Linear Algebra

The modified solutions of some nonlinear oscillators have been obtained based on the classical iteration procedure. In this article we have used the Fourier series and utilized all of its terms (sometimes approximately) in each iterative step. The third and fourth approximate frequencies of different nonlinear problems show good agreement with the exact values.

W3C's Semantic Web intents a common framework that allows data to be shared and reused across application and enterprise. The semantic web and its related technologies are the main directions of future web development where machine-processable information which supports user tasks. Ontologies are playing the vital role in Semantic Web. Researches on Ontology engineering had pointed out that an effective ontology application development methodology with integrated tool support is mandatory for its success. . Potential benefits are there to ontology engineering in making the toolset of Model Driven Architecture applicable to ontology modeling. Since Software and Ontology engineering are two complimentary branches, the scope of extension of the well proven methodologies and UML based modeling approaches used in software engineering to ontology engineering can bridge the gap between the engineering branches. This research paper is an attempt to suggest an exclusive hybrid methodology for ontology development from existing matured software engineering. Philosophical and engineering aspects of the newly derived methodology have been described clearly An attempt has been made for the application of proposed methodology with protégé editor. The full-fledged implementation of an domain ontology and its validation is the future research direction.

Resumen. El objetivo de este trabajo es argumentar la necesidad de investigar sobre el perfeccionamiento didáctico, a través del enfoque sistémico, en el estudio de los Espacios Vectoriales en el Álgebra Lineal. Como resultado se identifica y delimita un diseño metodológico, como propuesta de investigación, a través del cual se define el problema de investigación, objeto de estudio y el objetivo. Es un estudio descriptivo-interpretativo, se adopta la metodología cualitativa y se utiliza el método de revisión de literatura

Resumen. El objetivo de este trabajo es argumentar la necesidad de investigar sobre el perfeccionamiento didáctico, a través del enfoque sistémico, en el estudio de los Espacios Vectoriales en el Álgebra Lineal. Como resultado se identifica y delimita un diseño metodológico, como propuesta de investigación, a través del cual se define el problema de investigación, objeto de estudio y el objetivo. Es un estudio descriptivo-interpretativo, se adopta la metodología cualitativa y se utiliza el método de revisión de literatura

Álgebra Linear é o estudo dos espac ¸os vetoriais e das transformac ¸ões lineares entre eles. Quando os espac ¸os têm dimensões finitas, as transformac ¸ões lineares possuem matrizes. Também possuem matrizes as formas bilineares e, mais particularmente, as formas quadr áticas. Assim a Álgebra Linear, além de vetores e transformac ¸ões lineares, lida também com matrizes e formas quadr áticas. S ão numerosas e bastante variadas as situac ¸ões, em Matem ática e em suas aplicac ¸ões, onde esses objetos ocorrem. Daí a import ância central da Álgebra Linear no ensino da Matem ática. O presente livro apresenta uma exposic ¸ão introdutória de Álgebra Linear. Ele n ão pressupõe conhecimentos anteriores sobre o assunto. Entretanto convém lembrar que a posic ¸ão natural de um tal curso no currículo universit ário vem após um semestre (pelo menos) de Geometria Analítica a duas e três dimensões, durante o qual o estudante deve adquirir alguma familiaridade, em nível elementar, com a representac ¸ão algébrica de idéias geométricas e vice-versa. Tornou-se quase obrigatório, j á faz alguns anos, dedicar as primeiras sessenta ou mais p áginas de todo livro de Álgebra Linear ao estudo dos sistemas de equac ¸ões lineares pelo método da eliminac ¸ão gaussiana, motivando assim a introduc ¸ão das matrizes e dos determinantes. Somente depois disso s ão definidos os espac ¸os vetoriais. Esse costume n ão é seguido neste livro, cuja primeira sentenc ¸a é a definic ¸ão de espac ¸o vetorial. Mencionarei três razões para isso: (a) A definic ¸ão de Álgebra Linear dada acima; (b) N ão vejo vantagem em longas motivac ¸ões; (c) Sistemas lineares s ão entendidos mais inteligentemente depois que j á se conhecem os conceitos b ásicos de Álgebra Linear. De resto, esses conceitos (n úcleo, imagem, base, posto, subespac ¸o, etc), quando estudados independentemente, têm muitas outras aplicac ¸ões. Variando o conjunto X, obtêm-se diversos exemplos de espac ¸os vetoriais da forma F(X; R). Por exemplo, se X = {1, . . . , n} ent ão F(X; R) = R n ; se X = N ent ão F(X; R) = R ∞ ; se X é o produto cartesiano dos conjuntos {1, . . . , m} e {1, . . . , n} ent ão F(X; R) = M(m × n). Outros exemplos de espac ¸os vetoriais ocorrem como subespac ¸os, como veremos a seguir. Valem num espac ¸o vetorial, como conseq üências dos axiomas, as regras operacionais habitualmente usadas nas manipulac ¸ões numéricas. Vejamos algumas delas.

