Quenching for Discretizations of a Nonlocal Parabolic Problem with Neumann Boundary Condition

Boni, Théodore K; Nabongo, Diabaté

doi:10.4067/S0719-06462010000100004

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http://dx.doi.org/10.4067/S0719-06462010000100004

[8] Boni, T.K., On quenching of solution for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc., 7 (2000), 73-95.

[9] Boni, T.K., Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris, Sér. I, Math., 333 (2001), 795-800.

[10] Carrilo, C. and Fife, P., Spacial effects in discrete generation population models, J. Math. Bio., 50 (2005), 161-188.

[11] Chasseigne, E., Chaves, M. and Rossi, J.D., Asymptotic behavior for nonlocal diffusion equations whose solutions develop a free boundary, J. Math. Pures et Appl., 86 (2006), 271-291.

[12] Chen, X., Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Diff. Equat., 2 (1997), 128-160.

[13] Chen, X.Y. and Matano, H., Convergence, asymptotic periodicity and finite point blow up in one-dimensional semilinear heat equations, J. diff. Equat., 78 (1989), 160-190.

[14] Cortazar, C., Elgueta, M. and Rossi, J.D., A non&-local diffusion equation whose solutions develop a free boundary, Ann. Henry Poincaré, 6 (2005), 269&-281.

[15] Cortazar, C., Elgueta, M. and Rossi, J.D., How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 127-156.

[16] Cortazar, C., Elgueta, M., Rossi, J.D. and Wolanski, N., Boundary fluxes for nonlocal diffusion, J. Diff. Equat., 234 (2007), 360-390.

[17] Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, Springer, Berlin, (2003), 153-191.

[18] Fife, P. and Wang, X., A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Adv. Diff. Equat., 3 (1998), 85-110.

[19] Friedman, A. and McLeod, B., Blow-up of positive solution of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.

[20] Ignat, L.I. and Rossi, J.D., A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.

[21] Nabongo, D. and Boni, T.K., Quenching time of solutions for some nonlinear parabolicequations, An. St. Univ. Ovidius Constanta Math., 16 (2008), 87-102.

[22] Nabongo, D. and Boni, T.K., Quenching for semidiscretization of semilinear heat equation with Dirichlet and Neumann boundary conditions, Comment. Math. Univ. Carolinae, 49 (2008), 463-475.

[23] Nabongo, D. and Boni, T.K., Quenching for semidiscretization of a heat equation with singular boundary condition, Asympt. Anal., 59 (2008), 27-38.

[24] Nabongo, D. and Boni, T.K., Blow-up time for a nonlocal diffusion problem with Dirichlet boundary conditions, Comm. Anal. Geom., To appear.

[25] Nabongo, D. and Boni, T.K., Numerical quenching for a semilinear parabolic equation, Math. Modelling and Anal., To appear.

[26] Protter, M.H. and Weinberger, H.F., Maximum principle in differential equations, Prentice Hall, Englewood Cliffs, NJ, (1957)

[27] Perez-LLanos, M. and Rossi, J.D., Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonl. Anal. TMA, To appear.

[28] Walter, W., Differential-und Integral-Ungleucungen, Springer, Berlin., (1964).

Received: October, 2008.

Revised: October, 2009.

^rND^sAndren^nF^rND^sMazon^nJ.M^rND^sRossi^nJ.D^rND^sToledo^nJ ^rND^sBates^nP^rND^sChmaj^nA ^rND^sBates^nP^rND^sChmaj^nA ^rND^sBates^nP^rND^sHan^nJ ^rND^sBates^nP^rND^sHan^nJ ^rND^sBates^nP^rND^sFife^nP^rND^sWang^nX ^rND^sBoni^nT.K ^rND^sBoni^nT.K ^rND^sCarrilo^nC^rND^sFife^nP ^rND^sChasseigne^nE^rND^sChaves^nM^rND^sRossi^nJ.D ^rND^sChen^nX ^rND^sChen^nX.Y^rND^sMatano^nH ^rND^sCortazar^nC^rND^sElgueta^nM^rND^sRossi^nJ.D ^rND^sCortazar^nC^rND^sElgueta^nM^rND^sRossi^nJ.D ^rND^sCortazar^nC^rND^sElgueta^nM^rND^sRossi^nJ.D^rND^sWolanski^nN ^rND^sFife^nP^rND^sWang^nX ^rND^sFriedman^nA^rND^sMcLeod^nB ^rND^sIgnat^nL.I^rND^sRossi^nJ.D ^rND^sNabongo^nD^rND^sBoni^nT.K ^rND^sNabongo^nD^rND^sBoni^nT.K ^rND^sNabongo^nD^rND^sBoni^nT.K ^rND^1A01^nElena I^sKaikina^rND^1A02^nLeonardo^sGuardado-Zavala^rND^1A02^nHector F^sRuiz-Paredes^rND^1A02^nS^sJuarez Zirate ^rND^1A01^nElena I^sKaikina^rND^1A02^nLeonardo^sGuardado-Zavala^rND^1A02^nHector F^sRuiz-Paredes^rND^1A02^nS^sJuarez Zirate ^rND^1A01^nElena I^sKaikina^rND^1A02^nLeonardo^sGuardado-Zavala^rND^1A02^nHector F^sRuiz-Paredes^rND^1A02^nS^sJuarez Zirate

CUBO A Mathematical Journal Vol.12, N° 01, (41-58). March 2010

Korteweg-de Vries-Burgers Equation on a Segment

Elena I. Kaikina, Leonardo Guardado-Zavala, Hector F. Ruiz-Paredes, S. Juarez Zirate

Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, MEXICO email : ekaikina@matmor.unam.mx

Posgrado Electrica, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, MÉXICO emails: guardado@ps.itm.mx; hruiz@sirio.tsemor.mx; sjzirate@matmor.unam.mx

ABSTRACT

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on the interval (0, 1)

We prove that if the initial data , then there exists a unique solution ∪f the initial-boundary value problem (0.1). We also obtain the large time asymptotic of solution uniformly with respect to

Key words and phrases: Dissipative Nonlinear Evolution Equation, Large Time Asymptotics, Korteweg-de Vries-Burgers equation.

RESUMEN

Estudiamos el siguiente problema de valor inicial en la frontera para la ecuación de Kortewegde Vries-Burgers en el intervalo (0, 1)

Provamos que si el dato inicial , entonces existe una única solución ∪ del problema de valor inicial en la frontera (0.1).

También obtenemos comportamiento asintótico de la solución con respecto a

Math. Subj. Class.: 35Q35.

References

[1] Aikawa, H. and Hayashi, N., Holomorphic solutions of semilinear heat equations, Complex Variables, 16 (1991), pp. 115-125.

[2] Aikawa, H., Hayashi, N. and Saitoh, S., The Bergman space on a sector and the heat equation, Complex Variables, 15 (1990), pp. 27-36.

[3] Amick, C.J., Bona, J.L. and Schonbek, M.E., Decay of solutions of some nonlinear wave equations, J. Diff. Eq. 81 (1989), pp. 1-49.

[4] Biller, P., Asymptotic behavior in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation, Bull. Polish. Acad. Sc. Math., 32 (1984), pp. 275-282.

[5] Bona, J.L. and Luo, L., More results on the decay of the solutions to nonlinear dispersive wave equations, Discrete and Continuous Dynamical Systems, 1 (1995), pp. 151-193.

[6] Dix, D.B., The dissipation of nonlinear dispersive waves; the case of asymptotically weak nonlinearity, Comm. P.D.E., 17 (1992), pp. 1665-1693.

[7] Hayashi, N., Kaikina, E.I. and Manzo, R., Local and global existence of solutions to the nonlocal Whitham equation on half-line, Nonlinear Analisys, 48 (2002), pp. 53-75.

