All Questions

18
votes
2answers
945 views

Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next. It looks to me, that complex-analytic geometry has lost its relative positions ...
-3
votes
0answers
61 views

A continuous inverse function problem: search for an inconsistent theory approach to return a 'false' metric [on hold]

My problem is about metrizable space definition: I need to find a topology that can be described by a metric without this being, in fact, a 'true metric'. I search an alternative approach to avoid to ...
0
votes
0answers
45 views

Number of compositions of a positive number n, with factors between 1 and a certain number m

I'm trying to find the number of compositions of a positive number n, with factors between 1 and a certain number m. That is, all the combinations of limited numbers that add up to n $f(n, m)$ For ...
1
vote
1answer
59 views

connected and quasi-connected separators of a space

Does there exist a connected topological space $X$ and a subset $A\subseteq X$ such that no connected component of $A$ separates $X$, but some quasi-component of $A$ separates $X$? Meaning $X\...
2
votes
0answers
141 views

Does there exist $a_0$, such that $\{a_n\}_{n=0}^{\infty}$ is unbounded?

Suppose $\{a_n\}_{n=0}^{\infty}$ is a sequence, defined by the recurrence relation $$ a_{n+1} = \phi(a_n) + \sigma(a_n) - a_n, $$ where $\sigma$ denotes the divisor sum function and $\phi$ is Euler'...
3
votes
1answer
84 views

Conductor of quaternionic representation

Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{...
2
votes
0answers
62 views

Number of lattice points on spheres with center not at the origin

Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\...
4
votes
1answer
69 views

On the existence of a domination map of a finite polyhedron

A continuous map $d:X\to A$ is called domination if there exists a map $u:A\to X$ so that $d\circ u\simeq 1_A$. Is there a domination map $d:P\to P$ of a finite polyhedron $P$ so that $d$ is not a ...
6
votes
1answer
111 views

Continuous binary operations on $\beta\mathbb{N}$

It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone ...
2
votes
0answers
77 views

How to calculate tautological classes of some varieties?

I want to calculate tautological classes of some subvarieties sitting inside some bigger variety in the cohomology ring of that bigger variety. For example, suppose we have a chain of closed ...
1
vote
0answers
48 views

Associated subgroup of Weyl group

Let $\Phi$ be a root system. For a weight $\lambda\in\mathfrak{h}^*$, start by defining $ \Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \} $ and $ W_{[\...
1
vote
1answer
97 views

In search of a new isometric twisting invariant $ T= \tau_1.\tau_2 $

A curved line in $\mathbb R^3 $ has properties of curvature and torsion, and, on an $ \mathbb R^2 $ possesses surface scalar properties of normal curvature and geodesic torsion $ (\kappa_n, \tau_g).$ ...
-1
votes
1answer
51 views

analyze existence limit of sin(sqrt(x)) in infintiy [on hold]

i'm studying to test and i found a problem from last years: analyze the existence of limit $$sin(\sqrt x)$$ in infinity. Got some ideas about argument tending to infinity, but its especially hard for ...
4
votes
2answers
174 views

Ideals invariant under ring automorphisms

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties: $I$ is generated by two homogeneous elements; $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
1
vote
1answer
143 views

Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$

Let $\zeta$ be the zeta function of Riemann. Is the bound for $$I_{T}=\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$$ known ? It seems to me that $I_{T} \ll T\log T$ since $\log|\zeta(...
1
vote
0answers
27 views

On certain 2 dimensional foliation of $Gl(2,\mathbb{R})$ deleted by scalar matrices

We consider the following 4 dimensional open manifold $$M=Gl(2,\mathbb{R})\setminus \{\lambda I_2 \mid \lambda \in \mathbb{R}\}$$ where $I_2$ is the identity matrix. We consider the $2$ dimensional ...
3
votes
1answer
111 views

On a special type of subring of $\mathbb C[x_0,…,x_{q-1}]$

Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let $$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
-6
votes
0answers
59 views
0
votes
0answers
34 views

Concavity of a function after binomial transform and maximization

Suppose $f(x)$ is a one-dimensional discrete concave function, and $X(n,p)$ is Binomial random variable. Is the function $g(x)= \max_n \mathbb{E} f(x+X(n,p))$ also discrete concave?
-4
votes
0answers
63 views

Prove this mapping is bijective [on hold]

Prove that a map which defined by $P:\mathbb{N}\times \mathbb{N} \to \mathbb{N}$ where $$P(n,m) = \binom{n+m+1}{2} + m = \frac{(n+m+1)!}{2!(n+m-1)!} +m $$ is bijective.
4
votes
2answers
59 views

Complexity of Random Delaunay Triangulation in 3D

My question: Is the number of cells in a three-dimensional Poisson-Delaunay triangulation with $n$ vertices $\mathcal O(n)$ with probability one? which is equivalent to the question Is the ...
0
votes
0answers
161 views

On an exercise in section 4 of Chapter I from Hartshorne's Algebraic Geometry

It is about exercise 4.9: Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$ with $n\geq r+2$. Show that for suitable choice of $P \notin X$ and a linear $\mathbb{P}^{n-1}\subseteq \...
3
votes
2answers
107 views

Is a plane set still metrizable if two new subsets are declared open?

I am thinking of forming a finer topology on a particular subset of the plane. Let $X\subseteq \mathbb R ^2$ be endowed with the Euclidean topology $\tau$. Let $A,B\subseteq X$. Let $\tau'$ be the ...
7
votes
0answers
163 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
-1
votes
0answers
18 views

Measurement theory and sample size calculation for multivariate testing

Let $\mathbf{Y}$ be a vector of independent, normally-distributed random variables. Let $S_1$, $S_2$ and $S_3$ be three non-overlapping samples of sizes $N_1$, $N_2$ and $N_3$, respectively. Let $M_A$ ...
3
votes
0answers
48 views

What is the expected minimum total matching distance between two partitions of identically and independently distributed points?

Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, ...
6
votes
1answer
99 views

Terminology for expressing a graph as a sum of cliques (mod 2)

I am interested in the problem of expressing the edges of a given (undirected, simple) graph as the sum of edge sets of cliques modulo $2$. To be more concrete, given a graph $G=(V,E)$, I am seeking ...
1
vote
1answer
48 views

Literature on the controllability of networks under attack

I would like to request your advice on a problem arising from my research in the life sciences. Consider a modular, sparse weighted network which is partially controllable in the sense that some ...
2
votes
0answers
51 views

Are double cosets of cyclic subgroups separable in a special linear group?

Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set $$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$ Is $S$ closed in the profinite topology on $\mathrm{SL}_3(\mathbb{...
10
votes
0answers
191 views

Triangulation of the complex projective plane

In the 1983 paper ``The 9-vertex Projective Plane'' by W. Kuehnel and T.F. Banchoff (The Mathematical Intelligencer Vol 5.) the authors give a 9 vertex triangulation of the complex projective plane, ...
3
votes
1answer
60 views

Incompressible Navier-Stokes equation with heat conduction

How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
-4
votes
0answers
40 views

Proving a binomial series equal to a binomial expression [on hold]

I tried to prove $$\sum\limits_{r=0}^n\binom{n}{r}^2=\binom{2n}{n}$$ I used induction method: Assuming $$\sum\limits_{r=0}^n \binom{n}{r}^2 =\binom{2n}{n}.$$ Proving $$\sum\limits_{r=0}^{n+1} \...
13
votes
2answers
428 views

Is a matrix similar to its transpose over $\mathbb{Z}_p$?

Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$? On the one hand, I know the analogous fact is false ...
-3
votes
0answers
111 views

My question is about an article concerning p and t [on hold]

you say concerning p and t: An example of an element of p is the family of sets (indexed by k∈ℕ) defined by {m to the power of k :m∈ℕ}. But the second condition is not met because the set {2 to the ...
1
vote
0answers
55 views

Relate solutions to a polynomial system in complex numbers to solutions in a finite field

Suppose I have a system of polynomials which are homogeneous but of distinct degrees that I want to solve simultaneously: $$F_1(z_1,\ldots,z_n)=\cdots=F_m(z_1,\ldots,z_n)=0.$$ Let $X(\mathbb F)$ ...
5
votes
0answers
101 views

Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...
1
vote
1answer
85 views

Growth rate of bounded Lipschitz functions on compact finite-dimensional space

Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...
11
votes
1answer
130 views

Products of Cyclotomic Polynomials with Nonnegative Coefficients

I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients. Some ...
21
votes
1answer
284 views

Is there a short proof of the decidability of Kepler's Conjecture?

I've believed that the answer is "yes" for years, as suggested in various sources with reference to Tóth's work. For example, the Wikipedia article for Kepler Conjecture says: The next step toward ...
1
vote
0answers
28 views

Potential for a Monotone Operator

[Cross-posted from math.stackexchange] I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
4
votes
1answer
59 views

Reference for Shalika germs of GL(n)

I was reading the two Repka papers where he computes the leading and subleading Shalika germs for $GL_n$ and I was wondering, where are we since then? Have these germs (and the integrals) been ...
2
votes
0answers
140 views

Is the intersection of two function fields over finite fields again a function field?

I am interested in the question above. I know that the answer is NO if the base field is for instance $\mathbb{Q}$ (the intersection of $\mathbb{Q}(x^2)$ and $\mathbb{Q}((x-1)^2)$ is $\mathbb{Q}$ ...
2
votes
1answer
110 views

About independence spread

$A$, $B_{i}$ are some events. If $A$, $B_{i}$ are independent $\forall i \in \mathbb N$ and $A \cap B_{1}, A \cap B_{2}, ..., A \cap B_{k}, ...$ are independent in aggregate, how to show, that $\...
2
votes
0answers
38 views

On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
2
votes
0answers
53 views

Simple group with central extension by Prüfer group and an automorphism of infinite order

Do you know of any perfect group $G$ such that 1) $Z(G)$ is a Prüfer $p$-group for some prime $p$ 2) $G/Z(G)$ is infinite simple (or a direct product of infinitely many finite simple groups) 3) ...
-4
votes
0answers
246 views

Would this be a significant/publishable result on the Riemann zeta function? [on hold]

I recently proved that the Riemann zeta function $\zeta(s)$ has only finitely many zeros on any line $\Re(s)>1/2$. I've shown my work to a couple of professors (non-number theorists) and they both ...
1
vote
1answer
44 views

Randomly scaled random variables

Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX]\leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$? ...
0
votes
0answers
18 views

How to prove de Vries algebras morphisms are dense and full if their duals are into?

Well, this is a quite short question but I think it will require some explainations. Let's say that a de Vries (or compingent) algebra is a Boolean algebra $B=(B,0,1, \wedge, \vee, \neg) $ with a ...
1
vote
0answers
372 views

A new generalization of the dimension?

During my research, I came a cross on these notions : Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
1
vote
0answers
17 views

Existence of Costas array with specified displacment vectors?

Costas array is a set of $n$ points lying on the square of a $n×n$ checkerboard, such that each row or column contains only one point, and that all of the $n(n − 1)/2$ displacement vectors between ...

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