Resumen. En este trabajo presentamos un modelo de cálculo vectorial discreto sobre redes, estructuradas o no, siguiendo el guión proporcionado por la Geometría Diferencial. La clave para desarrollar un cálculo eficiente sobre variedades discretas que sea mimético al caso continuo reside en una adecuada definición del espacio tangente en cada vértice de la red. Esta estructura permite entonces considerar campos vectoriales, productos internos y tensores métricos generales, que conducen a las versiones discretas de los operadores, derivada, gradiente, divergencia, rotacional y laplaciano. Estos operadores satisfacen propiedades análogas a las verificadas por sus equivalente en el continuo y, en particular, permiten una formulación adecuada de problemas de contorno discretos generales.

La raíz cuadrada de-1, representada hoy por la unidad imaginaria (𝑖), es un concepto que transformó las matemáticas, pero su aceptación no fue inmediata. Su evolución pasó por etapas de escepticismo, resistencia y, finalmente, integración en la teoría matemática y sus aplicaciones científicas. Lo que inicialmente se consideró una idea ilógica o incluso imposible, terminó siendo una de las extensiones más influyentes de las matemáticas. La raíz cuadrada de -1 abrió un mundo de posibilidades para la resolución de ecuaciones, la física de ondas y la ingeniería moderna.

In a recent work by the author [Linear Algebra Appl. 298 (1999) 99] Krylov subspace methods were derived for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax = b, where A ∈ C N ×N is a singular and in general non-hermitian matrix that has an arbitrary index. One of these methods, modeled after the generalized conjugate residual method (GCR) and denoted DGCR, is considered in the present work again. It is shown that all of the approximations produced by DGCR exist, and a GMRES-like algorithm, denoted DGMRES, for its implementation is derived. Like GMRES, DGMRES too is economical computationally and storagewise.

Simultaneous iteration methods are extensions of the power method that were devised for approximating several dominant eigenvalues of a matrix and the corresponding eigenvectors. The convergence analysis of these methods has been given for both hermitian and nonhermitian matrices. In this paper we concentrate on normal matrices which include also the hermitian matrices, and provide a rigorous analysis for a common version of simultaneous iteration methods. We improve on some of the known results and derive some new ones as well. We also present a detailed analysis concerning the formation of spurious eigenvalue approximations.

Collective Identification Procedures (CIPs) model admission rules regulating membership in associations, communities and clubs: the Libertarian identification rule F l is the CIP which essentially relies on self-certification. This paper studies F l in an arbitrary finite atomistic lattice, allowing an unified treatment of collective identification problems with either exogenous or endogenous choice of classification labels. An axiomatic characterization of F l in that general setting is provided and contrasted with previously known characterizations which only work in more specialized (e.g. distributive) lattices, and are therefore confined to collective identification problems with exogenously fixed labels. Non-manipulability properties of F l on a certain simple restricted domain are also considered and shown to hold in any finite atomistic lattice. JEL Classification Numbers: D71, D72.

The theory of M- and H-matrices has become one of the basic tools in applied linear algebra and it has contributed to different areas of mathematical research and applications. Many results in numerical analysis, eigenvalue localization problems, analysis of iterative methods for solving systems of linear equations came from H-matrix theory. Also, many results in engineering rely on mathematical foundation that is, explicitly or implicitly, formulated in terms of H-matrices. In this talk, different matrix properties that guarantee nonsingularity of matrices and define different subclasses of H-matrices will be presented together with related results concerning Schur complement matrices, eigenvalue localization and bounds of the max-norm of the inverse matrix.

Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/ ... Guang-Hui Cheng, Xiao-Yu Cheng, Ting-Zhu Huang†, Tin-Yau Tam‡ ... Abstract Some bounds for the spectral radius of the Hadamard product of two nonneg-ative matrices are given. Some results involve M-...

In this paper, we refer to a asymptotic degree sequence as D = (d 1 , d 2 ,. .. , d n). The examination of topological indices on trees gives us a general overview through bounds to find the maximum and minimum bounds which reflect the maximum and minimum number of edges incident to every vertex in the graph, Albertson index known as uv∈E(G) |d u (G)d v (G)|, Sigma index σ(G) among D of tree T when d n • • • d 1. According to the first zegrb we show for a degree sequence of order n = 4, irr(T) = M 1 (T) 2-2 M 1 (T) + 4 i=1 |x ix i+1 |-(b + c)-1.

This paper deals with the numerical study of melting of phase change material around a horizontal circular cylinder in the presence of the natural convection in the melt phase. A two dimensional unsteady mathematical model has been formulated in terms of primitive variables and a coordinate transformation technique has been used to fix the moving front. The finite volume approach was used to discretize the system of governing equations, boundary and initial conditions and obtain a system of linear algebraic equations. In the numerical solution an implicit scheme was used for the momentum and energy equations and an explicit scheme for the energy balance at the interface. The numerical predictions were compared with available results to establish the validity of the model and the numerical approach.

The high energy physics community is discussing where investment is needed to prepare software for the HL-LHC and its unprecedented challenges. The ROOT project is one of the central software players in high energy physics since decades. From its experience and expectations, the ROOT team has distilled a comprehensive set of areas that should see research and development in the context of data analysis software, for making best use of HL-LHC's physics potential. This work shows what these areas could be, why the ROOT team believes investing in them is needed, which gains are expected, and where related work is ongoing. It can serve as an indication for future cooperation and research proposals.

ROOT is an object-oriented C++ framework conceived in the high-energy physics (HEP) community, designed for storing and analyzing petabytes of data in an efficient way. Any instance of a C++ class can be stored into a ROOT file in a machine-independent compressed binary format. In ROOT the TTree object container is optimized for statistical data analysis over very large data sets by using vertical data storage techniques. These containers can span a large number of files on local disks, the web, or a number of different shared file systems. In order to analyze this data, the user can chose out of a wide set of mathematical and statistical functions, including linear algebra classes, numerical algorithms such as integration and minimization, and various methods for performing regression analysis (fitting). In particular, the RooFit package allows the user to perform complex data modeling and fitting while the RooStats library provides abstractions and implementations for advanced statistical tools. Multivariate classification methods based on machine learning techniques are available via the TMVA package. A central piece in these analysis tools are the histogram classes which provide binning of one-and multi-dimensional data. Results can be saved in high-quality graphical formats like Postscript and PDF or in bitmap formats like JPG or GIF. The result can also be stored into ROOT macros that allow a full recreation and rework of the graphics. Users typically create their analysis macros step by step, making use of the interactive C++ interpreter CINT, while running over small data samples. Once the Preprint submitted to Elsevier 31 August 2015 development is finished, they can run these macros at full compiled speed over large data sets, using on-the-fly compilation, or by creating a stand-alone batch program. Finally, if processing farms are available, the user can reduce the execution time of intrinsically parallel tasks -e.g. data mining in HEP -by using PROOF, which will take care of optimally distributing the work over the available resources in a transparent way.

This book provides a comprehensive introduction to foundational mathematical concepts, structured across five chapters that seamlessly integrate set theory, category theory, and cohomology of groups. It is designed to serve as a resource for students, researchers, and professionals seeking a robust foundation in algebra, topology, and related mathematical structures. Chapter 1 explores the Fundamentals of Set Theory, delving into the nature and properties of sets, relationships between sets, and methods of representation. Set operations such as union, intersection, difference, and complement are discussed in detail, with a focus on their theoretical underpinnings through commutative, associative, and distributive laws, as well as De Morgan's laws. An exercise section consolidates understanding and facilitates practical application. Chapter 2 introduces the Theory of Categories, laying the groundwork for understanding objects, morphisms, and their relationships within categories and subcategories. Fundamental concepts such as monomorphisms, epimorphisms, and the universal properties of products, coproducts, pullbacks, and pushouts are explored. The role of kernels, cokernels, and zero morphisms in exact categories is also examined, emphasizing their importance in algebraic structures. Chapter 3 builds on these foundations with Advanced Concepts in Category Theory. It highlights functor categories, natural transformations, and quotient categories, providing a deeper insight into their construction and properties. Equalizers, coequalizers, and exact functors are characterized, and their applica

