Showing papers on "Algebraic expression published in 2018"

A correlation-based algebraic transition model:

Abstract: A correlation-based algebraic transition model that relies on local flow information is proposed. The model is qualified as an algebraic model, or a zero-equation model since it includes an intermi...

A Generalization of Algebraic Expression for Nonlinear Component of Symmetric Key Algorithms of Any Characteristic p

Abstract: Recently several block ciphers are proposed which are based on the inversion mapping over binary Galois field with n-input. These proposed block ciphers are Camellia, AES, Square and Hierocrypt in which S-box over binary Galois field with n-input is used. Now with the passage of time it is necessary to increase the security of these blocks ciphers by increasing the key space which can be increased by generalizing this concept over non-binary Galois field with n-input. In this paper, we have designed an innovative techniques through which we can easily find element of nonlinear component of block cipher namely S-box through a single expression instead of matrix algebraic computations. Our main idea here is to reduce the computational complexity while performing calculations for S-box which is one of the most important nonlinear components for any modern block ciphers. Also, we have transformed our existing problem of being using matrix algebra to a symbolic single expression algebra which reduces the tedious calculations of S-boxes.

Convex Formulations and Algebraic Solutions for Linear Quadratic Inverse Optimal Control Problems

Abstract: This paper presents convex formulations for inverse optimal control problems for linear systems to infer cost function matrices of a quadratic cost from both optimal and non-optimal closed-loop gains. It introduces an optimality measure which enables a formulation of the problem as a convex semidefinite program for the general case and a linear program for several special cases. We derive an explicit algebraic expression for general objective function matrices as well as conditions under which the solution to the inverse optimal control problem is unique. The result is derived by means of a vectorization and parametrization of the algebraic Riccati equation. A simulation example highlights the robust performance in the presence of noise on the measured closed-loop gain and the computational efficiency of the proposed problem formulations.

Three-cluster dynamics within the ab initio no-core shell model with continuum: How many-body correlations and α clustering shape He 6

Abstract: We realize the treatment of bound and continuum nuclear systems in the proximity of a three-body breakup threshold within the ab initio framework of the no-core shell model with continuum. Many-body eigenstates obtained from the diagonalization of the Hamiltonian within the harmonic-oscillator expansion of the no-core shell model are coupled with continuous microscopic three-cluster states to correctly describe the nuclear wave function both in the interior and asymptotic regions. We discuss the formalism in detail and give algebraic expressions for the case of core + $n+n$ systems. Using similarity-renormalization-group evolved nucleon-nucleon interactions, we analyze the role of $\phantom{\rule{0.28em}{0ex}}^{4}\mathrm{He}+n+n$ clustering and many-body correlations in the ground and low-lying continuum states of the Borromean $^{6}\mathrm{He}$ nucleus, and study the dependence of the energy spectrum on the resolution scale of the interaction. We show that $^{6}\mathrm{He}$ small binding energy and extended radii compatible with experiment can be obtained simultaneously, without resorting to extrapolations. We also find that a significant portion of the ground-state energy and the narrow width of the first ${2}^{+}$ resonance stem from many-body correlations that can be interpreted as core-excitation effects.

Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity

Abstract: We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined $S$-duality modularity transformation. We provide evidence for our formula by calculating virtual $\chi_y$-genera of moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined $\mathrm{SU}(r)$ Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3).

Geometry and Visual Reasoning

Abstract: The general aim of this chapter is to identify the potential learning opportunities provided by the introduction of historical geometric diagrams into student tasks. To this end, we examine some problem sets for secondary education students concerning situations to be solved with diagrams in which right triangles or solving second-degree equations are involved. In all cases the objective is that students should transfer linguistically expressed reasoning (second-degree algebraic expressions) to reasoning with visual diagrams (figures with squares and rectangles) that are the geometric interpretation of the second-degree algebraic expressions. The research is therefore focused on students’ learning process, and specifically, the results they achieve by the use of these diagrams.

