The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

Connessioni adattate a una struttura quasi prodotto

Sunto Un campo di vettori di una varietà riemanniana V_n+1 induce in essa mediante la decomposizione locale dello spazio tangente nel sottospazio parallelo a e nel sottospazio ad esso normale e supplementare una particolare struttura quasi prodotto. Vengono qui studiate le connessioni rispettanti tale struttura (connessioni vincolate) con particolare riguardo alla connessione vincolata canonicamente associata alla connessione riemanniana Γ. Si danno semplici formule che legano la derivazione covariante secondo alla derivazione covariante riemanniana. Lavoro eseguito nell'ambito dell'attività del Gruppo di ricerca matematica N^o 1 del C. N. R.     

Défaut d’estimation a priori pour un problème de dirichlet

Résumé  Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour montrer que les solutionsu ∈C ⁰ ∩C ¹ (Ω) du problème admettent une estimation a priori dansC ⁰ . Dans ce travail, on montre que, si les solutionsu sont juste supposéesC ⁰ , une telle estimation a priori n’existe plus. In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all the solutionsu ∈C ⁰ ∩C ¹ (Ω) of the problem admit an a priori estimate inC ⁰ . In this work, we show that, if one supposes the solutions are only inC ⁰ , such an a priori estimate does not hold.     

Holomorphic approximation on compact pseudoconvex complex manifolds

Let be a smoothly bounded compact pseudoconvex complex manifold of finite type in the sense of D’Angelo such that the complex structure of M extends smoothly up to bM. Let m be an arbitrary nonnegative integer. Let f be a function in H(M)∩ H_m(M), where H_m(M) is the Sobolev space of order m. Then f can be approximated by holomorphic functions on in the Sobolev space H_m(M). Also, we get a holomorphic approximation theorem near a boundary point of finite type.     

Construction of representations of the Poincaré Lie algebra by extension of direct integrals of representations ofSL(2,C)

Summary Continuing the program initiated in the first paper of this series, representations of the Lie algebra of the Poincaré group are constructed by forming suitable direct integrals of representations of SL(2, C) and extending them to by means of four additional momentum operators whose matrix elements are derived in a basis adapted to the direct integral structure. The representations obtained here include Wigner’s representations with positive mass and spin and Wigner’s representations with zero mass, spin and helicity ± . Lavoro eseguito con contributo del C.N.R. nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le applicazioni della Matematica alla Fisica ed all’Ingegneria. Entrata in Redazione il 12 marzo 1972.     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

What we know about the second adjunction mapping

Let be an algebraic submanifold of complex projective space ℙ ^N . Assume thatn:=dim and let denote the restriction of the hyperplane section bundle to . The meromorphic map associated to fork≥1 ties together the pluricanonical maps of the surface sections of . Known results show that the behavior of is far better than one would expect from experience with the pluricanonical mappings of algebraic surfaces. In this article we discuss the known results on the structure of the mappings and describe the open problems. Conferenza tenuta il 5 giugno 1997     

Problemi di curvatura su di una varietà quasi hermitiana

Sunto Su di una varietà quasi hermitiana V, la considerazione di alcune medie di curvature semplici e miste, relative a convenienti sistemi di 2-spazi associati ad un 2-spazio α, dà informazioni sulla curvatura semplice K_α, sulla media e sulla curvatura mista ( =Jα). Due delle medie accennate riescono nulle; le altre danno luogo a relazioni di dipendenza lineare. Considerate per la varietà V ipotesi di isotropia in media e di J-isotropia si ottengono teoremi del tipo di Schur. Altri risultati si stabiliscono nel caso k?hleriano.

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Summary Let V be an almost Hermite space, α a tangent 2-dimensional space of V; five systems of tangent 2-spaces of V are considered, suitably related to α. The sectional curvature K_α, the mean value , and the bisectional curvature are expressed in terms of mean values of sectional and bisectional curvatures, evalued over the above mentioned systems. Two of the mean values vanish; linear relationships hold among the other ones. Schur-like theorems are proven under the assumption that V be isotropic in mean or J-isotropic. Further results are obtained for the case that V be a K?hler space.

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Entrata in Redazione il 25 novembre 1971.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale Strutture Algebriche, Geometriche e loro Applicazioni. I primi risultati sono stati annunciati in una riunione di Gruppo a Bologna (dicembre 1970) e in una conferenza tenuta all'Ist. Mat. Univ. Perugia (aprile 1971).     

On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

LetM ⁿ be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM _n, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM ^2m+1(c) satisfies , whereH ² andg are the square mean curvature function and metric tensor onM ⁿ, respectively. The equality holds identically if and only if eitherM ⁿ is totally geodesic submanifold or n = 2 andM ⁿ is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM ⁿ ofM ²ⁿ⁺¹ (c) satisfies identically, then it is minimal.     

Loose block independence

A finite state stationary process is defined to be loosely block independent if long blocks are almost independent in the sense. We show that loose block independence is preserved under Kakutani equivalence and limits. We show directly that any loosely block independent process is the limit of Bernoulli processes and is a factor of a process which is Kakutani equivalent to a Bernoulli shift. The existing equivalence theory then yields that the loosely block independent processes are exactly the loosely Bernoulli (or finitely fixed) processes.     

Entropies with the branching property

Summary It is proved that every entropy with the property of branching has the form , where J is a compositive information measure under a regular binary operation. In particular all the classical entropies with the branching property such as that of Shannon have the form .

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Sunto Si dimostra che ogni entropia che goda della proprietà di diramazione ha necessariamente la forma dove J è una misura di informazione compositiva con una operazione binaria regolare. In particolare tutte le entropie classiche che godono della proprietà di diramazione quale quella di Shannon hanno la forma .

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Entrata in Redazione il 9 aprile 1973.     

A new class of contact riemannian manifolds

N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) ( for any unit -geodesic ( , where is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C). This work was supported in part by BSRI 98-1425.     

Mixing examples in the class of piecewise monotone and continuous maps of the unit interval

A process (T, P) is said to have the “ ” property if there is a uniform, positive lowerbound δ on the separation between theT-P names of (almost) every pair of pointsx≠y. A finite group rotation with partition into distinct points provides a trivial example. Given any process having the property we show that there exists a Bernoulli shiftB so thatT×B is measurably isomorphic to the natural extension of a piecewise monotone, continuous, and expanding map of the unit interval. This construction is applied to produce interval maps which are ergodic but not weak-mixing, weak-mixing but not mixing, and mixing but not exact with respect to their unique absolutely continuous invariant measures, in contrast with the results known for piecewiseC ^1+∈ expansive interval maps. In obtaining these examples we identify a number of nontrivial classes of automorphismsT which admit processes having the property. Supported by NSERC grant OGP0046586 90.     

Sobolev embedding in ℂn and the\overline \partial - equation

We prove a Sobolev embedding theorem for functions that are in a Sobolev space while their is in L^t, fort large enough. This allows us to deduceL ^p or Lipschitz estimates with loss from classical Sobolev estimates for the solution of in weakly pseudo-convex domains.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

On tensors which are functions of vectors

It has been shown that the most general constant tensor of any rank is the possible algebraic combination of Kronecker tensor and the totally antisymmetric tensor ... μ_n. Further, the tensor of a given rank r, which are functions of vectors p's, are suitable algebraic combinations of the tensor and the two constant tensors with coefficients which are functions of the scalar products formed by the vectors.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin