Symbolic-numeric interface: A review

Coxeter spectral classification of almost TP-critical one-peak posets using symbolic and numeric computations

Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop improved algorithms for matrix problems that arise in scientific computing, data science, machine learning, etc. Determinantal Point Processes (DPPs), a seemingly unrelated topic in pure and applied mathematics, is a class of stochastic point processes with probability distribution characterized by sub-determinants of a kernel matrix. Recent work has uncovered deep and fruitful connections between DPPs and RandNLA which lead to new guarantees and improved algorithms that are of interest to both areas. We provide an overview of this exciting new line of research, including brief introductions to RandNLA and DPPs, as well as applications of DPPs to classical linear algebra tasks such as least squares regression, low-rank approximation and the Nystrom method. For example, random sampling with a DPP leads to new kinds of unbiased estimators for least squares, enabling more refined statistical and inferential understanding of these algorithms; a DPP is, in some sense, an optimal randomized algorithm for the Nystrom method; and a RandNLA technique called leverage score sampling can be derived as the marginal distribution of a DPP. We also discuss recent algorithmic developments, illustrating that, while not quite as efficient as standard RandNLA techniques, DPP-based algorithms are only moderately more expensive.

The article offers the ways of modern information and communication technologies introduction in the process of teaching the discipline «Analytical geometry and linear algebra» for future teachers of physics and computer science. The authors highlight the actuality and experience of using information and communication technologies of training, innovative approaches to the organization of educational process in high school and determine the types of information and communication technologies of teaching analytic geometry and linear algebra at the pedagogical university. The possibilities of network technologies use in the process of teaching analytical geometry and linear algebra are considered, as well as determined advantages of their use for students’ in-class and out-of-class work. The authors demonstrate the ways of organizing students’ educational activity using the applied software for carrying out self-checking tasks correctness. It is presented the form of interactive learning with the use of AS Gran 3D, which effectively influences on the future teachers training and highlighted the advantages of using mobile technologies for checking students’ knowledge. The peculiarities of using mobile technologies in the process of preparation of future teachers of physics, computer science are pointed out. In particular, it is presented an example for testing students’ knowledge using the Plickers program which makes it possible to assess the level of students’ knowledge quickly. The examples show different ways of applying information and communication technologies, the use of which in traditional training contributes to the activation of cognitive activity of students, motivates them to work independently. It is noted that the educational process of using information and communication technologies of teaching intensifies students’ educational and cognitive activity, promotes the development of creative abilities and leadership qualities, forms the autonomy in gaining new knowledge.

2 - Algorithms, computations and structural information

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

This book developed from the need to teach a linear algebra course to students focused on data science and bioinformatics programs. These students tend not to realize the importance of linear algebra in applied sciences, since traditional linear algebra courses tend to cover mathematical contexts but not the computational aspect of linear algebra or its applications to data science and bioinformatics. The author presents the topics in a traditional course, yet offers lectures as well as lab exercises on simulated and empirical data sets. This textbook provides students a theoretical basis which can then be applied to the practical R and Python problems, providing the tools needed for real-world applications. Each section starts with working examples to demonstrate how tools from linear algebra can help solve problems in applied sciences. These exercises start from easy computations, such as computing determinants of matrices, to practical applications on simulated and empirical data sets with R so that students learn how to get started with R, along with computational examples in each section, and then students learn how to apply what they've learned to problems in applied sciences. This book is designed from first principles to demonstrate the importance of linear algebra through working computational examples with R and Python, including tutorials on how to install R in the Appendix. If a student has never seen R, they can get started without any additional help. Since Python is one of the most popular languages in data science, optimization, and computer science, code supplements are available for students who feel more comfortable with Python. R is used primarily for computational examples to develop students’ practical computational skills. About the Author: Dr. Ruriko Yoshida is an Associate Professor of Operations Research at the Naval Postgraduate School. She received her PhD in Mathematics from the University of California, Davis. Her research topics cover a wide variety of areas: applications of algebraic combinatorics to statistical problems such as statistical learning on non-Euclidean spaces, sensor networks, phylogenetics, and phylogenomics. She teaches courses in statistics, stochastic models, probability, and data science.

