Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX ₁, ,X _m .Assume thatf is quasiconvex and is the sum of nonconstant functionsf ₁, ,f _m defined on the respective factor sets. Then everyf _i is continuous; with at most one exception every functionf _i is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.We define the convexity index of a functionf _i appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off ₁ andf ₂, or, equivalently, in terms of the separation of the graphs off ₁ andf ₂ by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.     

Genetic algorithms for the selection of smoothing parameters in additive models

Summary  Additive models of the type y=f ₁(x ₁)+...+f _p(x _p)+ε where f _j , j=1,..,p, have unspecified functional form, are flexible statistical regression models which can be used to characterize nonlinear regression effects. One way of fitting additive models is the expansion in B-splines combined with penalization which prevents overfitting. The performance of this penalized B-spline (called P-spline) approach strongly depends on the choice of the amount of smoothing used for components f _j . In particular for higher dimensional settings this is a computationaly demanding task. In this paper we treat the problem of choosing the smoothing parameters for P-splines by genetic algorithms. In several simulation studies this approach is compared to various alternative methods of fitting additive models. In particular functions with different spatial variability are considered and the effect of constant respectively local adaptive smoothing parameters is evaluated.     

On some classes of modules and characterization of rings

An abstract class of R-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images and finite direct sums. In this paper we study the lattice properties of R-ad, the big lattice of all additive classes in R-mod for R any ring. Also, we study the lattice properties of R-ad when R is a Noetherian or semisimple ring. We describe a characterization of some rings through the additive classes and their properties. We study some properties of additive classes related with the associated linear filters.     

Exact number of solutions for a class of two-point boundary value problems with one-dimensional p-Laplacian

In this paper, exact number of solutions are obtained for the one-dimensional p-Laplacian in a class of two-point boundary value problems. The interesting point is that the nonlinearity f is general form: f(u)=λg(u)+h(u). Meanwhile, some properties of the solutions are given in details. The arguments are based upon a quadrature method.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

Polynomial operators and local approximation of solutions of pseudo-differential equations on the sphere

We study the solutions of an equation of the form Lu=f, where L is a pseudo-differential operator defined for functions on the unit sphere embedded in a Euclidean space, f is a given function, and u is the desired solution. We give conditions under which the solution exists, and deduce local smoothness properties of u given corresponding local smoothness properties of f, measured by local Besov spaces. We study the global and local approximation properties of the spectral solutions, describe a method to obtain approximate solutions using values of f at points on the sphere and polynomial operators, and describe the global and local rates of approximation provided by our polynomial operators. The research of this author was supported, in part, by grant DMS-0204704 from the National Science Foundation and grant W911NF-04-1-0339 from the U.S. Army Research Office     

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of -regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of -regular solutions is given. We also formulate sufficient conditions to construct a piecewise -regular solutions (continuation beyond maximal time of existence for -regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for in is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2mth order semilinear parabolic initial boundary value problem in a Hilbert space .     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

Existence, non-existence and asymptotic behavior of blow-up entire solutions of semilinear elliptic equations

We are mainly concerned with existence, non-existence and the behavior at infinity of non-negative blow-up entire solutions of the equation Δu=ρ(x)f(u) in RN. No monotonicity condition is assumed upon f and, in fact, we obtain solutions with a prescribed behavior both at infinity and at the origin. The method used to get existence is based upon lower and upper solutions techniques while for non-existence we explore radial symmetry, estimates on an associated integral equation and the Keller-Osserman condition.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

On the characterization of Pareto-optimal solutions in bicriterion optimization

For bicriterion optimization involving objective functionsf ₁ andf ₂ defined on a decision spaceX, a condition is presented under which the Pareto-optimal points can be characterized as solutions of the scalar optimization problems: Minimizef ₁(x), subject tof ₂(x),x X, which ranges over a certain interval. Using this condition, it is shown how the Pareto-optimal points can be so characterized in both convex and nonconvex situations.The author gratefully acknowledges Dr. Ivan Singer, Institute of Mathematics, Rumanian Academy of Sciences, Bucharest, Rumania, for pointing out the relevance of Remark 7 in Ref. 7 to the results of this note.     

Existence of Solutions to a Singular Initial Value Problem

Under the sign assumptions we investigate the global existence of solutions of the initial value problem x＇ =f（t, x, x＇）, x（0） = A, where the scalar function f（t, x,p） may be singular at x = A.     

Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function

This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem

where aC[0,1] changes sign. The proof of our main result is based upon bifurcation techniques.     

A note on positive evanescent solutions for a certain class of elliptic problems

This paper is devoted to the existence and properties of solutions of the following class of nonlinear elliptic differential equations Δu(x)+f(x,u(x))+g(‖x‖)x⋅∇u(x)=0, x∈Rn, ‖x‖>R. We prove existence of positive solutions vanishing at positive infinity. Our approach is based on the subsolution and supersolution method. The nonlinearity f covers both sublinear and superlinear cases and does not necessarily satisfy f(x,0)≡0. The asymptotic behavior of solutions is also described.     

Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations

The existence and uniqueness of positive radial solutions of the equations of the type [IML0001] in B_R, p>1 with Dirichlet condition are proved for λ large enough and f satisfying a condition[IML0002] is non-decreasing on [IML0003] It is also proved that all the positive solutions in C₁ ⁰(B^R) of the above equations are radially symmetric solutions for f satisfying [IML0004] and λ large enough.     

On Blow Up Solutions of a Quasilinear Elliptic Equation

Given a bounded regular domain Ω in ℝ^N, we study existence and asymptotic behaviour of the solutions of the equation Δu + |Du|^q = f(u) in Ω, which diverge on ∂Ω. We extend and complete some results contained in [4].     

Eisenstein series of 3/2 weight and one conjecture of Kaplansky

We shall prove thatf ₁ =x ² +y ⁴ + 7z ² represents all eligible numbers congruent to 2 mod 3 except 14 × 7^2k which was conjectured by Kanplansky. Our method is to use modular forms of weight 3/2. Our method can also be applied to other ternary quadratic forms.