Das asymptotische verhalten der greenschen funktion N-irregulaärer eigenwertprobleme mit zerfallenden randbedingungen

We prove asymptotic estimates for the Green's function of N-irregular eigenvalue problems My = λNγ with splitting boundary conditions. In contrast to the N-regular case the Green's function G(x,ζ,λ) grows exponentially for |λ| → ∞ if x > ζ. These estimates are fundamental for the expansion of functions into a series of eigenfunctions of N-irregular eigenvalue problems. In a subsequent paper it will be shown that this irregular behavior of G(x,ζ,λ) implies that only a very small class of functions can be expanded into a series of eigenfunctions of such problems.     

Forests of label-increasing trees

Label-increasing trees are fully labeled rooted trees with the restriction that the labels are in increasing order on every path from the root; the best known example is the binary case—no tree with more than two branches at the root, or internal vertices of degree greater than three—extensively examined by Foata and Schutzenberger in A Survey of Combinatorial Theory. The forests without branching restrictions are enumerated by number of trees by F_n(x) = x(x + 1)…(x + n ? 1), n >1 (F₀(x) = 1), whose equivalent: F_n(x) = Y_n(xT₁,…, xT_n), F_n(1)= T_{n + 1} = n!, is readily adapted to branching restriction.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Hypersurfaces containing flat subspaces

Lek k be an infinite field and suppose m.i. and n are positive integers such that t m We study the subset of k[x ₁,x ₂, … x_m ] which consists of 0 and the homogeneous members t of f of k[x ₁,x ₂, … x_m ] of fixed degree n such that there exists homogeneous F ₁, F ₂, … F_t in k[x ₁,x ₂, … x_m ] of degree one and homogenous g ₁ g ₂, …g_t , in k[x ₁,x ₂, … x_m ] such that f(x) = F ₁(x)g ₁(x) + F ₂(x)g ₂(x) + … + F _t (x)g _t (x) for each x in k ^m. In case k is algebrarcally closed we are able to prove that this set is an algebraic variety. Consequently. if k is also of characteristic 0 then we are able to prove that certain collections of symmetric k-valued multilinear functions are algebraic varieties.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

Positive ground state of coupled systems of Schrödinger equations in involving critical exponential growth

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations: where the nonlinearities f₁(x,s) and f₂(x,s) are superlinear at infinity and have exponential critical growth of the Trudinger‐Moser type. The potentials V₁(x) and V₂(x) are nonnegative and satisfy a condition involving the coupling term λ(x), namely, λ(x)²<δ²V₁(x)V₂(x) for some 0<δ<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and L^q‐estimates, we get regularity and asymptotic behavior.     

Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.     

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter-dependent boundary conditions

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h₁,h₂, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H₁,H₂, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ? be a prime ring of characteristic different from 2, 𝒬_r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [F(r), r]_kr ? r[G(r), r]_k = 0 for all r ∈ ?, then there exist a ∈ 𝒬_r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ?.     

The Point Spectrum and Spectral Geometry for Riemannian Manifolds with Cusps

The bifurcation of a solution of the equation f(x, λ) = 0 at the point (x₀, λ₀) is investigated. In the case that B: = –f_x(x₀, λ₀) is a FREDHOLM operator by the method of LJAPUNOV/SCHMIDT the original equation is equivalent to a system consisting of a locally uniquely solvable equation and an equation in a finit dimensional subspace, the so-called bifurcation equation. For analytical/recursion formulas are deduced to determine the locally unique solution. In the case of FREDHOLM operators B with index zero practicable criteria are given for the applicability of a theorem of IZE being a generalization of a well known theorem of KRASNOSELSKIJ.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

Recursive constructions of -polynomials over

This paper presents procedures for constructing irreducible polynomials over GF(2^s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F₀(x)GF(2^s) of degree n, polynomials F_k(x)GF(2^s) of degrees n2^k are constructed by iteratively applying the transformation x→x+x^-1, and their roots are shown to form a normal basis of GF(2^sn2k) over GF(2^s). In addition, the sequences are shown to be trace compatible, i.e., the trace map T_{GF(2^sn2k+1)/GF(2^sn2k)} fromGF(2^sn2k+1) onto GF(2^sn2k) maps the roots of F_k+1(x) onto those of F_k(x).     

Acomplete classification of bifurcation diagrams of classes of multiparameter p -Laplacian boundary value problems

We study the bifurcation diagrams of positive solutions of the multiparameter p -Laplacian problem (φ _p (u ′(x)))′ + f _λ,μ (u (x)) = 0, –1 < x < 1, u (– 1) = u (1) = 0, where p > 1, φ _p (y) = |y |^{p –2} y, (φ _p (u ′))′ is the one-dimensional p -Laplacian, f _λ,μ (u)= g (u, λ) + h (u, μ), and λ > λ₀ and μ > μ₀ are two bifurcation parameters, λ₀ and μ₀ are two given real numbers. Assuming that functions g and h satisfy hypotheses (H1)–(H4), for fixed μ > μ₀,we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (λ, ∥u ∥_∞)-plane, each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right. Hence the problem has at most two positive solutions for each λ > λ₀. More precisely, we prove the exact multiplicity of positive solutions. In addition, for all p > 1, we give interesting examples which show complete evolution of bifurcation diagrams as μ varies. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Picard iterates for positive solutions of a class of nonlinear eigenvalue problems

Summary We study positive solutions of the nonlinear eigenvalue problemF(u)=G(u) with some monotone operatorsF andG. In particular, we consider the case of nonlinear elliptic differential equations of second order and chooseF(u)=–divA(x, gradu)+b(x,u) and G(u)=g (x,u). Positive solutions are obtained by the Picard iterationsu ₀=0 andF(u _n+1)=G(u _n ).In order to get convergence of the sequenceu _n ,one has to study some comparison principles for the operatorF. Finally, the Picard iteration scheme allows a-priori estimates and bifurcation results for the admissible eigenvalue parameter.

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Zusammenfassung Für gewisse monotone OperatorenF andG untersuchen wir positive Lösungen des nichtlinearen EigenwertproblemsF(u)=G(u). Insbesondere betrachten wir nichtlineare elliptische Differentialgleichungen zweiter Ordnung und wählenF(u)=–divA(x, gradu)+b(x,u) sowieG(u)=g(x,u). Man erhält positive Lösungen durch das Picard-Iterationsverfahrenu ₀=0 undF(u _n+1)=G(u _n ).Um die Konvergenz der Folgeu _n nachzuweisen, benötigt man Vergleichsprinzipien fürF. Dann gestattet das Iterationsschema sogar a-priori Abschätzungen und Verzweigungsaussagen für die zulässigen Eigenwertparameter.

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Supported by the Deutscher Akademischer Austauschdienst (DAAD) and DICYT-University of Santiago de Chile.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Generalized derivations and left multipliers on Lie ideals

Let m, n be two fixed positive integers and let R be a 2-torsion free prime ring, with Utumi quotient ring U and extended centroid C. We study the identity F(x ^m+n+1) = F(x)x ^m+n  + x ^m D(x)x ⁿ for x in a non-central Lie ideal of R, where both F and D are generalized derivations of R and then determine the relationship between the form of F and that of D. In particular the conclusions of the main theorem say that if D is the non-zero map in R, then R satisfies the standard identity s ₄(x ₁, . . . , x ₄) and D is a usual derivation of R.