Book reviews

L.N. TREFETHEN and D. BAU, III,Numerical Linear Algebra,SIAM, Philadelphia, 1997G.-C. ROTA,Indiscrete Thoughts,Birkhäuser, Boston, 1997D.E. KEYES, A. SAMEH and V. VENKATAKRISHNAN, eds.Parallel Numerical Algorithms,Kluwer, Dordrecht, 1997A. KIRSCH,An Introduction to the Mathematical Theory of Inverse Problems,Springer, New York, 1996L.F. SHAMPINE, R.C. ALLEN, Jr. and S. PRUESS,Fundamentals of Numerical Computing,Wiley, New York, 1997C.W. UEBERHUBERNumerical Computation, 2 vols.Springer, Berlin, 1997W.G. McCALLUM et al.Multivariate Calculus,Wiley, New York, 1997ZHI-QUAN LUO, JONG-SHI PANG and D. RALPH,Mathematical Programs with Equilibrium Constraints,Cambridge University Press, Cambridge, 1996P.R. POPIVANOV and D.K. PALAGACHEV,The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations,Akademie Verlag, Berlin, 1997     

Book reviews

Devroye, L.; L. Györfi: Nonparametric Density Estimation: the L1 View. John Wiley & Sons New York, Chichester, Brisbane, Toronto, Singapore 1985, XI, 356 p., £ 45.90, ISBN 0-471816469

Gericke, H.: Mathematik in Antike und Orient. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1984, 140.Abb., 4 Kartenskizzen, XII, 292 S., DNI 98,-, ISBN 3-540-11647-8

K. ITö (ed.) StochasticAnalysis. Proceedings of the Traniguchi Internationaloal Symposium on Stochastic Analysis, Katata and Kyoto, 1982. North-Holland Mathematical Library, Vol. 32, North-Holland Publishing Company, Amsterdam 1984, 488 pages, Dfl. 250.- ISBN 0-444-87588-3

Weizenbaum, J.: Die Macht der Computer und dic Ohnmaeht der Vernuutt, Suhrkamp Taschenbuch Wissenschaf't 274. Suhrkamp Verlag Frankfurt/M. 1978, 368 S., DM 16,-, ISBN 3-518-27874-6

Krylov, N.V.; R. Sh. Lipiser; A.A Novikov (eds.): Statistics and Control of Stochastic Processes. Steklov Seminar, 1984. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1985, XIII, 507 S., DM 198, -, ISBN 3.540-96101-1

Sachs, H. (ed.): Graphs, Hypergraphs and Applications. Proceedings of the Conference on Graph Theory held in Eyba, Oct. 1984.. 'I'eubner-Texte zur Mathematik 73 Teubner Verlagsgesellschaft Leipzig 1985, 224 S., 23,-M, ISBN 666275-5

Bhat, U. Narayan; Elements of Applied Stochastic Processes. 2. Ed. Jonoh Wilet & Sons New York, Chichester, Brisbane, Toronto, Singapore 1984, IX, 685 p., £ 52.00, ISBN 0-471-87826-X

Methods of Operations Research. Bd, 66. Verlag Anton Hain Meisenheim, Königstein 1985, 410 S., DM 120,-, ISBN 3..445..02400..6

Maier, T.: Intervall-Input-Ontput-Rechnung. Mathernat.ical Systems in Economics. 101. Verlag Anton Rain, Meisenheim GmbH InigiteinTs. 1985, X, 338 S., DM 72,-, ISBN 3-445-02399-9.

Sawaragi, Y.; H. Nakayama; T. Tanino: Theory of Multi-objective Optimization. Mathematics in Science and Engineering. 176. Academic Press Orlando, San Diego, New York, London, Toronto 1985, 320 pp., $ 48.00, ISBN 0-12-620370-9.

Krämer, W.: Trend in ökonometrischen Modellen. Mathematical Systems in Economics. 93. Athenaum/Hain Konigstein 1985, 172 S., DM 48,-, ISBN 3-445-02386-7.     

