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     

Über Automorphismenbahnen von Elementen u≠O mit (ad u)3=αad u in endlichdimensionalen einfachen komplexen Lie-Algebren

In this paper we study certain semisimple elements in simple complex Koecher-Tits-constructions from Jordan-triplesystems. Let L be a finite dimensional simple complex Lie-Algebra and u O an element in L with (ad u)³=-ad u. Then there is a compact real form L of L, which contains u. The involutorial automorphism id_L+2 (ad_Lu)² of L induces a Cartan-decomposition of a real form L (u) of L and this gives us a criterion of conjugacy under Aut L for two such elements u₁, u₂L.Using this result, we show that the number of conjugacy classes of elements uL (u O) with (ad u)³=ad u (\{O}, under Aut L is equal to the number of similarity classes of Jordantriplesystems, the Koecher-Tits-construction of which is isomorphic to L. The corresponding data are finally listed for all possible types of L.     

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.     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

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.     

Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

On the congruence problem in infinite dimensional sesquilinear spaces

If two subspaces V and V of a sesquilinear space E are congruent (i.e., there is an isometry : E E with (V)=V) then their corresponding quadratic lattices V(V, E) and V(V, E) are isomorphic. It is shown that the converse holds for important types of sesquilinear spaces E, provided that dim(E) ₃. However, the converse generally fails if dim(E) ₃.     

Biclosed Binary Relations and Galois Connections

Given two closure spaces (E,) and (E,), a relation RE×E is said biclosed if every row of its matrix representation corresponds to a closed subset of E, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational applications are derived from this result.     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

On shifts of sequences

X ₂ ={x _k } _k=2 X={x _k } _k=1 . , , ( , ) . , , X— , , X ₂ H, , , , , X ₂ H.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

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.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

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.     

The geometric aspect of nuclearity

The main result of this paper is a necessary (and sufficient) condition for the nuclearity of a locally convex Hausdorff space E in terms of a geometric property of its topological dual E: there is a fundamental system of equicontinuous sets each of whose members has the following property: it is (E, E)-closed, convex and contains a well-positioned point, which admits a core representing measure.     

The Baum–Connes Conjecture and Discrete Group Actions on Trees

We introduce a property (BC) for discrete groups, which we prove to imply the Baum–Connes conjecture with coefficients and the K-amenability of the group. Then, we show that if is a discrete group which acts on a tree X such that X/ is compact, and the stabilizers of the vertices and the stabilizers of the edges satisfy (BC), then itself satisfies (BC) Finally, we indicate a couple of applications.     

Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

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.