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.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

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     

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.     

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.     

Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL _r (E) onto the centerZ(E) ofE. Let {T(t)} _t0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A))) ^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim _m D((R(, A)) ^m ) for any >0.     

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     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

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 .     

The application of an interpolational method

We consider an equation of infinite order in generalized derivatives in the sense of A. O. Gel'fond with a characteristic function of finite order L()=L₁()... L_n(). It is explained by a purely analytical method (by the application of an interpolational method) when any solution of the equation is the sum of the solutions of similar equations with the characteristic functions L₁(), ..., L_n(). In the case of an equation in ordinary derivatives, when L() is a function of exponential type, the problem is solved by the application of algebraic and functional methods of V. V. Napalkov [1].Translated from Matematicheskie Zametki, Vol. 18, No. 5, pp. 735–752, November, 1975.     

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ₁, ₂ satisfyingrk – ₂ v + 1 and ₂ = ₁ + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.Supported in part by Grant 59540043 (C), Japan.     

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.     

Quasiconvex uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a quasiconvex sequence _v of convergence factors transforms Fourier series of functions whose moduli of continuity do not exceed a given modulus of continuity(gd) into uniformly convergent series if and only if _n (1/n) log n 0 for n . The sufficiency of this condition is already known.Translated from Matematicheskie Zametki, Vol. 8, No. 5,pp. 619–623, November, 1970.     

Threshold theorems for a stochastic model of an epidemic with natural immunization

A stochastic model of an epidemic is investigated, taking account of the removal of ill members of the population (by death, by recovery with immunization, by isolation) and natural immunization. Limiting distributions are found for the size of the epidemic, the number immunized ₁, and their sum, under the assumption that the original number of susceptible individuals n and the number of ill individuals m , while n 1,n ₀< , where and are the coefficients for the contraction of the disease and of immunization respectively.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 385–392, September, 1970.     

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.     

Invariant equations defining coincident root loci

Let F(x) = xⁿ+₁ x^n-1+₂ x^n-2+ ··· +_n be a polynomial with complex coefficients, and suppose we are given a partition (₁,...,_r) of n. It is a classical problem to determine explicit algebraic conditions on the _i so that F may have roots with multiplicities ₁,...,_r. We give an invariant theoretic solution to this problem, to wit, we exhibit a set of covariants of F whose vanishing is a necessary and sufficient condition. The construction of such covariants is combinatorial, and involves associating a set of graphs on n vertices (called decisive graphs) to each .Received: 28 September 2003     

On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms

We study the first positive eigenvalue ^(p) ₁(g) of the Laplacian on p-forms for a connected oriented closed Riemannianmanifold (M, g) of dimension m. We show that for 2 p m – 2 a connected oriented closed manifold M admits three metrics g _i (i = 1, 2, 3) such that ^(p) ₁(g ₁)> ⁽⁰⁾ ₁(g ₁),^(p) ₁(g ₂) < ⁽⁰⁾ ₁(g ₂) and^(p) ₁(g ₃)= ⁽⁰⁾ ₁(g ₃).Furthermore, if (M, g) admits a nontrivial parallel p-form,then ^(p) ₁ ⁽⁰⁾ ₁ always holds.     

Stabilitätsuntersuchung an einer oszillierenden Couette-Strömung

Zusammenfassung Das Stabilitätsverhalten der Strömung einer zähen Flüssigkeit im Ringraum zwischen zwei unendlich langen koaxialen Zylindern wurde theoretisch untersucht; der innere Zylinder rotiert mit konstanter Winkelgeschwindigkeit, während der äussere in axialer Richtung schwingt. Die nichtlineare Analysis zu diesem Problem zeigt, daß diese Strömung stabil ist, wenn Frequenz und Amplitude der Schwingung einem gewissen BereichD der Frequenz-Amplituden-Ebene angehören.D hängt von der Reynoldszahl ab, die hier wie üblicherweise bei der Couette-Strömung definiert ist. Ist _cr die kritische Reynoldszahl der stationären Couette-Strömung, so zeigt die Untersuchung, daß das angegebene StabilitätsgebietD zur Achse verschwindender Ampltidue zusammenschrumpft, wenn gegen _cr strebt.D ist leer für> _cr .

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Summary The stability of the flow of a viscous fluid in the ringspace between two infinitely long coaxial cylinders is investigated; the inner cylinder is assumed to be rotating with a constant angular velocity while the outer one is oscillating in the axial direction. A nonlinear analysis shows that this flow is stable if the frequency and the amplitude of the oscillation are taken from a domainD of the frequency-amplitudeplane.D depends on the Reynolds number defined as usual in the case of Couette flow. When tends to _cr the critical Reynolds number of the steady Couette flow,D is shown to shrink to the axis of vanishing amplitude.D is empty for> _cr .

     

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ²ⁱ )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH ₀=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a _i (), 0) at 0 ^k , whereH (z)–z=a _i ()z ⁱ . Then one can find a parameter solutionS (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS (z)=z+b _i ()(z–z ₀) ⁱ with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 ² such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé     

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.