Regularisierte Faltung von Distributionen. Teil 2: Eine Tabelle von Fundamentallösungen

Zusammenfassung In der folgenden Liste sollen einige häufig vorkommende Differentialoperatoren und ihre Fundamentallösungen in der Art einer Integraltafel zusammengefaßt werden. Dabei ist zu beachten, daß Fundamentallösungen im allgemeinen (d. h. ohne Zusatzbedingungen) nicht eindeutig bestimmt sind. In den meisten Fällen sind Berechnungsverfahren zur Herleitung der angegebenen Fundamentallösungen skizziert. DAbei wurden sowohl neue Methoden (z. B. Regularisierung, Penalisierung, regularisierte Faltung: 3, 6, 8, 10, 15, 22, 23, 24, 28, 31, 38, 40, 44, 48, 50, 51, 52, 64, 65, 68, 69, 72, 73) als auch neue Fundamentallösungen (15, 16, 20, 22, 23, 24, 31, 68, 69, 70) angegeben, deren Berechnung im 1. Teil (Abschnitt 3) aus Platzgründen nicht aufgenommen wurde. Die Berechnung von Fundamentallösungen mittelspartieller Fourier- oder Laplacetrans-formation, oder durch Spezialisierung in gewissen allgemeinen, von F. J. Bureau oder G. Herglotz angegebenen Integralen wurde außer Betracht gelassen und wird nicht zitiert. Für die Operatoren 26, 66, 67, 71, 74 und 77 wurden keine Fundamentallösungen angeschrieben, da entweder die entsprechende Literatur nur auszugsweise zur Verfügung stand oder die Formeln zu umfangreich sind.

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Summary Similar to a table of integrals the following table contains some frequently occurring differential operators and their fundamental solutions. However one has to pay attention to the fact, that fundamental solutions are not given uniquely (i.e. without assuming further conditions). In most cases you find methods to derive the listed fundamental solutions. We give new methods (e.g. regularization, penalization, regularized convolution: 3, 6, 8, 10, 15, 22, 23, 24, 28, 31, 38, 40, 44, 48, 50, 51, 52, 64, 65, 68, 69, 72, 73) as well as new fundamental solutions (15, 16, 20, 22, 23, 24, 31, 68, 69, 70), whose computation is not contained in part one (section 3) for lack of space. The derivation of fundamental solutions bypartial Fourier- or Laplace Transformation, or by specializing certain general integrals given by F. J. Bureau or G. Herglotz is not taken into account. For the operators 26, 66, 67, 71, 74 and 77 no fundamental solutions are given either for the corresponding reference being only partially available or the formulas being too long.

     

On the fractional-order logistic equation

The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers (see [E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput. 160 (3) (2005) 683–699; A.M.A. El-Sayed, Fractional differential–difference equations, J. Fract. Calc. 10 (1996) 101–106; A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (2) (1998) 181–186; A.M.A. El-Sayed, F.M. Gaafar, Fractional order differential equations with memory and fractional-order relaxation–oscillation model, (PU.M.A) Pure Math. Appl. 12 (2001); A.M.A. El-Sayed, E.M. El-Mesiry, H.A.A. El-Saka, Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. Appl. Math. 23 (1) (2004) 33–54; A.M.A. El-Sayed, F.M. Gaafar, H.H. Hashem, On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations, Math. Sci. Res. J. 8 (11) (2004) 336–348; R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, pp. 223–276; D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application, vol. 2, Lille, France, 1996, p. 963; I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Solvak Academy of science-institute of experimental phys, ISBN: 80-7099-252-2, 1996. UEF-03-96; I. Podlubny, Fractional Differential Equations, Academic Press, 1999] for example). In this work we are concerned with the fractional-order logistic equation. We study here the stability, existence, uniqueness and numerical solution of the fractional-order logistic equation.     

