0
votes
0answers
14 views

Number of prime numbers in a range

Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$. Is it true that $A_n < const$? UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number ...
3
votes
0answers
10 views

Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq ...
-1
votes
0answers
29 views

Dividing a rectangle [on hold]

Okay, I need to divide a rectangle into 5 equal smaller rectangles. The overal size of the rectangle is 87" by 100". What is the length the smaller rectangles need to be approximately? Or even if you ...
1
vote
0answers
55 views

Ordinals separate from set theory

Is there an exposition and development of ordinals theory separate from set theory? That is, some first-order theory where terms are interpreted as ordinals, with constant $0$ (and maybe $\omega$), ...
3
votes
1answer
116 views

What are a couple of examples of finite sized but interesting categories?

I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting. The main one I know of is finite preorders -- I ...
0
votes
1answer
29 views

closed range bounded linear operators

Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?
0
votes
0answers
53 views

residue formula for connections on curves

Let $X$ be a smooth, projective curve over a field $k$ (characteristic zero is enough for me) and $E$ a line bundle on $X$. Assume that $E$ is equipped with an integrable logarithmic connection ...
-4
votes
0answers
44 views

sum and infinity [on hold]

If you have the sums $ (1+2+..+n) + (1+2+3+..+n-1)+ (1+2+3+..+n-2)+(1+2+3+..+n-3)+...+(1+2+3)+(1+2)+1$ for large enough $n$ $$\frac {n^3}{3!} \approx (1+2+..+n) + (1+2+3+..+n-1)+ ...
3
votes
1answer
74 views

gamma-factor of a primitive element of the Selberg class

Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor ...
3
votes
1answer
101 views

When are the powers of 2 sum-free mod n?

I've encountered the following question in my research: Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$. Let me call $A$ "sum-free" if there is no solution to $x+y=z$ for $x,y,z \in A$ with distinct ...
1
vote
2answers
113 views

Simultaneous lcms

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, ...
1
vote
1answer
29 views

How to determine an unitary operator involved in an unitary transformation?

Let two real matrices $A$ and $B$ are unitary equivalent. How to determine (computationally or theoritically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of ...
3
votes
2answers
40 views

Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$. Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths ...
1
vote
0answers
50 views

Do more generalizations of Schur's inequality exist?

I meet this following problem If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$ where $a_{i}$ are real numbers. when $n=3$, it is Schur's inequality so which $n$ such this ...
-1
votes
1answer
35 views

Question on real polynomial in projective space [on hold]

Hi all I was given this question and desperately in need of help as it is part of my graduate studiess I know it is true but my instructor told me to find the right way to do it and I am really ...
4
votes
1answer
77 views

Restricted Burnside Problem: Lower bound nilpotency class

Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$. Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$ of exponent $p$ has a maximal finite quotient ...
3
votes
1answer
41 views

$P_3$-factors for 3-regular, 3-connected cubic graphs

Suppose that $G=(V,E)$ is a simple graph. We know if $G$ is 3-regular, 3-connected and $|V|=4k$ for some $k\in \mathbb{N}$, then $G$ has a $P_4$-factor. Question. Let $G=(V,E)$ be 3-regular, ...
0
votes
0answers
85 views

Derivative of a group action [migrated]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
0
votes
0answers
33 views

Koch snowflake construction in many dimensions

I'm looking for some references to deal with the Koch snowflake construction. The construction basically says we can find a sequence $E_k$ such that $\sup_k|E_k|<\infty$ but $P(E_k) \rightarrow ...
-1
votes
0answers
27 views

How to understand the end invariant? [on hold]

In the Ranicki's book: Lower K- and L-theory, Chapter 3 discussing the end invariant, it is defined as the following: The end invariant of a finite G$_{X,Y}$(A)-contractible chain complex C in ...
-3
votes
0answers
16 views

total number of strongly connected 1 [on hold]

Given a square grid of 0's and 1's,i have to find the number of strongly connected 1's. A block of 1's is strongly connected if it is possible to move to any 1 in the block from any other 1 in the ...
1
vote
0answers
50 views

Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that $\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$ (stated, but not proved in "On ...
3
votes
0answers
46 views

What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?

The best reference I found is [Kac, Kazhdan '79] which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras. Theorem 1 of this paper gives the Shapovalov ...
26
votes
3answers
2k views

What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the ...
1
vote
1answer
80 views

About weak derivatives

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
-1
votes
2answers
245 views

What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [on hold]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
-2
votes
0answers
19 views

Determining the inside and outside of planar graphs by means of ray shooting [on hold]

Consider an embedding of a circle in the plane $\mathbb{R}^2$ splitting the plane into an outside and inside region (Jordan-Brouwer). Consider next a point $p$ in the plane. A standard procedure for ...
0
votes
0answers
68 views

Transversal intersection in the moving lemma

Let $X$ be a smooth projective variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of $X$. Chow's moving lemma which is proved in the book by Eisenbud and ...
1
vote
1answer
101 views

Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$

The following question was asked on math.stackexchange.com with no reply for the past week or so. Let $f : X \to Y$ be a morphism of smooth (integral) varieties over $\Bbb{C}$ with generic fiber equal ...
-1
votes
0answers
49 views

Finite dimensional invariant subspaces of $C^\infty(S^2)$ under rotations [on hold]

Characterize all smooth functions on $S^2$ for which the space generated by their rotations are finite dimensional
2
votes
1answer
48 views

Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum: $$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$ when random variables $X_i$ ar i.i.d. Are there any investigation ...
12
votes
2answers
492 views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
0
votes
1answer
87 views

Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group. Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
2
votes
1answer
24 views

Compact $R_1$-spaces

A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$. If $X$ is compact and $R_1$, does ...
0
votes
1answer
31 views

How to generate computational data in graph theory?

For a given number of nodes how many non-isomorphic graphs are available? Might be this is an open problem. For less number of vertices some computational statistics available. I want to get all ...
7
votes
1answer
140 views

E-infinity structure on singular cochains

Is there a transparent explanation of why the singular cochain complex of a topological space X is an $E_\infty$ algebra. There are combinatorial proofs using, say, the surjection operad, but is there ...
0
votes
1answer
117 views

A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...
2
votes
0answers
66 views

The Tensor product of algebra group

Let G is a locally compact group. Is the following true? The tensor product of $L^1(G)$ with $L^1(G)$ is $L^1(G \times G)$.
2
votes
0answers
29 views

A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...
10
votes
2answers
623 views

Is Every Holomorphic Near an Entire?

Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded. Let $U$ be the interior of $K$. Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction ...
1
vote
1answer
62 views

Differences of consecutive ordered fractional parts

Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...
1
vote
0answers
44 views

Can we prove the uniqueness of the local Artin map by using mostly global class field theory?

Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...
11
votes
0answers
185 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
0
votes
0answers
30 views

Explicit computation of a limit of a cosimplicial object

Let $\Delta$ be the simplex category. Let $T_{n}$ be the standard topological $n$ simplex, i.e. it is the set of points of $\mathbb{R}^{n}$ such that $0\leq t_{1}\leq \dots \leq t_{n}\leq 1$. Its ...
1
vote
1answer
105 views

The canonical bundle of an infinitesimal deformation

Let $X_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X_0$ over the ring of dual numbers. If the canonical bundle of $X_0$ is ample (resp. ...
2
votes
1answer
143 views

Convex polyhedron and its Gauß-curvature [on hold]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here. What I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. ...
0
votes
1answer
38 views

criterion for a differential of the third kind to be a logarithmic derivative of a function

Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential of the third kind with ...
2
votes
1answer
109 views

isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
6
votes
2answers
254 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
5
votes
1answer
149 views

Is it possible to evaluate Connect 4 positions with Combinatorial Game Theory?

The surreal numbers in Combinatorial Game Theory only work for certain classes of games (e.g. they must satisfy normal play convention). This rules out even reasonable games with fairly ...

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