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The blended solutions are more general than the unblended solutions for polynomial-like iterative equation with multivalued functions. In this paper we study the blended solutions of polynomial-like iterative equation with multivalued functions. [ABSTRACT FROM AUTHOR]

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation in a sense of Baire category. We reveal their differences in generic regularity and singular behavior. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper. [ABSTRACT FROM AUTHOR]

In this paper, we deduce new conditions for the existence of ground state solutions for the $p$-Laplacian equation $$\begin{equation*} \left \{ \begin{array}{@{}ll} -\mathrm {div}(\nabla u^{p-2}\nabla u)+V(x)u^{p-2}u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\[5pt] u\in W^{1, p}({\mathbb {R}}^{N}), \end{array} \right . \end{equation*}$$ which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function $t\mapsto f(x, t)/t^{p-1}$. In particular, both $tf(x, t)$ and $tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions. [ABSTRACT FROM AUTHOR]

In this paper, the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion. Here we introduce a cubic spline numerical model (Spline Model for short), which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/ bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh. A new algorithm of "fitting cubic spline--time step integration_fitting cubic spline--......" is developed to determine their first- and 2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model. And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model's mathematical foundation of numerical analysis. It is pointed out that the Spline Model has mathematical laws of "convergence" of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives. The "optimality" of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions. In addition, a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field. Besides, the slopes and curvatures of a central difference are identified respectively, with a smoothing coefficient of 1/3, three-point smoothing of that of a cubic spline. Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline, respectively. Furthermore, a global simulation case of adiabatic, non-frictional and "incompressible" model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model, whose initial condition comes from the NCEP reanalysis data, along with quasi-uniform latitude-longitude grids and the so-called "shallow atmosphere" Navier-Stokes primitive equations in the spherical coordinates. The Spline Model, which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme, provides an initial ideal case of global atmospheric circulation. In addition, considering the essentially non-linear atmospheric motions, the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation. [ABSTRACT FROM AUTHOR]

This paper concerns the regularity criterion of theweak solutions to the three-dimensional (3D) micropolar fluid equations in terms of the pressure. It is proved that if one of the partial derivatives of pressure satisfies ∂3 π ε Lp(0, T; Lq(R³)) with 2/p + 3/q ≤ 2, 3 < q < ∞,1 < p < ∞, then the weak solution of the micropolar fluid equations becomes regular on (0, T). [ABSTRACT FROM AUTHOR]

The Exp-function method combined with F-expansion method is employed to investigate the K(m,n) equation with t-dependent coefficients. The solitary wave solutions and periodic wave solutions to the equation are constructed analytically under certain circumstances. The results presented in this paper improve the previous results. [ABSTRACT FROM AUTHOR]

In this paper we obtain some sufficient conditions for Sakaguchi type functions of order β, Sn(β, s, t) defined as the class of normalized analytic functions in the open unit disc satisfying [Multiple line equation(s) cannot be represented in ASCII text], and 0 ≤β < 1. [ABSTRACT FROM AUTHOR]

In this paper we prove existence theorems for integro - differential equations xΔ(t) = f(t, x(t), ∫ 0 t k(t, s, x(s))Δs), x(0) = x0 t ∊ Ia = [0, a] ∩ T, a ∊ R+, where T denotes a time scale (nonempty closed subset of real numbers R), Ia is a time scale interval. Functions f, k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale. [ABSTRACT FROM AUTHOR]

This paper is concerned with rational Szegő quadrature formulas to approximate integrals of the form by a formula such as , where the weights λk are positive and the nodes zk are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szegő–Radau and rational Szegő–Lobatto quadrature formulas will be characterized. [ABSTRACT FROM PUBLISHER]

This paper discusses some common iterations of complex functions. The presentation is such that similar processes can easily be implemented and understood by undergraduate students. The aim is to illustrate some of the beauty of complex dynamics in an informal setting, while providing a couple of results that are not otherwise readily available in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, travelling wave solutions for a nonlinear variant of the PHI-four equation are studied by using the Weierstrass elliptic function method. As a result, some previously known solutions are recovered, and at the same time some new ones are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the long-time behavior of plate equation with nonlinear damping and critical nonlinearity. We prove the existence of a global attractor in the space . [Copyright &y& Elsevier]

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the “thorn in the side” of many mathematics students. In this paper we present a result intended to help students achieve a greater understanding of a familiar algebraic error. [ABSTRACT FROM AUTHOR]

Abstract: By using a new type of smoothing function, we first reformulate the generalized nonlinear complementarity problem over a polyhedral cone as a smoothing system of equations, and then develop a smoothing Newton-type method for solving it. For the proposed method, we obtain its global convergence under milder conditions, and we further establish its local superlinear (quadratic) convergence rate under the BD-regular assumption. Preliminary numerical experiments are also reported in this paper. [Copyright &y& Elsevier]