[8] Hayashi, N., Kaikina, E.I. and Paredez, F.R., Boundary-value problem for the Korteweg-de Vries-Burgers type equation, Nonlinear Differential Equations and Applications, 8, No. 4 (2001), pp. 439-463.

[9] Hayashi, N., Kaikina, E.I. and Shishmarev, I.A., Asymptotics of Solutions to the Boundary-Value Problem for the Korteweg-de Vries-Burgers equation on a Half-Line, Journal of Mathematical Analysis and Applications, 265 (2002), No. 2, pp. 343-370.

[10] Kaikina, E.I., Naumkin, P.I. and Shishmarev, I.A., Asymptotic behavior for large time of solutions to the nonlinear nonlocal Schrödinger equation on a half-line, SUT Journal of Mathematics, 35 (1) (1999), pp. 37-79.

[11] Naumkin, P.I. and Shishmarev, I.A., Asymptotic behavior for large time of solutions of Korteweg-de Vries equation with dissipation,Differential Equations, 29 (1993), pp. 253-263.

[12] Naumkin, P.I. and Shishmarev, I.A., Asymptotic relationship ast ∞ between solutions to some nonlinear equations, Differential Equations, 30 (1994), pp. 806-814.

[13] Shishmarev, I.A., Tsutsumi, M. and Kaikina, E.I., Asymptotics in time for the nonlinear nonlocal Schrödinger equations with a source, J. Math. Soc. Japan, 51 (1999), pp. 463-484.

Received: September, 2008.

Revised: October, 2009.

^rND^sAikawa^nH^rND^sHayashi^nN ^rND^sAikawa^nH^rND^sHayashi^nN^rND^sSaitoh^nS ^rND^sAmick^nC.J^rND^sBona^nJ.L^rND^sSchonbek^nM.E ^rND^sBiller^nP ^rND^sBona^nJ.L^rND^sLuo^nL ^rND^sDix^nD.B ^rND^sHayashi^nN^rND^sKaikina^nE.I^rND^sManzo^nR ^rND^sHayashi^nN^rND^sKaikina^nE.I^rND^sParedez^nF.R ^rND^sHayashi^nN^rND^sKaikina^nE.I^rND^sShishmarev^nI.A ^rND^sKaikina^nE.I^rND^sNaumkin^nP.I^rND^sShishmarev^nI.A ^rND^sNaumkin^nP.I^rND^sShishmarev^nI.A ^rND^sNaumkin^nP.I^rND^sShishmarev^nI.A ^rND^sShishmarev^nI.A^rND^sTsutsumi^nM^rND^sKaikina^nE.I ^rND^1A01^nValeriu^sPopa ^rND^1A01^nValeriu^sPopa ^rND^1A01^nValeriu^sPopa

CUBO A Mathematical Journal Vol.12, N° 01, (59-66). March 2010

Weakly Picard Pairs of Multifunctions

Valeriu Popa

Department of Mathematics, University of Bacu, Str. Spiru Haret nr. 8, 600114 Bacu, Romania email : vpopa@ub.ro

ABSTRACT

The purpose of this paper is to present a general answer for the following problem: Let be a metric space and two multifunctions. Determine the metric conditions which imply that is a weakly Picard pair of multifunctions and are weakly Picard multifunctions , for multifunctions satisfying an implicit contractive condition, generalizing some results from .

Key words and phrases: Multifunction, fixed point, implicit relation, weakly Picard multifunction, weakly Picard pair of multifunctions.

RESUMEN

El proposito de este artículo es presentar una respuesta general para el siguiente problema: Sea un espacio métrico y : dos multifunciones. Determine los condiciones metricas para las cuales sea un par de multifunciones de Picard debil y sean multifunciones satisfaziendo una condición contractiva implícita, generalizando algunos resultados de .

Math. Subj. Class.: 47H10, 54H25.

References

[1] Popa, V., Some fixed point theorems for contractive mappings, Stud.Cerc.St.Ser. Mat. Univ. Bacu, 7(1997), 127-133.

[2] Popa, V., Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math., 32(1999), 156-163.

[3] Rus, I.A., Petrusel, A. and Sintmrian, A., Data dependence of the fixed points set of multivalued weakly Picard operators, Studia Univ. "Babes Bolyai",Mathematica, 46(2)(2001), 111-121.

[4] Sintmrian, A., Weakly Picard pairs of multivalued operators, Mathematica, 45(2)(2003), 195-204.

[5] Sintmrian, A., Weakly Picard pairs of some multivalued operators, Mathematical Communications, 8(2003), 49-53.

[6] Sintmrian, A., Pairs of multivalued operators, Nonlinear Analysis Forum, 10(1)(2005), 55-67.

[7] Sintmrian, A., Some pairs of multivalued operators, Carpathian J. of Math. 21,1-2(2005), 115-125.

Received: May, 2008.

Revised: October, 2009.

^rND^sPopa^nV ^rND^sPopa^nV ^rND^sRus^nI.A^rND^sPetrusel^nA^rND^sSint

rian^nA ^rND^sSint

rian^nA ^rND^1A01^nNicolas^sRaymond ^rND^1A01^nNicolas^sRaymond ^rND^1A01^nNicolas^sRaymond

CUBO A Mathematical Journal Vol.12, N° 01, (67-81). March 2010

Uniform Spectral Estimates for Families of Schrödinger Operators with Magnetic Field of Constant Intensity and Applications

Nicolas Raymond

Laboratoire de Mathématiques, Université Paris-Sud 11, Bâtiment 425, F-91405, Francia email : nicolas.raymond@math.u-psud.fr

ABSTRACT

The aim of this paper is to establish uniform estimates of the bottom of the spectrum of the Neumann realization of on a bounded open set with smooth boundary when. This problem was motivated by a question occurring in the theory of liquid crystals and appears also in superconductivity questions in large domains.

Key words and phrases: Spectral theory, semiclassical analysis, Neumann Laplacian, magnetic field, liquid crystals.

RESUMEN

El objetivo de este artículo es establecer estimativas uniformes del espectro inferior de la realización de Neumann de sobre un conjunto abierto acotado con frontera suave cuando . Este problema fue motivado por una cuestión que ocurre en teoría de cristales líquidos y aparece también en cuestiones de supercontudividad en dominios grandes.

Math. Subj. Class.: 35P, 35J10, 35Q, 81Q20,

References

[Agm82] Agmon. S., Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, Princeton University Press, 1982.

[Alm02] Almog. Y., Non linear surface superconductivity for type ii superconductors in the large domain limit, Arch. Rat. Mech. Anal., 165:271-293, 2002.

[Alm08] Almog. Y., Thin boundary layers of chiral smectics, Calculus of variations and partial differential equations, to appear, 2008.

[BCLP02] Bauman, P., Carme Calderer, M., Liu, C. and Phillips, D., The phase transition between chiral nematic and smectic A* liquid crystals, Arch. Rational. Anal., 165:161-186, 2002.

[CFKS86] Cycon, H-L., Froese, R-G., Kirsch, W. and Simon, B., Schrödinger Operators, Springer-Verlag, 1986.

[dG95] de Gennes, P-G., The physics of liquid crystals, Clarendon Press, 2nd edition, 1995.

[DH93] Dauge, M. and Helffer, B., Eigenvalues variation. I. Neumann problem for SturmLiouville operators, Journal of Differential Equations, 104, 1993.

[FH08] Fournais, S. and Helffer, B., Spectral methods in surface superconductivity, To appear, 2008.

[HM01] Helffer, B. and Morame, A., Magnetic bottles in connection with superconductivity, J. Funct. Anal., 185(2):604-680, 2001.

[HM02] Helffer, B. and Morame, A., Magnetic bottles for the Neumann problem : The case of dimension 3, Proc. Indian. Sci., 112(1):71-84, 2002.

