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The notion of contact algebra is one of the main tools in mereotopology. This paper considers a generalisation of contact algebra (called extended distributive contact lattice) and the so called extended contact algebras which extend the language of contact algebras by the predicates covering and internal connectedness.
We introduce an algorithm that computes and counts the duals of finite G\"odel-Dummett algebras of k ≥ 1 elements. The computational cost of our algorithm depends on the factorization of k, nevertheless a Python implementation is sufficiently fast to compute the results for very large values of k.
In a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP(Γ)...
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parametrize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those...
There are many practical applications based on the Least Square Error (LSE) approximation. It is based on a square error minimization “on a vertical” axis. The LSE method is simple and easy also for analytical purposes. However, if data span is large over several magnitudes or non-linear LSE is used, severe numerical instability can be expected. The presented contribution describes a simple method...
In this paper a new 3-bit burst-error correcting code is proposed. Compared to a 1-bit error correcting Hamming code only two additional check bits are needed and compared to a 3-bit burst-error correcting Burton code the number of check bits can be reduced by 2. Since the proposed code is systematically designed by use of finite field algebra the code can be determined for an arbitrary word length...
The main aim of this article is to study Wd -fuzzy implication algebras which are subalgebra of fuzzy implication algebras. We showed that Wd -fuzzy implication algebras are regular fuzzy implication algebras, but the inverse is not true. The relations between Wd -fuzzy implication algebras and other fuzzy algebras are discussed. Properties and axiomatic systems for Wd -fuzzy implication algebras...
Final coalgebras as “categorical greatest fixed points” play a central role in the theory of coalgebras. Somewhat analogously, most proof methods studied therein have focused on greatest fixed-point properties like safety and bisimilarity. Here we make a step towards categorical proof methods for least fixed-point properties over dynamical systems modeled as coalgebras. Concretely, we seek a categorical...
Graph construction, a fundamental operation in a data processing pipeline, is typically done by multiplying the incidence array representations of a graph, E in and E out, to produce an adjacency array of the graph, A, that can be processed with a variety of algorithms. This paper provides the mathematical criteria to determine if the product A = E T\out E in will have the required structure of the...
A strong partial clone is a set of partial operations closed under composition and containing all partial projections. Let X be the set of all Boolean strong partial clones whose total operations are the projections. This set is of practical interest since it induces a partial order on the complexity of NP-complete constraint satisfaction problems. In this paper we study X from the algebraic point...
With the fast developing of computer hardware, many problems which are previously considered hard, will be solved practically. Number of theoretical problems that form the core of many public key cryptosystems are such of them. Therefore, new cryptosystems which are designed based on other mathematical techniques are of great demands. NTRU which is based on the hardness of lattice problems is one...
Algebra is one of the important fields of mathematics. It concerns in the study and manipulates of mathematical symbols. It also concerns with study of abstractions such as groups, rings and fields. Ring theory is the most attractive category of algebra in the area of cryptography. Recently, many algebraic cryptosystem protocols based on non-commutative algebraic structures such as; authentication,...
This paper presents a proposed structural framework for an assessment and CQI process that is both simple and effective. The process is based on the ideas that assessment should be done where it makes sense, it should not be necessarily redundant, and it should provide useful information for improving the program. Elements of the proposed process have been piloted in two separate programs, including...
This paper is an extended abstract of an invited talk of the samename, given at SYNASC 2016. It describes a sort of case study of howideas from computational logic (specifically Satisfiability ModuloTheory solving) provide new algorithms in symbolic computing. Inparticular, it describes how ideas from the NLSAT solver led to a newkind of Cylindrical Algebraic Decomposition.
Satisfiability Checking is a relatively young research area, aiming at the development of efficient software technologies for checking the satisfiability of existentially quantified logical formulas. Besides the success story of SAT solving for propositional logic, SAT-modulo-theories (SMT) solvers offer sophisticated solutions for different theories. When targeting arithmetic theories, SMT solvers...
This paper presents a system that automatically assesses multi-step answers to algebra questions. The system requires teacher involvement only during the question set-up stage. Two types of algebra questions are currently supported: questions with linear equations containing fractions, and questions with quadratic equations. The system evaluates each step of a student's answer and awards full/partial...
We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces into the space of matrices endowed with the trace norm with the property that the image of standard basis...
The University of Texas — Pan American (UTPA) is a minority serving institution in south Texas. The student population is predominantly made up of students from the local region, which includes two of the poorest counties in Texas. A significant number of high schools in the region offer some form of Calculus course, and in practice we find a number of incoming students with Calculus or Pre-Calculus...
We consider the use of the modified algebra of algorithms to formalize the grammars and outputs of context-free grammars. The inference rules and the derivations themselves of several applied grammars have been given in formulas of modified algebra of algorithms. We illustrate the use of operations of sequencing as well as operations of paralleling to describe the rules of output.
Since discovery of Goppa geometric codes, people have been asking the question: are there different constructions of block codes from algebraic curves that give the same parameters as Goppa geometric codes. Despite of great effort by researchers, no such constructions have been found so far. Although in literature, there are many constructions of block code from algebraic curves, most of them are...
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