Abstract: This paper investigates the regularity of solutions of convolution equations in the frame of classes of ultradifferentiable functions and ultradistributions. We improve previous work by Bonet, Chou, Fernández, Galbis, Meise and others. [Copyright &y& Elsevier]

Abstract: In this paper, a special lattice Boltzmann model is proposed to simulate two-dimensional unsteady Burgers’ equation. The maximum principle and the stability are proved. The model has been verified by several test examples. Excellent agreement is obtained between numerical predictions and exact solutions. The cases of steep oblique shock waves are solved and compared with the two-point compact scheme results. The study indicates that lattice Boltzmann model is highly stable and efficient even for the problems with severe gradients. [Copyright &y& Elsevier]

Abstract: In this paper, we derive two most general possible q-hypergeometric expansion formulae for 12 Φ 11(q) and r Φ s (z), respectively. The results are unique in the sense that no such results are available in the literature beyond 4 Φ 3(q) so far and have the advantage that these results are also applicable to the top-level 10 Φ 9(q) biorthogonal rational functions. [Copyright &y& Elsevier]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

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: Let be given integers. An undirected even-valent circulant graph, has n vertices , , and for each and j there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337–350]], is that where satisfies a linear recurrence relation of order . And, most recently, for odd-valent circulant graphs, a nice investigation on the number is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69–79]. In this paper, we explore further properties of the numbers from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for , we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337–350]. As examples, we describe our technique and the asymptotics of the numbers. [Copyright &y& Elsevier]

In the paper the ϕ-homogeneity equation almost everywhere is studied. Let and H be groups with zero. Assume that (X,G) is a G-space and (Y,H) is an H-space. We prove, under some assumption on (Y,H), that if the functions ϕ : G → H and F : X → Y satisfy the equation of ϕ-homogeneity F(αx) = ϕ (α)F(x) almost everywhere in G× X then either F is a zero function or there exists a homomorphism ... : G → H such that ϕ almost ... everywhere in G and there exists a function F : → Y such that ... (αx) = ...(α)... (x) for α η G\{0}, x ϵ X, and F = ... almost everywhere in X. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the integral transform containing the special function ϵ ((α, β)n;z) generalizing the Mittag-Leffler type function in the space ℒv,r (1 ⩽ r ⩽ ∞, v ϵ R) of Lebesgue measurble functions on R+ = (0, +∞) such that ∥ƒ∥v,r < ∞, where ∥ƒ∥v,r ≡ (Miltiple line equation cannot be converted to ASCII text) < ∞ (1 ⩽ r < ∞); ∥ƒ∥v,∞ ≡ esssupt>0∣tv ƒ(t)∣. Mapping properties such as the boundedness, the range, the representation and the inversion of the considered transform are proved. The results are based on the representation of the considered transform as the H-transform. [ABSTRACT FROM AUTHOR]

The additive Schwarz preconditioned inexact Newton (ASPIN) method was recently introduced [X.-C. Cai and D. E.\ Keyes, {\it SIAM J. Sci.\ Comput.}, 24 (2002), pp. 183--200] to solve the systems of nonlinear equations with nonbalanced nonlinearities. Although the ASPIN method has successfully been used to solve some difficult nonlinear equations, its convergence property has not been studied since it was proposed. In this paper, the convergence property of the ASPIN method is studied, and the obtained result shows that this method is locally convergent. Furthermore, the convergence rate for the ASPIN method is discussed and the obtained result is similar to that of the inexact Newton method. [ABSTRACT FROM AUTHOR]

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type -Δu + V(x)u = f(x,u), where x ∈ ℝN and lim|x| → ∞u(x) = 0. The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials V for which this splitting condition holds. [ABSTRACT FROM AUTHOR]

This paper develops two necessary conditions for a local minimum of an arbitrary extended real-valued function (Theorem 2.2) and, quite separately, two conditions, each sufficient for such a function to have a strict local minimum (Theorem 2.3). [ABSTRACT FROM AUTHOR]

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]

This paper deals with H-stability of Runge-Kutta methods with variable stepsize for the system of pantograph equations. It is shown that both Runge-Kutta methods with nonsingular matrix coefficient A and stiffly accurate Runge-Kutta methods are H-stable if and only if the modulus of stability function at infinity is less than 1. [ABSTRACT FROM AUTHOR]

In this paper a mathematical model of the transport and accretion of colloidal type particles in porous media is examined. Different from previously considered ones, the model also includes an inverse compaction factor. We present a numerical method for the solution of the coupled nonlinear differential system of equations, which models the above phenomenon. Accuracy of the presented method is tested on concrete examples. [Copyright &y& Elsevier]

