3
votes
2answers
211 views

Partitioning a binary tree into vertex-disjoint subtrees

Say we have a labeled, binary unrooted tree $T$, i.e. each node has either 1 or 3 neighbors. Denote by $L(T)$ the set of leaves (degree-one nodes) of $T$. For some $L \subseteq L(T)$, denote by ...
3
votes
1answer
148 views

Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q.2) plus some related, to which I don't know the answers nor reference. ...
3
votes
0answers
18 views

Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
1
vote
0answers
11 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
0
votes
1answer
25 views

If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space. Usually, we have a ...
18
votes
4answers
758 views

What is a Kelley ring?

I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...
5
votes
3answers
456 views

If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark: One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...
2
votes
1answer
68 views

Number of Plücker relations for a Grassmannian

Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: ...
0
votes
1answer
49 views

Variance of the normal CDF

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that $E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$. I'd like to compute ...
2
votes
0answers
86 views

Can we do better than zero padding of FFT?

My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
1
vote
0answers
21 views

Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that ...
5
votes
1answer
85 views

Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length?

Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2_\kappa$. $V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge ...
-5
votes
0answers
9 views

applications of systems of linear equations [on hold]

A person plans to invest a total of ​$260,000 in a money market​ account, a bond​ fund, an international stock​ fund, and a domestic stock fund. She wants 60​% of her investment to be conservative​ ...
6
votes
1answer
297 views

Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
15
votes
1answer
386 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
-1
votes
0answers
27 views

weakly p- summable sequence

Let $ (x_{n}) $ be a weakly $ p$-summable sequence in $ X $ and $ ( x^{\ast}_{n})$ a weakly null sequence in $ X^{\ast} $. Let $ i_{n} : Y_{n}\rightarrow X$ be the natural injection and $ p_{n} : ...
0
votes
0answers
36 views

p-groups with unique normal minimal subgroup

Is p-groups with unique normal minimal subgroup have been Classification? Is there any article on the subject?
1
vote
1answer
32 views

Pragmatic Test for Total Unimodularity

I want perform a simple check for total unimodularity. Question: what, if anything, can be concluded from the fact, that $$det(A)=1,\ a_{ij}\in\{-1,0,+1\}\ \wedge\ a_{ij}^{-1}\in\{-1,0,+1\}$$ ...
5
votes
2answers
247 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
5
votes
2answers
211 views

How to construct a free 2-group on a groupoid?

Let G be a groupoid. I'm wondering how to construct the free 2-group on G. By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$ equipped with a functor ...
0
votes
0answers
40 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ [on hold]

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
7
votes
1answer
181 views

Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$? It is a standard fact that Krull domains, and thus noetherian ...
0
votes
0answers
18 views

Stochastic Pontryagin Principle with a final state constraint

I am searching for information about the Stochastic Pontryagin Principle with a final state constraint. Someone knows a paper or a book where this case is treated in depth?
5
votes
1answer
131 views

The unique positive real root of summation function

update: add one condition according to answer below. I post this question in MSE a week ago. I thought this should be an easy freshman exercise, but it turns out not easy... The original question ...
-4
votes
0answers
40 views

What does spherical harmonics mean? [on hold]

I'm studying physical geodesy and I find this term spherical harmonics used quite much. Basically I know it's coordinate representation based on sphere rather than rectangular. Is that correct?
7
votes
0answers
38 views

Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
0
votes
0answers
37 views

Metric equivalence

Suppose we have two different metrics $g$ and $g'$ describing manifolds with the same dimension and isometry group. How can one determine if there is a such coordinate transformation $g\rightarrow ...
-3
votes
0answers
25 views

Global dimension of matrix algebra [on hold]

Let $k$ be a field, and $A=T_{n}(k)$. $gldim(A) = 1$, and if $B = A/rad(A)^{2}$, then $gldim(B) = n+1$. Some indication!! How can I prove that $gldim(A) = 1$, and $gldim(B) = n+1$ ? Thank you!
6
votes
1answer
167 views

Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question. For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...
5
votes
1answer
162 views

A solution for this equation with a certain condition

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ ...
3
votes
1answer
183 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
1
vote
2answers
55 views

Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
4
votes
2answers
99 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
41
votes
10answers
8k views

Nice proof of the Jordan curve theorem?

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof. What is the simplest known proof today? Is there an intuitive ...
1
vote
0answers
12 views

Upper bound on the number of convex connected components of the complement of the zero set of a polynomial

The classic Milnor-Thom upper bound on the sum of the Betti numbers of real algebraic sets (for a nice exposition and references, see e.g. N.R. Wallach, On a Theorem of Milnor and Thom, in S. Gindikin ...
3
votes
1answer
137 views

finitely presented subgroup and free solvable group of class 3

Let $F(n)$ be free group of rank $n\geq 2$. Denote by $F_d(n)$ the d-th derived subgroup, that is $F_d(n)=[F_{d-1}(n),F_{d-1}(n)]$ where $F_0(n)=F(n)$. The free solvable group of rank $n$ and ...
1
vote
0answers
18 views

Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...
2
votes
1answer
65 views

Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...
1
vote
0answers
37 views

What is the isomorphism type of the ring of invariants of the matrix algebra under the Klein four group?

If the group $G=\mathbb{Z}/2\mathbb{Z}$ acts on $\mathbb{C}[x]$ via $x\mapsto -x$, then we have $\mathbb{C}[x]^G=\mathbb{C}[x^2]$. If $G$ acts on $A=M_2(\mathbb{C}[x])$ via $\begin{bmatrix}a(x) & ...
-1
votes
0answers
59 views

Twisted Hodge numbers in a family

It is well known (e.g. Voisin's book) that for a smooth family $\pi: \mathcal{X} \to B$ of smooth projective varieties (and projectively normal) over $\mathbb{C}$, the Hodge numbers $h^{p,q}(X_b)$ are ...
0
votes
0answers
47 views

Additional condition to the Bollobas theorem in extremal set theory

The Bollobas'1965 theorem is the following: If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then ...
11
votes
0answers
337 views
+200

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
11
votes
0answers
88 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
4
votes
1answer
78 views

Negative population variable importance

I asked this question on stats.stackexchange and even elsewhere, but it never received an answer. I just state the probabilistic problem here. It is about the optimality of the conditional ...
4
votes
1answer
475 views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
1
vote
0answers
83 views

Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...
3
votes
0answers
87 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
0
votes
0answers
6 views
1
vote
0answers
57 views

Smooth points of the secant variety with a given tangent space [on hold]

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...
2
votes
0answers
19 views

Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...

15 30 50 per page