Periodic solutions of differential equations in Banach spaces

Let X be a Banach space, DX, f: [0,)xDX continuous and -periodic. In this paper we consider various conditions on D and f sufficient for existence of an -periodic solution of the differential equation u=f(t,u). In the main, we shall assume that D is closed bounded and convex and f satisfies a boundary condition at D such that D is flow invariant for u=f(t,u). The map f is assumed to be either compact or dissipative or a certain perturbation of such maps.     

Parties d'ensembles B-normaux

For a given subset A of the set R of real numbers, we defined M(A) as the infimum of all the lengths of the finite intervals I such that there exists a sequence of real numbers in I such that A is the associated normal set B(). We prove that if A is a subset of A, such that all the multiples k.a' belongs to A (for each non zero integer k and each element a of A), then M(A) is less or equal to 2.M(A). Thus the family of the subsets A such that M(A) is finite is closed under intersection and finite union.     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

Hasse-Witt matrices of Fermat curves

Let m be an integer with m3. Let K and K be perfect fields of characteristic p and p such that (p,m)=1 and (p,m)=1, respectively. Moreover let A and A be algebraic function fields over K and K defined by x^m+y^m=a(0, ak) and x^m+y^m=a(a0 ak), respectively. Put g=(m–1)(m–2)/2. Denote by M(K,p,a) and M(K,p,a) the Hasse-Witt matrices of A and A with respect to the canonical bases of holomorphic differentials. Then we show that if p+p0(mod.m) then rank M(K,p,a)+rank M(K,p,a)=g and if pp1 (mod.m) then rank M(K,p,a)=rank M(K,p,a).     

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.     

Zur Kennzeichnung der Kollineationen zwischen Quadriken

In projective spaces Pⁿ(K), Pⁿ(K) over commutative fields K, K of characteristic 2 we consider the quadrics Q^n–1Pⁿ(K), Q,^n–1Pⁿ(K) which have the same rank and index and each of which is different from its singularity space. A one-to-one correspondence of Q^n–1 onto Q'^n–1 can be extended uniquely to a collineation of Pⁿ(K) onto Pⁿ(K) if and only if satisfies the following condition: maps the (n–2)-dimensional quadrics of Q,^n–1 of maximal rankonto (n–2)-dimensional quadrics of Q^n–1 of maximal rank (Theorem 1).

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Herrn Prof. Dr. Helmut Karzel zum 60. Geburtstag gewidmet     

Support functions of convex compacta

The properties of the space (X_x) of all sublinear functionals, defined on a space X' (topologically adjoint to a Hausdorff locally convex barrelled space X) and continuous in the Arens topology × (X, X), equipped with topology of uniform convergence on bounded subsets of X are studied. It is shown that completeness and separability of a space X are hereditary for (X_x). Criteria for the compactness of subsets of (X_x) and conditions for the metrizability of compacta in (X_x) are given. The topological isomorphism between (X_x) and the space of all nonempty convex compacta in X with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.The author thanks S. S. Kutateladze for useful discussions regarding this article.     

Formula for the differentiation of operator-valued functions depending on a parameter

A study of the convergence of the differentiation formula (f(A))=f (A)A+f ^m(A)/21[AA]+f ^m(A)/3l[[AA]A]'+... where [XY]=XY–YX, and A=A(t) is a function of the real variable t with values in a Banach algebra.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 207–218, August, 1971.The author wishes to thank A. G. Aslanyan for his comments concerning this work.     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

Extensions and duality of finite geometric closure operators

If P and P are finite partially ordered sets such that P=P–e for some maximal element e P, all geometric closure operators on P are determined whose restriction to P equals a given closure operator on P.The classes Q(P) and Q(P^*) of all geometric closure operators on P and on its order dual P^* are shown to be anti-isomorphic partially ordered sets.     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

Über Untergruppen Endlicher Algebraischer Gruppen

Let k be a commutative ring, GG finite affine algebraic k-groups, and HH the dual Hopfalgebras of the affine algebras of G resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H-linear, HH-colinear, unitary, augmented isomorphism HHH H, where HH is the coalgebra belonging to G/G. (II) If the k-submodule of the fixelements of (HH)^* is isomorphic to k (e.g. k principal or semilocal), then HH is a Frobeniusextension of the second kind.     

On local uniqueness of the solution of boundary-value problems

In this paper we present conditions under which differentiability of the mappings F: ACⁿ(I) Lⁿ(I) and :ACⁿ(I) Rⁿ at x₀ ACⁿ(I) and the uniqueness of the solution of the boundaryvalue problem u = F(x₀)(u), (x₀)(u) = 0 imply local uniqueness of the solution x₀ of the boundary-value problem x = F(x), (x) = 0.Translated from Matematicheskie Zametki, Vol. 15, No. 6, pp. 891–895, June, 1974.The author thanks A. Ya. Lepin for the help he has given him in the preparation of this paper.     

On the dynamics of turbulent vortical flow

Zusammenfassung Die Wirbelströmung in einer turbulenten kompressiblen Flüssigkeit wird behandelt. Die Bewegungsgleichungen werden durch die Vernachlässigung der kleineren Glieder vereinfacht. Die experimentellen Ergebnisse erlauben eine Fehlerrechnung für einen Fall. Ein wichtiger Schritt in der Vereinfachung ist die Voraussetzung, dass die Radialgradienten viel grösser sind als die Axialgradienten. Für die kräftigen Wirbel erwies sich diese Annahme sowohl durch die Erfahrung als auch aus den wesentlichen theoretischen Betrachtungen heraus als begründet. Die vereinfachten Gleichungen zeigen einige wichtige Aspekte der schnelldrehenden Strömungen und geben einen Anhaltspunkt für die ausführlicheren Studien.

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Notation r radial co-ordinate - z axial co-ordinate - u, U, u radial velocity component, its time-mean value, and the turbulent perturbation to it - v, V, v tangential or swirl velocity component, etc. - w, W, w axial velocity component, etc. - p, P, p fluid pressure, etc. - ,R, fluid density, etc. - , , , numbers small with respect to unity - C, F, G, G ₀ constants of integration     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.     

On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×_C X especially.     

227-01227-01227-01-property and normal subgroups

Summary D-property (=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies D then its normal subgroups satisfy D-property as well. the Author shows that if the D and the D-properties (=set of the primes not in ) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties D and D is given in terms of the composition factors.     

Über einen Banachraum analytischer Funktionen

Let be a Banach-space of functions f analytical in a convex domain G with derivations f(z)0 in G and sup|arg f(z)|<. The distribution of the univalent, of the bounded and of the limit-continuous functions in B will be inquired.     

Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.