Related papers: Extremal set theory for the binomial norm
We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application,…
The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those…
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…
Statistical inference for extreme values of random events is difficult in practice due to low sample sizes and inaccurate models for the studied rare events. If prior knowledge for extreme values is available, Bayesian statistics can be…
In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erd\H{o}s-Stone-Bollob\'as theorem, several stability…
Combining a modular approach to the $abc$ conjecture developed by the second author with the classical method of linear forms in logarithms, we obtain improved unconditional bounds for two classical problems. First, for Szpiro's conjecture…
In this paper, we intend to generalize the classical set theory as much as possible. we will do this by freeing sets from the regular properties of classical sets; e.g., the law of excluded middle, the law of non-contradiction, the…
The extremal characteristics of random structures, including trees, graphs, and networks, are discussed. A statistical physics approach is employed in which extremal properties are obtained through suitably defined rate equations. A variety…
Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used…
Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied…
Necessary conditions for existence of normal extremals in optimal control of systems subject to nonholonomic constraints are derived as solutions of a constrained second order variational problems. In this work, a geometric interpretation…
We address an idea of applying generalized entropies in counting problems. First, we consider some entropic properties that are essential for such purposes. Using the $\alpha$-entropies of Tsallis-Havrda-Charv\'{a}t type, we derive several…
We generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices…
Different dependence scenarios can arise in multivariate extremes, entailing careful selection of an appropriate class of models. In bivariate extremes, the variables are either asymptotically dependent or are asymptotically independent.…
What if there is a teacher who knows the learning goal and wants to design good training data for a machine learner? We propose an optimal teaching framework aimed at learners who employ Bayesian models. Our framework is expressed as an…
We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to…
An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding Chernoff and Slud bounds, which not only demonstrates the quality of the presented…
We use the notion of generalized connection over a bundle map in order to present an alternative approach to sub-Riemannian geometry. Known concepts, such as normal and abnormal extremals, will be studied in terms of this new formalism. In…