A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Significato proiettivo di alcuni tipi di equazioni differenziali del secondo ordine

Sunto. Ogni elemento di 2^o ordine di curva di una superficie (ad asintotiche distinte e non rettilinee) determina duequadriche asintotiche osculatrici. Gli elementi di 2^o ordine tali che queste quadriche abbiano, in ogni punto della superficie, un contatto assegnato con una retta, conica o cubica sghemba associata al punto appartengono alle curve integrali di equazioni differenziali del tipov″P _i (v′)=P _i+3(v′) (i=1, 2) oveP _k (v′) è un polinomio di gradok inv′=dv/du a coefficienti funzioni diu ev. Determinazione della configurazione geometrica a partire dall'equazione.     

Equations d’evolution du second ordre associees a des operateurs monotones

This paper extends some recent results of V. Barbu. It is concerned with bounded solutions of the problem:u″∈Au, u′(0)∈ϖj(u(0)−a) whereA is a maximal monotone operator in a Hilbert spaceH, a∈D(A) andj is a strictly convex l.s.c. function fromH to [0,+∞]. An existence and uniqueness theorem for this problem is proved. Takingj to be the indicator function of a pointu ₀∈D(A), one obtains a bounded solutionu(t) of the initial value problem:u″∈Au, u(0)=u ₀. Denotingu(t)=S _1/2(t)u₀ one obtains a semi-group of contractions onD(A). The generator of this semigroup is denoted byA _1/2. Further properties ofS _1/2(t) andA _1/2 are studied.      

Perturbation and comparison of cosine operator functions

LetA be a closed linear operator such that the abstract Cauchy problemu″(t)=Au(t), t∈R; u(0)=x, u′(0)=y, is well-posed. We present some multiplicative perturbation theorems which give conditions on an operatorC so that the abstract Cauchy problems for differential equationsu″(t)=ACu(t) andu″(t)=CAu(t) also are well-posed. Some new or known additive perturbation theorems and mixed-type perturbation theorems are deduced as corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed. Research supported in part by the National Science Council of Taiwan.     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

On the non-negative non-increasing solutions of non-linear second order differential equations

Summary The sufficient conditions for the existence and uniqueness of solution u(t) of the differential equation u″=f(t, u, u′), are established, satisfying the condition u(t)= =u₀, u(t)≥0 and u′(t)≥0 for t ε (0,+∞). Entrata in Redazione il 26 aprile 1968.     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

A sum formula for the Casson-Walker invariant

We study the following question: How does the Casson-Walker invariant λ of a rational homology 3-sphere obtained by gluing two pieces along a surface depend on the two pieces? Our partial answer may be stated as follows. For a compact oriented 3-manifold A with boundary ∂A, the kernel L _A of the map from H ₁(∂A;Q) to H ₁(A;Q) induced by the inclusion is called the Lagrangian of A. Let Σ be a closed oriented surface, and let A, A′, B and B′ be four rational homology handlebodies such that ∂A, ∂A′, −∂B and −∂B′ are identified via orientation-preserving homeomorphisms with Σ. Assume that L _A = L _{A ′} and L _B = L _{B ′} inside H ₁(Σ;Q) and also assume that L _A and L _B are transverse. Then we express

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

Singular semi-linear equations inL 1(R)

Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L ¹(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫ _−∞ ^∞ . Some extensions of this result are also given. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS 75-05501.     

Analyticity of the sine family with L p -maximal regularity for second order Cauchy problems

If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

The increase of sums and products dependent on (y 1, …,y n ) by rearrangement of this set

LetF(u, v) be a symmetric real function defined forα<u, v<β and assume thatG(u, v, w)=F(u, v)+F(u, w)−F(v, w) is decreasing inv andw foru≦min (u, v). For any set (y)=(y ₁, …,y _n ),α<y _i <β, given except in arrangement Σ _i ^=1/n F(y _i ,y _i+1) wherey _n+1=y ₁) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the productΠ _i ^=1/n [(y2i−1y2i) ^m +α ₁(y _2i−1 y _2i ) ^m−1+ … +a _m ] wherea _k ≧0 and where the set (y)=(y ₁, ..,y _n ),y _i ≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman [1] and a lemma of Duffin and Schaeffer [2]. This paper is part of the author’s Master of Science dissertation at the Technion-Israel Institute of Technology. The author wishes to thank Professor B. Schwarz and Professor E. Jabotinsky for their help in the preparation of this paper.     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.