We consider the problem of allocating some amount of an infinitely divisible and homogeneous resource among agents having claims on this resource that cannot be jointly honored. A “rule” associates with each such problem a feasible division. Our goal is to uncover the structure of the space of rules. For that purpose, we study “operators” on the space, that is, mappings that associate to each r...

We study [2m−1, 2m]-binary linear codes whose weights lie between w0 and 2m−w0, where w0 takes the highest possible value. Primitive cyclic codes with two zeros whose dual satisfies this property actually correspond to almost bent power functions and to pairs of maximum-length sequences with preferred crosscorrelation. We prove that, for odd m, these codes are completely characterized by their ...

Binomial coefficients arise naturally in combinatorics. Recently the speaker initiated the study of certain divisibility properties of binomial coefficients, and products or sums of binomial coefficients. In this talk we introduce the speaker’s related results and various conjectures. The materials come from the author’s two preprints: 1. Z. W. Sun, Products and sums divisible by central binomi...

with C0 = 1. Their appearances occur in a dazzling variety of combinatorial settings where they are used to enumerate all manner of geometric and algebraic objects (see Richard Stanley’s collection [28, Chap. 6]; an online Addendum is continuously updated). Quite a lot is known about the divisibility of the Catalan numbers; see [2, 10]. They are obviously closely related to the middle binomial ...

The quantum regression formula for an open system consists in infinite hierarchy of conditions its multi-time correlation functions, thus requiring full access to the total "system+environment" evolution, and providing a stronger requirement than CP-divisibility. Here, we analyze CP-divisibility check validity beyond Born-Markov approximation (e.g. weak coupling limit) class generalized spin-bo...