreted relations, are there
(implicit) assumptions made about possible
operations on this set? For example, is it assumed to ...

**3**

votes

**0**answers

106 views

### Blowig-up a point in the singular locus

Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...

**0**

votes

**0**answers

19 views

### Practical Way to Detect a Markov Chain is Regular Given the Transition Matrix [migrated]

I understand that a Markov Chain is reducible if, given its transition matrix $P$, there exists $n$ such that every element of $P^n$ is greater than 0.
However, I am wondering that if there is an ...

**6**

votes

**2**answers

177 views

### Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...

**9**

votes

**1**answer

208 views

### When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?

Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...

**-2**

votes

**0**answers

30 views

### Backpropagation KL-divergence for training “one level neural network” [on hold]

Hi I hope that some one could help me.
I have a L matrix (25x1000) (I have 10000 works each work is represent by 25 bits)
I map each word to one of 5 classes {vary negative,negative, ...

**-3**

votes

**0**answers

50 views

### Strict partition of size n [on hold]

I want to know how to calculate how many strict partitions of X are with size n.
For example there are 22 partitions of number 8, and there are 6 strict partitions of 8 (partitions with distinct ...

**-2**

votes

**0**answers

68 views

### Analytic formula to evaluate the exact value of solid angle subtended by an ellipse at any arbitrary point lying on the vertical axis [on hold]

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center.
Standard equation ...

**0**

votes

**0**answers

67 views

### Relating overlapping simplicial complexes

Let $X$ be a simplicial complex and let $A,B\subset X$ be
subcomplexes such that $C=A\cap B$ is a non-empty simplicial
complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
...

**0**

votes

**1**answer

77 views

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= ...

**0**

votes

**0**answers

58 views

### Integral geometry and curvature of surfaces

I'll stick to 2-surfaces in $\mathbb{R}^3$ for simplicity. Higher dimensions generalizations welcome.
In classical integral geometry, we may obtain up to a scaling factor, for example, the surface ...

**0**

votes

**0**answers

29 views

### Find all integers x, y, and z such that 1/x + 1/y = 1/z [migrated]

Characterize all positive integers x, y, and z such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. For example, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$

**0**

votes

**0**answers

23 views

### contiguity and infinite complexes

Two simplicial maps $f: K^{(n)}\to L$, $g: K\to L$ are called contiguous if for every simplex $\tau\in K^{(n)}$ and it carrier $\sigma$ we have that $f(\tau)\cup g(\sigma)$ is a simplex in $L$.
It ...

**0**

votes

**1**answer

116 views

### A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other ...

**15**

votes

**3**answers

714 views

### Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.
Magnus, in his 1968 ...

**4**

votes

**1**answer

125 views

### Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...