Related papers: A note on the log-concavity of parking functions
Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's…
Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve $C$ with positive self-intersection. We prove that there exists a neighborhood $U\supset C$ such that any meromorphic…
We study operator log-convex functions on $(0,\infty)$, and prove that a continuous nonnegative function on $(0,\infty)$ is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditions related to…
Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we…
We conjecture that the Parisi functional in the SK model is convex in the functional order parameter and prove a partial result that shows the convexity along one-sided directions. A consequence of this result is log-convexity of L_1 norm…
We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…
A parking function of length $n$ is a sequence $\pi=(\pi_1,\dots, \pi_n)$ of positive integers such that if $\lambda_1\leq\cdots\leq \lambda_n$ is the increasing rearrangement of $\pi_1,\dots,\pi_n$, then $\lambda_i\leq i$ for $1\leq i\leq…
We prove sharp anti-concentration results for log-concave random variables on the real line in both the discrete and continuous setting. Our approach is elementary and uses majorization techniques to recover and extend some recent and not…
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we…
Graphs and hypergraphs combine expressive modeling power with algorithmic efficiency for a wide range of applications. Hedgegraphs generalize hypergraphs further by grouping hyperedges under a color/hedge. This allows hedgegraphs to model…
In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize…
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to…
We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.
Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an…
We propose and study a data-driven framework for identifying traffic congestion functions (numerical relationships between observations of traffic variables) at global scale and segment-level granularity. In contrast to methods that…
Parking in large metropolitan areas is often a time-consuming task with further implications toward traffic patterns that affect urban landscaping. Reducing the premium space needed for parking has led to the development of automated…
The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in…
We prove Vojta's generalized abc conjecture for algebraic tori over function fields with exceptional sets that can be determined effectively. Additionally, we establish a version of the conjecture for toric varieties. As an application, we…