Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations

The purpose of this paper is to establish strong lower energy estimates for strong solutions of nonlinearly damped Timoshenko beams, Petrowsky equations in two and three dimensions and wave-like equations for bounded one-dimensional domains or annulus domains in two or three dimensions. We also establish weak lower velocity estimates for strong solutions of the nonlinearly damped Petrowsky equation in two and three dimensions. The feedbacks in consideration have arbitrary growth close to the origin. These results improve the strong lower energy decay rates obtained in our previous papers (Alabau-Boussouira in J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010) for strong solutions of the nonlinearly locally damped wave equation and extend to systems and to Petrowsky equation the method of Alabau-Boussouira (J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010). These results are the first ones for Timoshenko beams and Petrowsky equations.     

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287–349, [2005]] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), [2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, [1985]] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, [1987]] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, [1965], [1974]] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245–246, [1976]] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, [1976]]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951–976, [1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.      

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.     

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.     

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.     

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.     

Detecting a conditional extreme value model

In classical extreme value theory probabilities of extreme events are estimated assuming all the components of a random vector to be in a domain of attraction of an extreme value distribution. In contrast, the conditional extreme value model assumes a domain of attraction condition on a sub-collection of the components of a multivariate random vector. This model has been studied in Heffernan and Tawn (JRSS B 66(3):497–546, 2004), Heffernan and Resnick (Ann Appl Probab 17(2):537–571, 2007), and Das and Resnick (2009). In this paper we propose three statistics which act as tools to detect this model in a bivariate set-up. In addition, the proposed statistics also help to distinguish between two forms of the limit measure that is obtained in the model.     

Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems

We extend the study of the integrability done by Leach and Miritzis (J Nonlinear Math Phys 13:535–548, 2006) on the classical model of competition between three species studied by May and Leonard (SIAM J Appl Math 29:243–256, 1975), to all real values of the parameters. Additionally, our results provide all polynomial, rational and analytic first integrals of this extended model. We also classify all the invariant algebraic surfaces of these models.     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

On integrability in elementary functions of certain classes of nonconservative dynamical systems

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries. As is known, the concept of integrability is sufficiently broad and undeterminate in general. In its construction, it is necessary to take into account in what sense it is understood (it is meant that a certain criterion according to which one makes a conclusion that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function classes the first integrals are sought for, etc. (see also [1, 4, 14, 17, 20–22, 35, 40–42, 47, 83–85, 117, 120]).     

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.     

A note on meromorphic functions that share a set with their derivatives

In this paper, we study a problem of meromorphic functions that share an arbitrary set having three elements with their derivatives. A uniqueness result is derived which is an improvement of some related theorems given by Fang and Zalcman (J. Math. Anal. Appl. 280 (2003), 273–283) and Chang, Fang, and Zalcman (Arch. Math. 89 (2007), 561–569). As an application, we generalize the famous Brück conjecture with the idea of sharing a set.     

Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error (Engl et al. in Regularization of inverse problems. Mathematics and its applications, vol. 375, Kluwer Academic, Dordrecht, 1996; Mathé in SIAM J. Numer. Anal. 42(3):968–973, 2004; Mathé and Pereverzev in Inverse Probl. 19(3):789–803, 2003; Vainikko in USSR Comput. Math. Math. Phys. 22(3): 1–19, 1982). In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003), mainly in the sense that the functions associated with orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification (e.g. truncated singular value decomposition (TSVD), Landweber’s method and Showalter’s method) also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced in Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003) are shown. In particular, SRMs having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally, several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown. This work was supported by DARPA/SPO, NASA LaRC and the National Institute of Aerospace under Grant VT-03-1, 2535, by AFOSR Grants F49620-03-1-0243 and FA9550-07-1-0273, by Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, and by Universidad Nacional del Litoral, U.N.L., Argentina, through Project CAI+D 2006, P.E. 236.     

A Generalized Comparison Theorem for BSDEs and Its?Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).     

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.     

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.     

Rough Sets Determined by Quasiorders

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible and its completely meet-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykała and J. A. Pomykała (Bull Pol Acad Sci, Math, 36:495–512, 1988) and M. Gehrke and E. Walker (Bull Pol Acad Sci, Math, 40:235–245, 1992) in case R is an equivalence.     

No end of the world in 2012 for 4OR

The purpose of this note is to give the readers of 4OR information on the state of the journal and its future, abiding by a triennial tradition which started with Bouyssou et al. (4OR 1(1):1–6, 2003) and continued with Bouyssou et al. (4OR 4(1):1–9, 2006) and Bouyssou et al. (4OR Q J Oper Res 7(1):15, 2009). In the 3 years that have passed since the last editorial note, three volumes of the journal have been published, each containing four issues: vol. 7 (2009), vol. 8 (2010) and vol. 9 (2011).     

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.