[HM04] Helffer, B. and Morame, A., Magnetic bottles for the Neumann problem : curvature effects in the case of dimension 3 (general case), Ann. Scient. E. Norm. Sup, 37(4):105-170, 2004.

[HP07] Helffer, B. and Pan, X-B., Reduced Landau-de Gennes functional and surface smectic state of liquid crystals, J. Funct. Anal., 255(11): 3008-3069, 2008.

[LP00] Lu, K. and Pan, X-B., Surface nucleation of superconductivity in 3-dimension, J. Differential Equations, 168:386-452, 2000.

[Pan03] Pan, X-B.., Landau-de Gennes model of liquid cristals and critical wave number, Communications in Math. Phys., 239:343-382, 2003.

[Pan04] Pan, X-B.., Surface superconductivity in 3 dimensions, Trans. Amer. Math. Soc., 356(10):3899-3937, 2004.

[Pan06] Pan, X-B.., Landau-De Gennes model of liquid crystals with small Ginzburg-Landau parameter, SIAM J. Math. Anal., 37:1616-1648, 2006.

Received: October, 2008.

Revised: October, 2009.

^rND^sAlmog^nY ^rND^sBauman^nP^rND^sCarme Calderer^nM^rND^sLiu^nC^rND^sPhillips^nD ^rND^sDauge^nM^rND^sHelffer^nB ^rND^sHelffer^nB^rND^sMorame^nA ^rND^sHelffer^nB^rND^sMorame^nA ^rND^sHelffer^nB^rND^sMorame^nA ^rND^sHelffer^nB^rND^sPan^nX-B ^rND^sLu^nK^rND^sPan^n-B ^rND^sPan^nX-B ^rND^sPan^nX-B ^rND^sPan^nX-B ^rND^1A01^nZead^sMustafa^rND^1A01^nHamed^sObiedat ^rND^1A01^nZead^sMustafa^rND^1A01^nHamed^sObiedat ^rND^1A01^nZead^sMustafa^rND^1A01^nHamed^sObiedat

CUBO A Mathematical Journal Vol.12, N° 01, (83-93). March 2010

A Fixed Point Theorem of Reich in G-Metric Spaces

Zead Mustafa and Hamed Obiedat

The Hashemite University, Department of Mathematics, P.O. Box 150459, Zarqa 13115, Jordan emails: zmagablh@hu.edu.jo, hobiedat@hu.edu.jo

ABSTRACT

In this paper we prove some fixed point results for mapping satisfying sufficient contractive conditions on a complete G-metric space, also we showed that if the G-metric space is symmetric, then the existence and uniqueness of these fixed point results follows from Reich theorems in usual metric space , where the metric induced by the G-metric space .

Key words and phrases: Metric space, generalized metric space, D-metric space, 2-metric space.

RESUMEN

En este artículo nosotros provamos algunos resultados de punto fijo para aplicaciones satisfaciendo condiciones suficientes de contractividad sobre un espacio G-métrico completo, también mostramos que si el espacio G-métrico es simétrico, entonces la existencia y unicidad de estos resultados de punto fijo siguen de teoremas de Reich en espacios métricos usuales , donde es la métrica inducida por el espacio G-métrico .

Math. Subj. Class.: 47H10, 46B20.

References

[1] Mustafa, Z. and Sims, B., Some Remarks Concerninig D-Metric Spaces, Proceedings of the Internatinal Conferences on Fixed Point Theorey and Applications, Valencia (Spain), July (2003). 189-198.

[2] Mustafa, Z., A New Structure For Generalized Metric Spaces - With Applications To Fixed Point Theory, PhD Thesis, the University of Newcastle, Australia, 2005.

[3] Mustafa, Z. and Sims, B., A New Approach to Generalized Metric Spaces, Journal of Nonlinear and Convex Analysis, Volume 7, No. 2 (2006). 289-297.

[4] Reich, S., Some Remarks concerning contraction mappings, Canad. Math. Bull. 14, (1971), 121-124. MR 49 #1501.

Received: September, 2008.

Revised: October, 2009.

^rND^sMustafa^nZ^rND^sSims^nB ^rND^sMustafa^nZ^rND^sSims^nB ^rND^sReich^nS ^rND^1A01^nIrena^sKosi-Ulbl^rND^1A02^nJoso^sVukman ^rND^1A01^nIrena^sKosi-Ulbl^rND^1A02^nJoso^sVukman ^rND^1A01^nIrena^sKosi-Ulbl^rND^1A02^nJoso^sVukman

CUBO A Mathematical Journal Vol.12, N° 01, (95-102). March 2010

An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹

Irena Kosi-Ulbl and Joso Vukman

Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, Maribor, Slovenia email : irena.kosi@uni-mb.si

Department of Mathematics and Computer Science, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, Maribor, Slovenia email : joso.vukman@uni-mb.si

ABSTRACT

In this paper we prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X, and let be a standard operator algebra. Suppose is a linear mapping satisfying the relation . In this case D is of the form and some , which means that D is a linear derivation. In particular, D is continuous. We apply this result, which generalizes a classical result of Chernoff, to semisimple H*- algebras.

This research has been motivated by the work of Herstein [4], Chernoff [2] and Molnár [5] and is a continuation of our recent work [8] and [9] .Throughout, R will represent an associative ring. Given an integer , a ring R is said to be n−torsion free, if for implies x = 0. Recall that a ring R is prime if for a, b R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B A is called a linear derivation in case holds for all pairs x, y R. In case we have a ring R an additive mapping D : R R is called a derivation if holds for all pairs x, y R and is called a Jordan derivation in case is fulfilled for all x R. A derivation D is inner in case there exists a R, such that holds for all x R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [4] asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. Cusack [3] generalized Herstein’s result to 2 -torsion free semiprime rings. Let us recall that a semisimple H*-algebra is a semisimple Banach * -algebra whose norm is a Hilbert space norm such that is fulfilled for all x, y, z A (see [1]). Let X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra A(X) L(X) is said to be standard in case F(X) A(X). Let us point out that any standard algebra is prime, which is a consequence of Hahn-Banach theorem.

Key words and phrases: Prime ring, semiprime ring, Banach space, standard operator algebra, H*-algebra, derivation, Jordan derivation.

RESUMEN

En este artículo nosotros provamos el seguiente resultado. Sea X un espacio de Banach real o complejo, sea L(X) a algebra de todos los operadores linares acotados sobre X, y sea una algebra de operadores estandar. Suponga una aplicación lineal verificando la relación . En este caso D es de la forma y algún , lo que significa que D es una deriviación lineal. En particual, D es continua. Nosotros aplicamos este resultado el cual generaliza un resultado clásico de Chernoff, para H*-algebras semisimple. Este trabajo fué motivado por un trabajo de Herstein [4], Chernoff [2] y Molnár [5] y este una continuación de nuestro reciente trabajo [8] y [9].

Notas

¹This research has been supported by the Research Council of Slovenia

References

[1] Ambrose, W., Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc., 57 (1945), 364-386.

[2] Chernoff, P.R., Representations, automorphisms, and derivations of some operator algebras, J. Funct. Anal., 2 (1973), 275-289

[3] Cusack, J., Jordan derivations on rings, Proc. Amer. Math. Soc., 53 (1975), 321-324.

[4] Herstein, I.N., Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104- 1119.

[5] Molnár, L., On centralizers of an H*-algebra, Publ. Math. Debrecen, 46, 1-2 (1995), 89-95.

[6] Šemrl, P., Ring derivations on standard operator algebras, J. Funct. Anal., 112 (1993), 318-324.

[7] Vukman, J., On automorphisms and derivations of operator algebras, Glasnik Mat., 19 (1984), 135-138.

[8] Vukman, J. and Kosi-Ulbl, I., A note on derivations in semiprime rings, Internat. J. Math. & Math. Sci., 20 (2005), 3347-3350.

