# All Questions

**0**

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3 views

### A quantitative version of Pelczynski's property ($V^{*}$)

Let me first fix some notations. Let $A$ be a bounded subset of a Banach space $X$. Set $$wk_{X}(A)=\widehat{d}(\overline{A}^{w^{*}},X)=\sup\{d(x^{**},X):x^{**}\in \overline{A}^{w^{*}}\},$$ where ...

**-1**

votes

**0**answers

8 views

### finite Projective plane

Let Z=PG(q,2) be a finite Projective plane over a finite field Fq, q a prime power. Show that there exists a commutative binary operation * in Z such
that
(i) x*y is neither x nor y for any x and y, ...

**1**

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**0**answers

6 views

### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Ito) "Stochastic calculus" defined on $L^1$ space, or some Olicz space between $L^2\, and\, L^1$

**1**

vote

**1**answer

28 views

### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...

**2**

votes

**1**answer

28 views

### Area of an irregular, n-sided, non-intersecting (edges) polygon algorithm

I need to generate an irregular, n-sided polygon of non-intersecting edges (n= 200, for example) with the smallest area possible. The position of the vertex is random and I've tried designing a couple ...

**-5**

votes

**0**answers

35 views

### Advice question? [on hold]

How to get to do paid mathematics reserach in graph theory pure mathematics in private by myself by getting some fund to help me support my family any advise?
At present I am doing my post doctoral ...

**-1**

votes

**1**answer

27 views

### Computing the inverse of a Cholesky decomposition

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...

**-9**

votes

**0**answers

65 views

### Please help me for answers to question. best regards [on hold]

Find the definition of a locale and its dual (i.e. a frame) and consider the
definition of a Grothendieck topology. Discuss the differences between this
concept and an ordinary topology on a set ...

**-4**

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**0**answers

51 views

### look for a right technique to solve logarithmic functional equations

I would like to solve this equation but can not find a standard technique
f(f(x)) = log(x)

**1**

vote

**1**answer

50 views

### Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to ...

**1**

vote

**2**answers

79 views

### Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s

I want to prove that in a sequence W of length n, consisting of 1s and 0s, $P$( in $W$ there is at most $\frac{\log_2n}2$ consecutive zeroes ) $\leq \frac{K}{n} $ for some constant K. Can anyone ...

**-4**

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**0**answers

142 views

### What area of maths have I reinvented? [on hold]

I was trying to solve the issue of calculating (total) utilities with infinite numbers of people. This is problematic, because many infinite series are divergent, so there does not appear to be a way ...

**3**

votes

**1**answer

34 views

### Converging to moments obeying Carleman's condition

I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that $\mu_N$ are a sequence of measures on $\mathbb{R}$. Let $m_{N,k}$ be the $k$-th ...

**8**

votes

**2**answers

178 views

### Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.

**1**

vote

**0**answers

43 views

### Counting growing tree trajectories

I am looking for help:
Beginning with a single node ($\circ$), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is ...

**-2**

votes

**0**answers

38 views

### concentric spheres with common radius [on hold]

I am trying to think of a way to solve this problem but I am not sure if it is doable or not. Here goes:
Assume we have n spheres that share a common radius (x0,y0,z0).
For each sphere we have one ...

**6**

votes

**2**answers

193 views

### Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space:
Therefore the quotient manifold
$$
\mathbb{HP}^{2}/\mathrm{U}(1)
$$
may be taken, writing $U(1)$ for the circle group. It has ...

**-1**

votes

**0**answers

16 views

### Probability of an event based on percentage in fixed lapse of time [on hold]

I am a software engineer. I am also a former triathlete that rides with a large group of friends every time we have a chance.
i am trying to come up with a little software to distribute among us ...

**-2**

votes

**0**answers

19 views

### Find the number of connected components in pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**-1**

votes

**0**answers

106 views

### The eigenvalue of operator $-\Delta$

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We know the eigenfunction of Laplacian operator $-\Delta$ is an orthonormal basis of $L^2$. Let $\{\omega_n\}$ denote the ...

**4**

votes

**2**answers

93 views

### Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = ...

**0**

votes

**0**answers

36 views

### 3-dimensional vectors satisfying certain equalities [on hold]

Question: Are there 5 distinct vectors $u,v,w,x,y \in \mathbb{R}^3$, all on the unit sphere (i.e. $||u||=||v||=||w||=||x||=||y||=1$), such that:
$||u+v+x||=||u+v+y||=||u+w+x||=||u+w+y||=1$
?
Also, ...

