A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

The spectra of closed interpolated operators

Let (E ₀,E ₁) be a compatible couple of Banach spaces, and letE : 0Re1 be the complex interpolation spaces ofE ₀,E ₁. LetT be a closed linear operator onE ₀+E ₁, then the restrictionT ofT to eachE is closed. If we denote by the extended spectrum ofT inE , then, under appropriate conditions, it is shown that the map is an analytic multifunction in the strip {C0<Re<1}. We use these results to give some applications to the spectral theory of semigroups.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

Topological influence of the presence of fully-λ-bases in locally convex spaces

This article reveals the topological impact of fully--bases in locally convex spaces where carries either the traditional normal topology or the fairly generalized-topology of Ruckle. It has been established that the generalized nuclearity of plays a significant role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully--base ( being equipped with normal topology or-topology) has been investigated.We acknowledge with thanks the suggestions of the referee.     

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.     

Minimality of derivative chains,corresponding to a boundary value problem on a finite segment

One investigates the minimality of derivative chains, constructed from the root vectors of polynomial pencils of operators, acting in a Hilbert space. One investigates in detail the quadratic pencil of operators. In particular, for L()=L₀+L₁+²L₂ with bounded operators L₀0, L₂0 and Re L₁0, one shows the minimality in the space_173-02 of the system {x_k, _kekx_k}, where x_k are eigenvectors of L(), corresponding to the characteristic numbers _kin the deleted neighborhoods of which one has the representation L^–1()=(–_k)^–1R_K+W_K() with one-dimensional operators R_k and operator-valued functions W_K(), k=1, 2, ..., analytic for =_k.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 195–205, February, 1990.     

Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System (≤ λ)/G/m

We consider a queuing system ()/G/m, where the symbol () means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed dt. The case where the queue length first attains the level r m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity ₁(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, ₁(t) = O( ^{r+ 1}₁ ^{m– 1} _{r– m+ 1}), where _k is the kth moment of the service-time distribution.     

Ein Störungssatz für abgeschlossene,lineare Operatoren

Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6].     

Irregular but nonfractal drums andn-dimensional weyl conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set R ⁿ (n 1) with irregular but nonfractal boundary. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN _i ()(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ _i ()=() –N _i () for large, where() is the Weyl term. This implies that for the irregular but nonfractal drum , not only the volume || _n is spectral invariant but also the area of boundary || _n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Quasi-symmetric 3-designs with triangle-free graph

The following result is proved: Let D be a quasi-symmetric 3-design with intersection numbers x, y(0x<y<k). D has no three distinct blocks such that any two of them intersect in x points if and only if D is a Hadamard 3-design, or D has a parameter set (v, k, ) where v=(+2)(²+4+2)+1, k=²+3+2 and =1,2,..., or D is a complement of one of these designs.