Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometric and o-minimal Littlewood-Offord problems
168
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The classical ErdH{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,dots,a_nin mathbb{R}^d$, any $xin mathbb{R}^d$, and uniformly random $(xi_1,dots,xi_n)in{-1,1}^n$, we have $Pr(a_1xi_1+dots+a_nxi_n=x)=O(n^{-1/2})$. In this paper we show that $Pr(a_1xi_1+dots+a_nxi_nin S)le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.
rate research
Read More
In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.
We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of Schur $P$- and $Q$-function identities. We also demonstrate the relationship between the shifted tableau switching process and the shifted $J$-operation due to Worley.
For any triple $(i,a,mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $mu$, we define a quiver current $H^{(i,a)}_mu$ acting on the tensor power $Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(mathbf{i},mathbf{a},mu(bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{mathbf{i},mathbf{a}}_{mu(bullet)}$ as the result of acting on $1inLambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{mathbf{i},mathbf{a}}_{mu(bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztigs convolution diagram determined by the sequences $mathbf{i},mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztigs convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saitos vertex representation of the quantum toroidal algebra of type $mathfrak{sl}_r$.
Generalized Turan problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.
We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hiverts quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alphas. The F-expansion of P_lambda/mu is facilitated by introducing starred tableaux.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