Comparison of linear and nonlinear analysis in estimating the Thomas model parameters for methylene blue adsorption onto natural zeolite in fixed-bed column

Large linear systems are ubiquitous in modern computational science and engineering. Their efficient solution is frequently challenging, especially in conjunction with problems described by parametric PDEs, where many such systems have to be solved. The primary approach for solving large linear systems is the use of Krylov subspace iterative methods with well-designed preconditioners. Recently, graph neural networks (GNNs) have been used in conjunction with solving parametric PDEs, and have been shown to be promising tool for designing preconditioners. Nevertheless, the GNN-based preconditioners reported in the literature have worse effect on system’s spectrum than analogous preconditioners from classical linear algebra. Here we employ well-established preconditioners from linear algebra as starting point for training GNN to obtain preconditioners that yield a more substantial reduction in the condition number of the systems when compared to classical preconditioners. Numerical experiments demonstrate the efficiency of our approach in comparison to classical and known neural network-based methods for parametric PDEs. In addition, a heuristic justification for the loss function employed in this study is provided, and we demonstrate that preconditioners obtained by learning with this loss function reduce the condition number in a way that is more desirable for the conjugate gradient iterative method.

Computational complexity is regarded by many Computer Science students as extremely difficult and as a topic to be avoided. However, the concepts of an algorithm and of computational complexity as a means of characterising the resource consumption of algorithms are fundamental in Computer Science and are included in all curricula for it. To better motivate students and to increase their interest in computational complexity, this paper suggests introducing it by examining algorithms, a.k.a. recipes, for making noodles. This paper describes several traditional algorithms for making Chinese and Italian noodles and classifies each according to its computational complexity. It compares the power of the algorithms. It considers the nature of variations of the traditional algorithms. It examines machines that implement some of the algorithms. It cites a world speed record for making a large number of noodles using the algorithm with the maximal complexity. It shows how computational thinking and other topics can be introduced in the same manner. It concludes by mentioning avenues for further studies.

We showcase a modular, graphical language—graphical linear algebra—and use it as high-level language to reason calculationally about linear algebra. We propose a minimal framework of six axioms that highlight the dualities and symmetries of linear algebra, and use the resulting diagrammatic calculus as a convenient tool to prove a number of diverse theorems. Our work develops a relational approach to linear algebra, closely connected to classical relational algebra.

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes. The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential role in Eilenberg's variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3. The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schutzenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schutzenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4. The final theme is that of equations. These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

: Numerically intensive calculations are not well supported by Prolog, yet there are important applications that require tightly coupled symbolic and numeric calculations. The Aquarius Numeric Processor (ANP) is an extended numeric Instruction Set Architecture (ISA) based on the Berkeley Programmed Logic Machine (PLM) to support integrated symbolic and numeric calculations. This extension expands the existing numeric data type to include 32- and 64-bit integers, single and double precision floating-point numbers conforming to the IEEE Standard P754. A new class of data structure, numeric arrays, is added to represent matrices and arrays found in most scientific programming languages. Powerful numeric instructions are included to manipulate the new data types. Dynamic Operand Coercion (DOC) provides dynamic type checking and coercing of operands. The ANP and PLM together provide for the efficient execution of symbolic and numeric operations written in AI languages such as Prolog and Lisp. Simulated performance results indicate the system will achieve about 10 MFLOPs on the Prolog version of some Whetstone and Linpack benchmarks and close to 20 MFLOPs on some matrix operations (all in double precision).

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

Based on the author's course at NYU, Linear Algebra and Probability for Computer Science Applications gives an introduction to two mathematical fields that are fundamental in many areas of computer science. The course and the text are addressed to students with a very weak mathematical background. Most of the chapters discuss relevant MATLAB functi

Advances in Group Processes publishes theoretical analyses, reviews, and theory based empirical chapters on group phenomena. The series adopts a broad conception of “group processes.” This includes work on groups ranging from the very small to the very large, and on classic and contemporary topics such as status, power, trust, justice, social influence, identity, decision-making, intergroup relations and social networks. Previous contributors have included scholars from diverse fields including sociology, psychology, political science, economics, business, philosophy, computer science, mathematics and organizational behavior. Volume 42 assembles papers related to a variety of topics in small groups and organizational research. The volume includes papers that address theoretical and empirical issues including 1) literature reviews in the areas of race, gender, masculinity and DEI initiatives, 2) how political party affiliations impact social forces such as situational affective definitions and levels of institutional trust, and 3) whether or not group membership affects the tendency to intervene in fraternity hazing, if gender ideologies are related to support for legalized abortion, and the emotional responses to acts of intergroup forgiveness and revenge. Overall, the volume includes papers that reflect a wide range of theoretical approaches from leading scholars who work in the general area of group processes.