Related papers: Samuelson's Webs
The paper begins by exploring the various definitions of norms on semigroups and then presents a new definition of a normed semigroup. The properties of normed semigroups in the new sense are investigated. The new definition of the norm is…
We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the…
A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the…
We define two words in a language to be connected if they express similar concepts. The network of connections among the many thousands of words that make up a language is important not only for the study of the structure and evolution of…
Google's PageRank method was developed to evaluate the importance of web-pages via their link structure. The mathematics of PageRank, however, are entirely general and apply to any graph or network in any domain. Thus, PageRank is now…
We review theoretical approaches to the understanding of food webs. After an overview of the available food web data, we discuss three different classes of models. The first class comprise static models, which assign links between species…
In this paper, we introduce a new class of stochastic multilayer networks. A stochastic multilayer network is the aggregation of $M$ networks (one per layer) where each is a subgraph of a foundational network $G$. Each layer network is the…
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite fields and we establish new asymptotical and not asymptotical upper bounds about it.
This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the…
For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for…
A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…
We employ tools from the fields of symbolic computation and satisfiability checking---namely, computer algebra systems and SAT solvers---to study the Williamson conjecture from combinatorial design theory and increase the bounds to which…
The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that…
We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does…
Downarowicz and Maass (2008) have defined the topological rank for all Cantor minimal homeomorphisms. On the other hand, Gambaudo and Martens (2006) have expressed all Cantor minimal continuous surjections as the inverse limits of certain…
The minimum rank of a simple graph $G$ is the smallest possible rank over all symmetric real matrices $A$ whose nonzero off-diagonal entries correspond to the edges of $G$. Using the zero forcing number, we prove that the minimum rank of…
We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof…
We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with $(i)$ unconstrained interdependent…
We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…
Consider a collection of m competing machine learning algorithms. Given their performance on a benchmark of datasets, we would like to identify the best performing algorithm. Specifically, which algorithm is most likely to ``win'' (rank…