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.     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

The spectrum of an elliptic operator of second order

Under minimal requirements on the coefficients and the boundary of the domain it is proved that the spectrum of the first boundary-value problem for an elliptic operator of second order always lies in the half-plane Re , where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. On the line Re = , there are no other points of the spectrum.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 351–358, September, 1976.     

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.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

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.     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

Completeness of the plans of sequential estimation for Wiener spaces with a drift and some uniqueness theorems

One considers a family of n-dimensional Wiener processess x(t) =(t) +t, depending on a drift parameter , where is the standard Wiener process. Let be a closed subset of the space of trajectories ⁿ×₊ (plan) and assume that the measure , defined by the first occurrence of the trajectory of the process x in the set , is a probability measure. One gives conditions which the plan has to satisfy in order that from the equality for any there should follow that f 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 69–72, 1983.The authors are grateful to V. P.Khavin for discussions on certain questions of potential theory.     

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.     

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 Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

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.     

Asymptotic behavior of the spectrum of the Maxwell operator

For the distribution functions of the positive and the negative eigenvalues of the operator in a domain with a smooth boundary, one obtains the asymptotic formula N^±()=(3²)^–1 mes ·³+0(²). Under additional assumptions on the properties of the geodesic billiard in , one shows that N±()= (3²)^–1 mes ·³+0(²).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 169–180, 1983.     

Morphisms of the Heat Equation

In this article we consider a certain class of mappings between Riemannian manifolds of the type M × R^{+ *}, which preserve two objects associated to the heat equation. First we show that maps which pull back local solutions of the heat equation to local solutions of the heat equation, called heat equation morphisms have to be the product of a homothetic submersion and an affine map of R^{+ *}. Then using the above characterisation, we study maps which pull back the heat kernel to the heat kernel. We call such maps heat kernel morphisms. We show in Theorem 3 that, in the case of compact manifolds, a map : M × M × R^{+ *} N × N × R^{+ *} of the form (x,y,t) = ( (x), (y),h(t)), with surjective, is a heat kernel morphism if and only if is a homothetic covering of N()-sheets and constant dilation such that N() = ^m (m = dim M) and h(t) = ²t. In particular, if is bijective then it must be an isometry and h(t) = t. A similar problem was considered from a different point of view in [6].     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

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.     

Statistical topography. I. Fractal dimension of coastlines and number-area rule for Islands

Summary Statistical topography involves the study of the geometrical properties of the iso-sets (contour lines or surfaces) of a random potential (r). Previous work [1,2] has addressed coastlines on a random relief (x, y) that possess a single characteristic spatial scale with topography belonging to the universality class of the random percolation problem. In the present paper this previous analytical approach is extended to the case of a multiscale random function with a power spectrum of scales, ^H , in a wide range of wavelengths, ₀ < < _m . It is found that the pattern of the coastline differs significantly from that of a monoscale landscape provided that –3/4 <H < 1, with the case –3/4 <H < 0 corresponding to the long-range correlated percolation and 0 <H < 1 to the fractional Brownian relief. The expression for the fractal dimension of an individual coastline is derived,D _h = (10 – 3H)/7, the maximum value of whichD _h = 7/4, corresponds to the monoscale relief. The distribution functionF(a) of level lines over their sizea is calculated:F(a) a ^–4(1-H)/7, for ₀ a _m . A comparison of the theoretical results with geographical data is presented.     

Reflecting stationary sets and successors of singular cardinals

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a which is ⁺ⁿ -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the bad stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for many PT(, ₁). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular which is a limit of supercompacts the bad stationary set concentrates on the right cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular {<⁺ : cf<} is good. It is proved in ZFC that if=cf>₁ then {<⁺ : cf<} is the union of sets on which there are squares.     

Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights

An arbitrary Muckenhoupt A₂-weight w² on the special contour ( 1/2) generates a function Y_,w (, t), which for =1, w²(z) 1 coincides with the exponential exp{it}. In the paper, with the aid of B. S. Pavlov's geometric approach, one obtains criteria for the unconditional basis property of families of functions of the form {y,w(_k,t):_k} in the space L₂(0, ). The analytic foundation of the constructions is a generalization of M. M. Dzhrbashyan's certain results (power weight) to the case of arbitrary Muckenhoupt A₂- weights.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 34–80, 1991.     

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.