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We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show…
Let $\mathbb V$ be an arbitrary linear space and $f:\mathbb V \times \ldots \times \mathbb V \to \mathbb V$ an $n$-linear map. It is proved that, for each choice of a basis ${\mathcal B}$ of $\mathbb V$, the $n$-linear map $f$ induces a…
In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon codes and algebraic geometry codes. This lemma states that two vector…
The lambda calculus is a widely accepted computational model of higher-order functional pro- grams, yet there is not any direct and universally accepted cost model for it. As a consequence, the computational difficulty of reducing lambda…
In analogy with the regularity lemma of Szemer\'edi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials \calF =…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
We present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for the involutive uninorm logic IUL is proved. Then the standard completeness of IUL follows as a lemma by…
We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [OSV12], we establish an equivalence between matrix inversion and…
We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference…
This paper describes a general framework for automatic termination analysis of logic programs, where we understand by ``termination'' the finitenes s of the LD-tree constructed for the program and a given query. A general property of…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of electronic storage…
This paper introduces the continuous tensor abstraction, allowing indices to take real-number values (for example, A[3.14]). It also presents continuous tensor algebra expressions, such as C(x,y) = A(x,y) * B(x,y), where indices are defined…
In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172--1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to…
In this report, we summarize the set partition enumeration problems and thoroughly explain the algorithms used to solve them. These algorithms iterate through the partitions in lexicographic order and are easy to understand and implement in…
We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed, using Kuo's graphical condensation method. Additionally, we prove a formula for a weighted version of…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
This paper introduces the concept of abstracted model reduction: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models. To effectively reduce high-order,…
Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably…
Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and it is shown that the corresponding invariants lead to very…