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In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of dilute electrons and plasma can be modeled by Vlasov–Poisson–Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is , by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245–316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro–micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components. [Copyright &y& Elsevier]

Abstract: This paper derives an explicit formula for Branson''s Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions with their first derivatives having the same fixed points with the same multiplicities. The results in this paper improve those given by K. Tohge, Xiao-Min Li and Hong-Xun Yi. [Copyright &y& Elsevier]

Abstract: In this paper, by means of a series of counterexamples, we study in a systematic way the relationships among (pseudo, quasi) α-preinvexity, (strict, strong, pseudo, quasi) α-invexity and (strict, strong, pseudo, quasi) αη-monotonicity. Results obtained in this paper can be viewed as a refinement and improvement of the results of Noor and Noor [M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316 (2006) 697–706]. [Copyright &y& Elsevier]

Abstract: Let R be a local ring essentially of finite type over a field k of characteristic . In the paper [M. Furuya, H. Niitsuma, Regularity criterion of Noetherian local rings of prime characteristic, J. Algebra 247 (2002) 219–230], we constructed some regularity criterion for such a local ring R in terms of the higher differential algebra and the -basis. In this paper, we introduce the concept of a reduced index of a Noetherian ring and we show the sharpened result of the above criterion and further we also give a geometric regularity criterion in terms of the higher differential algebra and the -basis. The latter criterion yield the sharpened result of a part of Orbanz''s theorem [U. Orbanz, Höhere Derivationen und Regularität, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2]. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the distributional Chébli–Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337–354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945–957] about distributional Hankel transforms. We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented. [Copyright &y& Elsevier]

Abstract: Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples. [Copyright &y& Elsevier]

Abstract: Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy. [Copyright &y& Elsevier]

Abstract: In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order. [Copyright &y& Elsevier]

Abstract: In this paper, the property of linearly dependence of solutions and for higher order linear differential equation where () are entire periodic coefficients and () is the dominant coefficient, is investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we mainly discuss the infinite loss of regularity and μ-loss for a σ-evolution type model with oscillating in time coefficients. On the one hand, an explicit counter-example has been constructed in the frequency space to show the precise infinite loss of regularity. On the other hand, due to the finite propagation speed property for , we construct the counter-example of a sequence of solutions in by applying state of art techniques. [Copyright &y& Elsevier]

Abstract: In this paper, essentially strongly order-preserving and conditionally set-condensing semiflows are considered. Obtained is a new type of generic quasi-convergence principles implying the existence of an open and dense set of stable quasi-convergent points when the state space is order bounded. The generic quasi-convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations, giving some results of theoretical and practical significance. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine–Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47–84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257–289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101–132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large. [Copyright &y& Elsevier]

Abstract: In this paper, we improve contractivity conditions of solutions for the positive equilibrium of the following differential equation with piecewise constant arguments: where is a nonnegative continuous function on , , , , , and . In particular, for the case and , we really improve the known three type conditions of the contractivity for solutions of this model (see for example, [Y. Muroya, A sufficient condition on global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 32 (2003) 75–83]). For the other case and , under the condition , the obtained result partially improves the known results on the contractivity of solutions for the positive equilibrium of this model given by the author [Y. Muroya, Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270 (2002) 602–635] and others. [Copyright &y& Elsevier]

Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic . Assume that p is good for G. Pommerening''s theorem [K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening''s theorem, which is a further simplification of Premet''s first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet''s proof of the existence theorem for good transverse slices to the nilpotent -orbits in . [Copyright &y& Elsevier]

Abstract: Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given signal. The shrinkage method is applied in different areas, including data compression, signal processing and statistics. The almost everywhere convergence of resulting wavelet series has been established in [T. Tao, On the almost everywhere convergence of wavelet summation methods, Appl. Comput. Harmon. Anal. 3 (1996) 384–387] and [T. Tao, B. Vidakovic, Almost everywhere behavior of general wavelet shrinkage operators, Appl. Comput. Harmon. Anal. 9 (2000) 72–82]. With a representation of in terms of wavelet coefficients of f, we are interested in considering the influence of wavelet thresholding to f on its derivative . In this paper, for the representation of differential operators in nonstandard form, we establish the almost everywhere convergence of estimators as threshold tends to zero. [Copyright &y& Elsevier]

Abstract: This paper focuses on a nonlinear equation from thin plate theory of the form . We obtain maximum principles for certain functions defined on the solution of this equation using P-functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles, J. Anal. Math. 30 (1976) 421–433] and Schaefer [P.W. Schaefer, Solution, gradient, and laplacian bounds in some nonlinear fourth order elliptic equations, SIAM J. Math. Anal. 18 (1987) 430–434]. [Copyright &y& Elsevier]

