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Quartic scaling second-order approximate coupled cluster singles and doubles via tensor hypercontraction: THC-CC2

JOURNAL OF CHEMICAL PHYSICS 2013年第12期138卷 124111-124111页

作者： Hohenstein, Edward G. Kokkila, Sara I. L. Parrish, Robert M. Martinez, Todd J. Stanford Univ Dept Chem Stanford CA 94305 USA Stanford Univ PULSE Inst Stanford CA 94305 USA SLAC Natl Accelerator Lab Menlo Pk CA 94025 USA Georgia Inst Technol Sch Chem & Biochem Ctr Computat Mol Sci & Technol Atlanta GA 30332 USA

The second-order approximate coupled cluster singles and doubles method (CC2) is a valuable tool in electronic structure theory. Although the density fitting approximation has been successful in extending CC2 to larger molecules, it cannot address the steep O(N-5) scaling with the number of basis functions, N. Here, we introduce the tensor hypercontraction (THC) approximation to CC2 (THC-CC2), which reduces the scaling to O(N-4) and the storage requirements to O(N-2). We present an algorithm to efficiently evaluate the THC-CC2 correlation energy and demonstrate its quartic scaling. This implementation of THC-CC2 uses a grid-based least-squares THC (LS-THC) approximation to the density-fitted electron repulsion integrals. The accuracy of the CC2 correlation energy under these approximations is shown to be suitable for most practical applications. (C) 2013 American Institute of Physics. [http://***/10.1063/1.4795514]

关键词： ELECTRONIC structure LEAST squares APPROXIMATION theory TENSOR algebra ALGORITHMS ELECTRONS

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Computational resolution of Diophantine equations by means of alpha-dense curves

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KYBERNETES 2012年第1-2期41卷 51-67页

作者： Mvondo, Esther Claudine Bitye Cherruault, Yves Mazza, Jean-Claude Univ Paris 06 Paris France

Purpose - The purpose of this paper is to use alpha-dense curves for solving some Diophantine equations, such as Pythagorean triples, Linear Diophantine equations, the Pell Fermat equation, the Mordell equation for positive values. Design/methodology/approach - The paper's aim is to present the applications in number theory of a new method based on alpha-dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. The alpha-dense curves generalize the space filling curves (Peanocurves,. . .) and fractal curves. This technique can be used for solving all problems of operational research in a simple way. The main idea consists in expressing n variables by means of a single one. Findings - Apply the method to number theory. One of the most important applications is related to global optimization. Multivariable optimization problems coming from operational research or from industry can be quickly and easily solved. Originality/value - The paper presents a new method based on alpha-dense curves for solving Diophantine equations.

关键词： polynomials algebra Operational research Global optimization Integer programming number theory Diophantine equations

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Poisson-type limit theorems for the generalised linear inclusion

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DISCRETE MATHEMATICS AND applications 2012年第4期22卷 477-491页

作者： Kopyttsev, V. A. Mikhailov, V. G.

Let A be a T x n random matrix over the finite field GF (q) let D be some set of nonzero elements of an n-dimensional space V-n over GF (q),and for every x epsilon D let a set B (x) of elements of a T -dimensional space V-T over the same field be given. We find sufficient conditions for convergence, as n;T -> infinity, of the distribution of the number of solutions of the system of inclusions x epsilon D, Ax epsilon B(x) to Poisson-type distributions. The obtained results allow us to investigate asymptotic properties of the number of solutions of a system of the form x epsilon D, Ax + S(x) epsilon B, where S(x) is an arbitrary mapping of V-n into V-T, and hence, properties of the distribution of the number of solutions of random polynomial equations.