[9] Vukman, J., On derivations of standard operator algebras and semisimple H*-algebras, Studia Sci. Math. Hungar., 44 (2007), 57-63.

Received: June, 2008.

Revised: October, 2009.

^rND^sAmbrose^nW ^rND^sChernoff^nP.R ^rND^sCusack^nJ ^rND^sHerstein^nI.N ^rND^sMolnár^nL ^rND^semrl^nP ^rND^sVukman^nJ ^rND^sVukman^nJ^rND^sKosi-Ulbl^nI ^rND^sVukman^nJ ^rND^1A01^nE^sBallico ^rND^1A01^nE^sBallico ^rND^1A01^nE^sBallico

CUBO A Mathematical Journal Vol.12, N° 01, (103-114). March 2010

Brill-Noether Theories for Rank 1 Sheaves on

E. Ballico*

Dept. of Mathematics, University of Trento, 38050 Povo (TN), Italy email : ballico@science.unitn.it

ABSTRACT

Here we discuss some Brill-Noether problems for rank 1 sheaves on stable curves.

Key words and phrases: Stable curve, reducible curve, Brill-Noether theory.

RESUMEN

Nosotros discutimos aquí algunos problemas de Brill-Noether para hazes de rango 1 sobre curvas estables.

Math. Subj. Class.: 14H10, 14H51.

References

[1] Artamkin, I.V., Canonical maps of pointed nodal curves, Sb. Math, 195 (2004), no. 5,615-642.

[2] Ballico, E., Low degree spanned sheaves with pure rank 1 on reducible curves, International Journal of Pure and Applied Mathematics, 55 (2009), no. 1, 109-120.

[3] Ballico, E., Gonality for stable curves and their maps with a smooth curve as their target, Central European Journal of Mathematics, 7 (2009), n. 1, 54-58.

[4] Bayer, D. and Eisenbud, D., Graph curves, with an appendix by Sung Won Park, Adv. Math., 86 (1991), no. 1, 1-40.

[5] Buchweitz, R.-O. and Greuel, G.-M., The Milnor number and deformations of complex curve singularities, Invent. Math., 58 (1980), no. 3, 241-281.

[6] Caporaso, L., A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc., 7 (1994), no. 3, 589-660.

[7] Caporaso, L., Brill-Noether theory of binary curves, arXiv:math/0807.1484.

[8] Catanese, F., Pluricanonical - Gorenstein - curves, Enumerative geometry and classical algebraic geometry, 51-95, Birkhäuser, Basel, 1982.

[9] Catanese, F., Franciosi, M., Hulek, K. and Reid, M., Embeddings of curves and surfaces, Nagoya Math. J., 154 (1999), 185-220.

[10] Ciliberto, C., Harris, J. and Miranda, R., On the surjectivity of the Wahl map, Duke Math. J., 57 (1988), no. 3, 829-858.

[11] Eisenbud, D. and Harris, J., Limit linear series: basic theory, Invent. Math., 85 (1986), no. 2, 337-371.

[12] Eisenbud, D., Koh, J. and Stillman, M., (Appendix with J. Harris), Amer. J. Math., 110 (1988), no. 3, 513-539.

[13] Esteves, E. and Medeiros, N., Limit canonical systems on curves with two components, Invent. Math., 149 (2002), 267-338.

[14] Farkas, G., Birational aspects of Mg, arXiv:math/08100702.

[15] Greuel, G.-M. and Knörrer, H., Einfache Kurvensingularitäten und torsionfreie Moduln, Math. Ann., 270 (1985), 417-425.

[16] Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves, Invent. Math., 67 (1980), no. 1, 23-86.

[17] Melo, M., Compactified Picard stacks over Mg, arXiv:math/0710.3008, Math. Z. (to appear) .

[18] Osserman, B., Linear series and the existence of branched covers, Compositio Math., 144 (2008), no. 1, 89-106.

[19] Pandharipande, R., A compactification of the universal moduli space of slope-semistable vector bundles over , J. Amer. Math. Soc., 9 (1996), no. 2, 425-471.

[20] Sernesi, E., Deformations of algebraic schemes, Springer, Berlin, 2006.

[21] Seshadri, C., Fibrés vectoriels sur les courbes algébriques, Astérisque, 96, 1982.

Received: October, 2008.

Revised: October, 2009.

^rND^sArtamkin^nI.V ^rND^sBallico^nE ^rND^sBallico^nE ^rND^sBayer^nD^rND^sEisenbud^nD ^rND^sBuchweitz^nR.-O^rND^sGreuel^nG.-M ^rND^sCaporaso^nL ^rND^sCatanese^nF^rND^sFranciosi^nM^rND^sHulek^nK^rND^sReid^nM ^rND^sCiliberto^nC^rND^sHarris^nJ^rND^sMiranda^nR ^rND^sEisenbud^nD^rND^sHarris^nJ ^rND^sEsteves^nE^rND^sMedeiros^nN ^rND^sGreuel^nG.-M^rND^sKnörrer^nH ^rND^sHarris^nJ^rND^sMumford^nD ^rND^sOsserman^nB ^rND^sPandharipande^nR ^rND^sSeshadri^nC ^rND^1A01^nGrigori^sRozenblum^rND^1A02^nNikolay^sShirokov ^rND^1A01^nGrigori^sRozenblum^rND^1A02^nNikolay^sShirokov ^rND^1A01^nGrigori^sRozenblum^rND^1A02^nNikolay^sShirokov

CUBO A Mathematical Journal Vol.12, N° 01, (115-132). March 2010

Entire Functions in Weighted L₂ and Zero Modes of the Pauli Operator with Non-Signdefinite Magnetic Field

Grigori Rozenblum and Nikolay Shirokov

Department of Mathematics, Chalmers University of Technology, and Department of Mathematics University of Gothenburg, S-412 96 Gothenburg, Sweden email : grigori@math.chalmers.se

Department of Mathematics and Mechanics,St. Petersburg State University, Russia Email: nikolai.shirokov@gmail.com

ABSTRACT

For a real non-signdefinite function B(z), z ࢠC, we investigate the dimension of the space of entire analytical functions square integrable with weight e ^±2F , where the function F(z) = F(x1, x2) satisfies the Poisson equation ΔF = B. The answer is known for the function B with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions B, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the Pauli operator.

Key words and phrases: Pauli operators, Zero modes, Entire functions.

RESUMEN

Para una función no signo definida B(z), z ࢠ C, investigamos la dimensión del espacio de funciones analíticas enteras de cuadrado integrable con peso e ^±2F , donde la función F(z) = F(x1, x2) verifica la ecuación de Poisson ΔF = B. La respuesta es conocida para la función B con signo constante. Discutimos algunas clases de funciones B no signo definida e positivamente homogéneas, donde dimensión zero y infinita pueden ocurrir. En el caso anterior nosotros presentamos un método de construir funciones enteras con un comportamiento en infinito prescrito en diferentes direcciones. El tópico es estrechamente relacionado con la cuestión de la dimensión del subespacio de energía zero para el operador de Pauli.

Math. Subj. Class.: 30D15, 81Q10, 47N50, 35Q40.

References

[1] Aharonov, Y. and Casher, A., Ground state of a spin- 1/2 charged particle in a twodimensional magnetic field, Phys. Rev. A (3), 19 (1979), no.6, 2461-2462.

[2] Erdös, L. and Vugalter, V., Pauli operator and Aharonov-Casher theorem for measure valued magnetic fields, Comm. Math. Phys., 225 (2002), no. 2, 399-421.

[3] Hörmander, L., An introduction to complex analysis in several variables, North Holland, Princeton, Amsterdam, 1973.

[4] Rozenblum, G. and Shirokov, N., Infiniteness of zero modes for the Pauli operator with singular magnetic field, J. Funct. Anal., 233 (2006), no. 1, 135-172.