We introduce a new abstract domain, namely the domain of Interval Linear Equalities (itvLinEqs), which generalizes the affine equality domain with interval coefficients by leveraging results from interval linear algebra. The representation of itvLinEqs is based on a row echelon system of interval linear equalities, which natively allows expressing classical linear relations as well as certain topologically non-convex (even unconnected or non-closed) properties. The row echelon form limits the expressiveness of the domain but yields polynomial-time domain operations. Interval coefficients enable a sound adaptation of itvLinEqs to floating-point arithmetic. itvLinEqs can be used to infer and propagate interval linear constraints, especially for programs involving uncertain or inexact data. The preliminary experimental results are encouraging: itvLinEqs can find a larger range of invariants than the affine equality domain. Moreover, itvLinEqs provides an efficient alternative to polyhedra-like domains.

We introduce a new numerical abstract domain, so-called interval polyhedra (itvPol), to infer and propagate interval linear constraints over program variables. itvPol, which allows to represent constraints of the form k [a k , b k ]x k ≤ c, is more expressive than the classic convex polyhedra domain and allows to express certain non-convex (even unconnected) properties. The implementation of itvPol can be constructed based on interval linear programming and an interval variant of Fourier-Motzkin elimination. The preliminary experimental results of our prototype are encouraging, especially for programs affected by interval uncertainty, e.g., due to uncertain input data or interval-based abstractions of disjunctive, non-linear, or floating-point expressions. To our knowledge, this is the first application of interval linear algebra to static analysis.

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect $k$-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Many combinatorial optimization problems can be modeled concisely with a system of polynomial equations. Examples include the detection of k-colorings, stable sets, flows, matchings, and satisfiability (see and the references therein). It follows that solving general systems of polynomial equations is at least NP-hard. For this reason, mathematicians have rarely used nonlinear polynomials for practical computation or to provide complexity bounds (although they can be very useful otherwise ). In this article, we discuss four iterative algorithms tailored to solve combinatorial systems of polynomial equations. We explain how these algebraic procedures can be applied to integer hull approximation and also the recognition of combinatorial properties such as k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. We report on computational complexity bounds, structural results, and computer experiments. When the field of coefficients is the real numbers our methodology closely resembles other iterative procedures such as Lovász-Schrijver, Sherali-Adams, the Lasserre hierarchy, and others that are used in integer programming and optimization over semialgebraic sets . The algorithms we present are also related to the solvability methods of Laurent, Lasserre and Rostalski . The key difference is that we work over arbitrary fields of coefficients which allows a wider range of modeling.

In this paper we prove the following negative result: Product Logic [9] does not have the Deduction Theorem, that is,_ there is no binary defined connective in the language of Product Logic such that the Ded uction Theorem is satisfied with respect to it. We prove this theorem mainly by using algebraic methods: we prove that Product Logic is algebraizable, that the variety of Product Algebras is its equivalent quasivariety semantics and that this variety has no equationally definable principal congruences.

In this study, initially the geometric meanings of the curvatures of the curves parametrized with the arc length are given in E 5 . This is followed by the calculation of the Frenet vectors and curvatures of any curve. After these, some results have been given for the state of evolute curve X being a W-curve and the Frenet vectors and curvatures of involute curve Y have been calculated in terms of Frenet vectors and curvatures of the curve X. At last, the differential equation of the spherical curves, the equation of the radius and the center of the osculating hyperspheres have been achieved in E 5 .

