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.     

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.     

Spin Wavelets on the Sphere

In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose needlet-type spin wavelets as an extension of the needlet approach recently introduced by Narcowich et al. in SIAM J. Math. Anal. 38, 574–594 (2006) and J. Funct. Anal. 238, 530–564 (2006) and then considered for more general manifolds by Geller and Mayeli in Math. Z. 262, 895–927 (2009), Math. Z. 263, 235–264 (2009), and Indiana Univ. Math. J. (2009). We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data.     

Bézier variant of Baskakov-Beta-Stancu operators

In the year 1994, Gupta (Approx Theory Appl (N.S.) 10(3):74–78, 1994) introduced the integral modification of well known Baskakov operators with weights of Beta basis functions and obtained better approximation over the usual Baskakov Durrmeyer operators. The rate of convergence for Bézier variant of these operators for functions of bounded variations were discussed in Gupta (Int J Math Math Sci 32(8):471–479, 2002). The present paper is the extension of the previous work, here we consider the Bézier variant of Baskakov-Beta-Stancu operators. We estimate the rate of convergence of these operators for the bounded functions. In the end of the paper we suggest an open problem.     

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.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Well-posedness for a mean field model of Ginzburg–Landau vortices with opposite degrees

In this paper we analyze the hydrodynamic equations for Ginzburg–Landau vortices as derived by E (Phys. Rev. B. 50(3):1126–1135, 1994). In particular, we are interested in the mean field model describing the evolution of two patches of vortices with equal and opposite degrees. Many results are already available for the case of a single density of vortices with uniform degree. This model does not take into account the vortex annihilation, hence it can also be seen as a particular instance of the signed measures system obtained in Ambrosio et al. (Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2):217–246, 2011) and related to the Chapman et al. (Eur. J. Appl. Math. 7(2):97–111, 1996) formulation. We establish global existence of L ^p solutions, exploiting some optimal transport techniques introduced in this context in Ambrosio and Serfaty (Commun. Pure Appl. Math. LXI(11):1495–1539, 2008). We prove uniqueness for L ^∞ solutions, as expected by analogy with the incompressible Euler equations in fluidodynamics. We also consider the corresponding Dirichlet problem in a bounded domain. Moreover, we show some simple examples of 1-dimensional dynamic.     

A Fréchet derivative-free cubically convergent method for set-valued maps

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.      

Aggregation in age and space structured population models: an asymptotic analysis approach

In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).     

An Index Theorem for Band-Dominated Operators with Slowly Oscillating Coefficients (after Deundyak and Shteinberg)

We provide a proof of an index theorem for band-dominated operators with slowly oscillating coefficients. The statement is essentially the same as the main result of the announcement of Deundyak and Shteinberg (Funct Anal Appl 19(4):321–323, 1985), but our methods are very different from those hinted at there. The index theorem we prove can also be seen as a partial generalization to higher dimensions of the main result of the article of Rabinovich et al. (Integr Equ Oper Theory 49:221–238, 2004).     

Small Support Spline Riesz Wavelets in Low Dimensions

In Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006), a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in Han and Shen (J. Fourier Anal. Appl. 11:615–637, 2005). Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006) were used in a surface fitting algorithm of Johnson et al. (J. Approx. Theory 159:197–223, 2009), and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in Khodakovsky et al. (Proceedings of SIGGRAPH, 2000).     

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065–3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491–509, 2010) and many others.     

One-dimensional chemotaxis kinetic model

In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar–Dunbar–Alt system (Othmer in J Math Biol 26(3):263–298, 1988). This version was exhibited in Calvez in Amer Math Soc, pp 45–62, 2007 for the macroscopic well-known Keller–Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009, Chalub, Math Models Methods Appl Sci, 16(7 suppl):1173–1197, 2006, Chalub, Monatsh Math, 142(1–2):123–141, 2004, Chalub, Port Math (NS), 63(2):227–250, 2006. The model we study here behaves in a similar way to the original model in two dimensions with the spherical symmetry assumption on the initial data which is described in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009. We prove the existence and uniqueness of solutions for this model, as well as a convergence result for a family of numerical schemes. The advantage of this model is that numerical simulations can be easily done especially to track the blow-up phenomenon.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow of closed curves in \mathbbR^d, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.     

Local and parallel algorithms for fourth order problems discretized by the Morley–Wang–Xu element method

This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley–Wang–Xu (MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881–909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas in (Xu and Zhou (Math Comp 69:881–909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al. (Appl Numer Math 37:519–533, 2001); Huang et al. (Sci China Ser A 49:109–120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes of methods are of the sizes O(h + H ²) and O(h + H ²(H/h)^(d−1)/2), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.     

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.     

An FIO calculus for marine seismic imaging, II: Sobolev estimates

We establish sharp L ²-Sobolev estimates for classes of pseudodifferential operators with singular symbols [Guillemin and Uhlmann (Duke Math J 48:251–267, 1981), Melrose and Uhlmann (Commun Pure Appl Math 32:483–519, 1979)] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in \mathbb R²{\mathbb R^2} with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics [Felea (Comm PDE 30:1717–1740, 2005), Felea and Greenleaf (Comm PDE 33:45–77, 2008), Nolan (SIAM J Appl Math 61:659–672, 2000)].     

On Sears’s general transformation formula for basic hypergeometric series

We present a new proof of a general transformation formula for basic hypergeometric series that was discovered by D.B. Sears (Proc. Lond. Math. Soc. 53(2): 181–191, 1951) and discuss some special cases. Next we apply Sears’s general transformation to give an extension of a result of Andrews et al. (Duke Math. J. 108: 395–419, 2001).