[5] Shigekawa, I., Spectral properties of Schrödinger operators with magnetic fields for a spin 1/2 particle, J. Func. Anal., 101 (1991), 255-285.

Received: October, 2008.

Revised: October, 2008.

^rND^sAharonov^nY^rND^sCasher^nA ^rND^sErdös^nL^rND^sVugalter^nV ^rND^sRozenblum^nG^rND^sShirokov^nN ^rND^sShigekawa^nI ^rND^1A01^nStanislas^sOuaro ^rND^1A01^nStanislas^sOuaro ^rND^1A01^nStanislas^sOuaro

CUBO A Mathematical Journal Vol.12, N° 01, (133-148). March 2010

Well-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L¹-Data

Stanislas Ouaro

Laboratoire d'Analyse Mathématique des Equations (LAME), UFR. Sciences Exactes et Appliquées, Université de Ouagadougou 03 BP 7021 Ouaga 03 Ouagadougou, Burkina Faso email : souaro@univ-ouaga.bf

ABSTRACT

We study the anisotropic boundary value problem on , where is a smooth open bounded domain in . We prove the existence and uniqueness of an entropy solution for this problem.

Key words and phrases: Anisotropic; variable exponent; entropy solution; electrorheological fluids.

RESUMEN

Estudiamos el problema de valores en la frontera anisotropico en , u = 0 sobre , donde es un dominio abierto suave do . Proveamos la existencia y unicidad de una solución de entropía para este problema.

Math. Subj. Class.: 35J20, 35J25, 35D30, 35B38, 35J60

References

[1] Alvino, A., Boccardo, L., Ferone, V., Orsina, L. and Trombetti, G., Existence results for non-linear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl., 182 (2003), 53-79.

[2] Bendahmane, M. and Karlsen, K.H., Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres, Electron. J. Differential Equations 2006, No. 46, 30 pp.

[3] Bénilan, Ph., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J.L., An L¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 22 (1995), 241-273.

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[5] Chen, Y., Levine, S. and Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66, no. 4 (2006), 1383-1406.

[6] Diening, L., Theoretical and Numerical Results for Electrorheological Fluids, PhD. thesis, University of Frieburg, Germany, 2002.

[7] Diening, L., Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L^p(.) and W^1,p(.), Math. Nachr., 268 (2004), 31-43.

[8] Edmunds, D.E. and Rakosnik, J., Density of smooth functions in W^k,p(x)(Ω), Proc. R. Soc. A, 437 (1992), 229-236.

[9] Edmunds, D.E. and Rakosnik, J., Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267-293.

[10] Edmunds, D.E. and Rakosnik, J., Sobolev embeddings with variable exponent, II, Math. Nachr., 246-247 (2002), 53-67.

[11] El Hamidi, A., Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl., 300 (2004), 30-42.

[12] Fan, X. and Zhang, Q., Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.

[13] Harjulehto, P., Hästö, P., Koskenova, M. and Varonen, S., The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal., 25 (2006), 79-94.

[14] Hudzik, H., On generalized Orlicz-Sobolev space, Funct. Approximatio Comment. Math., 4 (1976), 37-51.

[15] Koné, B., Ouaro, S. and Traoré, S., Weak solutions for anisotropic nonlinear elliptic equations with variable exponent, Electron J. Differ. Equ., 144 (2009), 1-11.

[16] Kovacik, O. and Rakosnik, J., On spaces L^p(x) and W^1,p(x), Czech. Math. J., 41 (1991), 592-618.

[17] Leray, J. and Lions, J.L., Quelques résultats de Visik sur les problémes elliptiques nonlinéaires par les méthodes de Minty et Browder, Bull. Soc. Math. France., 93 (1965), 97-107.

[18] Mihailescu, M., Pucci, P. and Radulescu, V., Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J. Math. Anal. Appl., 340 (2008), no. 1, 687-698.

[19] Mihailescu, M. and Radulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A, 462 (2006), 2625- 2641.

[20] Musielak, J., Orlicz Spaces and modular spaces, Lecture Notes inMathematics, 1034 (1983), springer, Berlin.

[21] Orlicz, W., Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-212.

[22] Pfeiffer, C., Mavroidis, C., Bar-Cohen, Y. and Dolgin, B., Electrorheological fluid based force feedback device, in Proc. 1999 SPIE Telemanipulator and Telepresence Technologies VI Conf. (Boston, MA), 3840 (1999), pp. 88-99.

[23] Rajagopal, K.R. and Ruzicka, M., Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78.

[24] Ruzicka, M., Electrorheological fluids: modelling and mathematical theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin, 2000.

[25] Sanchon, M. and Urbano, J.M., Entropy solutions for the p(x)-Laplace Equation, Trans. Amer. Math. Soc., 361 (2009), no 12, 6387-6405.

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[27] Tsenov, I.V., Generalization of the problem of best approximation of a function in the space L⁸, Uch. Zap. Dagestan Gos. Univ., 7 (1961), 25-37.

[28] Winslow, W.M., Induced Fibration of Suspensions, J. Applied Physics, 20 (1949), 1137-1140.

Received: June, 2008.

Revised: november, 2008.

^rND^sAlvino^nA^rND^sBoccardo^nL^rND^sFerone^nV^rND^sOrsina^nL^rND^sTrombetti^nG ^rND^sBendahmane^nM^rND^sKarlsen^nK.H ^rND^sBénilan^nPh^rND^sBoccardo^nL^rND^sGallouët^nT^rND^sGariepy^nR^rND^sPierre^nM^rND^sVazquez^nJ.L ^rND^sBénilan^nPh^rND^sBrezis^nH^rND^sCrandall^nM.G ^rND^sChen^nY^rND^sLevine^nS^rND^sRao^nM ^rND^sDiening^nL ^rND^sEdmunds^nD.E^rND^sRakosnik^nJ ^rND^sEdmunds^nD.E^rND^sRakosnik^nJ ^rND^sEdmunds^nD.E^rND^sRakosnik^nJ ^rND^sEl Hamidi^nA ^rND^sFan^nX^rND^sZhang^nQ ^rND^sHarjulehto^nP^rND^sHästö^nP^rND^sKoskenova^nM^rND^sVaronen^nS ^rND^sHudzik^nH ^rND^sKoné^nB^rND^sOuaro^nS^rND^sTraoré^nS ^rND^sKovacik^nO^rND^sRakosnik^nJ ^rND^sLeray^nJ^rND^sLions^nJ.L ^rND^sMihailescu^nM^rND^sPucci^nP^rND^sRadulescu^nV ^rND^sMihailescu^nM^rND^sRadulescu^nV ^rND^sOrlicz^nW ^rND^sPfeiffer^nC^rND^sMavroidis^nC^rND^sBar-Cohen^nY^rND^sDolgin^nB ^rND^sRajagopal^nK.R^rND^sRuzicka^nM ^rND^sSanchon^nM^rND^sUrbano^nJ.M ^rND^sSharapudinov^nI ^rND^sTsenov^nI.V ^rND^sWinslow^nW.M ^rND^1A01^nIoannis K^sArgyros ^rND^1A01^nIoannis K^sArgyros ^rND^1A01^nIoannis K^sArgyros

CUBO A Mathematical Journal Vol.12, N° 01, (149-159). March 2010

An Improved Convergence and Complexity Analysis for the Interpolatory Newton Method

Ioannis K. Argyros

Cameron University, Department of Mathematical Sciences, Lawton, OK 73505, USA email : iargyros@cameron.edu

ABSTRACT

We provide an improved compared to local convergence analysis and complexity for the interpolatory Newton method for solving equations in a Banach space setting. The results are obtained using more precise error bounds than before and the same hypotheses/computational cost.

