We give a new combinatorial de nition of a sort of weak !category originally devised by J. Baez and J. Dolan in nite dimensional cases. Our de nition is a mixture of both inductive and coinductive de nitions, and suitable for `computational category theory.' Keyword: weak n-category, bicategory, tricategory, formalized mathematics

In this paper, we are concern with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics.

We study metric conditions implied by a combinatorial curvature condition on a finite cubical 2-complex. Our main result is that any finite, simply connected, cubical 2-complex with the 8-property, admits a strongly convex metric. 2000 Mathematics Subject Classification: 53C21.

We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver. 2010 Mathematics Subject Classification: Primary 16G20, secondary 05E05, 14N35

We apply the new theory of cluster algebras of Fomin and Zelevinsky to study some combinatorial problems arising in Lie theory. This is joint work with Geiss and Schröer (§3, 4, 5, 6), and with Hernandez (§8, 9). Mathematics Subject Classification (2000). Primary 05E10; Secondary 13F60, 16G20, 17B10, 17B37.

The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.

One of the most important processes involved in securing a cryptographic system is establishing the keys on which the system will rely. In this article we review the significant contribution of combinatorial mathematics to the development of the theory of cryptographic key establishment. We will describe relevant applications, review current research and, where appropriate, identify areas where...

We study the proof theoretic strength of several infinite versions of finite combinatorial theorem with respect to the standard Reverse Mathematics hierarchy of systems of second order arithmetic. In particular, we study three infinite extensions of the stable marriage theorem of Gale and Shapley. Other theorems studied include some results on partially ordered sets due to Dilworth and to Dushn...

We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

Gaussian numbers, also called Gaussian polynomials or q-binomial coefficients are the q-analogs of common binomial coefficients. First introduced by Euler these polynomials have played an important role in many different branches of mathematics. Sylvester discovered the unimodality of their coefficients, using invariant theory. Gauss recognized the connection of the coefficients to proper integ...