Conflict-directed A and its role in model-based embedded systems

Artificial Intelligence has traditionally used constraint satisfaction and logic to frame a wide range of problems, including planning, diagnosis, cognitive robotics and embedded systems control. However, many decision making problems are now being re-framed as optimization problems, involving a search over a discrete space for the best solution that satisfies a set of constraints. The best methods for finding optimal solutions, such as A^*, explore the space of solutions one state at a time. This paper introduces conflict-directed A^*, a method for solving optimal constraint satisfaction problems. Conflict-directed A^* searches the state space in best first order, but accelerates the search process by eliminating subspaces around each state that are inconsistent. This elimination process builds upon the concepts of conflict and kernel diagnosis used in model-based diagnosis [J. de Kleer, B.C. Williams, Diagnosing multiple faults, Artif. Intell. 32(1) (1987) 97-130; J. de Kleer, A. Mackworth, R. Reiter, Characterizing diagnoses and systems, Artif. Intell. 56 (1992) 197-222] and in dependency-directed search [R. Stallman, G.J. Sussman, Forward reasoning and dependency-directed backtracking in a system for computer-aided circuit analysis, Artif. Intell. 9 (1977) 135-196; J. Gaschnig, Performance measurement and analysis of certain search algorithms, Technical Report CMU-CS-79-124, Carnegie-Mellon University, Pittsburgh, PA, 1979; J. de Kleer, B.C. Williams, Back to backtracking: controlling the ATMS, in: Proceedings of AAAI-86, 1986, pp. 910-917; M. Ginsberg, Dynamic backtracking, J. Artif. Intell. Res. 1 (1993) 25-46]. Conflict-directed A^* is a fundamental tool for building model-based embedded systems, and has been used to solve a range of problems, including fault isolation [J. de Kleer, B.C. Williams, Diagnosing multiple faults, Artif. Intell. 32(1) (1987) 97-130], diagnosis [J. de Kleer, B.C. Williams, Diagnosis with behavioral modes, in: Proceedings of IJCAI-89, 1989, pp. 1324-1330], mode estimation and repair [B.C. Williams, P. Nayak, A model-based approach to reactive self-configuring systems, in: Proceedings of AAAI-96, 1996, pp. 971-978], model-compilation [B.C. Williams, P. Nayak, A reactive planner for a model-based executive, in: Proceedings of IJCAI-97, 1997] and model-based programming [M. Ingham, R. Ragno, B.C. Williams, A reactive model-based programming language for robotic space explorers, in: Proceedings of ISAIRAS-01, 2001].     

Cyclic compositions and trisotopies

Cyclic compositions in the sense of Springer [T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Universität Göttingen, 1963] and Knus, Merkurjev, Rost and Tignol [M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. With a preface in French by J. Tits] are investigated by means of cyclic trisotopies, a concept originally due to Albert [A.A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958) 1–28]. Using the quadrupling of composition algebras, we enumerate cyclic trisotopies and compositions in a rational manner, i.e., without extending the base field. We relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the Tits process of cubic Jordan algebras.     

Nonzero-sum differential games (the noncooperative variant)

One presents the fundamental results of the theory of noncooperative differential games: necessary and sufficient equilibrium conditions, existence theorems, properties of equilibrium solutions, numerical methods. One indicates applications to concrete problems in economics, the mechanics of controlled motions, and to strategic games.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 15, pp. 199–266, 1977.The authors are grateful to É. M. Vaisbord, A. F. Koaonenko, V. S. Molostvov, V. P. Patsyukov, V. A. Plotnikov, V. V. Podinovskii, R. A. Polyak, and R. T. Yanushevskii for the problems in Sec. 11 and for the remarks which have been taken into account in the final editing of the survey.     

Transmission problems in a thin layer set in the framework of the Hölder spaces: Resolution and impedance concept

We consider a family of singular transmission problems depending on some small positive parameter δ set in the juxtaposition of two rectangular domains. They are written in the form of abstract elliptic equations and the study, here, is given in the Hölder spaces completing in this way the work in Lp cases given in [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143-176]. In this first part, we present a new approach for the resolution of these problems by using the concept of impedance operator. This method is different of the one performing a rescaling in the thin layer, see [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143-176]. It leads to obtain direct and simplified problems. We use the Dunford calculus and some techniques similar to that in [R. Labbas, Problèmes aux limites pour une équation différentielle abstraite de type elliptique, Thèse d'état, Université de Nice, 1987; A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, Unified study of elliptic problems in Hölder spaces, C. R. Math. Acad. Sci. Paris 134 (2005); G. Dore, A. Favini, R. Labbas, K. Lemrabet, S. Maingot, A transmission problem in a thin layer, Part I, Sharp estimates, in press], to prove existence, uniqueness, results and some specific estimates on the impedance operator. This study will allow us, in a forthcoming work, to obtain respectively optimal regularities and the limit problem when δ→0.     

