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The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…
We answer a question of Simental by providing a combinatorial interpretation of a formula which generalizes rational Catalan numbers and which appears in the study of Springer fibers. We provide an interpretation in terms of binary…
We derive formulas for $(i)$ the number of toroidal $n\times n$ binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and $(ii)$ the number of toroidal $n\times n$ binary arrays, allowing rotation and/or…
The purpose of this paper is to introduce the idea of triangular Ramsey numbers and provide values as well as upper and lower bounds for them. To do this, the combinatorial game Mines is introduced; after some necessary theorems about…
We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation…
The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be…
We highlight a striking difference in behavior between two widely used variants of coordinate ascent variational inference: the sequential and parallel algorithms. While such differences were known in the numerical analysis literature in…
The paper describes a parser for Categorial Grammar which provides fully word by word incremental interpretation. The parser does not require fragments of sentences to form constituents, and thereby avoids problems of spurious ambiguity.…
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof…
Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan…
We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give…
I review the three principal methods to assign meaning to recursion in process algebra: the denotational, the operational and the algebraic approach, and I extend the latter to unguarded recursion.
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results…
There is inherent information captured in the order in which we write words in a list. The orderings of binomials --- lists of two words separated by `and' or `or' --- has been studied for more than a century. These binomials are common…
This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and…
A tanglegram is a pair of binary trees with the same set of leaves. Unlabeled tanglegrams were counted recently by Billey, Konvalinka, and Matsen, who also proposed the problem of counting several variations of unlabeled tanglegrams…
Consider a binary classification problem solved using a feed-forward artificial neural network (ANN). Let the ANN be composed of a ReLU layer and several linear layers (convolution, sum-pooling, or fully connected). We assume the network…
In this paper we present two original methods for recognizing textual inference. First one is a modified resolution method such that some linguistic considerations are introduced in the unification of two atoms. The approach is possible due…
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…