Key words and phrases: Newton's method, local convergence, Banach space, interpolatory Newton method, complexity, radius of convergence.

RESUMEN

Nosotros entregamos aquí un análisis de convergencia local y complejidad para el método de interpolación de Newton para resolver ecuaciones en espacios de Banach. Los resultados mejoran los de e son obtenidos usando mas precisas cotas de error y las mismas hipotesis y costo computacional.

Math. Subj. Class.: 65G99, 65H10, 65B05, 47H17, 49M15.

References

[1] Argyros, I.K., A unifying local-semilocal convergence analysis and applications for twopoint Newton-like methods in Banach space, J. Math. Anal. Applic., 298 (2004), 374-397.

[2] Argyros, I.K., Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., River Edge, New Jersey, 2005.

[3] Argyros, I.K., An improved convergence and complexity analysis of Newton's method for solving equations, (to appear) .

[4] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Moscow, 1959.

[5] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[6] Traub, J.F. and Wozniakowski, H., Strict lower and upper bounds on iterative computational complexity . In Analytic Computational Complexity, J.F. Traube, Ed., Academic Press, New York, 1976, pp. 15-34.

[7] Traub, J.F. and Wozniakowski, H., Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Machinery, 26, No. 2 (1979), 250-258

Received: October, 2008. Revised: January, 2009.

^rND^sArgyros^nI.K ^rND^sTraub^nJ.F^rND^sWozniakowski^nH ^rND^sTraub^nJ.F^rND^sWozniakowski^nH ^rND^1A01^nIoannis K^sArgyros^rND^1A02^nSaïd^sHilout ^rND^1A01^nIoannis K^sArgyros^rND^1A02^nSaïd^sHilout ^rND^1A01^nIoannis K^sArgyros^rND^1A02^nSaïd^sHilout

CUBO A Mathematical Journal Vol.12, N° 01, (161-174). March 2010

Convergence Conditions for the Secant Method

Ioannis K. Argyros and Saïd Hilout

Department of Mathematics Sciences, Lawton, OK 73505, USA email : iargyros@cameron.edu

Poitiers university, Laboratoire de Mathématiques et Applications, 86962 Futuroscope Chasseneuil Cedex, France email : said.hilout@math.univ-poitiers.fr

ABSTRACT

We provide new sufficient convergence conditions for the convergence of the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided in this study.

Key words and phrases: Secant method, Banach space, majorizing sequence, divided difference, Fréchet-derivative.

RESUMEN

Son dadas nuevas condiciones suficientes para la convergencia del método de la secante para una solución localmente única de una ecuación no lineal en un espacio de Banach. Estas ideas nuevas usan funciones recurrentes, tipo-Lipschitz y tipo centro-Lipschitz sobre la diferencia dividida de los operadores envolvidos. Resulta que esta manera las cotas de errores son mas precisas que las anteriores y bajo nuestras hipótesis de convergencia nosotros podemos cubrir casos donde las condiciones previas eran violadas. Ejemplos numéricos son dados en este estudio.

Math. Subj. Class.: 65H10, 65B05, 65G99, 65N30, 47H17, 49M15.

References

[1] Argyros, I.K., The theory and application of abstract polynomial equations, St.Lucie/CRC/ Lewis Publ. Mathematics series, 1998, Boca Raton, Florida, U.S.A.

[2] Argyros, I.K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169 (2004), 315-332.

[3] Argyros, I.K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298 (2004), 374-397.

[4] Argyros, I.K., New sufficient convergence conditions for the Secant method, Chechoslovak Math. J., 55, (2005), 175-187.

[5] Argyros, I.K., Convergence and applications of Newton-type iterations, Springer-Verlag Publ., New-York, 2008.

[6] Argyros, I.K. and Hilout, S., Efficient methods for solving equations and variational inequalities, Polimetrica Publ. Co., Milano, Italy,

[7] Bosarge, W.E. and Falb, P.L., A multipoint method of third order, J. Optimiz. Th. Appl., 4 (1969), 156-166.

[8] Chandrasekhar, S., Radiative transfer, Dover Publ., New-York, 1960.

[9] Dennis, J.E., Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Analysis and Applications (L.B. Rall, ed.), Academic Press, New York, (1971), 425-472.

[10] Gutiérrez, J.M., A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math., 79 (1997), 131-145.

[11] Hernández, M.A., Rubio, M.J. and Ezquerro, J.A., Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., 115 (2000), 245-254.

[12] Huang, Z., A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math., 47 (1993), 211-217.

[13] Kantorovich and L.V., Akilov, G.P., Functional Analysis, Pergamon Press, Oxford, 1982.

[14] Laasonen, P., Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser I, 450 (1969), 1-10.

[15] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[16] Potra, F.A., Sharp error bounds for a class of Newton-like methods, Libertas Mathematica, 5 (1985), 71-84.

[17] Schmidt, J.W., Untere Fehlerschranken fur Regula-Falsi Verhafren, Period. Hungar., 9 (1978), 241-247.

[18] Yamamoto, T., A convergence theorem for Newton-like methods in Banach spaces, Numer. Math., 51 (1987), 545-557.

[19] Wolfe, M.A., Extended iterative methods for the solution of operator equations, Numer. Math., 31 (1978), 153-174

Received: October, 2008.

Revised: January, 2009.

^rND^sArgyros^nI.K ^rND^sArgyros^nI.K ^rND^sArgyros^nI.K ^rND^sBosarge^nW.E^rND^sFalb^nP.L ^rND^sGutiérrez^nJ.M ^rND^sHernández^nM.A^rND^sRubio^nM.J^rND^sEzquerro^nJ.A ^rND^sHuang^nZ ^rND^sLaasonen^nP ^rND^sPotra^nF.A ^rND^sSchmidt^nJ.W ^rND^sYamamoto^nT ^rND^sWolfe^nM.A ^rND^1A01^nRubén A^sHidalgo ^rND^1A01^nRubén A^sHidalgo ^rND^1A01^nRubén A^sHidalgo

CUBO A Mathematical Journal Vol.12, N° 01, (175-179). March 2010

A Short Note On M-Symmetric Hyperelliptic Riemann Surfaces *

Rubén A. Hidalgo

Departamento de Matemáticas, Universidad Técnica Federico Santa María, Valparaíso, Chile email : ruben.hidalgo@usm.cl

ABSTRACT

We provide an argument, based on Schottky groups, of a result due to B. Maskit which states a necessary and sufficient condition for the double oriented cover of a planar compact Klein surface of algebraic genus at least two to be a hyperelliptic Riemann surface.

Key words and phrases: Schottky groups, Hyperelliptic Riemann surfaces.

RESUMEN

Damos un argumento, basado en grupos de Schottky, de un resultado debido a B. Maskit el cual establece una condición necesária y suficiente para el cubrimiento duplo orientado de una superficie de Klein compacta planar de genero algebrico al menos dos ser una superficie de Riemann hipereliptica.

Math. Subj. Class.: 30F10, 30F40.

Notas

*Partially supported by projects Fondecyt 1070271 and UTFSM 12.09.02

References

[1] Ahlfors, L. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton NJ, 1960.

[2] Farkas, H. and Kra, I., Riemann Surfaces, Second edition. Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992.

[3] Keen, L., On Hyperelliptic Schottky groups, Ann. Acad. Sci. Fenn. Series A.I. Mathematica, 5, 1980.

[4] Maskit, B., Remarks on m-symmetric Riemann surfaces, Contemporary Math., 211 (1997), 433-445.

Received: September 2008.

Revised: January 2009.