Abstract: In this paper, it is shown that some necessary characteristic conditions for unique extremality obtained by Zhu and Chen are also sufficient and some sufficient ones by them actually imply that the uniquely extremal Beltrami differentials have a constant modulus. In addition, some local properties of uniquely extremal Beltrami differentials are given. [Copyright &y& Elsevier]

Abstract: This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis of () in which is not necessarily invertible, and thus the system () could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting. [Copyright &y& Elsevier]

Abstract: Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree , with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l. [Copyright &y& Elsevier]

Abstract: The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions. [Copyright &y& Elsevier]

Abstract: In this paper, characterizations of the existence of the directional derivative and second-order parabolic directional derivative of a locally Lipschitzian function are established. These characterizations involve the adjacent cone and second-order adjacent set of the graph of the function. [Copyright &y& Elsevier]

Abstract: Let and be analytic in the unit disk. The Hadamard product of f and g is defined by . This paper gives some characterizations of functions in spaces in terms of the Hadamard products. [Copyright &y& Elsevier]

An analytical model to simulate wheel/rail interaction using the Green''s functions method is proposed in this paper. The model consists of a moving wheel on a discretely supported rail. Particularly for this model of rail, the bending and the longitudinal displacement are coupled due to the rail pad and a complex model of the rail pad is adopted. An efficient method for solving a time-domain analysis for wheel/rail interaction is presented. The method is based on the properties of the rail''s Green functions and starting to these functions, a track''s Green matrix is assembled for the numerical simulations of wheel/rail response due to three kinds of vertical excitations: the steady-state interaction, the rail corrugation and the wheel flat. The study points to influence of the worn rail—rigid contact—on variation in the wheel/rail contact force. The concept of pinned–pinned inhibitive rail pad is also presented. [Copyright &y& Elsevier]

Abstract: In an earlier paper [D. Richards, Q. Zheng, Determinant formulas for multidimensional hypergeometric period matrices, Adv. in Appl. Math. 29 (2002) 137–151] on the determinants of certain period matrices, we formulated a conjecture about the determinant of a certain hypergeometric matrix. In this article, we establish this conjecture by constructing a system of linear equations in which that determinant is one of the variables. As a consequence, we obtain the value of an integral which generalizes the well-known multidimensional beta integral of A. Selberg [A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 26 (1944) 71–78] and some hypergeometric determinant formulas of A. Varchenko [A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1206–1235, English translation, Math. USSR-Izv. 35 (1990) 543–571; A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990) 146–158, English translation, Math. USSR-Izv. 36 (1991) 155–167]. [Copyright &y& Elsevier]

Abstract: In this paper, we present lower and upper bounds for the independence number and the clique number involving the Laplacian eigenvalues of the graph G. [Copyright &y& Elsevier]

Abstract: In electrical impedance tomography, one tries to recover the conductivity inside a body from boundary measurements of current and voltage. In many practically important situations, the object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. It has been shown that the inhomogeneities can be characterized by the factorization technique if the conductivity coefficient jumps to a higher or lower value on the boundaries of the inclusions. In this paper, we extend the results to the case of weaker inclusions: If the inhomogeneities inside the body are more (or less) conductive than the known background, if the conductivity coefficient and its lowest normal derivatives are continuous over the inclusion boundaries, and if the mth normal derivative of the conductivity jumps on the inclusion boundaries, then the factorization method provides an explicit characterization of the inclusions. [Copyright &y& Elsevier]

Abstract: In this paper, we provide oscillation properties of every solution of the neutral differential equation with positive and negative coefficients where , , , . [Copyright &y& Elsevier]

Abstract: In this paper, we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions, which improve and generalize the related results of Gu, Bergweiler and Lin. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of periodic solutions of the second order differential equations . Using continuation lemma, we obtain the existence of periodic solutions provided that () is sublinear when x tends to positive infinity and satisfies a new condition where M, d are two positive constants. [Copyright &y& Elsevier]

Abstract: In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in -norm and weighted -norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke. The equation is of reaction–diffusion type in a cylinder of radius R and of diffusion and absorption type outside of the cylinder. We prove the nonexistence of a travelling front when R is small enough and the existence if R is large enough using a recent energy method. We construct the travelling front as the limit in time of a solution with a well-chosen initial condition, in a travelling referential. [Copyright &y& Elsevier]

Abstract: In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, , decays sufficiently fast at infinity at two different times, then . [Copyright &y& Elsevier]

