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We review our recently developed methods for large-scale electronic structure calculations, both in one-electron theory and many-electron theory. The method are based on the density matrix representation, together with the Wannier state representation and the Krylov subspace method, in one-electron theory of a-few-tens nm scale systems. The hybrid method of quantum mechanical molecular dynamical simulation is explained.The Krylov subspace method, the CG (conjugate gradient) method and the shifted-COCG (conjugate orthogonal conjugate gradient) method, can be applied to the investigation of the ground state and the excitation spectra in many-electron theory. The mathematical foundation of the Krylov subspace method for large-scale matrix computation is focused and the key technique of the shifted-COCG method, e.g. the collinear residual and seed switching, is explained. A wide variety of applications of these extended novel algorithm is also explained. These are the fracture formation and propagation, liquid carbon and formation process of gold nanowires, together with the application to the extend Hubbard model.

Includes bibliographies

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.​

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.​

GATE CSE Resources Theory of Computation These notes/pdfs are created by Applied AI, and I have uploaded here so that everyone can easily access them. You can also download the complete GATE CSE syllabus notes at this link:  https://archive.org/details/gate-applied-ai-notes NOTE: Complete syllabus file is huge (530 MB) If you need some more resources for GATE, you can check out my GitHub repo:  https://github.com/AnuragAnalog/GateResources

GATE CSE Resources Theory of Computation These notes/pdfs are created by Applied AI, and I have uploaded here so that everyone can easily access them. You can also download the complete GATE CSE syllabus notes at this link:  https://archive.org/details/gate-applied-ai-notes NOTE: Complete syllabus file is huge (530 MB) If you need some more resources for GATE, you can check out my GitHub repo:  https://github.com/AnuragAnalog/GateResources

GATE CSE Resources Theory of Computation These notes/pdfs are created by Applied AI, and I have uploaded here so that everyone can easily access them. You can also download the complete GATE CSE syllabus notes at this link:  https://archive.org/details/gate-applied-ai-notes NOTE: Complete syllabus file is huge (530 MB) If you need some more resources for GATE, you can check out my GitHub repo:  https://github.com/AnuragAnalog/GateResources

Tri 3 – 2016/2017

Tri 3 – 2017/2018

Tri 3 – 2017/2018

Tri 3 – 2017/2018

Tri 3 – 2017/2018

ksiazka

Bangalore University students can refer to this book or students who are interested in the field of CS, IT, MCA, BTECH, and many others field can refer this book

The work shows that it is possible to formalize all properties regularly observed in (deterministic and non-deterministic) algorithms in second-order predicate calculus. Moreover, it is shown that for any given algorithm it suffices to know how to formalize its 'partial correctness' by a second-order formula in order to formalize all other properties by second-order formulas. The result is of special interest since 'partial correctness' has already been formalized in second-order predicate calculus for many classes of algorithms.

This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.

A generalization of the semiclassical cubic spline known in the literature as the exponential spline is discussed. In actuality, the exponential spline represents a continuum of interpolants ranging from the cubic spline to the linear spline. A particular member of this family is uniquely specified by the choice of certain tension parameters. The theoretical underpinnings of the exponential spline are outlined. This development roughly parallels the existing theory for cubic splines. The primary extension lies in the ability of the exponential spline to preserve convexity and monotonicity present in the data. Next, the numerical computation of the exponential spline is discussed. A variety of numerical devices are employed to produce a stable and robust algorithm. An algorithm for the selection of tension parameters that will produce a shape preserving approximant is developed. A sequence of selected curve-fitting examples are presented which clearly demonstrate the advantages of exponential splines over cubic splines.