In this study, initially the geometric meanings of the curvatures of the curves parametrized with the arc length are given in $E^5$. This is followed by the calculation of the Frenet vectors and curvatures of any curve. After these, some results have been given for the state of evolute curve $X$ being a W-curve and the Frenet vectors and curvatures of involute curve $Y$ have been calculated in terms of Frenet vectors and curvatures of the curve X. At last, the differential equation of the spherical curves, the equation of the radius and the center of the osculating hyperspheres have been achieved in $E^5$.

We present our experience with the design and implementation of a mathematical extension for an object-oriented programming language. Primarily this includes support for multidimensional matrices, indexing by ranges and vectors, sparse data structures. As first class citizens in the language, they lead to a more natural code in scientific programs and enable powerful compiler optimizations. We discuss our concept of smoothly integrating mathematical constructs into the language and compiler on top of .NET. Finally we show that this approach results in a reasonably good performance and thereby we prove the suitability of the (fully managed) .NET platform for high performance computing applications.

In this study we present some modifications in the semiempirical quantum chemistry MOPAC2009 code that accelerate single-point energy calculations (1SCF) of medium-size (up to two thousand five hundred atoms) molecular systems using GPU co-processors and multi-threaded shared-memory CPUs. Our modifications consisted of using a combination of highly-optimized linear algebra libraries for both CPU (LAPACK and BLAS from Intel MKL) and GPU (MAGMA and CUBLAS) to hasten time-consuming parts of MOPAC such as the pseudo-diagonalization, full diagonalization and density matrix assembling. We have shown that it is possible to obtain large speedups just by using CPU serial linear algebra libraries in the MOPAC code. As a special case we show a speedup of up to 14 times for a methanol simulation box containing 2400 atoms and 4800 basis functions, with even greater gains in performance when using multi-threaded CPUs (2.1 times in relation to the single-threaded CPU code using linear algebra libraries) and GPUs (3.8 times). This degree of acceleration opens new perspectives for modeling larger structures which appear in inorganic chemistry (such as zeolites and MOFs), biochemistry (such as polysaccharides, small proteins and DNA fragments) and materials science (such as nanotubes and fullerenes). In addition, we believe that this parallel (GPU-CPU) MOPAC code will make it feasible to use semiempirical methods in lengthy molecular simulations using both hybrid QM/MM and QM/QM potentials.

We are concerned about a view in undergraduate computer science education, especially in the early courses, that it's okay to be math-phobic and still prepare oneself to become a computer scientist. Our view is the contrary: that any serious study of computer science requires students to achieve mathematical maturity (especially in discrete mathematics) early in their undergraduate studies, thus becoming well-prepared to integrate mathematical ideas, notations, and methodologies throughout their study of computer science. A major curricular implication of this theme is that the prerequisite expectations and conceptual level of the first discrete mathematics course should be the same as it is for the first calculus course --- secondary school pre-calculus and trigonometry. Ultimately, calculus, linear algebra, and statistics are also essential for computer science majors, but none should occur earlier than discrete mathematics. This paper explains our concerns and outlines our re...

Given an edge-weighted graph where all weights are nonnegative reals, an edge reweighting is an assignment of nonnegative reals to edges such that, for each vertex, the sums of given and new weights assigned to the edges incident on the vertex do coincide. An edge is then said to be invariant if its weight is the same for any edge reweighting. We show that the set of invariant edges of an arbitrary edge-weighted graph can be determined in time linear in the size of the underlying graph. Moreover, an application to the security of statistical data is discussed.

By elementary methods of combinatorial mathematics and uniqueness of solutions systems of linear algebraic equations for non degenerate cases proved a theorem about the number and structure of the singular points of n-dimensional dynamical system of population a dynamics Lotka-Volterra model. Showed that the number of singular points for this system is equal to 2n and their structure on a combination of zero and nonzero coordinates coincides with the binomial coefficients.

Absfrucr-A constrained transform coding procedure is developed which is a combination of transform coding with differential pulse code modulation. The algorithm avoids block boundary mismatch errors, yet retains the coding efficiency of transform coding. A general theory of constrained transform coding is developed which includes the discrete cosine transformation and tensor products of splines as special cases. Results using the cosines and splines are given for two images. A complete discussion of the necessary linear algebra background is also given.

