# All Questions

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### boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$

Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$ Now pick up one such bundle $\xi$,we have the long exact sequence ...
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1answer
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### 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 ...
0answers
31 views

### selecting balls w.p. 1/2 [on hold]

Suppose there are $n$ balls in the beginning. In each round, each ball is retained with probability $1/2$ and discarded with probability $1/2$. What is the probability that there is still a ball (or ...
3answers
78 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 ...
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30 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 ...
0answers
67 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 ...
0answers
8 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 ...
0answers
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### 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 ...
1answer
89 views

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

I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach): Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of ...
1answer
163 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. ...
0answers
27 views

### Complex Financial Formula [on hold]

In the process of putting together a component of our project which takes the following variables: 1) Yearly quantity of income 2) Down payment in dollars 3) Monthly debt total in terms of minimum ...
1answer
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### 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 ...
1answer
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### 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 ...
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29 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 $A := \{(x, y) : x ≤ y\}$, $A^{−1} := \{(x, y) : (y, x) ∈ A\}$.
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22 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 : ...
0answers
57 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 : ...
0answers
59 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 ...
0answers
38 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 $C \cap (-C) = \{ 0 \}$. ...
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### 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!
1answer
41 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 ...
0answers
37 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 ...
0answers
27 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? ...
1answer
50 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 ...
0answers
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 ...
1answer
56 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 ...
0answers
37 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 ...
0answers
95 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 ...
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### 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. ...
0answers
51 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 ...
1answer
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### 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 ...
2answers
125 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 ...
0answers
38 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 ...
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### 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 ...
0answers
33 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) = ...
3answers
155 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 $H \langle g\rangle = G$. ...
0answers
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### 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 ...
0answers
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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 $\pi:Y\rightarrow ... 0answers 19 views ### Practical Way to Detect a Markov Chain is Regular Given the Transition Matrix [migrated] I understand that a Markov Chain is reducible if, given its transition matrix$P$, there exists$n$such that every element of$P^n$is greater than 0. However, I am wondering that if there is an ... 1answer 112 views ### Homological criterion for$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}$, ... 1answer 191 views ### When is$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)$as a ring) ... 0answers 30 views ### Backpropagation KL-divergence for training “one level neural network” [on hold] Hi I hope that some one could help me. I have a L matrix (25x1000) (I have 10000 works each work is represent by 25 bits) I map each word to one of 5 classes {vary negative,negative, ... 0answers 45 views ### Strict partition of size n [on hold] I want to know how to calculate how many strict partitions of X are with size n. For example there are 22 partitions of number 8, and there are 6 strict partitions of 8 (partitions with distinct ... 0answers 54 views ### Analytic formula to evaluate the exact value of solid angle subtended by an ellipse at any arbitrary point lying on the vertical axis [on hold] Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ... 0answers 61 views ### Relating overlapping simplicial complexes Let$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)\$, ...

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