Exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction–diffusion terms

The problem of mean square exponential stability for a class of impulsive stochastic fuzzy cellular neural networks with distributed delays and reaction–diffusion terms is investigated in this paper. By using the properties of M-cone, eigenspace of the spectral radius of nonnegative matrices, Lyapunov functional, Itô’s formula and inequality techniques, several new sufficient conditions guaranteeing the mean square exponential stability of its equilibrium solution are obtained. The derived results are less conservative than the results recently presented in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008). In fact, the systems discussed in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008) are special cases of ours. Two examples are presented to illustrate the effectiveness and efficiency of the results.     

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338–353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682–687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80–98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69–81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313–334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169–186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741–747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.      

A posteriori error analysis of time-dependent Stokes problem by Chorin-Temam scheme

We present a posteriori-residual analysis for the approximate time-dependent Stokes model Chorin-Temam projection scheme (Chorin in Math. Comput. 23:341–353, 1969; Temam in Arch. Ration. Mech. Appl. 33:377–385, 1969). Based on the multi-step approach introduced in Bergam et al. (Math. Comput. 74(251):1117–1138, 2004), we derive error estimators, with respect to both time and space approximations, related to diffusive and incompressible parts of Stokes equations. Using a conforming finite element discretization, we prove the equivalence between error and estimators under specific conditions.     

Ranking of fuzzy numbers by a new metric

In this paper, a new approach for comparison among fuzzy numbers based on new metric distance (D _TM) is proposed. All reasonable properties of ranking function are proved. At first, the distance on the interval numbers based on convex hall of endpoints is proposed. The existing distance measures for interval numbers, (Bardossy and Duckstein in Fuzzy rule-based modeling with applications to geophysical, biological and engineering systems. CRC press, Boca Raton, 1995; Diamond in Info Sci 46:141–157, 1988; Diamond and Korner in Comput Math Appl 33:15–32, 1997; Tran and Duckstein in Fuzzy Set Syst 130:331–341, 2002; Diamond and Tanaka Fuzzy regression analysis. In: Slowinski R (ed) Fuzzy sets in decision analysis, operations research and statistics. Kluwer, Boston, pp 349–387, 1998) do not satisfy the properties of a metric distance, while the proposed distance does. It is extended to fuzzy numbers and its properties are proved in detail. Finally, we compare the proposed definition with some of the known ones.     

On Covering Problems of Rado

T. Rado conjectured in 1928 that if ℱ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ℐ⊆ℱ of pairwise disjoint squares, such that ℐ covers at least 1/4 of the area covered by ℱ. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ℱ. The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (in Proc. Lond. Math. Soc. 51:232–264, 1949; 53:243–267, 1951; and J. Lond. Math. Soc. 42:127–130, 1968) on this subject. After 45 years, Ajtai (in Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 21:61–63, 1973) came up with a surprising example disproving T. Rado’s conjecture. We revisit Rado’s problem and present improved upper and lower bounds for squares, disks, convex bodies, centrally symmetric convex bodies, and others, as well as algorithmic solutions to these variants of the problem.     

Some Properties of Digital Covering Spaces

In this paper, we study digital versions of some properties of covering spaces from algebraic topology. Among our results are some that correct or improve upon the presentation of assertions in earlier papers (Boxer and Karaca in J. Math. Imaging Vis. 32:23–29, 2008; Han in Inf. Sci. 177:3731–3748, 2007; Han in Inf. Sci. 178:550–561, 2008).     

The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Open

We offer evidence in the disproof of the continuity of the length of minimum inner spanning trees with respect to a parameter vector having a zero component. The continuity property is the key step of the proof of the conjecture in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Therefore the Steiner ratio conjecture proposed by Gilbert-Pollak (SIAM J. Appl. Math. 16(1):1–29, 1968) has not been proved yet. The Steiner ratio of a round sphere has been discussed in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) by assuming the validity of the conjecture on a Euclidean plane in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Hence the results in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) have not been proved yet.     

Disjoint essential sets of implicates of a CQ Horn function

In this paper we study a class of CQ Horn functions introduced in Boros et al. (Ann Math Artif Intell 57(3–4):249–291, 2010). We prove that given a CQ Horn function f, the maximal number of pairwise disjoint essential sets of implicates of f equals the minimum number of clauses in a CNF representing f. In other words, we prove that the maximum number of pairwise disjoint essential sets of implicates of f constitutes a tight lower bound on the size (the number of clauses) of any CNF representation of f.     

The individual ergodic theorem on the IF-events with product

The ergodic theory and particularly the individual ergodic theorem were studied in many structures. Recently the individual ergodic theorem has been proved for MV-algebras of fuzzy sets (Riečan in Czech Math J 50(125):673–680, 2000; Riečan and Neubrunn in Integral, measure, and ordering. Kluwer, Dordrecht, 1997) and even in general MV-algebras (Jurečková in Int J Theor Phys 39:757–764, 2000). The notion of almost everywhere equality of observables was introduced by Riečan and Jurečková (Int J Theor Phys 44:1587–1597, 2005). They proved that the limit of Cesaro means is an invariant observable for P-observables. In Lendelová (Int J Theor Phys 45(5):915–923, 2006c) showed that the assumption of P-observable can be omitted. In this paper we prove the individual ergodic theorem on family of IF-events and show that each P {\mathcal{P}} -preserving transformation in this family can be expressed by two corresponding P^\flat,P^\sharp {\mathcal{P}}^\flat,{\mathcal{P}}^\sharp -preserving transformations in tribe T. {\mathcal{T}}.     

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.     

Thwarting intercept-and-resend attack on Zhang’s quantum secret sharing using collective rotation noises

This work deliberately introduces collective-rotation noise into quantum states to prevent an intercept-resend attack on Zhang’s quantum secret sharing scheme over a collective-noise quantum channel (Zhang in Phys A 361:233–238, 2006). The noise recovering capability of the scheme remains intact. With this design, the quantum bit efficiency of the protocol is doubled when compared to Sun et al.’s improvement on Zhang’s scheme (Sun et al. in Opt Commun 283:181–183, 2010).     

Julia 集和Mandelbrot 集的反演变换

论文讨论了居里叶集与曼德尔布罗特集的反演变换问题,通过扩充复平面 上关于任意定点的反演变换,获得了两类共轭函数。使得这两类共轭函数的居里叶集与曼德 尔布罗特集,恰好是原居里叶集与曼德尔布罗特集关于定点的反演变换,并运用逃逸时间算 法绘制居里叶集和曼德尔布罗特集的反演图。     

State BL-algebras

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna (An algebraic approach to states on MV-algebras. In: Novák V (ed) Fuzzy logic 2, proceedings of the 5th EUSFLAT conference, September 11–14, Ostrava, vol II, pp 201–206, 2007; Int J Approx Reason 50:138–152, 2009) as an MV-algebra with internal state as a unary operation. Di Nola and Dvurečenskij (Ann Pure Appl Logic 161:161–173, 2009a; Math Slovaca 59:517–534, 2009b) gave a stronger version of a state MV-algebra. In the present paper, we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.     

All-Pairs Shortest Paths with Real Weights in <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis><Superscript>3</Superscript>/log <Emphasis Type="Italic">n</Emphasis>) Time

We describe an O(n ³/log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices. This slightly improves a series of previous, slightly subcubic algorithms by Fredman (SIAM J. Comput. 5:49–60, 1976), Takaoka (Inform. Process. Lett. 43:195–199, 1992), Dobosiewicz (Int. J. Comput. Math. 32:49–60, 1990), Han (Inform. Process. Lett. 91:245–250, 2004), Takaoka (Proc. 10th Int. Conf. Comput. Comb., Lect. Notes Comput. Sci., vol. 3106, pp. 278–289, Springer, 2004), and Zwick (Proc. 15th Int. Sympos. Algorithms and Computation, Lect. Notes Comput. Sci., vol. 3341, pp. 921–932, Springer, 2004). The new algorithm is surprisingly simple and different from previous ones. A preliminary version of this paper appeared in Proc. 9th Workshop Algorithms Data Struct. (WADS), Lect. Notes Comput. Sci., vol. 3608, pp. 318–324, Springer, 2005.     

Characteristic-Based Schemes for Multi-Class Lighthill-Whitham-Richards Traffic Models

In this paper we provide the full spectral decomposition of the Multi-Class Lighthill Whitham Richards (MCLWR) traffic models described in (Wong et al. in Transp. Res. Part A 36:827–841, 2002; Benzoni-Gavage and Colombo in Eur. J. Appl. Math. 14:587–612, 2003). Even though the eigenvalues of these models can only be found numerically, the knowledge of the spectral structure allows the use of characteristic-based High Resolution Shock Capturing (HRSC) schemes. We compare the characteristic-based approach to the component-wise schemes used in (Zhang et al. in J. Comput. Phys. 191:639–659, 2003), and propose two strategies to minimize the oscillatory behavior that can be observed when using the component-wise approach.     

Newton法对应单参有理函数族的广义M-J集

讨论了Newton法对应单参数有理函数族的广义Mandelbrot集和Julia集,给出了它们的构造算法,证明了其广义Mandelbrot集的有界性,并给出了其周期点个数的计算公式.利用数学实验的方法研究了广义Mandelbrot集周期芽苞分布规律,并通过对比分析得到了它们与z~n+c的Mandelbrot集和Julia集之间的族相似性类似的新的族相似关系.文中算法为Mandelbrot集和Julia集的发展提供了新的思路.     

A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration

In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009, CAM Report 09-28, UCLA, March 2009; Yin, CAAM Report, Rice University, 2009) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ ¹ basis pursuit, TV−L ² minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.     

A survey of recursive analysis and Moore’s notion of real computation

The theory of analog computation aims at modeling computational systems that evolve in a continuous space. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by Turing (Proc Lond Math Soc 2(42):230–265, 1936), Grzegorczyk (Fundam Math 42:168–202, 1955), and Lacombe (Compt Rend l’Acad Sci Paris 241:151–153, 1955). Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero-finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them.     

A note on double splittings of different monotone matrices

Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners. For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different monotone matrices.     

Tailored Finite Point Method for First Order Wave Equation

Following the idea of the tailored finite point method proposed in Han and Huang (J. Comput. Math. 26:728–739, 2008) and Huang (Netw. Heterog. Media 4:91–106, 2009), a series of efficient numerical schemes are developed for the one dimensional scalar wave equation within various types of media. Stability and accuracy are analyzed and numerically verified. In particular we can obtain unconditionally stable implicit schemes that can be solved explicitly for boundary value problems. We can also deal with the propagation of discontinuity and highly oscillatory waves efficiently. The generalization to higher order schemes is straightforward.