0
votes
0answers
24 views

Galois group of equation [on hold]

Let the equation $5x^6-12x^5-12x^4+204x^3+81x^2-792x+414=0$ The Galois group of $P(x)=5x^6-12x^5-12x^4+204x^3+81x^2-792x+414$ have solvable to be or not?
-3
votes
0answers
20 views

How to verify a vector has a steady gradual increase mathematically? [on hold]

I rank the values inside my vector then look to verify the values have a gradual but steady increase. Ideally if I had 10 numbers the smallest would occur first and the second smallest would rank ...
2
votes
0answers
13 views

Hyperbolicity of Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$ (with trivial generic stabilizer if necessary). This question is about "hyperbolicity" and it is motivated ...
3
votes
0answers
16 views

Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true? (S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...
0
votes
0answers
23 views

Identification of Hilbert space with dual follows from another identification?

Let $X$ be a Hilbert space with inner product $(\cdot,\cdot)_X$, and let $Y$ be another Hilbert space with inner product $(\cdot,\cdot)_Y$. Suppose there is a bijective continuous linear operator $F:X ...
10
votes
0answers
101 views

Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?

Possibly this has already been asked, but it came up again in this question of Daniel Litt. Does every smooth, projective morphism $f:Y\to \mathbb{C}P^1$ admit a section, i.e., a morphism ...
4
votes
0answers
50 views

Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples: Sets and functions, due to Lawvere. Modules over some ...
0
votes
0answers
76 views

A question on the Euclidean domain $\mathbb{Z}[\omega]$ [on hold]

Let $\omega=\frac{-1+i\sqrt{3}}{2}=e^{\frac{2 \pi i}{3}}$ be a complex cube root of unity, and $\mathbb{Z}[\omega]$ the Euclidean domain. In view of that $\int_0^\infty e^{ix} ...
-2
votes
0answers
30 views

Existence of minimizer in Sobolev space $H^{1,2}(\Omega)$ [on hold]

I am looking at a functional $$\frac{\int_{\partial \Omega} u^2 \mathrm{dx}}{ \left(\int_{\Omega} u^q \mathrm{dx} \right)^{2/q} }$$ And i want to know if the minimizer exists in the space ...
7
votes
1answer
74 views

A linear category with objects of infinite length but which is otherwise finite?

Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...
0
votes
0answers
14 views

Computing row-sum scaled unsigned Stirling nos. of the 1st kind

As the title suggests, I would like to compute probability vectors with elements proportional to (unsigned) Stirling numbers of the first kind by row. For easy reference, here is the Wiki page. For ...
1
vote
0answers
13 views

Semigroup nilpotents and compostional inversion

The integer coefficients of a general partition formula for the compositional inverse of a function are a refined version of the coefficients of the generating series for the number of nilpotents in a ...
0
votes
0answers
57 views

Is holomorphic 2-form on Moduli of Higgs bundle in Biswas' paper is non-degenerate

In Biswas' paper, Geometry of moduli of Higgs bundles, he defined a holomorphic 2-form on moduli of stable Higgs bundles, using Kodaira-Spencer map and Petersson-Weil metric. I want to know whether ...
3
votes
0answers
35 views

Probability that a sum of intependent random variables hits a point

Let $X_1,\ldots,X_n$ be independent random elements of a normed space $X$. Suppose that $\sup_{x\in X}\mathbb{P}(X_i=x)=p_i$. What is the best known upper bound for $$\sup_{x\in X} ...
0
votes
0answers
90 views

Conjecture Reference Request

If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found ...
2
votes
1answer
64 views

a question about minimal non-abelian groups

Let $G$ be a minimal non-abelian group with cyclic Sylow $p$-subgroup $P$ and normal Sylow $q$-subgroup $Q$ , see [ Huppert, Endlich Gruppen I, Aufgaben III, 5.14]. My quesion is, if there is another ...
9
votes
2answers
298 views

Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

This is a cross-posted question, originally active here on math.stackexchange. For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function $$ F_G:S\times ...
0
votes
0answers
60 views

Is the parallelogram rule an axiom or a theorem in euclidean geometry? [on hold]

I am aware of the proof of the rule in inner product spaces. Excluding the geometry of Descartes, is it possible to prove parallelogram rule or is it an axiom?
1
vote
0answers
23 views

lower bound of a trace quadratic form

i want to find a lower bound on the following expression: $tr(AXA^T)$ in terms of $tr(X)$ where A is real $n\times n$ matrix and $X>0$ is positive symmetric. It seems that the following bound is ...
0
votes
1answer
147 views

Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function?

