Related papers: Normal Binary Hierarchical Models

A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover…

We introduce the family of graphical Hermite simplices and study the Smith normal forms of their matrices of vertex vectors, which is equivalent to studying the group structure of the cokernels for these matrices. Our motivation is to study…

Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…

We give essentially unique ``normal forms'' for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity which is a p:q resonant-saddle. Hence each vector field of that type is conjugate, by a…

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…

One property of networks that has received comparatively little attention is hierarchy, i.e., the property of having vertices that cluster together in groups, which then join to form groups of groups, and so forth, up through all levels of…

We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…

Let $Q=(q_n)_{n=1}^\infty$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both…

We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in…

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and…

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…

We investigate the class of bisymmetric and quasitrivial binary operations on a given set $X$ and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations.…

For studying intrusion detection data we consider data points referring to individual IP addresses and their connections: We build networks associated with those data points, such that vertices in a graph are associated via the respective…

We study the computational complexity of Feedback Vertex Set on subclasses of Hamiltonian graphs. In particular, we consider Hamiltonian graphs that are regular or are planar and regular. Moreover, we study the less known class of…

In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence…

Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the…

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, this…

Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend…