A generalization of Tverberg's Theorem

The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (1 < k < r+1) what is the minimal number of points we need in order to guarantee that there exists an r partition of them such that any k of the r convex hulls intersect. This minimal number is denoted by T(d,r,k). Reay conjectured that T(d,r,k) = T(d,r) for all d,r and k. In this article we prove that this is true for the following cases: when k > [ (d+3)/2 ]-1 or when d < rk/(r-k)-1 and for the specific values d = 3; r = 4; k = 2 and d = 5; r = 3; k = 2

On the first row map GLd+1(R)⟶Umd+1(R)Ed+1(R)GL_{d+1}(R)\longrightarrow \frac{Um_{d+1}(R)}{E_{d+1}(R)}

In this paper, we prove that there is group homomorphism from general linear group over a polynomial extension of a local ring to the W. van der Kallen group

Regularity and continuity of the multilinear strong maximal operators

Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}^d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}^d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}_{\mathcal{R}}$ by $\mathcal {M}_{\mathcal{R}}$.Then, $\mathcal {M}_{\mathcal{R}}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W^{1,p_1}(\mathbb{R}^d)\times \cdots\times W^{1,p_m}(\mathbb{R}^d)$ to $W^{1,p} (\mathbb{R}^d)$, from the Besov spaces $B_{s}^{p_1,q} (\mathbb{R}^d)\times\cdots\times B_s^{p_m,q}(\mathbb{R}^d)$ to $B_s^{p,q}(\mathbb{R}^d)$, from the Triebel-Lizorkin spaces $F_{s}^{p_1,q}(\mathbb{R}^d)\times\cdots\times F_s^{p_m,q}(\mathbb{R}^d)$ to $F_s^{p,q}(\mathbb{R}^d)$. As a consequence, we further showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p_1}(\mathbb{R}^d)\times \cdots\times W^{s,p_m}(\mathbb{R}^d)$ to $W^{s,p}(\mathbb{R}^d)$ for $0<s\leq 1$ and $1<p<\infty$. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the $p$-quasicontinuity of $\mathscr{M}_{\mathcal{R}}$. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.Comment: 49 page

A stratification of the moduli space of vector bundles on curves

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define {\se}_k(E):=k\deg E-r\max\deg F. where the maximum is taken over all subbundles $F$ of rank $k$ of $E$. The ${s}_k$ gives a stratification of the moduli space ${\cal M}(r,d)$ of stable vector bundles of rank $r$ and degree on $d$ on $C$ into locally closed subsets {\calM}(r,d,k,s) according to the value of $s$ and $k$. There is a component ${\cal M}^0(r,d,k,s)$ of ${\cal M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\cal M}^0(r,d,k,s)$ admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it For $g\ge \frac{r+1}{2}$ and $0<s\leq k(r-k)(g-1) +(r+1)$, $s\equiv kd \mod r,$ ${\cal M}^0(r,d,k,s)$ is non-empty,and its component ${\cal M}^0(r,d,k,s)$ is of dimension} $$\dim {\cal M}^0(r,d,k,s)=\left\{\begin{array}{lcl} (r^2+k^2-rk)(g-1)+s-1& &s<k(r-k)(g-1) &{\rm if}& r^2(g-1)+1& & s\ge k(r-k)(g-1)\end{array}\right.$$Comment: Latex, Permanent e-mail L. Brambila-Paz: [email protected] Classification: 14D, 14

Efficient generation of ideals in a discrete Hodge algebra

Let $R$ be a commutative Noetherian ring and $D$ be a discrete Hodge algebra over $R$ of dimension $d>\text{dim}(R)$. Then we show that (i) the top Euler class group $E^d(D)$ of $D$ is trivial. (ii) if $d>\text{dim}(R)+1$, then $(d-1)$-st Euler class group $E^{d-1}(D)$ of $D$ is trivial.Comment: To appear in J. Pure and Applied Algebr