Given a series with integral coefficiens as following: $$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...
4
votes
0answers
55 views

Does $\mathsf{fReR}_0$ prove the existence of the cartesian product of two sets

$\mathsf{fReR}_0$ is the set-theoretical system whose axioms consist of: (1) Axiom of extensionality: $\forall z\in x\ (z\in y)\wedge\forall z\in y\ (z\in x)\rightarrow x=y$ (2) Axiom of empty set: ...
1
vote
0answers
82 views

Equations over $\mathbb{Z}[[T]]$ vs. equations over $\mathbb{Z}_p$

This question might be deemed totally unanswerable, unless there is an obvious counterexample. Answers to either effect would be welcome. Question. Let $X$ be a finite-type scheme over ...
1
vote
0answers
23 views

Cohomological dimension of transcendental p-adic extensions

Let $k = \mathbb{Q}_p$ for any prime $p$ and set $L = k(t_1,..,t_n)$. The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$. It is ...
2
votes
2answers
117 views

A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another metric $d_{f}$ on $Y$ With $$ d_{f}(a,b)=Hd(f^{-1}(x), f^{-1}(y))$$ ...
2
votes
0answers
78 views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...
0
votes
0answers
63 views

Zero-mean assumptions concerning r.d.'s when reading graduate-level probability texts

I'm reading through T. Taos book on random matrices (to be found here), and it is frequent to make without-loss-of-generality certain reductions when proving theorems as for example Hoeffdings ...
0
votes
1answer
53 views

Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed? What is the distribution of euclidean norm of complex Nagakami? Cheers!
0
votes
1answer
32 views

Help in finding distribution of the following function of random variable

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
-5
votes
0answers
46 views

What is induction to n^(n-1) >= n! for n=9,10 [on hold]

Can somebody help me? How do I prove n^(n-1) >= n! for n=9,10..... by induction?
4
votes
0answers
89 views

A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint): Taking $A$ to be a point and iterating this collapsing lemma, this ...
5
votes
1answer
232 views

Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...
4
votes
2answers
92 views

Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist an $m \times n$ matrix $A$ and a vector $x \in \mathbb{R}^m$ such that: The entries of $A$ are $\in \{0, 1\}$. For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are ...
5
votes
1answer
225 views

Dimension of Hilbert scheme?

I have a subvariety $V \subset X$, and I want to compute the dimension of the connected component of $\textrm{Hilb}(X)$ containing $[V]$. I can give explicit deformations of $V$ showing that the ...
4
votes
3answers
147 views

Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?

Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is ...
1
vote
0answers
60 views

Spectral sequence and HOM functor

I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules ...
0
votes
0answers
26 views

Number of graphs with n vertices and k edges up to isomorphy [duplicate]

Is there a way to figure out the number of individual graphs with n vertices and k edges up to isomorphy? I'm particularly looking at graphs with: n = 25, k = 50 n = 50, k = 170 n = 100, k = 700
0
votes
0answers
30 views

Rate of convergence in narrow convergence

Does anyone help me in the following question? I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
0
votes
0answers
10 views

Bounding a function pointwise from bound on the expectation [migrated]

Suppose we have a Lipschitz continuous positive function with bounded expectation Eg < a. What can be said about the point-wise bound ?
2
votes
0answers
124 views

What are the minimal degrees of the real and imaginary part of an algebraic complex number? [on hold]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...
1
vote
0answers
71 views

Finitely co/continuous monad induced by an operad

It is well known that any operad on a nice monoidal category induces a monad. I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...
3
votes
0answers
113 views

What is God's number for the WrapSlide puzzle?

WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
-4
votes
0answers
26 views

First-order nonlinear ordinary differential eqauation [on hold]

can someone help me to solve this equation? I have been trying a few methods. Thanks. y'=(y/x)*((xy + 1)/(xy - 1))
1
vote
1answer
53 views

Wave equation with linear coefficients

The following pde came up in a physics problem: $$ (Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y), $$ A,B,C,D are fixed ...
-2
votes
0answers
33 views

T is not compact operator [migrated]

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...
-3
votes
0answers
34 views

Find the integral [on hold]

How can we find the integral of the 1/(1+x^4) in the interval -infinity to +infinity.I tried to find and got it to be pi/sqrt(2). Am I correct? Please help me with an appropriate method. I tried to ...
3
votes
1answer
104 views

When are all sums of the elements of a set different?

Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that \begin{equation} \sum_{i \in I} x_i \neq \sum_{j \in ...
1
vote
0answers
22 views

Recognizing bridgeless cubic graph with special 2-factor

A 2-factor of graph $G(V, E)$ is a set of vertex-disjoint cycles that cover $V$. It is known that every connected bridgeless cubic graph contains a 2-factor (and a perfect matching). I conjecture ...
6
votes
2answers
206 views

Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...
0
votes
1answer
75 views

What is the difference between the moduli space of curves and the moduli space of orbi-curves?

I feel that I should already know the answer to this, but it never sits quite right in my head. Let's look at something relatively concrete. One way you can look at the moduli of hyperelliptic curves ...
1
vote
1answer
115 views

Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory. I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...

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