三角Banach代数的等距同构

Let (X) and (X) be Banach algebras.Let (X) be a Banach (X),(X)-module with bounded 1.Then (X) is a Banach algebra with the usual operations and the norm ‖［AOMB］‖=‖A‖+‖M‖+‖B‖.Such an algebra is called a triangular Banach algebra.In this paper the isometric isomorphisms of triangular Banach algebras are characterized.     

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.     

Isoperimetric inequalities for infinite hyperplane systems

Let be an infinite discrete system ofk-dimensional flats inn-dimensional Euclidean space. If the totalk-dimensional volume of the flats in intersected with the ball of center 0 and radiusr, divided by the volume of that ball, tends to a limit forr→∞, then this limit is called thedensity of . We consider isoperimetric problems of the following kind. If is a hyperplane system of positive density, find sharp upper bounds for the density of the system ofk-flats (k∈{0, ...,n−2}) that are generated as intersections of hyperplanes in . Ideas from the theory of uniform distribution of sequences are employed to define a large class of hyperplane systems, calleduniform, for which all necessary densities exist, isperimetric inequalities can be proved, and systems with extremal intersection densities can be characterized.     

Recurrent and almost-periodic sequences

A sequence g : N → C is called almost-periodic if it belongs to the completion of the C-linear space spanned by the sequences e _ϑ with ϑ ∈ R/Z, where e _ϑ(n) = e ^2πiϑn for n ∈ N, under the semi-norm . Every has a mean value . A sequence g: N → C is called recurrent if it satisfies a linear recurrence equation of the form

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.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

Counting the relatively finite factors of a Bernoulli shift

LetT acting on (Ω, ℱ, μ) be afinite entropy Bernoulli shift. AT invariant factor is “relatively finite” if a.e. fiber of has a finite, hence constantk, number of points. We say two factors , “sit the same” if there is a measurable measure preserving map ϕ with and . We show here that up to sitting the same there are only finitely many relatively finite factors withk point fibers in a Bernoulli shift, and that they are classified by a certain algebraic structure in the symmetric group onk-points. This work was supported in part by the Miller Institute for Basic Research, U. C. Berkeley.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

Bounding the piercing number

It is shown that for everyk and everyp≥q≥d+1 there is ac=c(k,p,q,d)<∞ such that the following holds. For every familyℋ whose members are unions of at mostk compact convex sets inR ^d in which any set ofp members of the family contains a subset of cardinalityq with a nonempty intersection there is a set of at mostc points inR ^d that intersects each member ofℋ. It is also shown that for everyp≥q≥d+1 there is aC=C(p,q,d)<∞ such that, for every family of compact, convex sets inR ^d so that among andp of them someq have a common hyperplane transversal, there is a set of at mostC hyperplanes that together meet all the members of . This research was supported in part by a United States-Israel BSF Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

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.     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

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 .     

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.     

On minimum and maximum spanning trees of linearly moving points

In this paper we investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Here, a transition means a change on the combinatorial structure of the spanning trees. Suppose that we are given a set ofn points ind-dimensional space,S={p ₁,p ₂, ...p _n }, and that all points move along different straight lines at different but fixed speeds, i.e., the position ofp _i is a linear function of a real parametert. We investigate the numbers of transitions of MinST and MaxST whent increases from-∞ to +∞. We assume that the dimensiond is a fixed constant. Since there areO(n ²) distances amongn points, there are naivelyO(n ⁴) transitions of MinST and MaxST. We improve these trivial upper bounds forL ₁ andL _∞ distance metrics. Letk _p (n) (resp. ) be the number of maximum possible transitions of MinST (resp. MaxST) inL _p metric forn linearly moving points. We give the following results in this paper: κ₁(n)=O(n ^5/2 α(n)),κ _∞(n)=O(n ^5/2 α(n)), , and where α(n) is the inverse Ackermann's function. We also investigate two restricted cases, i.e., thec-oriented case in which there are onlyc distinct velocity vectors for movingn points, and the case in which onlyk points move.     

Local rules for pentagonal quasi-crystals

The existence of different kinds of local rules is established for many sets of pentagonal quasi-crystal tilings. For eacht∈ℝ there is a set of pentagonal tilings of the same local isomorphism class; the caset=0 corresponds to the Penrose tilings. It is proved that the set admits a local rule which does not involve any colorings (or markings, decorations) if and only ift=m+nτ. In other words, this set of tilings is totally characterized by patches of some finite radius, orr-maps. When the set admits a local rule which involvescolorings. For the set of Penrose tilings the construction here leads exactly to the Penrose matching rules. Local rules for the caset=1/2 are presented.     

Free discontinuity problems with unbounded data: The two dimensional case

We prove the existence of a minimizing pair for the functionalG defined for every closed setK ⊂R ² and for every functionu ∈C ¹(ω/K) by where ω is an open set inR ², λ, μ>0,q≥1,g ∈L ^q (ω) ∩L ^p (ω) withp>2q andH ¹ is the 1-dimensional Hausdorff measure.     

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.     

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.     

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.