关键词： LIMIT theorems (Probability theory) POISSON'S equation DIFFERENTIAL inclusions THEOREMS (Mathematics) FINITE fields (algebra) polynomials

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An orthogonal polynomials transform-based variable block size adaptive vector quantisation for colour image coding

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IET IMAGE PROCESSING 2012年第6期6卷 635-646页

作者： Krishnamoorthy, R. Punidha, R. Anna Univ Technol Comp Vis Lab Dept CSE Tiruchirappalli 620024 Tamil Nadu India

In this paper, a new adaptive vector quantisation with variable block size based on orthogonal polynomials is proposed for colour image coding. The input colour image is partitioned into fixed larger block sizes and the homogeneity/complexity of the local region is tested with the mean-normalised variance classifier. The homogeneous regions are encoded with fixed larger blocks and the complex regions are further divided into smaller block sizes for encoding as it gives the wealth of low-level information such as the high detail and high number of contrast edges. The integer-based orthogonal polynomials transform is then applied on the variable block sizes utilising the inter-correlation property of individual colour planes and interactions among the colour planes so as to reduce the vector dimension and hence the less computational cost. The proposed technique generates two dynamic codebooks and two backup codebooks adaptively and use rate-distortion cost criteria to encode both the homogeneous and complex regions separately. Since the proposed method encodes all the three colour components in single pass, the colour image encoding time is slightly higher than that of grey scale image coding but in contrast to three times in the existing colour image coding techniques.

关键词： backup codebooks grey scale image coding colour image coding techniques polynomials image colour analysis inter-correlation property computational cost input colour image computer vision and image processing techniques integer programming Image and video coding image coding Integral transforms complex regions variable block size adaptive vector quantisation low-level information algebra colour image encoding time image classification mean-normalised variance classifier contrast edges integer-based orthogonal polynomials transform dynamic codebooks homogeneous regions rate distortion theory homogeneous redions transforms vector dimension rate-distortion cost criteria vector quantisation

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Skew-adjacency matrices of graphs

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LINEAR algebra AND ITS applications 2012年第12期436卷 4512-4529页

作者： Cavers, M. Cioaba, S. M. Fallat, S. Gregory, D. A. Haemers, W. H. Kirkland, S. J. McDonald, J. J. Tsatsomeros, M. Queens Univ Dept Math & Stat Kingston ON K7L 3N6 Canada Univ Calgary Dept Math & Stat Calgary AB T2N 1N4 Canada Univ Delaware Dept Math Sci Newark DE 19716 USA Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Tilburg Univ Dept Econometr & Oper Res NL-5000 LE Tilburg Netherlands Natl Univ Ireland Maynooth Hamilton Inst Maynooth Kildare Ireland Washington State Univ Dept Math Pullman WA 99164 USA

The spectra of the skew-adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs. This leads to the following topics: graphs whose skew-adjacency matrices are all cospectral: relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew-adjacency matrices;skew-spectral radii and an analogue of the Perron-Frobenius theorem;and the number of skew-adjacency matrices of a graph with distinct spectra. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Skew-adjacency matrices Graph spectra Odd-cycle graphs Matchings polynomials Pfaffians

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Real computation with least discrete advice: A complexity theory of nonuniform computability with applications to effective linear algebra

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ANNALS OF PURE AND APPLIED LOGIC 2012年第8期163卷 1108-1139页

作者： Ziegler, Martin Tech Univ Darmstadt FB4 D-64289 Darmstadt Germany

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f : X (sic) x bar right arrow f (x) is an element of Y, additional structural information about the input x is an element of X (e.g. any kind of promise that x belongs to a certain subset X' subset of X, or does not) should be taken advantage of. In several examples from real number computation, such advice even makes the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A.(x) over bar for a given singular real n x n-matrix A is possible when knowing rank(A) is an element of {0, 1, ..., n - 1);and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors to) a given real symmetric n x n-matrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n-fold (i.e. roughly log n bits of) additional information is indeed necessary in order to render this problem (continuous and) computable;whereas for finding some single eigenvector of A, providing the truncated binary logarithm of the dimension of the least-dimensional eigenspace of A-i.e. [1 + log(2) n]-fold advice-is sufficient and optimal. Our proofs employ, in addition to topological considerations common in Recursive Analysis, also combinatorial arguments. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Recursive Analysis Topological complexity Linear algebra

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Markov measures and extended zeta functions

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2012年第1-2期38卷 305-323页

作者： Jorgensen, P. E. T. Paolucci, A. M. Univ Iowa Dept Math Iowa City IA 52224 USA Max Planck Inst Math D-53111 Bonn Germany

In this paper we study a family of representations of the Cuntz algebras O-p where p is a prime. These algebras are built on generators and relations. They are C*-algebras and their representations are a part of non-commutative harmonic analysis. Starting with specific generators and relations we pass to an ambient C*- algebra, for example in one of the Cuntz-algebras. Our representations are motivated by the study of frequency bands in signal processing: We construct induced measures attached to those representations which turned out to be related to a class of zeta functions. For a particular case those measures give rise to a class of Markov measures and q-Bernoulli polynomials. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter q, and an associated family of q-relations where the number q is a "quantum-deformation," and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this q-zeta function.

