All Questions
0
votes
0answers
13 views
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 ...
0
votes
0answers
21 views
Orthogonal vector to arbitrary vector in R3 [on hold]
I got a vector $(0, 0, 0)^T \neq v \in R^3$. Now I want a closed formula for some orthogonal vector to $w$ (I don't care which).
My problem is that if I, for instance, fix $w_1$ and $w_2$ then $w_3 = ...
0
votes
0answers
24 views
How can one define inaccessibility and beyond explicitly?
I would like to see quite explicit definitions of notions of inaccessibility and hyperinaccessibility and higher in the primitive language of set theory; to simplify some we avail ourselves with set ...
-1
votes
0answers
26 views
Prove an equation is always false [on hold]
How can I prove an equation is always false?
For example:
b = b + 1
is false for all values of b. Very simple to see.
Now given a more complicated equation, such as:
b = sin(sin(b) - .56))
...
0
votes
1answer
23 views
How to minimize this sparse quadratic function?
There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
3
votes
1answer
44 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 ...
-1
votes
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 ...
8
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
-1
votes
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 ...
-1
votes
0answers
12 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
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 ...
3
votes
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. ...
-5
votes
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 ...
2
votes
1answer
30 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 ...
0
votes
1answer
92 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
0answers
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 $A := \{(x, y) : x ≤ y\}$, $A^{−1} := \{(x, y) : (y, x) ∈ A\}$.
0
votes
0answers
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 :
...
-5
votes
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 :
...
-4
votes
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 ...
1
vote
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
$C \cap (-C) = \{ 0 \}$. ...
-4
votes
0answers
42 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
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 ...
0
votes
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 ...
-6
votes
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? ...
2
votes
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 ...
0
votes
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 ...
3
votes
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 ...
-5
votes
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 ...
1
vote
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 ...
1
vote
0answers
44 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
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 ...
1
vote
1answer
143 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
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 ...
0
votes
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 ...
5
votes
0answers
130 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 ...
5
votes
1answer
127 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
0answers
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
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) = ...
2
votes
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 $H \langle g\rangle = G$. ...
-2
votes
0answers
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
0answers
81 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
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 ...
5
votes
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 $\operatorname{Tor}$, ...
9
votes
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) ...
-2
votes
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, ...
-3
votes
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 ...
-1
votes
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 ...
0
votes
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 $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
...