In the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty. These methods are applied to the fragmentation equation with mass loss, yielding sufficient conditions for the existence of conservative and shattering solutions. Our results provide a mathematical framework that clarifies the discussion of [Phys. Rev. A 43 (1991) 656, Phys. Rev. A 41 (1990) 5755, J. Phys. A 24 (1991) 3967] on shattering fragmentation in such models showing, among others, that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss, contrary to the conjecture of [Phys. Rev. A 43 (1991) 656, Phys. Rev A 41 (1990) 5755]. [Copyright &y& Elsevier]

We consider a positive C²-function u in Rn which is equal to two different spherical solutions respectively in two disjoint domains, u satisfies the equation Δu + n(n - 2)Kun+2/n-2 = 0 in Rn. By a result of Gidas, Ni and Nirenberg, K ... 1 in Rn. In this paper we discuss lower bounds on supRn K -1. The estimates are applied to study blow-up solutions and interaction of bubbles. [ABSTRACT FROM AUTHOR]

In this paper we study genus 2 function fields K with degree 3 elliptic subfields.We compute an equation for the locus of these fields K in the moduli space of genus 2 curves. From this we derive that the number of Aut .. K ... -classes of such subfields of fixed K is 0,1,2 or 4. [ABSTRACT FROM AUTHOR]

In this paper we study the solution of the equation ⋄c1k⋄c2ku(x)=f(x,Δc1k−1□c1k⋄c2ku(x)) where ⋄c1k⋄c2k is the product of the Diamond operators defined by⋄c1k=1/c14∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj22kand⋄c2k=1/c24∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj2∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj22kand⋄c2k=1/c24∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj22−∑j=p+1p+q∂2/∂xj22kand⋄c2k=1/c24∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj22k and ⋄c2k=1/c24∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj2∑i=1p∂2/∂xi22−∑j=p+1p+q∂2/∂xj22−∑j=p+1p+q∂2/∂xj22k,where c1 and c2 are positive constants, k is a non-negative integer, p+q=n is the dimension of the n-dimensional Euclidean space Rn, x=(x1,…,xn)∈Rn, u(x) is an unknown and f(x,Δc1k−1□c1k⋄c2k) is a given function. It is found that, the existence of the solution u(x) of such equation depending on the conditions of f and Δc1k−1□c1k⋄c2ku(x). Moreover such solution u(x) related to the elastic wave equation depending on the conditions of p, q and k. [Copyright &y& Elsevier]

The central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function b... of a set ... with thin boundary under the influence of a velocity field. We relate it to equations and constructions used in the context of level set methods. We further introduce a new moving narrow-band method which not only can be readily implemented to solve our evolution equation, but could also be used for equations of motion by curvatures. In the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach (e.g., local semiconvexity in an extended sense of the oriented distance function of the closure of the set). For W2,p-Sobolev domains a new characterization and a compactness theorem are given in terms of the Laplacian of the oriented distance function rather than its whole Hessian matrix. [ABSTRACT FROM AUTHOR]

The nonlinear complementarity problem can be reformulated as nonsmooth equation. In this paper we propose a new smoothing. Newton algorithm for the solution of the nonlinear complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper. a general class of loss functions based on the inversion of the standard beta probability density function (pdf) is examined. The extension of this loss function from a standard beta pdt ranging from (0, 1) to the general beta pdf ranging from (p. q) is examined through the scale invariance property under a linear transformation. An industrial application in quality assurance is used to demonstrate this general class of Loss functions. Mathematical derivations are attached in the Appendices. [ABSTRACT FROM AUTHOR]

This paper presents a quantitative discussion on the stability of the second-order periodic difference equation, which characterizes the discrete periodic time-varying system. The periodic parameter discrete system, which corresponds to Mathieu's equation in the continuous system, is represented by a second-order linear difference equation with a small parameter E in the varying term. The parameter E plays an important role in the determination of the stability. By applying McLachlan's method, the expression for the boundary between the stability and the instability can be determined analytically with regard to the parameters contained in the equation. For the case where the periodic parameter of the difference equation is represented as a sum of two even functions, the boundary between stability and instability is determined. The stability can be analyzed in a similar way for the case where the periodic parameter is represented as a sum of N even functions in Fourier series. [ABSTRACT FROM AUTHOR]

The article presents a discussion about the use of pencils to help solve quadratic equations. It offers a diagram relative to curves plotted from various quartic formulas which shows a resemblance to a pencil. It mentions that there is a mathematical notion that a pencil is being used in geometry and topology in previous years. It emphasizes that one of the amazing things about mathematics is that ideas that at first seem unrelated are often related in surprising ways such as the use of pencils as mathematical models for the quadratic equation.