5
votes
0answers
31 views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
2
votes
0answers
13 views

References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
5
votes
0answers
49 views

What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?

Given $x\in\mathbb{R}^n$, $x_i$ denotes its $i$-th coordinate. My question is: What is $Vol(\{x\in[0,1]^n|\sum_{i=1}^n x_i\le t\})$ for $t\in\mathbb{R}$ ? Is there some kind of "easily computable" ...
0
votes
0answers
14 views

Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
1
vote
0answers
27 views

Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
5
votes
0answers
102 views

Do we know that 'most' finite groups are Galois groups of number fields?

The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...
3
votes
1answer
56 views

Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$

Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this ...
4
votes
0answers
33 views

Elementary embeddings with the same critical point

Question: Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j_n \circ k_n$ for ...
-1
votes
0answers
29 views

A general Formula for calculating Bezout's identity? [on hold]

I know how to calculate Bezout's identity by using the extended Euclidian algorithm (running the regular algorithm "backwards" where each step is calculated before I proceed to the next one, like so ...
3
votes
2answers
108 views

another question about connected open sets in $R^2$

Before posting this question,I just asked a similar question:a question about connected open sets in $R^2$. I got several nice answers.Now I want to ask: Let $U$ be a nonempty connected open set in ...
4
votes
1answer
53 views

Site dependance of the Cech weak equivalences on simplicial sheaves

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site. One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start ...
0
votes
0answers
121 views

Areas of Mathematics [on hold]

Can anyone please provide me with the main areas of math and how each area branches out with its categories, subsets, and tasks as in a knowledge graph or provide me with some useful contacts? I know ...
0
votes
1answer
74 views

Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...
2
votes
1answer
51 views

A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection

$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...
0
votes
0answers
12 views

Intuition behind shrinking and subsampling in gradient boosted regression/classification?

I'm using gradient boosted decision trees from here: http://scikit-learn.org/stable/modules/generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier In ...
10
votes
2answers
183 views

a question about connected open sets in $R^2$

Let $U,V$ be two nonempty connected open sets in $\mathbb{R}^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty ...
2
votes
0answers
58 views

On the local root number(or local $\epsilon$-factor)

I want to ask some question related to the local root number. Let $E/F$ be a quadratic extension of p-adic local fields and $\psi:E \to \mathbb{C}$ is an additive character of $E$. Let $\phi:WD(E) ...
-2
votes
1answer
90 views

finitely generated subgroups of SO(3) [on hold]

Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each ...
2
votes
0answers
36 views

Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...
1
vote
0answers
27 views

Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group. I am ...
1
vote
1answer
56 views

n-cube connectivity problem

Given $n\geq 2$, let us consider the n-cube $H_n=(V,E)$, i.e. vertex set $V$ is $\{0,1\}^n$. Here, the edges of $H_n$ are directed, oriented by set inclusion, i.e., $(x,y)\in E$ iff $x\subseteq y$ and ...
1
vote
0answers
133 views

On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems". Proposition 1.4: When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
3
votes
2answers
247 views

Twin primes for polynomials in $\Bbb Z[X]$

The following paper provides result on an analog of twin primes conjecture for $\Bbb F_q[X]$ http://www.math.uga.edu/~pollack/twins.pdf Is there an analog of twin primes conjecture for $\Bbb Z[X]$? ...
3
votes
1answer
145 views

Localizations or quotients of categories?

Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category ...
0
votes
0answers
69 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
1
vote
1answer
99 views

Does the associated Lie algebra determine a group?

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
6
votes
2answers
88 views

Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs. Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
-4
votes
0answers
37 views

Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$ [migrated]

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...
-4
votes
0answers
28 views

Distributions properties [on hold]

Let $\varphi\in\mathcal{D}(\mathbb{R})$ the set of functions $\mathcal{C}^\infty$ with compact support, $\delta_n$ is the Dirac in $n$ and the functionals : $$ T = \sum_{n=0}^{+\infty} e^n ...
2
votes
0answers
98 views

Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
4
votes
1answer
266 views

Generalization of a property of $A_n; n\geq 5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_n; n\geq 5$. Then there exists a maximal subgroup $M$ of $A_n$ such that $H\not\leq M$ and $K\not\leq M$. To see this ...
-1
votes
0answers
36 views

Random Spanning Tree Edge Probability [on hold]

I am working on a problem with a Loop Erased Random Walk used to create random spanning trees. The problem has many parts, but there are two hints to help with the more complicated problems Figure ...
2
votes
0answers
64 views

Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties: For every two points in the plane there exists a unique geodesic joining them. Every geodesic determines exactly two points on the ...
4
votes
1answer
135 views

Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a simple (non-self-intersecting) closed curve of length $L$ on the unit-radius sphere $S$. So if $L=2\pi$, $\gamma$ could be a great circle. I am seeking the most equitably ...
-1
votes
0answers
57 views

Are there any special properties of graph eigenvalues of perfect matchings?

Say if the graph is just a perfect matching of its vertices OR if it is an union of a few perfect matchings OR its an union of a few perfect matchings and a part of another? Anything if one further ...
4
votes
0answers
83 views

Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
1
vote
0answers
46 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ ...
-4
votes
0answers
27 views

Irreflexivity of relations on sets [on hold]

How can I know if the relations: xy >= 1 and x=y+1 or x=y-1 Are irreflexive on Z(all integers)? Thank you!
4
votes
2answers
176 views

Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether ( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...
13
votes
0answers
124 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to \mathbb{R}$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
3
votes
1answer
216 views

Does this function have any exponential growth?

Has anyone seen any function of the following type? $$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$ The question is whether for some constant ...
-2
votes
0answers
75 views

What is the characterization of a graph Laplacian? [on hold]

Given a matrix, what properties must it have so that its ensured that there exists a graph whose Laplacian it would be? (...may be you can consider weighted and unweighted cases separately...) And ...
0
votes
0answers
16 views

the ratio between product of two trace functions maximization

Consider the following Optimization [\begin{array}{l} \mathop {\max }\limits_{\bf{X}} \,\frac{{trace\left( {{\bf{XA}}} \right)trace\left( {{\bf{XB}}} \right)}}{{trace\left( {{\bf{XC}}} ...
0
votes
0answers
71 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ ...
0
votes
0answers
34 views

Random Matrix Theory: question on a quantity on page 87 in book of Bai and Silverstein 2010

Anyone else is reading carefully the book of Bai and Silverstein 2010, titled "Spectral Analysis of Large Dimensional Random Matrices"? On page 87 of this book, when they state the final step in the ...
4
votes
2answers
430 views

Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence. I'm a bit ...
0
votes
1answer
62 views

A decomposition of incompressible vector fields

In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved: Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$. Call $w$ its ...
2
votes
1answer
128 views

A question about generalized Dyck words

I am interested in counting the following. How many words using $n-1$ copies of $u$ and ${n \choose 2} - n+1$ copies of $d$ begin with $uu$ and, in general, the $k^{th}$ $u$ is among the first ${k ...
2
votes
1answer
128 views

Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...
0
votes
0answers
116 views

Self homotopy equivalence

Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ...

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