Calcul Fonctionnel Holomorphe en Dimension Infinie dans une Algebre de Banach Quotient

L. Waelbroeck a montré dans [10] que le calcul fonctionnel holomorphe de plusieurs variables dans une algèbre de BanachA se généralisait à une algèbre de Banach quotientA|α (i.e:α un idéal de Banach deA tel que l’injectionα→A soit continue) et que c’est dans ce cadre qu’il est naturel et a de nombreuses applications. Dans [13], [1] et [4] les auteurs ont construit des calculs fonctionnels holomorphes en dimension infinie dans une algèbre de Banach. Nous étendons ici leurs calculs au cas d’une algèbre de Banach quotient ainsi que le théorème d’Arens-Calderon.      

Sur le Degré Minimum des Connexions Polynomiales Entre les Projections Dans une Algèbre de Banach

Résumé  L’insiemeP degli elementi idempotenti di un’algebra di BanachA é in generale non connesso. J. Zemanek [7] e B. Aupetit [1] hanno dimostrato che le componenti connesse di tale insieme sono connesse per archi. Inoltre J. Esterle [4] ha dimostrato che due elementi diP appartenti alla stessa componente connessa possono essere collegati attraverso un cammino polinomiale. In questo lavoro si studia il minimo grado di tali polinomi seA è un’algebra di Banach di dimensione finita oppure seA è l’algebra degli operatori limitati su uno spazio di Banach.     

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy. In this way, we reduce the study ofC ^r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ^∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.     

Automorphy for some <Emphasis Type="Italic">l</Emphasis>-adic lifts of automorphic mod <Emphasis Type="Italic">l</Emphasis> Galois representations. II

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

Simple proof of the nonrationality of three-dimensional quartics

The rigidity theorem for smooth three-dimensional quartics [1] is reproved following an idea of Corti [2] and using Shokurov’s connectedness theorem. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 667–673, May, 1999.     

E-unitaryR-unipotent semigroups

By making use of McAlister’s P-theorem [4] O’Carroll proved in [5] that every E-unitary inverse semigroup can be embedded into a semidirect product of a semilattice by a group. Recently an alternative proof of this result was published by Wilkinson [10]. In this paper we generalize this theorem by proving that every E-unitaryR-unipotent semigroup S can be embedded into a semidirect product of a band B by a group where B belongs to the variety of bands generated by the band of idempotents of S.     

A representation formula of sum rules for zeros of polynomials

A representation formula in terms of Lucas polynomials of the second kind in several variables (see formula (4.3)), for the sum rulesJ _s ⁽ⁱ⁾ introduced by K.M. Case [1] and studied by J.S. Dehesa et al. [2]–[3] in order to obtain informations about the zeros’ distribution of eigenfunctions of a class of ordinary polynomial differential operator, is derived. Lavoro eseguito nell’ambito del G.N.I.M. del C.N.R.     

The Blok–Ferreirim theorem for normal GBL-algebras and its application

Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras. Presented by J. G. Raftery. Received May 23, 2007; accepted in final form February 20, 2008.     

Old and new moving-knife schemes

In general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?     

Sur le spectre du laplacien d’une métrique invariante à gauche sur un groupe de Lie

On montre que le spectre du laplacien d’une métrique invariante à gauche sur un groupe de Lie non compact, unimodulaire, est un intervalle [σ, ∞] avec, éventuellement, des valeurs propres de multiplicité infinies. Dans certains cas particuliers, on montre aussi qu’il est absolument continu; en général l’intervalle [σ, ∞] est le spectre essentiel. Cette étude a débuté pour la première fois avec le travail Furutani et al. (Commun Part Differ Equ 18, 1993).     

Splitting ρκλ into maximally many stationary sets

Letκ >ω be a regular cardinal and λ >κ a cardinal. Solovay’s classical result for κ[So] led Menas [Me] to conjecture that a stationary subset ofP _κλ would split into λ^<κ stationary set of size κ⁺ (see[BT]), the conjecture implies that the size is (κ⁺)^<κ as well. Part of this work was done during the author’s stay at Boston University as one of the Japanese Overseas Research Fellows. He gratefully acknowledge Professor Akihiro Kanamori’s hospitality. He also wishes to thank members of the set theory seminar at Waseda University for their interest at the early stage.     

Favard’s interpolation problem in one or more variables

Given scattered data on the real line, Favard [4] constructed an interpolant which depends linearly and locally on the data and whose nth derivative is locally bounded by the nth divided differences of the data times a constant depending only on n. It is shown that the (n —1)th derivative of Favard’s interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corresponding divided differences. In this sense, Favard’s univariate interpolant is the best possible. Favard’s result has been extended [8] to a special case in several variables, and here the extent to which this can be repeated in a more general setting is proven exactly.     

Valiron summability of the derived fourier series and its coniugate series

Summary Valiron [12] obtained very extensive generalisations ofBorel’s method. In this paper the author has consideredValiron summability. The results proved are equivalent toBorel’s for k=1.     

On steiner systems

Steiner’s “combinatorial problems” have so far been solved only fork=3 [5, 3] and fork=4 [1,2]. In this paper a complete solution of the problem is given for “closed” Steiner systems, i.e. systems havingn=2 ^k−1−1 elements. Use is made of methods developed by Zaremba [7] for abelian groups. This research was supported by the United States Air Force under Grant No. AF-EOAR-6360 and monitored by the European Office of Aerospace Research.     

Homotopy types of box complexes

In [14] Matoušek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovász’s original bound [9] can be restated as X(G) ≥ ind(B(G)) + 2. Sarkaria’s bound [15] can be formulated as X(G) ≥ ind(B₀(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that up to ℤ₂-homotopy the box complex B(G) can be any ℤ₂-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14]. Supported by the joint Berlin/Zürich graduate program Combinatorics, Geometry, and Computation, financed by ETH Zürich and the German Science Foundation (DFG).     

Some common fixed point theorems for selfmappings satisfying two contractive conditions of integral type in a uniform space

In this paper, we establish some common fixed point theorems for selfmappings of a uniform space by employing both the concepts of an A—distance and an E—distance introduced by Aamri and El Moutawakil [1] and two contractive conditions of integral type. Our results are generalizations and extensions of the classical Banach’s fixed point theorem of [2, 3, 19], some results of Aamri and El Moutawakil [1], Theorem 2.1 of Branciari [5] as well as a result of Jungck [7].      

Gap series constructions for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

Highly oscillatory bounded solutions of div(∇u|∇u| ^p−2) = 0 are constructed when p > 2. Fatou’s theorem is shown to fail for this equation. Tom Wolff wrote this paper in 1984, but he never published it. With his family’s permission, we have edited it for publication here. Except for the shorter proof of Lemma 2.1 and the citations of [1] and [12], our alterations to the paper have mostly been typographical. We thank Juan Manfredi for help on Section 3.     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.