Examination of the reasons for these mistakes made by elementary school 7thgrade students in relation to algebraic expressions

Abstract: The aim of this study is to investigate reasons for these mistakes made by secondary school 7th-grade students in relation to algebraic expressions. In accordance with this aim, the sample of the study consists of 150 7th-grade students studying at secondary schools in Trabzon city centre in the 2013-2014 academic year. In this study, the case study method was used. In the study, a knowledge test was applied in order to determine the mistakes made by the students in the subject of algebra, and a semi-structured interview was held in order to determine the reasons for the mistakes made. The data obtained from these interviews were analysed using the content analysis method. From the data obtained from the study, it was observed that the students’ knowledge of algebraic expressions was not at the desired level, and many mistakes were made in the algebra knowledge test applied. Some of these mistakes made are the student’s ignoring the variable, solving the given algebraic expression by transforming it into an equation, using the x variable instead of the variable given in the question, and establishing the equation for the given problem incorrectly. From the interviews held, the reasons for these mistakes were determined as the student’s failure to attribute a meaning to the variable in the operation, failure to distinguish the concepts of unknown and variable, identifying the expression of available with the x expression, the lack of knowledge in mathematical operations, and the insufficiency of the time allocated to the subject of algebra.

Towards scalable pattern-based optimization for dense linear algebra.

Abstract: Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient machine-level representations is far from trivial: developers should be assisted by automatic optimization tools so that they can focus their attention on high-level problems, rather than low-level details. The tractability of these optimizations is highly dependent on the choice of the primitive constructs in terms of which the computations are to be expressed. In this work we propose to describe operations on multi-dimensional arrays using a selection of higher-order functions, inspired by functional programming, and we present rewrite rules for these such that they can be automatically optimized for modern hierarchical and heterogeneous architectures. Using this formalism we systematically construct and analyse different subdivisions and permutations of the dense matrix multiplication problem.

Cobra: A Framework for Cost-Based Rewriting of Database Applications

Abstract: Database applications are typically written using a mixture of imperative languages and declarative frameworks for data processing. Data processing logic gets distributed across the declarative and imperative parts of a program. Often, there is more than one way to implement the same program, whose efficiency may depend on a number of parameters. In this paper, we propose a framework that automatically generates all equivalent alternatives to a given program using a given set of program transformations, and chooses the least cost alternative. We use the concept of program regions as an algebraic abstraction of a program and extend the Volcano/Cascades framework for optimization of algebraic expressions, to optimize programs. We illustrate the use of our framework for optimizing database applications. We show through experimental results, that our framework has wide applicability in real-world applications and provides significant performance benefits.

Towards scalable pattern-based optimization for dense linear algebra.

Abstract: Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient machine-level representations is far from trivial: developers should be assisted by automatic optimization tools so that they can focus their attention on high-level problems, rather than low-level details. The tractability of these optimizations is highly dependent on the choice of the primitive constructs in terms of which the computations are to be expressed. In this work we propose to describe operations on multi-dimensional arrays using a selection of higher-order functions, inspired by functional programming, and we present rewrite rules for these such that they can be automatically optimized for modern hierarchical and heterogeneous architectures. Using this formalism we systematically construct and analyse different subdivisions and permutations of the dense matrix multiplication problem.

Algebra: What Should We Teach and How Should We Teach It?

Abstract: The "new math" movement of the 1960s attempted in introducing some new ideas and new approaches into algebra instruction. But the changes that have persisted into today's curriculum have been more cosmetic than substantial. The emergence of sophisticated calculators and computers as tools in both computational and abstract mathematics have changed the way that mathematicians do mathematics and the way that scientists, engineers, and social scientists use mathematics. All mathematics instruction and algebra instruction in particular, should be designed to promote understanding of concepts and to encourage thinking. The chapter discusses the three criteria: intrinsic value, pedagogical value, and intrinsic excitement or beauty. In algebra, exponential growth and decay, especially if taught within the context of population dynamics or pollution control, can fit the criterion of intrinsic excitement. Existing software packages will simplify algebraic expressions; add, subtract, multiply, and divide polynomials and polynomial fractions.

Cobra: A Framework for Cost Based Rewriting of Database Applications

Abstract: Database applications are typically written using a mixture of imperative languages and declarative frameworks for data processing. Application logic gets distributed across the declarative and imperative parts of a program. Often, there is more than one way to implement the same program, whose efficiency may depend on a number of parameters. In this paper, we propose a framework that automatically generates all equivalent alternatives of a given program using a given set of program transformations, and chooses the least cost alternative. We use the concept of program regions as an algebraic abstraction of a program and extend the Volcano/Cascades framework for optimization of algebraic expressions, to optimize programs. We illustrate the use of our framework for optimizing database applications. We show through experimental results, that our framework has wide applicability in real world applications and provides significant performance benefits.