In this paper we describe the numerical analysis underlying our efforts to develop an accurate and reliable code for simulating flame propagation using complex physical and chemical models. We discuss our spatial and temporal discretization schemes, which in our current implementations range in order from two to six. In space we use staggered meshes to define discrete divergence and gradient operators, allowing us to approximate complex diffusion operators while maintaining ellipticity. Our temporal discretization is based on the use of preconditioning to produce a highly efficient linearly implicit method with good stability properties. High order for time accurate simulations is obtained through the use of extrapolation or deferred correction procedures. We also discuss our techniques for computing stationary flames. The primary issue here is the automatic generation of initial approximations for the application of Newton's method. We use a novel time-stepping procedure, which allows the dynamic updating of the flame speed and forces the flame front towards a specified location. Numerical experiments are presented, primarily for the stationary flame problem. These illustrate the reliability of our techniques, and the dependence of the results on various code parameters.

In this paper we describe the numerical analysis underlying our efforts to develop an accurate and reliable code for simulating flame propagation using complex physical and chemical models. We discuss our spatial and temporal discretization schemes, which in our current implementations range in order from two to six. In space we use staggered meshes to define discrete divergence and gradient operators, allowing us to approximate complex diffusion operators while maintaining ellipticity. Our temporal discretization is based on the use of preconditioning to produce a highly efficient linearly implicit method with good stability properties. High order for time accurate simulations is obtained through the use of extrapolation or deferred correction procedures. We also discuss our techniques for computing stationary flames. The primary issue here is the automatic generation of initial approximations for the application of Newton's method. We use a novel time-stepping procedure, which allows the dynamic updating of the flame speed and forces the flame front towards a specified location. Numerical experiments are presented, primarily for the stationary flame problem. These illustrate the reliability of our techniques, and the dependence of the results on various code parameters.

The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases, the function may belong to one of various nondeterministic function classes) polynomial-time function f such that when at least one of x and y belongs to L, then f(x,y) \in L \cap \{x,y\}. The intuition here is that it is saying: ``Regarding membership in L, if you put a gun to my head and forced me to bet on one of x or y as belonging to L, my money would be on f(x,y).'' In this article, we present a number of open problems from the theory of semifeasible algorithms. For each we present its background and review what partial results, if any, are known.

The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases, the function may belong to one of various nondeterministic function classes) polynomial-time function f such that when at least one of x and y belongs to L, then f(x,y) \in L \cap \{x,y\}. The intuition here is that it is saying: ``Regarding membership in L, if you put a gun to my head and forced me to bet on one of x or y as belonging to L, my money would be on f(x,y).'' In this article, we present a number of open problems from the theory of semifeasible algorithms. For each we present its background and review what partial results, if any, are known.

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.

The aim of this research effort was to extend earlier work airfoil leading-edge vortex (LEV) shedding to finite-wing flows. The current research shows that leading-edge suction, which was shown in the earlier work to govern LEV formation in airfoils with rounded leading edges, also governs LEV formation on finite wings. It is shown that when the maximum value of the span wise distribution of leading-edge suction on a finite wing reaches a critical value, LEV initiation takes place. The critical value is the same as that for the corresponding airfoil section, allowing it to be determined from 2D experiments or computations. Further, the critical value is independent of motion kinematics so long as LEV formation is not preceded by significant trailing-edge flow reversal. This insight was used to augment an in viscid unsteady vortex lattice method (UVLM) to handle LEV shedding from finite wings by using a vortex sheet to model the LEV shedding along the span. By convecting the LEV sheet using local velocity, the UVLM was able to predict vortex-sheet roll up, which agreed well with high-order computations. The current research has brought to light important insights in the initiation of LEV shedding on finite wings, which can be used in low-order modeling and flow control.

One of the major problems encountered in trying to formally verify the correctness of computer programs that use real arithmetic (hereinafter referred to as mathematical programs) is that the mathematical properties of real arithmetic operations in computers are much more complicated and much harder to work with than the mathematical properties of the corresponding ideal mathematical operations. This occurs because the real number type implemented on a finite computer is not the same as the ideal, mathematical real number type. A finite machine can only represent finitely many different real numbers, whereas there are infinitely many ideal real numbers. The idea behind the theory of asymptotic computing is to develop techniques to prove that the accuracy of a mathematical program goes to infinity (e.g., larger and larger numbers of representation bits for mantissas and exponents used in binary floating point arithmetic). The theory of asymptotic computing, then, is essentially a general formalization of the notions of accuracy and accuracy going to infinity, but without having to show how fast convergence happens (a major source of difficulty in numerical analysis). Keywords: Software verification.

