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We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L. K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we construct a bijection through different combinatorial objects,…
We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…
Several results about the union-closed sets conjecture are presented.
We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version…
We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…
In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in…
We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's "realizability disjunction" connective does not admit…
In this note we refine the alternativity in some bifurcation theorems of Rabinowitz type, and then improve a few of results in Lu (2022) [17].
We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature.…
In this paper, we prove a conjecture of Schnell in the surface case.
Approximations to the Kruskal-Katona theorem are stated and proven. These approximations are weaker than the theorem, but much easier to work with numerically.
We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.
We use the Taylor-Wiles-Kisin patching method to prove some new cases of the Breuil-Schneider conjecture.
If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…
We use the geometric reformulation of Markov's uniqueness conjecture in terms of the simple length spectrum on the modular torus to rewrite the conjecture in combinatorial terms by explicitly describing this set of lengths.
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
We consider the time-bounded reachability problem for continuous-time Markov decision processes. We show that the problem is decidable subject to Schanuel's conjecture. Our decision procedure relies on the structure of optimal policies and…
We obtain new partial results supporting the spectral set conjecture in dimension 1.
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error…