Perturbational non-canonical theory of molecular orbitals and its applications

Abstract: The article contains a summary of fundamentals of the perturbational non- canonical molecular orbital (PNCMO) theory formerly developed by the author. In some respects, the PNCMO theory is a generalization of the well-known simple PMO theory: First, the usual diagonalization problem (and/or the eigenvalue equation) for a certain model Hamiltonian matrix ($\mathbf{H}$) is now replaced by two interrelated non-canonical one-electron problems, namely by the block-diagonalization problem for the matrix $\mathbf{H}$\ following from the Brillouin theorem and determining non-canonical (localized) MOs (NCMOs) and by the commutation equation for the respective one-electron density matrix (charge-bond order (CBO)) matrix. Second, perturbative solutions of the above-specified alternative problems are sought in terms of entire submatrices (blocks) of the matrix $\mathbf{H}$\ instead of usual matrix elements (e.g. of Coulomb and resonance parameters). Third, a generalized version of the perturbation theory (PT) is used in place of the standard Rayleigh-Schr\"{o}dinger PT (RSPT), wherein non-commutative quantities stand for the usual (commutative) ones (cf. the so-called non-commutative RSPT (NCRSPT)). As a result, algebraic expressions are derived for the principal quantum-chemical characteristics (including the CBO matrix, the NCMO representation matrix and the total energy) that embrace definite classes of Hamiltonian matrices and thereby of molecules. To illustrate the point, saturated and conjugated hydrocarbons are taken as examples. Arguments are given that the PNCMO theory possibly forms the basis of a novel way of qualitative chemical thinking.

Time-based querying of graph databases

Abstract: A method includes receiving, via a processor, a query comprising pathway variables and at least one evaluation time and determining an anchor set based on at least one of the pathway variables The method also includes translating the pathway variables into a pathway algebraic expression based on the anchor set and the at least one evaluation time, and executing the pathway algebraic expression on a graph database to return a pathway set

Embodied cognition of the student mathematical imaginations in conceptual understanding of algebraic expression

Abstract: Development of imagination in education creates a continuous education process and always new. The aim of research to describe of the student mathematical imagination and embedied cognition in conseptual understanding of algebraic expression. Type of this research is explorative with qualitative descriptive approach. The study was conducted in SMP Negeri 2 Semarang, at the end of 2017 for 3 months. Subjects involved in the study is a student who has a visual learning style. Methods of data collection using tests, interviews and observations. Test of data validity of research result using time triangulation technique. The results of this study are as follows: 1) subject imagine variables as objects that she recognized, such as the number of objects in a box or tin; 2) subject imagines an example and not an example of algebraic expression when it will define the concept of algebraic expression; 3) subject using gesture representation of variables, coefficients or constants; gesture pointing and gestures of writing as embodied cognition of the mathematical imaginations used; 4) subjects using the utterance as embodied cognition of mathematical imagination used or as the embodiment of social interaction that he did to get a confirmation or approval of researchers.

Some Algebraic Approach for the Second Painlevé Equation Using the Optimal Homotopy Asymptotic Method (OHAM)

Abstract: The study of Painleve equations has increased during the last years, due to the awareness that these equations and their solutions can accomplish good results both in the field of pure mathematics and in theoretical physics. In this paper we introduced the optimal homotopy asymptotic method (OHAM) approach to propose analytic approximate solutions to the second Painleve equation. The advantage of this method is that it provides a simple algebraic expression that can be used for further developments while maintaining good performance and fitting closely the numerical solution.

Algebraic Reasoning in Early Grade: Promoting through Lesson Study and Open Approach

Abstract: The objective of this study is to analyze what characteristics are of early grade students’ algebraic reasoning in context of open approach and lesson study. Ethnographic study was employed to conduct in this qualitative study. The study was carried out in one mathematics classroom which is a case study of this study and it is the case of classroom which has been using open approach and lesson study since 2006. The 3 teachers are as a member of school lesson study team participated the study as informant and so were 10 students from the class. The data were collected through 9 consecutive lessons by observation with audio-video tape recording, interview, students’ written works and daily field notes. The lessons were designed by carrying into 4 steps of open approach: posing problem, students’ self-learning, whole class discussion and comparison, and summing-up by connecting students’ emergent mathematical ideas. All activities were guided by Thai version of 1st grade Japan mathematics textbook. The results showed that characteristics of first grade students’ algebraic reasoning are as follows: 1) using algebraic expressions to represent addition situation and posing situation to represent given expressions, 2) constructing and using a tool to find problem results more easily, 3) extending solutions to another domain of number, 4) using various representations to justify their ways of thinking, and 5) reasoning about relations among numbers. The algebraic reasoning occurred under the condition that teachers and students had connected among 3 worlds oriented to Inprasitha’s approach: real world, semi-concrete world, and mathematics world.