Multilinear estimates for Calder\'on commutators

In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, including that Calder\'on commutator maps the product of Lorentz spaces $L^{d,1}(\mathbb{R}^d)\times\cdots\times L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, which is the higher dimensional nontrivial generalization of the endpoint estimate that the $n$-th order Calder\'on commutator maps $L^{1}(\mathbb{R})\times\cdots\times L^{1}(\mathbb{R})\times L^1(\mathbb{R})$ to $L^{\frac{1}{1+n},\infty}(\mathbb{R})$. When considering the target space $L^{r}(\mathbb{R}^d)$ with $r<\frac{d}{d+n}$, some counterexamples are given to show that these multilinear estimates may not hold. The method in the present paper seems to have a wide range of applications and it can be applied to establish the similar results for Calder\'on commutator with a rough homogeneous kernel.Comment: 33 pages, 1 figure, this text overlap with arXiv 1710.09664. International Mathematics Research Notices, to appea

The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r

We use the Robinson-Schensted correspondence, followed by symbol-crunching, in order to derive explicit expressions for the quantities mentioned in the title. We follow it by number crunching, in order to compute the first terms of these sequences. As an encore, we cleverly implement Ira Gessel's celebrated determinant formula for the generating functions of these sequences, to crank out many terms. This modest tribute is dedicated to one of the greatest enumerators alive today (and definitely the most modest one!), Ira Martin Gessel, who is turning 64 years-old todayComment: 5 page

The Morphism Induced by Frobenius Push-Forwards

Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$ and $F_{X/k}:X\rightarrow X^{(1)}$ be the relative Frobenius morphism. Let $\mathfrak{M}^{s(ss)}_X(r,d)$ (resp. $\mathfrak{M}^{s(ss)}_{X^{(1)}}(r\cdot p,d+r(p-1)(g-1))$) be the moduli space of (semi)-stable vector bundles of rank $r$ (resp. $r\cdot p$) and degree $d$ (resp. $d+r(p-1)(g-1)$) on $X$ (resp. $X^{(1)}$). We show that the set-theoretic map $S^{ss}_{\mathrm{Frob}}:\mathfrak{M}^{ss}_X(r,d)\rightarrow\mathfrak{M}^{ss}_{X^{(1)}}(r\cdot p,d+r(p-1)(g-1))$ induced by [\E]\mapsto[{F_{X/k}}_*(\E)] is a proper morphism. Moreover, if $g(X)\geq 2$, the induced morphism $S^s_{\mathrm{Frob}}:\mathfrak{M}^s_X(r,d)\rightarrow\mathfrak{M}^s_{X^{(1)}}(r\cdot p,d+r(p-1)(g-1))$ is a closed immersion. As an application, we obtain that the locus of moduli space $\mathfrak{M}^{s}_{X^{(1)}}(p,d)$ consists of stable vector bundles whose Frobenius pull back have maximal Harder-Narasimhan Polygon is isomorphic to Jacobian variety \Jac_X of $X$

The Chow Ring of the Non-Linear Grassmannian

Let M_{P^k}(P^r, d) be the moduli space of unparameterized maps \mu:P^k -> P^r satisfying \mu^*(O(1))= O(d). M_{P^k}(P^r,d) is a quasi-projective variety, and, in case k=1, M_{P^1}(P^r,d) is the fundamental open cell of Kontsevich's space of stable maps \bar{M}_{0,0}(P^r,d). It is shown that the Q-coefficient Chow ring of M_{P^k}(P^r,d) is canonically isomorphic to the Chow ring of the Grassmannian Gr(P^k, P^r)= M_{P^k}(P^r,1).Comment: 17 pages, AMSLate

Existence and uniqueness of Wloc1,rW^{1,r}_{loc}-solutions for stochastic transport equations

We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,r}({\mathbb R}^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\mathbb R}^d))$.In particular, when $r=\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of $L^q(0,T;L^p({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,p}({\mathbb R}^d)$ initial data may induce non-existence for strong solutions (in $W^{1,p}_{loc}({\mathbb R}^d)$), we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. It is an interesting example of a deterministic PDE that becomes well-posed under the influence of a multiplicative Brownian type noise. We extend the existing results \cite{FF2,FGP1} partially