7
votes
0answers
143 views

What is the smallest density of a metrizable space without countable separation?

A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...
2
votes
1answer
87 views

If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...
0
votes
0answers
27 views

Basic Definition and Notations in RWRE [on hold]

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...
0
votes
0answers
112 views

Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
-1
votes
0answers
21 views

Closed form formula for fill rate given a discrete distribution? [on hold]

I'm wondering whether there is a closed form way to obtain good estimates for fill rate given a discrete distribution of demand. I created a simple monte carlo simulation to see if I could see any ...
-1
votes
0answers
62 views

A rash guess about distribution of primes based on meager empirical evidence?

Elementary number theory is a field in which imbeciles can ask questions that experts cannot answer (and I wonder if discrete geometry is a similar subject in that respect?) and herewith I submit ...
2
votes
1answer
146 views

Resolution of the ideal of the Abel-Jacobi image of a curve?

Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the ...
-3
votes
0answers
26 views

function mapping odd numbers to counting numbers [on hold]

Mapping even numbers to counting numbers is straight forward. Without introducing any other variable: i = 0, 2, 4, 6, . . . if i > 0: count = i/2 what about ...
-4
votes
0answers
26 views

similarity metric for geometries [on hold]

I'm searching for a method to calculate the degree of similarity of two given geometries. These geometries can be of any type and can have an arbitrary shape. For the sake of simplicity, I primarily ...
0
votes
0answers
161 views

Can some exotic sphere be diffeomorphically embedded into some $R^n$?

Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding? ...
-5
votes
0answers
64 views

A property of minimal prime ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $p\subseteq I_1 $ or $...
0
votes
1answer
73 views

Exact formula for computing n-step transition probability of random walks with self-transitions

Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...
5
votes
1answer
166 views

Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality. Does there exist a bijection $f : B_V \to B_W$ such that, for each $b_V \in B_V$,...
2
votes
1answer
161 views

Cambridge Mathematical Tripos papers from late 19th century

Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?
4
votes
2answers
96 views

Variation of Radon transform for probability measures on $\mathbb C$

Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...
1
vote
1answer
145 views

Could I affirm that $f$ is not identically 0?

Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric: $ d(x,y)=\sum_{i\geq 1}\frac{|...
3
votes
0answers
49 views

When is a functorial coverage a sheaf, and what universal property does it have?

In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is a function assigning to each object $A$ of $\...
2
votes
0answers
77 views

Measures on a unit sphere of a Hilbert space

Consider a real separable infinite-dimensional Hilbert space $H$. Let $S=\{h\in H\mid \|h\|=1\}$ be a unit sphere in $H$. What are the most natural measures on $S$? Is there a (Borel) measure $\mu$ on ...
1
vote
0answers
66 views

Reference quest: variety of lines and variety of planes

Let $X\subset \mathbb P_{\mathbb C}^n$ be a smooth projective variety, $F(X)\subset G(2,n+1)$ its Fano variety of lines and $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ ...
1
vote
0answers
70 views

Ellipticity of Bott-Chern Laplacian

I want to prove that Bott-Chern Laplacian $$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...
0
votes
0answers
84 views

graduate study in graph theory and combinatorics in canada [on hold]

I'm looking for any graduate programs related to graph theory or combinatorics in canada like in waterloo or simon fraser universities. any other suggestions?
6
votes
0answers
160 views

getting papers published when you're not affiliated to a university [on hold]

I graduated in Maths 20 years ago, spent a long time away from the subject and recently returned to it. I work entirely alone right now but after a refresher phase, I'm starting to look at some very ...
2
votes
2answers
751 views

Unreasonable application of mathematics to the other areas [on hold]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science? I found ...
0
votes
1answer
69 views

Exterior derivative on principal bundle [on hold]

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
-5
votes
0answers
32 views

Cardinality of infinite sets' subtraction [on hold]

$A = \mathbb{N}$; $B = \{\frac{p}{q} : p, q \in \mathbb{Z}\setminus\{0\}\}$. Find the cardinality of $P(A\setminus B)$. We know that the cardinality of $\mathbb{N}$ is $\aleph_0$, so as the ...
4
votes
0answers
204 views

Description of connecting maps of Derived functors

Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
2
votes
1answer
90 views

Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose there is additional axiom, or constraint if you prefer, ...
0
votes
0answers
47 views

is it possible to solve this nonlinear differential equation analytically? [on hold]

I have a hard time trying to solve a first order nonlinear equation, I posted it in math.stackexchange but I did not get any answer. I have the following first order differential equation: $$y^2\...
0
votes
0answers
27 views

optimize a Quadratic Matrix Programming with multi-spherical constraints

I have got the following quadratic problem restricted on the Cartesian product of Euclidean spheres. $\underset{X \in \mathbb{R}^{n\times 3}}{\text{min}}$ $Q(X) = \frac{1}{2} Tr(X^TA X) + Tr(B^T X)$ ...
9
votes
0answers
72 views

On spatial tensor products of von Neumann algebras

Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras. We write $\odot$ for the algebraic tensor product, and $\bar\otimes$ for the spatial tensor ...
1
vote
0answers
72 views

Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ...
0
votes
0answers
29 views

$nD$ rotation around a general $(n-2)$-dimensional subspace [on hold]

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...
9
votes
0answers
337 views

Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...
-4
votes
0answers
30 views

A number divided by 2 n times and sum each division [on hold]

Is there a function that can complete n number of divisions by 2 and sum? If your base is 400 for example and n=5: answer ...
3
votes
0answers
48 views

commutativity of a diagram in cohomology of $C^*$-algebras

The setting is the same as in my last question commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras) : Let $A$ be in the bootstrap category (=N in the other ...
1
vote
1answer
83 views

Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
2
votes
1answer
82 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
0
votes
0answers
67 views

some strange sums ramanujan type

i found sum as $$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...
-1
votes
0answers
26 views

product distinct prime factors of prime(n)-1 and prime(n)+1 [on hold]

The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are such primes common? Can a value of ...
1
vote
0answers
70 views

The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
-1
votes
0answers
65 views

Finding the unique Nash equilibrium [on hold]

$$ m^2(1-m)[(1+m)^2 - R] x^3 + [6m^2R + 12mR - 2m(1-m^2)(6+2m) - 4tm^2(1+m)^2] x^2 + [(1-m)(6+2m)^2 + 8tm(1+m)(6+2m)] x - 4t(6+2m)^2=0 $$ where: m ∈ (0, 0.5), R ∈ [0, 0.25], t ∈ [0, 1], x ∈ [0, 2]...
2
votes
0answers
58 views

On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...
-1
votes
0answers
13 views

Markov Chain: Number of communicating classes of a power of the irreducible transition matrix [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P_k$. In terms of $d$ and $k$, how many communicating classes does $P_k$ have, and what is the period ...
6
votes
0answers
120 views

Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...

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