One of the major problems encountered in trying to formally verify the correctness of computer programs that use real arithmetic (hereinafter referred to as mathematical programs) is that the mathematical properties of real arithmetic operations in computers are much more complicated and much harder to work with than the mathematical properties of the corresponding ideal mathematical operations. This occurs because the real number type implemented on a finite computer is not the same as the ideal, mathematical real number type. A finite machine can only represent finitely many different real numbers, whereas there are infinitely many ideal real numbers. The idea behind the theory of asymptotic computing is to develop techniques to prove that the accuracy of a mathematical program goes to infinity (e.g., larger and larger numbers of representation bits for mantissas and exponents used in binary floating point arithmetic). The theory of asymptotic computing, then, is essentially a general formalization of the notions of accuracy and accuracy going to infinity, but without having to show how fast convergence happens (a major source of difficulty in numerical analysis). Keywords: Software verification.

This report results from a contract tasking Computing Center of Academy of Sciences as follows: The description of the mechanical characteristics and prediction of the behavior of composite materials reinforced by short carbon fibers and fillers is the goal of the project. The development of the theoretical techniques of the research of reinforced composite materials with taking into account of interphasic interactions between carbon fibers and the matrix represents the task of the project. The experimental validation of the theoretical research by using the measured mechanical properties of reinforced composite materials is also the task of the project. The accomplishment of such tasks has been possible thanks to the recent program of mathematical description and modeling of interfacial interactions and also to the appropriate experimental test. To describe the effective medium characteristics with nanoscale structures the homogenization problem is solved. The methods of solutions of the identification problems are developed within framework of the Project for designed models with nanoscale structures. A strong impact on the knowledge of composite theoretical behavior is expected as a result of this Project. The obtained experimental results should support the extension of the mathematical description to other kinds of composites, mainly fiber/epoxy, carbon/carbon etc.

Tri 1 2022/2023

Tri 1 2022/2023

Tri 1 2022/2023

Tri 1 2022/2023

This booklet contains a variety or comments regarding J. Glenn Brookshear's text "Theory of Computation: Formal Languages, Automata, and Complexity" (ISBN 0-8053-0143-7) along with some ideas and transparency masters that a teacher or a course based on that text may want to use. The booklet contains a chapter corresponding to each chapter in the parent text and concludes with a section of transparency masters. Within each chapter are sections entitled General Comments, Comments Regarding Chapter Review Problems, and Additional Problems. ISBN 0-8053-0144-5

Includes bibliographical references (p. 381-385) and index

Includes bibliographical references (p. 381-385) and index

Includes bibliographical references (p. 381-385) and index

Includes bibliographical references (p. 381-385) and index

Includes bibliographical references (p. 381-385) and index

Includes bibliographical references (p. 381-385) and index

The theory of finite Fourier transforms is developed from the definitions of infinite transforms and applied to the computation of convolutions, correlations, and power spectra. Detailed procedures for these computations are given, including listings and writeups of FORTRAN subroutines.

Continuum reduction and Monte Carlo simulation are used to calculate the heavy quark potential and the string tension in large N Yang-Mills theory in four dimensions. The potential is calculated out to a separation of nine lattice units on a lattice with extent six in each direction.

