Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].     

Fixed points in intuitionistic fuzzy metric spaces

The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96], we define the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek [Kramosil O, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 1975;11:326–34]. Further the well-known fixed point theorems of Banach and Edelstein are extended to intuitionistic fuzzy metric spaces with the help of Grabiec [Grabiec M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1988;27:385–9].     

Global existence of periodic solutions on a simplified BAM neural network model with delays

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799–838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equations 1994;106:27–39]. Finally, computer simulations are performed to illustrate the analytical results found.     

A note on k-generalized projections

In this note, we investigate characterizations for k-generalized projections (i.e., A^k = A^*) on Hilbert spaces. The obtained results generalize those for generalized projections on Hilbert spaces in [Hong-Ke Du, Yuan Li, The spectral characterization of generalized projections, Linear Algebra Appl. 400 (2005) 313–318] and those for matrices in [J. Benítez, N. Thome, Characterizations and linear combinations of k-generalized projectors, Linear Algebra Appl. 410 (2005) 150–159].     

Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li [A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng [B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge [A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].     

On the theory of differential games in systems with aftereffect : PMM vol. 42, no. 6, 1978 pp.969–977

An encounter-evasion differential game is studied for control systems with aftereffect [1–4]. A feature of the system being analyzed is that it has a time-lag effect with respect to the controls which provides the system with important new peculiarities. Using the investigations in [1–4], conditions for the solvability of the problem are indicated and the required control procedures are constructed.     

Unimodularity of the Clar number problem

We study the generalization to bipartite and 2-connected plane graphs of the Clar number, an optimization model proposed by Clar [E. Clar, The Aromatic Sextet, John Wiley & Sons, London, 1972] to compute indices of benzenoid hydrocarbons. Hansen and Zheng [P. Hansen, M. Zheng, The Clar number of a benzenoid hydrocarbon and linear programming, J. Math. Chem. 15 (1994) 93–107] formulated the Clar problem as an integer program and conjectured that solving the linear programming relaxation always yields integral solutions. We establish their conjecture by proving that the constraint matrix of the Clar integer program is always unimodular. Interestingly, in general these matrices are not totally unimodular. Similar results hold for the Fries number, an alternative index for benzenoids proposed earlier by Fries [K. Fries, Uber Byclische Verbindungen und ihren Vergleich mit dem Naphtalin, Ann. Chem. 454 (1927) 121–324].     

On stable position control in differential games : PMM vol. 42, no. 6, 1978, pp 963–968

A procedure for control with a guide (1–3], yielding a solution, stable with respect to small data disturbances, of game problems of dynamics is proposed for a class, more general in comparison with [1–3], of conflict-controlled systems, i.e., systems requiring uniqueness of the program motions and their uniform boundedness. The terminology and notation follow [2].     

On the equivalence of the convergence criteria between modified Mann–Ishikawa and multi-step iterations with errors for successively strongly pseudo-contractive operators

In this paper, the equivalence of the convergence between the modified Mann–Ishikawa and multi-step Noor iterations with errors is proven for the successively strongly pseudo-contractive operators without Lipschitzian assumption. Our results generalize the recent results of the paper [B.E. Rhoades, S.M. Soltuz, The equivalence between Mann–Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219–228; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudo-contractive maps, J. Math. Anal. Appl. 289 (2004) 266–278] by extending to the more generalized multi-step iterations with errors and hence improve the corresponding results of all the references in bibliography by providing the equivalences of convergence between all of these up-to-date iteration schemes.     

On strong consistency of the fuzzy generalized nearest neighbor rule

The k nearest neighbor rule (k-NNR) is a well-known nonparametric decision rule in pattern classification. Fuzzy set theory has been widely used to represent the uncertainty of class membership. Several researchers extended conventional k-NNR to fuzzy k-NNR, such as Bezdek et al. [Fuzzy Sets and Systems 18 (1986) 237–256], Keller et al. [IEEE Trans. Syst. Man, and Cybern. 15(4) (1985) 580–585], Béreau and Dubuisson [Fuzzy Sets and Systems 44 (1991) 17–32]. In this paper, we describe a fuzzy generalized k-NN algorithm. This algorithm is a unified approach to a variety of fuzzy k-NNR's. Then we create the strong consistency of posterior risk of the fuzzy generalized NNR.     

A note on the article “Fuzzy alpha-sets and alpha-continuous maps”

In this note we show that some results which appeared in the article by M.K. Singal and N. Rajvanshi [Fuzzy Sets and Systems 48 (1992) 383–390] are incorrect.     

Crisis-induced chaos in the Rose-Hindmarsh model for neuronal activity

The bifurcation diagrams for the Rose-Hindmarsh model are obtained from the Poincaré maps which govern the dynamics of this differential system. The Lyapunov spectra for this model are estimated from time series. The transition from periodicity to crisis-induced chaos. and back to periodicity is presented for I [2.5, 2.69]. and is qualitatively different from the transitions described for different parameter regions [A. V. Holden and Yinshui Fan, Chaos, Solitons & Fractals 2, 221–236 (1992); Chaos, Solitons & Fractals 2, 349–369 (1992)]. A piecewise smooth, one-dimensional map is constructed to simulate the dynamics of the model and to reproduce the process of crisis-induced chaos.     

Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index

We introduce a new approach to the algorithmic computation of the Conley index for continuous maps. We use the technique of splitting an index pair into two layers which is inspired by the work of Mrozek, Reineck and Srzednicki [M. Mrozek, J.F. Reineck, R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000) 4171–4194]. The main advantage of our construction over the approach based directly on the one introduced by Mischaikow, Mrozek and Pilarczyk [K. Mischaikow, M. Mrozek, P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005) 199–229] is that our cubical sets have the excision property. Moreover, our solution has some advantages in comparison to the approach recently proposed by Mrozek [M. Mrozek, Index pair algorithms, Found. Comput. Math. 6 (2006) 457–493].     

Periodicity and repetitions in parameterized strings

One of the most beautiful and useful notions in the Mathematical Theory of Strings is that of a Period, i.e., an initial piece of a given string that can generate that string by repeating itself at regular intervals. Periods have an elegant mathematical structure and a wealth of applications [F. Mignosi and A. Restivo, Periodicity, Algebraic Combinatorics on Words, in: M. Lothaire (Ed.), Cambridge University Press, Cambridge, pp. 237–274, 2002]. At the hearth of their theory, there are two Periodicity Lemmas: one due to Lyndon and Schutzenberger [The equation a^M=b^Nc^P in a free group, Michigan Math. J. 9 (1962) 289–298], referred to as the Weak Version, and the other due to Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109–114]. In this paper, we investigate the notion of periodicity and the closely related one of repetition in connection with parameterized strings as introduced by Baker [Parameterized pattern matching: algorithms and applications, J. Comput. System Sci. 52(1) (1996) 28–42; Parameterized duplication in strings: algorithms and an application to software maintenance, SIAM J. Comput. 26(5) (1997) 1343–1362]. In such strings, the notion of pairwise match or “equivalence” of symbols is more relaxed than the usual one, in that it rests on some mapping, rather than identity, of symbols. It seems natural to try and extend notions of periods and periodicities to encompass parameterized strings. However, we know of no previous attempt in this direction. Our preliminary investigation yields results as follows. For periodicity, we get (a) a generalization of the Weak Version of the Periodicity Lemma for parameterized strings, showing that it is essential that the two mappings inducing the periodicity must commute; (b) a proof that an analogous of the Lemma by Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109–114] cannot hold for parameterized strings, even if the mappings inducing the periodicity “commute”, in a sense to be specified below; (c) a proof that parameterized strings over an alphabet of at least three letters may have a set of periods which differ from those of any binary string of the same length—whereby the parameterized analog of a classic result by Guibas and Odlyzko [String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A 30 (1981) 183–208] cannot hold. We also derive necessary and sufficient conditions characterizing parameterized repetitions, which are patterns of length at least twice that of the period, and show how the notion of root differs from the standard case, and highlight some of the implications on extending algorithmic criteria previously adopted for string searching, detection of repetitions and the likes. Finally, as a corollary of our main results, we also show that binary parameterized strings behave much in the same way as non-parameterized ones with respect to periodicity and repetitions, while there is a substantial difference for strings over alphabets of at least three symbols.     

Deriving amplitude equations for weakly-nonlinear oscillators and their generalizations

Results by physicists on renormalization group techniques have recently sparked interest in the singular perturbations community of applied mathematicians. The survey paper, [Phys. Rev. E 54(1) (1996) 376–394], by Chen et al. demonstrated that many problems which applied mathematicians solve using disparate methods can be solved using a single approach. Analysis of that renormalization group method by Mudavanhu and O’Malley [Stud. Appl. Math. 107(1) (2001) 63–79; SIAM J. Appl. Math. 63(2) (2002) 373–397], among others, indicates that the technique can be streamlined. This paper carries that analysis several steps further to present an amplitude equation technique which is both well adapted for use with a computer algebra system and easy to relate to the classical methods of averaging and multiple scales.     

Asymptotics of a certain integral

The asymptotic behaviour for t → ∞ of ^∞₀ exp[tx–c(x)]dx is studied. The function c is positive and ′(x) → ∞ (x → ∞). Sufficient conditions on c are given in order that the method of Laplace is applicable.     

Weakly clopen functions

We introduce a new class of functions called weakly clopen function which includes the class of almost clopen functions due to Ekici [Ekici E. Generalization of perfectly continuous, regular set-connected and clopen functions. Acta Math Hungar 2005;107:193–206] and is included in the class of weakly continuous functions due to Levine [Levine N. A decomposition of continuity in topological spaces. Am Math Mon 1961;68:44–6]. Some characterizations and several properties concerning weakly clopenness are obtained. Furthermore, relationships among weak clopenness, almost clopenness, clopenness and weak continuity are investigated.     

Oscillations of mechanical systems that do not become linear when the parameter vanishes : PMM vol. 42, no. 6, 1978, pp. 1039–1048

The results of investigations in [1] are extended to multidimensional systems that become nonlinear at μ = 0. Two-dimensional mechanical systems were investigated in [2,3]. The characteristic equations of systems considered here contain in the critical system either a pair of pure imaginary roots or two zero roots with one or two groups of solutions and n roots with negative real parts in the adjoint system. It is shown that the investigation of such systems necessitates the imposition on the system of some constraints that supplement those specified in [1], The auxilliary function u⁽¹⁾_k (θ) used in the determination of Liapunov's function is derived by a different method than in [1 – 3], In two of the three investigated cases the problem is reduced to the determination of roots of some integral real irrational function. An example is presented.     

Estimating market shares in each market segment using the information entropy concept

Sales data of a certain product for the various competitors are usually available at the aggregate level. However these data give no clue to the heterogeneities in the sales pattern across different market segments. Heterogeneities are caused by different purchasing behavior in each market segment; as a purchaser in a segment will be attracted to the attributes of the product most important to that segment. This concept can be formalized via a simple attraction model that utilizes an elasticity measure for each quality or price attribute [G.S. Carpenter, L.G. Cooper, D.M. Hanssens, D.F. Midgley, Modeling asymmetric competition, Marketing Science 7 (4) (1998) 393–412]. Assessment of these elasticities is not difficult since customer response – in each market segment – to perception of quality and price is tracked by most firms [J. Ross, D. Georgoff, A survey of productive and quality issues in manufacturing. The state of the industry, Industrial Management 3 (5) (1991) 22–25]. This paper attempts to formulate a generic framework based on the information entropy concept that utilizes such an attraction model to estimate competitors’ sales in each market segment.