基于ISM的地震灾后恢复重建影响因素分析

地震灾后恢复重建是地震灾害受灾国的一项艰巨而繁重的工作任务.地震灾害的破坏性使恢复重建具有复杂性、不确定性和开放性的特点,因此地震灾后恢复重建影响因素不但具有复杂性,而且影响范围和程度也各不相同.引入ISM模型解决了地震灾后恢复重建影响因素多而杂的问题,在文献资料统计基础上构建地震灾后恢复重建影响因子解析结构模型.通过分析确定地震灾后恢复重建关键影响区素共16个:重建主体、重建经验、施工条件、物价、资金、工程成本、建材质量、施工工期、标准施工、检验维修、工程利用、公共服务、自然环境、运行经费、社会饵障、居民激励.该结论为政府优化地震灾后恢复重建工作,保证重建效果具有重要意义.     

q-Discretization of the two-dimensional Toda equations

q-Discrete versions of the two-dimensional Toda molecule equation and the two-dimensional Toda lattice equation are proposed through the direct method. The Bäcklund transformation and the Lax pair of the former are obtained. Moreover, the reduction to theq-discrete cylindrical Toda equations is also discussed.Department of Electronical Engeneering, Doshisha University, Kyoto 610-03, Japan. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. (On leave from Department of Applied Mathematics, Faculty of Engineering, Hiroshima University.) Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153, Japan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 390–398, June, 1994.     

Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves

Continuing a line of investigation initiated in [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361–421] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which appears to offer applications in the multi-dimensional case.A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [G.J. Lord, D. Peterhof, B. Sandstede, A. Scheel, Numerical computation of solitary waves in infinite cylindrical domains, SIAM J. Numer. Anal. 37 (2000) 1420–1454] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361–421] and answering a question of [J. Niesen, Evans function calculations for a two-dimensional system, presented talk, SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, USA, May 2007] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.     

BOOK REVIEWS

Book reviewed in this article: Textbook of Quantitative Inorganic Analysis, by I. M. Kolthoff, Ph.D., Professor of Analytical Chemistry, University of Minnesota, and E. B. Sandell, Ph.D., Assistant Professor of Analytical Chemistry, University of Minnesota. A Chemical-Technical Dictionary (German-English-French-Russian), by A. W. Mayer. (Translation under direction of B. N. Menschutkin and M. A. Bloch.) Preservice Course in Automotive Mechanics, by James V. Frost, Instructor in Automotive Trades, Brooklyn High School of Automotive Trades, Brooklyn, New York. Automotive Mechanics, in two volumes by Clarence G. Barger Elements of Food Biochemistry, by William H. Peterson, Ph.D., Professor of Biochemistry, University of Wisconsin, Madison, Wisconsin; John T. Skinner, Ph.D., Assistant Chemist, Kentucky Agricultural Experiment Station, Lexington, Kentucky; and Frank M. Strong, Ph.D., Associate Professor of Biochemistry, University of Wisconsin, Madison, Wisconsin. The Chemical Formulary. A Collection of Valuable, Timely, Practical Commercial Formulae and Recipes for Making Thousands of Products in Many Fields of Industry. Volume VI. Editor-in-Chief, H. Bennett. Air Navigation. Flight Preparation Series. Published under the Supervision of the Training Division, Bureau of Aeronautics, U. S. Navy, Part One, Introduction to Earth. Radio—I, Written to Conform to the Pre-induction Training Course in Fundamentals of Radio as Prepared by the War Department, by R. E. Williams, Manager, School Service, Westinghouse Electric and Manufacturing Company, Pittsburgh, Pennsylvania, and Charles A. Scarlott, Editor, Westinghouse Engineer. Minerals, Their Identification, Uses, and How to Collect Them, by Herbert S. Zim, and Elizabeth K. Cooper. Basic Radio Principles, by Maurice Grayle Suffern, Captain, Signal Corps, Army of the United States.     

BOOK REVIEW

CHAOS IN BIOLOGICAL SYSTEMS, H. Degn, A.V. Holden, L.F. Olsen, eds., N.A.T.O. Advanced Science Institutes Series, No. 138, Plenum Press, New York and London, 1987     