关键词： Commutator Quantum theory Signal processing Zeta functions Hilbert space Spectrum

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Belief functions on distributive lattices 26

Belief functions on distributive lattices

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26th AAAI Conference on Artificial Intelligence and the 24th Innovative applications of Artificial Intelligence Conference, AAAI-12 / IAAI-12

作者： Zhou, Chunlai Department of Computer Science and Technology School of Information Renmin University of China Beijing 100872 China

ISBN: (纸本)9781577355687

The Dempster-Shafer theory of belief functions is an important approach to deal with uncertainty in AI. In the theory, belief functions are defined on Boolean algebras of events. In many applications of belief functions in real world problems, however, the objects that we manipulate is no more a Boolean algebra but a distributive lattice. In this paper, we extend the Dempster-Shafer theory to the setting of distributive lattices, which has a mathematical theory as attractive as in that of Boolean algebras. Moreover, we apply this more general theory to a simple epistemic logic the first-degree-entailment fragment of relevance logic R, provide a sound and complete axiomatization for reasoning about belief functions for this logic and show that the complexity of the satisfiability problem of a belief formula with respect to the class of the corresponding Dempster-Shafer structures is NP-complete. Copyright © 2012, Association for the Advancement of Artificial Intelligence. All rights reserved.

关键词： Boolean algebra

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Least common container of tree pattern queries and its applications

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ACTA INFORMATICA 2012年第3期49卷 173-202页

作者： Wang, Junhu Yu, Jeffrey Xu Pang, Chaoyi Liu, Chengfei Griffith Univ Gold Coast Qld Australia Chinese Univ Hong Kong Sha Tin Hong Kong Peoples R China CSIRO Brisbane Qld Australia Swinburne Univ Technol Melbourne Vic Australia

Tree patterns represent important fragments of XPath. In this paper, we show that some classes C of tree patterns exhibit such a property that, given a finite number of compatible tree patterns P-1, ... , Pn. C, there exists another pattern P such that P-1, ... , Pn are all contained in P-1 and for any tree pattern Q. C, P-1, ... , Pn are all contained in Q if and only if P is contained in Q. We experimentally demonstrate that the pattern P is usually much smaller than P-1, ... , Pn combined together. Using the existence of P above, we show that testing whether a tree pattern, P-1 is contained in another, Q. C, under an acyclic schema graph G, can be reduced to testing whether PG, a transformed version of P-1 is contained in Q without any schema graph, provided that the distinguished node of P is not labeled *. We then show that, under G, the maximal contained rewriting (MCR) of a tree pattern Q using a view V can be found by finding the MCR of Q using VG without G, when there are no *-nodes on the distinguished path of V and no *-nodes in Q.

关键词： QUERYING (computer science) MODULES (algebra) EXISTENCE theorems GRAPH theory TREE graphs PATHS & cycles (Graph theory)

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Lattice based tools for cryptanalysis in various applications

Lattice based tools for cryptanalysis in various application...

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2nd International Conference on computer Science and Information Technology, CCSIT 2012

作者： Santosh Kumar, R. Narasimham, C. Pallam Setty, S. Department of Information Technology MVGR College of Engg. Vizianagaram India Department of Information Technology VR Siddhartha Engineering College Vijayawada-7 India Dept. of Computer Science and Systems Engineering Andhra University Vishakhapatnam India

ISBN: (纸本)9783642272981

Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in number theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this survey paper, we are focusing on point lattices and then describing an introduction to the theoretical and practical aspects of lattice reduction. Finally, we describe the applications of lattice reduction in cryptanalysis like subset sum problem of low density, modular equations, Attacking RSA with small e by knowing parts of the message and Diophantine Approximation using LLL algorithm. © Institute for computer Sciences, Social Informatics and Telecommunications Engineering 2012.

关键词： number theory

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