On bands in which every finitely generated subband is an endomorphic image

A semigroup S is calledE-semigroup ( -semigroup) if every (finitely generated) subsemigroup of S is an endomorphic image of S and Ē-semigroup ( -semigroup) if every subsemigroup of S is an E-semigroup -semigroup. All classes X ε {Ē, , E, } are distinct even in the case of semilattices. It is established when a free band (semilattice) is an X-semigroup. -, - and Ē-chains and E-chains with identity or zero, Ē-and , X-bands with identity and X-semigroups with identity and zero are found.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.     

Structures of polyhedra determined by submodular functions on crossing families

The present paper shows that for any submodular functionf on a crossing family with , if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique submodular function with such thatB(f) coincides with the base polyhedron associated with the submodular system . Here, iff is integer-valued, thenf ₁ is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.     

Stochastic dilations of uniformly continuous completely positive semigroups

For an arbitrary uniformly continuous completely positive semigroup ( _t :t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.     

The automorphism group of the Toeplitz algebra onH2(T2)

The automorphism group of the C^* -algebra generated by the Toeplitz operatos with symbols being continuous functions on the bicircle is characterized completely. The investigation is based on the analysis of the behaviour of an automorphism of the Toeplitz algebra on itsC* -ideal chains, and the state of the closed ideals in .     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.     

Intersections of curve systems and the crossing number ofC 5 ×C 5

It and are two families of pairwise disjoint simple closed curves in the plane such that each curve in intersects each curve in , then the total number of points of intersection in is at least 2(m−1)n, where , and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.     

Sul contatto di terz’ordine di due superficie algebriche lungo curve

Sunto Si dànno delle condizioni necessarie affinchè due superficie algebriche F, G d’una V ₃ algebrica non singolare possano presentare contatto d’ordine assegnato q −1 lungo una curva priva di punti multipli. Si approfondisce sopratuito il caso q=3, con applicazione al problema di determinare le superficie algebriche G_n di S ₃ che presentano contatto del 3^o ordine lungo una ⁿ con una superficie quartica F ₄ .     

A Schwarz lemma for a domain related to μ-synthesis

We prove a Schwarz lemma for a domain in ℂ³ that arises in connection with a problem in H^∞ control theory. We describe a class of automorphisms of and determine the distinguished boundary of We apply our Schwarz lemma to a special case of the μ-synthesis problem.     

Enumeration ofQ-acyclic simplicial complexes

Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( _k ⁿ⁻¹ )k-faces, (3)H _k (C)=0. We prove that for ,H _k−1(C) is a finite group, and our main result is: . This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.     

Operator analogue of the Krein-Milman theorem in the generalized state spaces

We discuss the Krein-Milman-type problems in the C* -convexity theory for the generalized state space of C*-algebraA. The main results are that every BW-compact, C*-convex subset of possesses a C*-extreme point and every BW-compact, C* -convex subset of is the C^*-convex hull of its C*-extreme points.     

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingx _i +x _j ≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope with integral vertices. We triangulate with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupB _d . This labeling gives rise to a shelling of the triangulation of , where theh-vector of (and the Ehrharth ^*-vector of can be computed as a descent statistic on a subset ofB _d defined in terms ofG. A recursive way of computing theh-vector of is also given, and a recursive formula for the volume of . This work was partially supported by grants from the Icelandic Council of Science and the Royal Swedish Academy of Sciences, respectively.     

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ. Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.     

Some intersection theorems on two-valued functions

Let be a family of two-valued functions defined on ann-element set in which each pair of functions in satisfy a given intersection condition. For certain intersection conditions we determine the maximal value of .     

Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

On a property of the moebius group

Summary It is shown that the group of all Moebius transformations (M. T.) contains a subgroup such that every non-involutory M. T. not in can be transformed into each of its conjugates outside the subgroup by a unique element of this subgroup. Hence every non-integral and non-involutory M. T. can be reduced into a normal form by means of a unique element of . Also a subgroup *, can be determined such that elements of * reduce all non-involutory M. T. not in *, into their classical (integral) normal forms. There is also a discussion of the question as to other fields over which the results remain valid. To Enrico Bompiani on his scientific Jubilee.     

Hankel-Schur multipliers and multipliers of the space H1

In this paper there is given a sufficient condition for a Hankel matrix _F to belong to the space of Schur multipliers of all bounded operators in ² (or, what is the same, to the tensor algebra V²). It is shown that ifw is a nonnegative function on T, such that is a sequence of integers, {F_i}_j1 is a sequence of polynomials,) and, then _FV². It follows from this that under these conditions F is a multiplier of the space H¹, i.e.,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 113–119, 1984     

Formal language theory and the geometry of 3-manifolds

Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3-manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3-manifold which satisfies Thurston's geometrization conjecture. Toward this end, the class of asynchronously groups is introduced and studied, where is an arbitrary full abstract family of languages. For example may be the family of regular languagesReg, context-free languagesCF, or indexed languagesInd. The class consists of precisely those groups which are asynchronously automatic. It is proved that contains all of the above fundamental groups, but that does not. Indeed a virtually nilpotent group belongs to if and only if it is virtually abelian. The first author was partially supported by NSF grant DMS-9203500 and FNRS (Suisse). He also wishes to thank the University of Geneva for its hospitality while this paper was being written. The second author thanks the Institute for Advanced Study for its hospitality while this paper was being written.     

Sull'apertura di luoghi negli spettri affine e projettivo di un dominio graduato noetheriano

Riassunto Sia una proprietà degli anelli discende per fedele piattezza, ascende quando le fibre sono buone, di carattere locale e per cui vale il criterio di Nagata. SeS è un dominio graduato noetheriano valgono alcune implicazioni per l'apertura del in SpecS, ProjS e altri schemi collegati adS.

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Summary Let be a property meaningful for rings which descends by faithful flatness, ascends when there are good fibers, with local character and for which Nagata's criterion holds. IfS is any graded noetherian domain some implications hold for the openness of the in SpecS, ProjS and other schemes related toS.

     

State-constrained control problems with neither coercivity norL 1 bounds on the controls

Summary A state-constrained, nonlinear, minimum problem is considered with dynamics depending sublinearly on a control which is not bounded in theL ¹ norm. Because of the lack of coercivity, the value map fails to be continuous, even in the unconstrained case. However, we prove that under suitable assumptions—which guarantee the continuity of the value maps of the problems withL ¹-bounded controls—the value map is upper semicontinuous and solves a Bellman equation with a continuous Hamiltonian. Moreover, the map obtained by by replacing its values at the horizon t=T with the values of the cost function turns out to be the maximal subsolution of the corresponding value problem. Entrata in Redazione il 31 dicembre 1997.