Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

A note on a nonlinear eigenvalue problem

Summary In this paper a necessary and sufficient condition for the existence of negative eigenvalues for the problem-u – u=(x)¦u¦^p–2u in u¦=0 is given. Here Rⁿ is supposed a smooth bounded domain, 0 a bounded nonnegative function, (₁, ₂), ₁ and ₁ being the first and the second eigenvalue of - in with zero Dirichlet boundary data, p2 and, if n 3, p < 2n¦(n–2). Moreover in the linear case (p=2) a uniqueness result is proved.Work supported by G.N.A.F.A. and by M.P.I, of Italy Fondi 40% Equazioni Differenziali e Calcolo delle Variazioni and Fondi 60% Analisi matematica.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

Excesses of systems of exponential functions

Conditions on the closeness of real sequences {_n} and {_n} are studied which imply the equality of the excesses of the systems {exp(i_nx)} and {exp(i_nx)} in the space L²(–a, a). A theorem is formulated in terms of the difference of the sequences {_n} and {_n} enumerating the functions. In the corollaries of the theorem, conditions are given in terms of the behavior of the difference _n–_n0. An example is constructed showing that the condition _n–_n0 alone is not sufficient for equality of the excesses.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 803–814, December, 1977.     

275-01275-01275-01Regularity of the spatial derivatives of the solutions to parabolic systems in divergence form

Sunto Si dimostra che, nelle ipotesi: fⁱ L^2,(Q, ^N), 0<< n+2, i=0,1,..., n, uL^2,(Q,^N)H _–T ^*0,1/2() (Q,^N), D_iuL^2,(Q,^N),i=1,2,...,n, la soluzione v: Q ^N del problema di Cauchy-Dirichlet: ha derivate spaziali D_iv appartenenti a L^2,(Q,^N) e che sussiste la maggiorazione:.

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Lavoro eseguito con contributo finanziario del M.U.R.S.T. e nell'ambito del G.N.A.F.A. del C.N.R.     

A minimax principle for the critical value of a boundary value problem with positive and increasing nonlinearity

Summary For a non-linear boundary value problem with a positive and increasing non-linearity there exists a critical value* of the parameter, beyond which there are no solutions. We give a minimax characterization of*.

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Zusammenfassung In der Randwertaufgabe –u(x)=f(x, u(x)), u(a)=u(b)=0, seif positiv und wachsend im zweiten Argument. Dann gibt es einen Wert*, so dass keine Lösung existiert für>*. In dieser Arbeit wird* durch ein Minimaxprinzip charakterisiert. Der Beweis beruht auf der Anwendung von Ober- und Unterlösungen und monotonen Iterationen.

     

A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

Locally forced critical surface waves in channels of arbitrary cross section

Critical long surface waves forced by locally distributed external pressure applied on the free surface in channels of arbitrary cross section are studied in this paper. The fluid under consideration is inviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumed to be near critical, i.e.,u ₀=u _c ++0(²), where 0<1 andu _c is the critical velocity determined by the geometry of the channel. The external pressure applied on the free surface as the forcing is ² P(x). Then the first order perturbation of the free surface elevation satisfies a forced Korteweg-de Vries equation (fK-dV). It is shown in this paper that: (i) If (supercritical), the stationaryfK-dV has two cusped solitary wave solutions; (ii) if (subcritical), the stationaryfK-dV has a downstream cnoidal wave solution; (iii) when= _L , the unique stationary solution of thefK-dV is a wave free hydraulic fall; (iv) if= _d =– _L , thefK-dV has a jump solution; and (v) if _L << _c , thefK-dV does not have stationary solutions. Some free surface profiles and bifurcation diagrams are presented.     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

An existence result for variational problems and applications to some nonlinear hyperbolic and elliptic equations involving critical Sobolev exponents

Summary We prove the existence of nontrivial solutions for nonlinear equations of the type Lu=g(x, u) + ¦u¦^¯p–2u, ¯ p > 2, where L is a continuous self- adjoint linear operator in a Hilbert space H and ug(x, u) is a lower order perturbation of ¦u¦^¯p–1. We assume that ¯p is the critical exponent in the sense that the embedding H L^p, If is compact for 1p<¯p and is continuous (not necessarily compact) for p=¯p. From this result we deduce, for example, that u_tt -u- u=¦u¦^2/Nu, u L²(S^N×S¹) has at least one pair (–u, u) of solutions nonconstant with respect to t, provided that is sufficiently close to some eigenvalue of _tt–.Work supported by M.P.I. Italy (fondi 40%, 60%) and by G.N.A.F.A. of C.N.R.     

Nonlinear eigenvalue approximation

Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.     

On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,S _T e _h, ^–1 I (hR, T) whereS _T is the Cauchy singular integral operator ande _h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra and its matrix analogue . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.     

On blocking sets in affine planes

Denote by _q an affine plane of order q. In the desarguesian case _q=AG(2,q), q 5(q= p^h, p prime), we prove that the smallest cardinality of a blocking set is 2q–1. In any arbitrary affine plane _q (desarguesian or not) with q5, for any integer k with 2q–1 k(q–1)², we construct a blocking set S with ¦S¦=k. For an irreducible blocking set S of _q we determine the upper bound S [qq]+1. We prove that if _q contains a blocking set S which is irreducible with its complementary blocking set, then necessarily _q=AG(2, 4) and S is uniquely determined. Finally we introduce techniques to obtain blocking sets in AG(2, q) and in PG(2, q).Research partially supported by G.N.S.A.G.A. (CNR)