Diagonale Netze aus Schmieg- und Krümmungslinien I

In this paper we develop the theory of nets of curves in a regular C^r-2-surface Eⁿ (r1, n2) using the concept C^s-net (of curves); the term diagonal nets of curves defined by W. BPLASCHKE [2] in E² is generalized accordingly. A regular C^r-surface E³ (r2) of negative GAUSSian curvature is called a (C^r-)D_SK-surface if its asymptotic lines (S-lines) and lines of curvature (K-lines) locally form a pair of diagonal nets. For the C³-D_SK-surfaces a criterion is given and distinct categories are determined, in particular all those C³-D_SK-surfaces in which the S- and K-lines can be arranged as (curvilinear) kites, respectively parallelograms and their diagonals.

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Auszugsweise vorgetragen auf der Geometrietagung in Oberwolfach (1.10.l974).     

A general theory of almost factoriality

Let R be a commutative integral domain with 1. The non-zero elements a,b of R may be calledv-coprime if aRbR=abR. A Krull domain is calledalmost factorial if for all f,g in R there is nN such that fⁿRgⁿR is principal. From this it is easy to establish that if R is almost factorial then for all x in R there is nN such that xⁿ=p₁p₂...p_r where p_i are mutually v-coprime primary elements and that this expression is unique. In this article we drop the requirement that R be Krull and replace the primary elements by elements calledprime blocks and develope a theory of almost factoriality, a special case of which is the theory of almost factorial Krull domains.     

Subalgebras that are cyclic as submodules

Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism : _R ⁿ AA (n2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dim_kA<, as well as in case R=, dimS<, and n0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the generic algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring D_n,m, and thus acquires an idempotent element in some integral domain extending D_n,m.Partially supported by National Science Foundation Grant GP-38229.     

Extremal problems in classes of univalent functions,omitting prescribed values

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f()=c₁+c₂²+... regular and univalent in the circleU={||<1} and such that f(₁)f(₂)=1 for ₁₂U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f(₀)| in the class of functions f()R with a fixed value f(₀, where ₀ is an arbitrary point U, and of the maximum of |f(₀)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Eine Klasse nicht einbettbarer reeller Laguerre-Ebenen

Replace in the parabolic model of the classical Laguerre-Plane the parabolas y=a(x–b)²+c, a0, by the curves y=af(x–b)+c with f(x)= if x0, and f(x)=(–x)^r ₂ if x<0. For each pair r₁, r₂>1 we obtain again a Laguerre-Plane (r₁,r₂).(r₁,r₂) can be embedded only if r₁=r₂=2.     

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 conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

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 rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

On semianalytic subsets of rigid analytic varieties

We show some topological properties of semianalytic subsets of rigid analytic varieties: curve selection lemma, the closure of a semianalytic subsetS is semianalytic, for every quasi-compact morphismf. As an application we show that a morphismf: X Y of rigid analytic varieties is open at a pointx X if and only if SpecO _X,x SpecO _Y,f(x) is surjective.     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Skew Brownian motion-type of extensions

We consider a class of possible extensions of a given symmetric Feller processZ _t fromR{0} to the entire real lineR, depending on a parameter [0, 1]. It is proved that the proposed extension exists if =1/2; for 1/2, exists if and only ifZ _t does not jump over 0 (e.g., ifZ _t is a diffusion).     

Positivity notions for coherent sheaves over compact complex spaces

LetX be a reduced compact complex space, X a coherent sheaf, andV=V() its associated linear fiber space. LetV _R be the reduction ofV, letA be the analytic set inX over which is not locally-free, and letV be the closure inV _R ofV _R |(X–A). is (primary) weakly positive if the zerosection ofV (V) is exceptional. is (primary) cohomologically positive if, for any coherent sheaf X, for all 0,k1. Then is (primary) weakly positive if and only if is (primary) cohomologically positive.LetX be a normal irreducible compact complex space. ThenX is Moishezon if and only if it carries a primary weakly positive, and hence primary cohomologically positive, coherent sheaf.Several other positivity notions are also discussed.     

Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

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.