**-2**

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**0**answers

10 views

### Best algorithm to compute the first eigenvector of symmetric matrix [migrated]

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following :
$$\mathbf{A}=\mathbf{N}-\mathbf{P},$$
with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...

**3**

votes

**0**answers

50 views

### When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is a smooth algebraic variety with free divisor $D$ so that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a ...

**1**

vote

**2**answers

98 views

### Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?

**-3**

votes

**0**answers

49 views

### Does this numerical series have any special name? [on hold]

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers.
Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...

**2**

votes

**1**answer

88 views

### First proof of the integral representation of the hypergeometric function $F(a,b,c;\cdot)$

Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true
$$F(a,b,c;x)=\sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n} x^n=\frac{\Gamma ( c ...

**1**

vote

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54 views

### 2nd order partial differential equation with non-constant coefficients

During my research I came across the following differential equation:
$$f(x,y) = \left(y^2 \partial_{x}^{2} + x^2 \partial_{y}^{2} \right)f(x,y)$$
Any ideas how to solve it without using series ...

**5**

votes

**0**answers

68 views

### Sets of matrices which are irreducible but not strongly irreducible

A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger ...

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127 views

### One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following :
Let $K$ be an algebraically closed field. ...

**-1**

votes

**0**answers

20 views

### 2D convolution property [on hold]

If I have three square matrices a,b, and c of equal size. say each of them are 3x3 matrices. then practically it is possible that
d = (a.b) * c .....(1)
= a * (b.c) .....(2)
that is 2D convolution ...

**0**

votes

**0**answers

48 views

### Extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is
$v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...

**6**

votes

**1**answer

178 views

### Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$

A145722 is
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
Using the pari program and offset 0, up to ...

**1**

vote

**1**answer

122 views

### Does every ultrafilter contain sets of sup-measure $0$?

Let $\mathbb{N}$ be the set of positive integers and for $A\subseteq {\mathbb{N}}$ set $$m(A) = \text{lim sup}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Does every ultrafilter ${\cal U}$ on ...

**0**

votes

**0**answers

23 views

### Isomorphisms of well ordered sets [migrated]

Let $A$, $B$ be well ordered sets. If there exist order-preserving functions $f : A \to B$ and $g : B \to A$ (do not need to preserve initial segments), are $A$ and $B$ isomorphic? I know this is not ...

**2**

votes

**0**answers

110 views

### Questions about configuration spaces on the sphere and higher loop spaces

In the paper Configuration spaces on the sphere and higher loop spaces, Paolo Salvatore, Mathematische Zeitschrift
November 2004, Volume 248, Issue 3, pp 527-540, I have some questions about ...

**2**

votes

**1**answer

85 views

### Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...

**1**

vote

**0**answers

29 views

### On modulus of powers

Given $n\in\Bbb N$ is there an $a_n\in\Bbb N$ such that for every $a>a_n$ there are $b,c\in\Bbb N$ such that $b^i\bmod a,c^i\bmod a\in(\sqrt a,\sqrt a\log a)$ for every $i\in\{1,\dots,n\}$? If so ...

**0**

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**0**answers

9 views

### What is the 4D axis of rotation for Necker cube inversion? [migrated]

See the figure on top of page 47 of Rudy Rucker's book. ...

**-1**

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**0**answers

30 views

### Research on unique 2d geometric structures - terminology and resources [on hold]

First of all, please note that I am not a professional mathematician, but this topic probably touches some non-obvious areas, so I hope to find assistance here. Also note that it is very hard to ...

**6**

votes

**0**answers

79 views

### The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme ...

**0**

votes

**1**answer

43 views

### Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ ...

**7**

votes

**0**answers

77 views

### Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...

**5**

votes

**0**answers

83 views

### Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...

**6**

votes

**3**answers

172 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...

**2**

votes

**1**answer

120 views

### Number of Plücker relations for a Grassmannian

Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: ...

**-5**

votes

**0**answers

11 views

### applications of systems of linear equations [on hold]

A person plans to invest a total of $260,000 in a money market account, a bond fund, an international stock fund, and a domestic stock fund. She wants 60% of her investment to be conservative ...

**3**

votes

**0**answers

51 views

### Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$.
Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that ...

**1**

vote

**1**answer

69 views

### If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space.
Usually, we have a ...

**1**

vote

**0**answers

60 views

### p-groups with unique normal minimal subgroup

Is p-groups with unique normal minimal subgroup have been Classification? Is there any article on the subject?