Abstract: Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones. [Copyright &y& Elsevier]

Abstract: In this paper, we study the biharmonic equation in the half-space , with . We prove in theory, with , existence and uniqueness results. We consider data and give solutions which live in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459–472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278–2295]. Finally, we give two examples to illustrate the obtained results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of stationary solutions to the Vlasov–Poisson–Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as . In fact, the stationary Vlasov–Poisson–Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being for some , it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov–Poisson–Fokker–Planck system, J. Math. Anal. Appl. 202 (1996) 1058–1075] where the decay rate is assumed. [Copyright &y& Elsevier]

Abstract: In this paper we construct generalized solutions (weak solutions) to a linear discontinuous differential equation which appears as mathematical model in mechanics and for which usual methods are not quite adequate. The result is applied to the equation of an elastic beam on a Winkler type of foundation. [Copyright &y& Elsevier]

Abstract: An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarz’s study of a map carrying invariant operators to operators on the orbifold and on an algorithm to compute rings of invariants using Gröbner bases due to Derksen [Derksen, Harm, 1999. Computation of invariants for reductive groups. Adv. Math. 141 (2), 366–384]. It is also possible to avoid using Derksen’s algorithm, instead relying on the Reynolds operator and the Molien series. [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]

Abstract: We give here a new proof of the non-degeneracy of the fundamental bilinear form for -m-Quasi-Invariants and for m-Quasi-Invariants of classical Weyl groups. We also indicate how our approach can be extended to other Coxeter groups. This bilinear form plays a crucial role in the original proof [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, arXiv: math.QA/0106175 v1, June 2001] that m-Quasi-Invariants are a free module over the invariants as well as in all subsequent proofs [Y. Berest, P. Etingof, V. Ginsburg, Cherednik algebras and differential operators on quasi-invariants, math.QA/0111005; A. Garsia, N. Wallach, Some new applications of orbit harmonics, Sém. Lothar. Combin. 50 (2005), Article B50j]. However, in previous literature this non-degeneracy was stated and used without proof with reference to some deep results of Opdam [E.M. Opdam, Some applications of shift operators, Invent. Math. 98 (1989) 1–18] on shift-differential operators. This result hinges on the validity of a deceptively simple identity on Dunkl operators which, at least in the case, begs for an elementary painless proof. An elementary but by all means not painless proof of this identity can be found in a paper of Dunkl and Hanlon [C. Dunkl, P. Hanlon, Integrals of polynomials associated with tableaux and the Garsia–Haiman conjecture, Math. Z. 228 (1998) 537–567. 71]. Our proof here is not elementary but hopefully it should be painless and informative. [Copyright &y& Elsevier]

Abstract: In this paper we study the problem where are positive radial functions, and . We give existence and nonexistence results, depending on the parameters p and λ. It turns out that is a critical value for the existence of solutions. [Copyright &y& Elsevier]

Abstract: It is conjectured by Erdős, Graham and Spencer that if with , then this sum can be decomposed into n parts so that all partial sums are ⩽1. This is not true for as shown by , , . In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for . In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for . Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if and no partial sum (certainly not a single term) is the inverse of an positive integer. [Copyright &y& Elsevier]

Abstract: The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition. [Copyright &y& Elsevier]

Abstract: Two fixed point theorems for the sum of contractive and compact operators are obtained in this paper, which generalize and improve the corresponding results in [H. Schaefer, Über die methode der a priori-Schranken, Math. Ann. 129 (1955) 415–416; T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383–2390; V.I. Istrǎtescu, Fixed Point Theory, an Introduction, Reidel, Dordrecht, 1981; T.A. Burton, K. Colleen, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 23–31; D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980]. As the applications for the results, we obtain the existence of periodic solutions for some evolution equations with delay, which extend the corresponding results in [T.A. Burton, B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations 90 (1991) 357–396]. [Copyright &y& Elsevier]

Abstract: This paper pursues Takeuchi''s Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard–Vessiot theory, J. Algebra 122 (1989) 481–509] to the Picard–Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra. [Copyright &y& Elsevier]

Abstract: One approach for computations with special functions in computer algebra is the systematic use of analytic functions whenever possible. This naturally leads to problems of how to answer questions about analytic functions in a fully effective way. Such questions comprise the determination of the radius of convergence or the evaluation of the analytic continuation of the function at the endpoint of a broken line path. In this paper, we propose a first definition for the notion of an effective analytic function and we show how to effectively solve several types of differential equations in this context. We will limit ourselves to functions in one variable. [Copyright &y& Elsevier]