Representation of consistent recursive rules

This paper develops the recursive model for connective rules (as proposed in V. Cutello, E. Molina, J. Montero, Associativeness versus recursiveness, in: Proceedings of the 26th IEEE International Symposium on Multiple-valued Logic, Santiago de Compostela, Spain, 29–31 May, 1996, pp. 154–159; V. Cutello, E. Molina, J. Montero, Binary operators and connective rules, in: M.H. Smith, M.A. Lee, J. Keller, J. Yen (Eds.), Proceedings of NAFIPS 96, North American Fuzzy Information Processing Society, IEEE Press, Piscataway, NJ, 1996, pp. 46–49), where a particular solution in the Ordered Weighted Averaging (OWA) context (see V. Cutello, J. Montero, Recursive families of OWA operators, in: P.P. Bonissone (Ed.), Proceedings of the Third IEEE Conference on Fuzzy Systems, IEEE Press, Piscataway, NJ, 1994, pp. 1137–1141; V. Cutello, J. Montero, Recursive connective rules, International Journal of Intelligent Systems, to appear) was translated into a more general framework. In this paper, some families of solutions for the key recursive equation are obtained, based upon the general associativity equation as solved by K. Mak (Coherent continuous systems and the generalized functional equation of associativity, Mathematics of Operations Research 12 (1987) 597–625). A context for the representation of families of binary connectives is given, allowing the characterization of key families of connective rules.     

Some examples of the generalized Borel transform approach to the complex WKB method

We present here an illustration of our general approach and the results of [1] for the special case of a secondorder differential equation.Institute of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, E-mails: gurarii@icph.sherna.msk.su, gurarii@g1684.chg.free.net.; Tel-Aviv University, E-mail:matsaev@math.tau.ac.il. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 332–341, September, 1994.     

On a class of Newton-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of Newton-like methods considered also in [I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Computational theory of iterative methods, in: C.K. Chui, L. Wuytack (Eds.), Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co, New York, USA, 2007; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971], in order to approximate a locally unique solution of an equation in a Banach space.     

Solitary waves of the generalized KdV equation with distributed delays

In this paper, we study an integro-differential equation based on the generalized KdV equation with a convolution term which introduces a time delay in the nonlinearity. Special attention is paid to the existence of solitary wave solutions. Motivated by [M.J. Ablowitz, H. Seger, Soliton and Inverse Scattering Transform, SIAM, Philadelphia, 1981; C.K.R.T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Math., vol. 1609, Springer, New York, 1995; T. Ogawa, Travelling wave solutions to perturbed Korteweg-de Vries equations, Hiroshima Math. J. 24 (1994) 401-422], we prove, using the linear chain trick and geometric singular perturbation analysis, that the solitary wave solutions persist when the average delay is suitably small, for a special convolution kernel.     

BOOK REVIEW

LARGE MARINE ECOSYSTEMS: PATTERNS, PROCESSES AND YIELDS, K. Sherman, L.M. Alexander, and B.D. Gold, AAAS, 1991, 247 pp., soft cover, ISBN 0-87168-348-9, $34.95, $27.95 members     

Reverse–Complement Similarity Codes

We discuss a general notion of similarity function between two sequences which is based on their common subsequences. This notion arises in some applications of molecular biology [A.G. D'yachkov, P.L. Erdos, A.J. Macula, V.V. Rykov, D.C. Torney, C.-S. Tung, P.A. Vilenkin, and P.S. White, Exordium for DNA codes, Journal of Combinatorial Optimization 7 (4) (2003)]. We introduce the concept of similarity codes and study the logarithmic asymptotics for the size of optimal codes. Our mathematical results announced in [A.G. D'yachkov, D.C. Torney, P.A. Vilenkin, and P.S.White, On a class of codes for the insertion-deletion metric, Proc. of ISIT–2002, Lausanne, Switzerland, July 2002] correspond to the longest common subsequence (LCS) similarity function [V.I. Levenshtein, Binary codes capable of correcting deletions, insertions, and reversals, J. Soviet Phys.—Doklady, 10, 707–710, 1966] which leads to a special subclass of these codes called reverse-complement (RC) similarity codes. RC codes for additive similarity functions have been studied in previous papers [A.G. D'yachkov and D.C. Torney, On similarity codes, IEEE Trans. Inform. Theory 46 (4) (2000) 1558–1564], [A.G. D'yachkov, D.C. Torney, P.A. Vilenkin, and P.S. White, Reverse– complement similarity codes for DNA sequences, Proc. of ISIT–2000, Sorrento, Italy, July 2000], [P.A. Vilenkin, Some asymptotic problems of combinatorial coding theory and information theory (in Russian), Ph.D. dissertation, Moscow State University, 2000], [V.V. Rykov, A.J. Macula, C.M.Korzelius, D.C. Engelhart, D.C. Torney, and P.S. White, DNA sequences constructed on the basis of quaternary cyclic codes, Proc. of 4-th World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, July 2000].