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We provide an interpretation of entanglement based on classical correlations between measurement outcomes of complementary properties: states that have correlations beyond a certain threshold are entangled. The reverse is not true, however.…
We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.…
The law of likelihood underlies a general framework, known as the likelihood paradigm, for representing and interpreting statistical evidence. As stated, the law applies only to simple hypotheses, and there have been reservations about…
This paper is concerned with unreachable pawn diagrams and the subset of which can be generated using Hall's Marriage Theorem. The result is 1 in 23 diagrams are unreachable by applying the theorem.
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane…
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in…
The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a…
An technically interesting proof of a known theorem.
We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which…
Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By…
Computational materials design often profits from the fact that some complicated contributions are not calculated for the real material, but replaced by results of models. We turn this approximation into a very general and in principle…
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…
We prove a combination theorem for PD(n)-pairs.
A very simple but useful almost sure convergence theorem of probability is given.
We present a new kind of normalization theorem: linearization theorem for skew products. The normal form is a skew product again, with the fiber maps linear. It appears, that even in the smooth case, the conjugacy is only H\"older…
We discuss two approaches to producing generalized parity proofs of the Kochen-Specker theorem. Such proofs use contexts of observables whose product is $I$ or $-I$; we call them constraints. In the first approach, one starts with a fixed…