^rND^sKeen^nL ^rND^sMaskit^nB ^rND^1A01^nA.A^sShaikh^rND^1A02^nC.S^sBagewadi ^rND^1A01^nA.A^sShaikh^rND^1A02^nC.S^sBagewadi ^rND^1A01^nA. A^sShaikh^rND^1A02^nC. S^sBagewadi

CUBO A Mathematical Journal Vol.12, N° 01, (181-193). March 2010

On N(k)-Contact Metric Manifolds

A.A. Shaikh and C.S. Bagewadi

Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India email : aask2003@yahoo.co.in

Department of Mathematics, Kuvempu University, Jana Sahyadri, Shankaraghatta-577 451, Karnataka, India email : prof_bagewadi@yahoo.com

ABSTRACT

The object of the present paper is to study a type of contact metric manifolds, called contact metric manifolds admitting a non-null concircular and torse forming vector field. Among others it is shown that such a manifold is either locally isometric to the Riemannian product or a Sasakian manifold. Also it is shown that such a contact metric manifold can be expressed as a warped product is a dimensional manifold.

Key words and phrases: Contact metric manifold, k-nullity distribution, contact metric manifold, concircular vector field, torse forming vector field, n-Einstein, Sasakian manifold, warped product.

RESUMEN

El objetivo del presente artículo es estudiar un tipo de variedades métricas de contacto, llamadas variedades métricas de contacto admitiendo un campo de vectores concircular y forma torse. Es demostrado también que tales variedades son o localmente isométricas a productos Riemannianos o una variedade Sasakian. Es demostrado que tales variedades métricas de contacto pueden ser expresadas como un producto deformado es una variedad dimensional.

Math. Subj. Class.: 53C05, 53C15, 53C25.

References

[1] Adati, T., On Subprojective spaces III, Tohoku Math. J., 3(1951), 343-358.

[2] Blair, D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, 1976.

[3] Blair, D.E., Two remarks on contact metric structure , Tohoku Math. J., 29(1977), 319-324.

[4] Blair, D.E., Koufogiorgos, T. and Papantoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel J. of Math., 19(1995), 189-214.

[5] Shaikh, A.A. and Baishya, K.K., On (k, µ)-contact metric manifolds, J. Diff. Geom. and Dyn. Sys., 11(1906), 253-261.

[6] Schouten, J.A., Ricci Calculas (Second Edition), Springer-Verlag, 1954, 322.

[7] Tanno, S., Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40(1988), 441-448.

[8] Yano, K., On the torse forming direction in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20(1940), 340-345.

[9] Yano, K., Concircular geometry I-IV, Proc. Imp. Acad. Tokyo, 16(1940), 195- 200, 354-350.

Received: July, 2008.

Revised: January, 2009.

^rND^sAdati^nT ^rND^sBlair^nD.E ^rND^sBlair^nD.E^rND^sKoufogiorgos^nT^rND^sPapantoniou^nB.J ^rND^sShaikh^nA.A^rND^sBaishya^nK.K ^rND^sTanno^nS ^rND^sYano^nK ^rND^sYano^nK ^rND^1A01^nJacqueline^sRojas^rND^1A02^nRamón^sMendoza^rND^1A03^nEben da^sSilva ^rND^1A01^nJacqueline^sRojas^rND^1A02^nRamón^sMendoza^rND^1A03^nEben da^sSilva ^rND^1A01^nJacqueline^sRojas^rND^1A02^nRamón^sMendoza^rND^1A03^nEben da^sSilva

CUBO A Mathematical Journal Vol.12, N° 01, (195-217). March 2010

Projective Squares in and Bott's Localization Formula

Jacqueline Rojas*, Ramón Mendoza and Eben da Silva

UFPB-CCEN - Departamento de Matemática, Cidade Universitária, 58051-900, Joo Pessoa-PB - Brasil email : jacq@mat.ufpb.br

UFPE-CCEN - Departamento de Matemática, Cidade Universitária, 50740-540, Recife-PE - Brasil email : ramon@dmat.ufpe.br

UFRPE/UAST-CCEN - Departamento de Matemática e Física, Fazenda Saco, caixa postal 63, 56900-000, Serra Talhada-PE - Brasil email : eben@uast.ufrpe.br

ABSTRACT

We give an explicit description of the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in the projective plane that allows us to afford a natural embedding in a product of Grassmann varieties. We also use this description to explain how to apply Bott's localization formula (introduced in 1967 in Bott's work [2]) to give an answer for an enumerative question as used by the first time by Ellingsrud and Strmme in [8] to compute the number of twisted cubics on a general Calabi-Yau threefold which is a complete intersection in some projective space and used later by Kontsevich in [16] to count rational plane curves of degree d passing through 3d - 1 points in general position in the plane.

Key words and phrases: Hilbert scheme, Bott's localization formula.

RESUMEN

En este trabajo, damos una descripción explícita del esquema de Hilbert, que parametriza los subesquemas cerrados de dimensión cero y grado 4 del plano proyectivo, esto nos permite mapear este esquema en un producto de variedades de Grassmann. Usamos dicha construcción, para explicar como se utiliza la fórmula de localización de Bott (introducida en 1967 por Bott en [2]) para responder una pregunta de Geometria Enumerativa, tal como lo hicieron Ellingsrud y Strmme en [8], para calcular cuantas cúbicas torcidas existen en una variedad de CalabiYau tridimensional, que es una intersecci´on completa en algún espacio proyectivo, y que fue usada posteriormente por Kontsevich en [16], para contar curvas planas racionales de grado d pasando por 3d - 1 puntos en posición general en el plano.

Math. Subj. Class.: 14C05, 14N05.

Notas

Dedicated to Israel Vainsencher on the occasion of his 60th birthday.

*Partially supported by CNPq (Edital Casadinho No 620108/2008-

References

[1] Avritzer, D. and Vainsencher, I., Hilb4P2, in Proceedings of the Conference at Sitges, Spain (1987), ed. S. Xambó, Springer-Verlag Lect. Notes Math. 1436 (1987), 30-59.

[2] Bott, R., A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967), 311-330.

[3] Brion, M., Equivariant cohomology and equivariant intersection theory, in Representation theory and algebraic geometry, Kluwer (1998), 1-37.

[4] Cox, D. and Katz, S., Mirror symmetry and algebraic geometry, Math. Surv. 68 - Amer. Math. Soc., Providence, RI, 1999.

[5] Edidin, D. and Graham, W., Equivariant intersection theory Invent. Math. 131, no. 3 (1998), 595-634.

[6] Edidin, D. and Graham, W., Localization in equivariant intersection theory and the Bott residue formula Am. J. Math. 120, No. 3 (1998), 619-636.

[7] Ellingsrud, G. and Gttsche, L., Hilbert schemes of points and Heisenberg algebras, School on Algebraic Geometry (Trieste, 1999), 59-100, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000.

[8] Ellingsrud, G. and Strmme, S.A., Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), 175-193.

[9] Fogarty, J., Algebraic Families on an Algebraic Surface, Amer. J. Math., 90 (1968), 511- 521.

[10] Fulton, W., Intersection Theory, Graduate Texts in Math, Springer-Verlag, 1977.

[11] Grünberg, D. and Moree, P., Sequences of enumerative geometry: congruences and asymptotics, arXiv:math/0610286v1 [math.NT] (10-2006) .

[12] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1985.

[13] Kleiman, S.L., Intersection theory and enumerative geometry: a decade in review, with the collaboration of Anders Thorup on §3. Proc. Sympos. Pure Math., 46, Part 2, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 321-370, Amer. Math. Soc., Providence, RI, 1987.

[14] Kleiman, S.L. and Laksov, D., Schubert Calculus, The American Mathematical Monthly, Vol. 79, No. 10 (1972), 1061-1082.

[15] Kock, J. and Vainsencher, I., An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves, Progress in Mathematics, Birkhäuser Boston, 2006.

[16] Kontsevich, M., Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994), 335-368, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995.