Algebraic Expression of Spatial and Temporal Pattern

Abstract: Universal learning machine is a theory trying to study machine learning from mathematical point of view. The outside world is reflected inside an universal learning machine according to pattern of incoming data. So, machine will have its subjective view and such subjective view is adapting/learning. In \cite{paper2, cpaper}, we discussed subjective spatial pattern, and established a powerful tool -- X-form, which is an algebraic expression for subjective spatial pattern. However, as the initial stage of study, we only discussed spatial pattern there and leave temporal pattern for later study. Here, we will discuss spatial and temporal patterns, and establish algebraic expression for subjective patterns, spatial and temporal.

Structure Sense in Algebraic Expressions and Equations of Groups of Students

Abstract: High school Algebra is a critical subject that bridges students’ ability to reaching a higher level of mathematical knowledge, attitudes, and skills. This study determined the “structure sense” in algebraic expressions and equations of groups of students. Using in-depth analysis on the written outputs of the Grades 8,9,and 10 students, the study explored the algebraic structure of their solutions. The students portrayed different ways of solving Algebraic expressions and equations. Results show that those who have adequate knowledge and skills were successful in carrying out the works while those who are not familiar with the structural properties of the tasks found difficulty in solving, which yielded an ambiguous solution, which led to incorrect answers. The students lacked the conceptual understanding of the given problem, and they have poor skills in manipulating expressions, calculation mistakes, and technical errors. They even displayed different methods of solving algebraic expressions and equations, from simple, to more detailed, and to more complicated but understandable solutions. Recognition of the “structure sense” of every term in algebraic expressions and equations are necessary for students to perform appropriate manipulations.

Algebraic Expression of Subjective Spatial and Temporal Patterns.

Abstract: Universal learning machine is a theory trying to study machine learning from mathematical point of view. The outside world is reflected inside an universal learning machine according to pattern of incoming data. This is subjective pattern of learning machine. In [2,4], we discussed subjective spatial pattern, and established a powerful tool -- X-form, which is an algebraic expression for subjective spatial pattern. However, as the initial stage of study, there we only discussed spatial pattern. Here, we will discuss spatial and temporal patterns, and algebraic expression for them.

Modeling and Stability Analysis for Markov Jump Networked Evolutionary Games

Abstract: This paper investigates the algebraic formulation and stability analysis for a class of Markov jump networked evolutionary games by using the semitensor product method and presents a number of new results. Firstly, a proper algorithm is constructed to convert the given networked evolutionary games into an algebraic expression. Secondly, based on the algebraic expression, the stability of the given game is analyzed and an equivalent criterion is given. Finally, an example works out to support the new results.

Combinatorial Aspects of the Quantized Universal Enveloping Algebra of $$\mathfrak {sl}_{n+1}$$ sl n + 1

Abstract: Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon $$\mathscr {U}_h(\mathfrak {sl}_2)$$ , the quantized universal enveloping algebra of the Lie algebra $$\mathfrak {sl}_2$$ . In this paper, combinatorial structure in $$\mathscr {U}_h(\mathfrak {sl}_2)$$ is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $$n=1$$ . We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s $$sR$$ -matrix, but also for the arguably mysterious ribbon elements of $$\mathscr {U}_h(\mathfrak {sl}_2)$$ . Finally, we extend these techniques to the higher-dimensional algebras $$\mathscr {U}_h(\mathfrak {sl}_{n+1})$$ . While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.

Generating Algebraic Expressions for Labeled Grid Graphs

Abstract: The paper investigates relationship between algebraic expressions and labeled graphs. We consider directed grid graphs having m rows and n columns. Our intent is to simplify the expressions of these graphs. With that end in view, we describe two methods which generate expressions for directed grid graphs. For both methods, lengths of the expressions grow polynomially with n while m is determined as a constant parameter. Besides, we apply these methods to a square grid graph in which the number of rows is equal to the number of columns. We prove that the lengths of the expressions derived by the methods depend exponentially and quasi-polynomially, respectively, on the size of the graph.