УДК 512.7+512.9, 688.321 Р. В. Скуратовський, викладач МАУП, Інститут комп'ютерних та інформаційних технологій, м. Київ Запропонований нами метод факторизації, на відміну від більшості різновидів методу GNFS [1, 2] окрім kGNFS, володіє всіма властивостями для успішного застосування паралельних обчислень. Ключові слова: факторизація числа, паралельні обчислення.

We discuss the solution of regularized linear programs using a primal-dual barrier method. Our implementation is based on indefinite Cholesky-type factorizations of full and reduced KKT systems. Regularization improves the stability of the Cholesky factors and allows infeasibility to be detected, but it affects the efficiency of Cross-over to Simplex (to obtain a basic solution to the original problem). We explore these effects by running OSL on the larger Netlib examples.

Background: Clouds and MapReduce have shown themselves to be a broadly useful approach to scientific computing especially for parallel data intensive applications. However they have limited applicability to some areas such as data mining because MapReduce has poor performance on problems with an iterative structure present in the linear algebra that underlies much data analysis. Such problems can be run efficiently on clusters using MPI leading to a hybrid cloud and cluster environment. This motivates the design and implementation of an open source Iterative MapReduce system Twister. Results: Comparisons of Amazon, Azure, and traditional Linux and Windows environments on common applications have shown encouraging performance and usability comparisons in several important non iterative cases. These are linked to MPI applications for final stages of the data analysis. Further we have released the open source Twister Iterative MapReduce and benchmarked it against basic MapReduce (Hadoop) and MPI in information retrieval and life sciences applications. Conclusions: The hybrid cloud (MapReduce) and cluster (MPI) approach offers an attractive production environment while Twister promises a uniform programming environment for many Life Sciences applications. Methods: We used commercial clouds Amazon and Azure and the NSF resource FutureGrid to perform detailed comparisons and evaluations of different approaches to data intensive computing. Several applications were developed in MPI, MapReduce and Twister in these different environments.

In thG paper the "convergence" of an iterative refinement procedure for solving triangular linear algebraic systems is proved. The application of this procedure to the "accurate" LU decomposition of near-singular systems is descm'bed. A simple nunaerical example illustrates the theory.

Введение. За последние несколько десятилетий язык программирования Python превратился в первоклассный инструмент для научных вычислений, включая анализ и визуализацию больших наборов данных. Это может удивить давних поклонников Python: сам по себе этот язык не был создан в расчете на анализ данных или научные вычисления. В 2014 г. Фернандо Перес (Fernando Perez) анонсировал дополнительный уникальный проект от IPython под названием Project Jupyter (). Он похож на интерфейс блокнота других программ, таких как Maple, Mathematica и SageMath, стиль вычислительного интер-УДК 004.43 + 519.

A new definition of the Fibonacci Matrix is given. The elements of the matrix consist of the Fibonacci numbers. First a preliminary knowledge on Fibonacci sequences and their properties are given. Then the new definition of the matrix is given together with some properties. The difference from the common definition is also discussed. The determinant of the matrix and its properties are posed and proven. Applications to systems of algebraic equations are also outlined.

We develop highly efficient parallel pricing methods on Graphics Processing Units (GPUs) for multi-asset American options via a Partial Differential Equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Finite difference methods on uniform grids are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The discrete nonlinear penalized equations at each timestep are solved using a penalty iteration. A GPU-based parallel Alternating Direction Implicit Approximate Factorization technique is employed for the solution of the linear algebraic system arising from each penalty iteration. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets.

Sparse storage formats are techniques for storing and processing the sparse matrix data efficiently. The performance of these storage formats depend upon the distribution of non-zeros, within the matrix in different dimensions. In order to have better results we ...

The coupled Navier-Stokes and Cahn-Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics. One is the ease in simulating topological changes such as interfacial rupture and coalescence. Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

This is a study about the didactical organization of a research based group of activities designed using APOS theory to help university students make constructions needed to understand and graph two-variable functions, but found to be lacking in previous studies. The model of the “moments of study” of the Anthropological Theory of Didactics is applied to analyze the activities in terms of their institutional viability.

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets into the grooves. On comparing with some molecular-dynamics simulation measures, it is pointed out that the applied pressure, which has to be less than the capillary pressure in the superhydrophobic state, can be correlated with the penetration depth of the no-shear interface.