The aim in this note is to define an algorithm to carry out minimal curvature spherical harmonics interpolation, which is then used to calculate the Laplacian for multi-electrode EEG data analysis. The approach taken is to respect the data. That is, we implement a minimal curvature condition for the interpolating surface subject to the constraints determined from the multi-electrode data. We implement this approach using spherical harmonics interpolation. In this elegant example we show that minimization requirement and constraints complement each other to fix all degrees of freedom automatically, as occurs in gauge theories. That is, the constraints are respected, while only the orthogonal subspace minimization constraints are enforced. As an example, we discuss the application to interpolate control data and calculate the temporal sequence of laplacians from an EEG Mismatch Negativity (MMN) experiment (using an implementation of the algorithm in IDL).

The focus of this Basic Research Initiative has been on the fundamental aspects of ultrafast laser-matter interactions that are involved in femtosecond laser micromachining. The main scientific results of the program are as follows:1. We have introduced a new universal criterion of femtosecond laser ablation that is independent on the angle of incidence and polarization of the laser beam.2. We have conducted experiments on laser-driven microexplosions in sapphire and on the interface of silicon and fused silica. We have discovered several new super-dense phases of aluminum and silicon.3. We have quantified different ionization channels in femtosecond laser ablation and devised an improved rate equation for ionization for several technologically important transparent dielectrics (sapphire, fused silica, soda-lime glass).

This report presents two numerical methods considered for the computation of fuel-optimal, low-thrust orbit transfers in large numbers of burns. The origins of these methods are observations made with the extremal solutions of transfers in small numbers of burns; there seems to exist a trend such that the longer the time allowed to perform an optimal transfer the less fuel that is used. These longer transfers are obviously of interest since they require a motor of low thrust; however, we also find a trend that the longer the time allowed to perform the optimal transfer the more burns are required to satisfy optimality. Unfortunately, this usually increases the difficulty of computation. Both of the methods described use small-numbered burn solutions to determine solutions in large numbers of burns. One method is a homotopy method that corrects for problems that arise when a solution requires a new burn or coast arc for optimality. The other method is to simply patch together long transfers from smaller ones. An orbit correction problem is solved to develop this method. This method may also lead to a good guidance law for transfer orbits with long transfer times.

This report presents two numerical methods considered for the computation of fuel-optimal, low-thrust orbit transfers in large numbers of burns. The origins of these methods are observations made with the extremal solutions of transfers in small numbers of burns; there seems to exist a trend such that the longer the time allowed to perform an optimal transfer the less fuel that is used. These longer transfers are obviously of interest since they require a motor of low thrust; however, we also find a trend that the longer the time allowed to perform the optimal transfer the more burns are required to satisfy optimality. Unfortunately, this usually increases the difficulty of computation. Both of the methods described use small-numbered burn solutions to determine solutions in large numbers of burns. One method is a homotopy method that corrects for problems that arise when a solution requires a new burn or coast arc for optimality. The other method is to simply patch together long transfers from smaller ones. An orbit correction problem is solved to develop this method. This method may also lead to a good guidance law for transfer orbits with long transfer times.

Tri 2 – 2015/2016

Tri 2 – 2015/2016

Tri 2 – 2015/2016

We have studied the formation of Hall-qubit (LLL state) in Quantum Hall effect due to the Aharonov-Bhom oscillation of quasiparticles.The spin echo method plays the key role in the topological entanglement of qubits. The proper ratio of fluxes for maximally entangling qubits has also been pointed out. The generation of higher Quantum Hall state may be possible with the help of quantum teleportation.

In these lectures the authors sketches a theory of functions and relations that are computable and specifiable, uniformly over a class of many sorted algebras. Such a class K of algebras is a mathematical model of the semantics of software module; in particular, a module defining an abstract data type. The theory of computation, specification and verification over K has many applications; those I will describe concern the following subjects: Program specification and verification; logic programming modules and abstract data types; and synchronous concurrent algorithms and their application in hardware design. The author's research on these apparently disparate applications is part of a program based on the algebraic theory of abstract data types, that aims at the formulation and analysis of the many interesting notions of computability, specifiability, and verifiability that exist in different areas of computer science and mathematics. (KR)

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems.

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems.