Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

The Chow Rings of Generalized Grassmannians

Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol. 56 (part 1), pp. 1–26, AMS, Providence, 1994) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, 2005) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, 2006), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces).     

Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model

This is a companion paper to Li and Zhao (Queueing Syst. 63:355–381, 2009) recently published in Queueing Systems, in which the classical preemptive priority queueing system was considered. In the current paper we consider the classical non-preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through an analysis of generating functions. This is done for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and also for the distribution of the total number of customers in the system, respectively. This complete characterization is supplemental to the existing literature, which would be useful to researchers.     

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

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

Some new common fixed point theorems in fuzzy metric spaces

The purpose of this paper is to prove some new common fixed point theorems in (GV)-fuzzy metric spaces. While proving our results, we utilize the idea of compatibility due to Jungck (Int J Math Math Sci 9:771–779, 1986) together with subsequentially continuity due to Bouhadjera and Godet-Thobie (arXiv: 0906.3159v1 [math.FA] 17 Jun 2009) respectively (also alternately reciprocal continuity due to Pant (Bull Calcutta Math Soc 90:281–286, 1998) together with subcompatibility due to Bouhadjera and Godet-Thobie (arXiv:0906.3159v1 [math.FA] 17 Jun 2009) as patterned in Imdad et al. (doi:) wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect of any one of the involved maps are relaxed. Our results substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in metric as well as fuzzy metric spaces which include some relevant results due to Imdad et al. (J Appl Math Inform 26:591–603, 2008), Mihet (doi:), Mishra (Tamkang J Math 39(4):309–316, 2008), Singh (Fuzzy Sets Syst 115:471–475, 2000) and several others.     

Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt, Proc Natl Acad Sci USA 42:43–47, 1956 and Parzen, Ann Math Stat 33:1965–1976, 1962) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg’s method rather than on Bernstein’s small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the strong mixing coefficients which do not depend on the bandwidth.     

Feynman’s Operational Calculi: Disentangling Away from the Origin

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ⁿ . The method of disentangling follows Feynman’s heuristic rules from in (Feynman in Phys. Rev. 84:18–128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ⁿ while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.     

Efficiency and equilibria in games of optimal derivative design

In this paper, optimal derivative design when multiple firms compete for heterogenous customers is studied. Ties in the agents’ best responses generate discontinuous payoffs. Efficient tie-breaking rules are considered: In a first step, the model presented by Carlier et al. (Math Financ Econ 1:57–80, 2007) is extended, and results of Page and Monteiro (J Math Econ 39:63–109, 2003, J Econ Theory 134:566–575, 2007, Econ Theory 34:503–524, 2008) are used to prove the existence of (mixed-strategies) Nash equilibria. In a second step, the case of risk minimizing firms is studied. Socially efficient allocations are introduced, and their existence is proved. In particular, the entropic risk measure is considered.     

On Lévy-driven vacation models with correlated busy periods and service interruptions

This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Lévy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server’s preceding busy period. Regarding the former point: the Lévy process active during the busy period is assumed to have no negative jumps, whereas the Lévy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537–555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.     

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C ₀-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).     

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992). Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable. Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper. To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001). Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.      

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.     

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3:162–170, 1980) of adaptive type based on the Newton–Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17:207–217, 1991; ACM Trans. Math. Softw. 17:218–232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner’s ones (Computation of numerical quadrature formulas. In: Rice, J.R. (ed.) Mathematical Software, 229–259. Academic, Orlando, FL, 1971).      

Generalized Eisenstein series and several modular transformation formulae

B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper, we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00003). This work also partially supported by BK21-Postech CoDiMaRo.     

Multi-step nonlinear conjugate gradient methods for unconstrained minimization

Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of matrices. Recently, seeking fast convergence of these methods, Dai and Liao (Appl. Math. Optim. 43:87–101, 2001) proposed a conjugate gradient method based on the secant condition of quasi-Newton methods, and later Yabe and Takano (Comput. Optim. Appl. 28:203–225, 2004) proposed another conjugate gradient method based on the modified secant condition. In this paper, we make use of a multi-step secant condition given by Ford and Moghrabi (Optim. Methods Softw. 2:357–370, 1993; J. Comput. Appl. Math. 50:305–323, 1994) and propose two new conjugate gradient methods based on this condition. The methods are shown to be globally convergent under certain assumptions. Numerical results are reported.     

Gaussian Stationary Processes: Adaptive Wavelet Decompositions,Discrete Approximations,and Their Convergence

We establish particular wavelet-based decompositions of Gaussian stationary processes in continuous time. These decompositions have a multiscale structure, independent Gaussian random variables in high-frequency terms, and the random coefficients of low-frequency terms approximating the Gaussian stationary process itself. They can also be viewed as extensions of the earlier wavelet-based decompositions of Zhang and Walter (IEEE Trans. Signal Process. 42(7):1737–1745, [1994]) for stationary processes, and Meyer et al. (J. Fourier Anal. Appl. 5(5):465–494, [1999]) for fractional Brownian motion. Several examples of Gaussian random processes are considered such as the processes with rational spectral densities. An application to simulation is presented where an associated Fast Wavelet Transform-like algorithm plays a key role. The second author was supported in part by the NSF grant DMS-0505628.