[17] Meireles, A. and Vainsencher, I., Equivariant Intersection Theory and Bott's Residue Formula, notes for the Escola de Álgebra, UnB-2000, Mat. Contemporanea-SBM, vol. 20, I (2001), 1-70 .

[18] Mumford, D., Lectures on curves on an algebraic surface, Annals of Math. Studies 59, Princeton Univ. Press, Princeton, 1966.

[19] Nakajima, H., Lectures on Hilbert Schemes of points on Surfaces, AMS University Lecture Series, Volume 18, Providence RI, 1999.

[20] Schubert, H., Die n-dimensionale Verallgemeinerung der Anzahlen fr die vielpunktig verührenden Tangenten einer punktallgemeinen Flche m-ten Grades, Math. Annalen 26 (1886), 52-73.

[21] da Silva, E.A., Uma Compactificação do Espaço das Quádruplas de Pontos em P2, master dissertation, DM -UFPB, 2005.

[22] Witten, E., Topological sigma models, Comm. Math. Phys. 118, no. 3 (1988), 411-449.

Received: July, 2008.

Revised: January, 2009.

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ttsche^nL ^rND^sEllingsrud^nG^rND^sStr

mme^nS.A ^rND^sFogarty^nJ ^rND^sKleiman^nS.L ^rND^sKleiman^nS.L^rND^sLaksov^nD ^rND^sMeireles^nA^rND^sVainsencher^nI ^rND^sMumford^nD ^rND^sNakajima^nH ^rND^sSchubert^nH ^rND^sWitten^nE ^rND^1A01^nA.P^sFarajzadeh^rND^1A02^nA^sAmini-Harandi^rND^1A03^nD^sO’Regan^rND^1A04^nR.P^sAgarwal ^rND^1A01^nA.P^sFarajzadeh^rND^1A02^nA^sAmini-Harandi^rND^1A03^nD^sO’Regan^rND^1A04^nR.P^sAgarwal ^rND^1A01^nA. P^sFarajzadeh^rND^1A02^nA^sAmini-Harandi^rND^1A03^nD^sO’Regan^rND^1A04^nR. P^sAgarwal

CUBO A Mathematical Journal Vol.12, N°01, (219-230). March 2010

Strong Vector Equilibrium Problems in Topological Vector Spaces Via KKM Maps

A.P. Farajzadeh, A. Amini-Harandi*, D. O'Regan and R.P. Agarwal

Department of Mathematics, Razi University, Kermanshah, 67149, Iran email : ali-ff@sci.razi.ac.ir

Department of Mathematics, University of Shahrekord, Shahrekord, 88186-34141, Iran email : aminih_a@yahoo.com

Department of Mathematics, National University of Ireland, Galway, Ireland email : donal.oregan@nuigalway.ie

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA email : agarwal@fit.edu

ABSTRACT

In this paper, we establish some existence results for strong vector equilibrium problems (for short, SVEP) in topological vector spaces. The solvability of the SVEP is presented using the Fan-KKM lemma. These results give a positive answer to an open problem proposed by Chen and Hou and generalize many important results in the recent literature.

Key words and phrases: Strong vector equilibrium, Upper sign continuity, Pseudomonotone bifunction, Quasimonotone bifunction.

RESUMEN

En este artículo, establecemos algunos resultados de existencia para problemas de equilibrio strong vector en espacios vectoriales topológicos (abreviadamente, SVEP). La salubilidad del SVEP es presentada usando el lema de Fan-KKM. Estos resultados dan una respuesta positiva a problemas abiertos propuestos por Chen y Hon y generalizan varios resultados importantes en la literatura reciente.

Math. Subj. Class.: PLEASE INFORM

Notas

*The second author was in part supported by a grant from IPM (No. 85470015)

References

[1] Ansari, Q.H., Vector equilibrium problems and vector variational inequalities, In: F. Giannessi(ed) Vector variational inequalities and vector equilibria, Mathematical theories, Kluwer, Dordrecht, (2000), 1-16

[2] Bianchi, M. and Pini, R., Coercivity conditions for equilibrium problems, J. Optim. Theory Appl., 124 (2005), 79-92.

[3] Chen, G.Y. and Hou, S.H., Existence of solutions for vector variational inequalities, in: F.Giannessi(Ed), Vector variational inequalities and vector equilibria, kluwer publishers, Dordrecht, Holland, (2000), 73-86.

[4] Fakhar, M. and Zafarani, J., Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions, J. Optim. Theory Appl., 126 (2005), 109-124.

[5] Fan, K., Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.

[6] Fang, Y.P. and Huang, N.J., On the strong vector variational inequalities, Research Report, Department of Mathematics, Sichuan University, 2002.

[7] Fang, Y.P. and Huang, N.J., Strong vector variational inequalities in Banach spaces, Appl. Math. Lett., 19 (2006), 362-368.

[8] Giannessi, F., Vector variational inequalities and vector equilibria, Mathematical theories, Kluwer, Dordrecht, 2000.

[9] Giannessi, F., Theorems of alternative, quadratic programs, and complementarity problems, in: R. W. Cottle, Giann, J.L. Lions (Eds), Variational inequality and Complementarity problems, John Wiley and Sons, New York, (1980), 151-186.

[10] Iusem, A.N. and Sosa, W., New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635.

[11] Oettli, W. and Schlager, D., Existence of equilibria for monotone multivalued mappings, Math. Meth. Oper. Res., 48 (1998), 219-228.

[12] Park, S., Fixed points, intersections theorem, variational inequalities, and equilibrium theorems, Inter. J. Math. Math. Sci., 2 (2000), 73-93.

[13] Yang, F., Wu, C. and He, Q., Applications of Ky Fan's inequality on -compact set to variational inclusion and n- person game theory, J. Math. Anal. Appl., 319 (2006), 177-186.

Received: January, 2009 .

Revised: March, 2009.

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CUBO A Mathematical Journal Vol.12, N° 01, (23-40). March 2010

Quenching for Discretizations of a Nonlocal Parabolic Problem with Neumann Boundary Condition

Théodore K. Boni¹ and Diabaté Nabongo²

¹Institut National Polytechnique Houphouët-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, (Côte d'Ivoire) email : theokboni@yahoo.fr

²Université d'Abobo-Adjamé, UFR-SFA, Département de Mathématiques et Informatiques, 16 BP 372 Abidjan 16, (Côte d'Ivoire) email : nabongo_diabate@yahoo.fr

ABSTRACT

In this paper, under some conditions, we show that the solution of a discrete form of a nonlocal parabolic problem quenches in a finite time and estimate its numerical quenching time. We also prove that the numerical quenching time converges to the real one when the mesh size goes to zero. Finally, we give some computational results to illustrate our analysis.

Key words and phrases: Nonlocal diffusion, quenching, numerical quenching time.

RESUMEN

En este artículo mostramos, bajo algunas condiciones, que la solución de una forma discreta de un problema parabólico no local se sofoca en tiempo finito y estimamos su tiempo de sofocamiento numérico. Probamos también que el tiempo de sofocamiento numérico converge par un real cuando el tamaño de la malla tiende a cero. Finalmente damos algunos resultados computacionales para ilustrar nuestros análisis.

References

[1] Andren, F., Mazon, J.M., Rossi, J.D. and Toledo, J., The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equat., 8 (2008), 189-215.

[2] Andren, F., Mazon, J.M., Rossi, J.D. and Toledo, J., A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, Preprint.

[3] Bates, P. and Chmaj, A., An intergrodifferential model for phase transitions: stationary solutions in higher dimensions, J. Statistical Phys., 95 (1999), 1119-1139.

[4] Bates, P. and Chmaj, A., A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.

[5] Bates, P. and Han, J., The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312.

[6] Bates, P. and Han, J., The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Diff. Equat., 212 (2005), 235-277.

[7] Bates, P., Fife, P. and Wang, X., Travelling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.