Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

Schauder type theorems for differentiable and holomorphic mappings

Denoting byC _wu ^p (E) the algebra of allC ^p-real-valued functions on the real Banach spaceE such that the functions and the derivatives are weakly uniformly continuous on bounded subsets, it is known that the algebra homomorphismsA:C _wu ^q (F)C _wu ^p (E) are induced by differentiable mappingsg:EF ^**. We prove that, for 1p+1q, the following are equivalent: (a)A is compact; (b)g and its derivatives are compact; (c)gC _wu ^p (E,F ^**) (the authors had proved that, forp=q<,A is [weakly] compact if and only ifg is a constant mapping, and it is known that ifq<p, thenA is always induced by a constant mapping and is therefore compact). Also, for an entire function of bounded typegH _b (U,F), where is a balanced open subset, andE,F are complex Banach spaces, lettingA:H _b (F)H _b (U) be the homomorphism given byA(f)=fg for allfH _b (F), we prove thatA is compact if and only ifg is compact.Supported in part by DGICYT Grant PB 94-1052 (Spain).Supported in part by DGICYT Grant PB 93-0452 (Spain).     

Differential calculus founded on an isomorphism

This paper founds calculus on a natural isometric linear isomorphism. Once this Foundational Isomorphism is proved (with elementary Banach space methods) several familiar calculus properties of continuous curves drop out as quick corollaries. Then the calculus can be further developed in very general setting via categorical methods. VSERC aided     

Complex calculus via foundation theorems

It was shown recently that continuous differentiation theory can be founded on a natural isometric linear isomorphism. We develop in this paper an analogous theory for complex differentiation; however, this requires different foundational isomorphism.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

Fredholm-valued holomorphic mappings on a Banach space

In this article we show that the pointwise existence of a regulariser for holomorphic Fredhom-valued mappings defined on pseudo-convex domains in Banach spaces with an unconditional basis implies the existence of a holomorphic regulariser.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

A new proof of Fréchet differentiability of Lipschitz functions

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ₂ (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w ^* closure of the set of its points of Gateaux differentiability is norm separable. Received May 31, 1999 / final version received February 16, 2000?Published online April 19, 2000     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

A method for computing symmetries and conservation laws of integra-differential equations

A method for computing symmetries and conservation laws of integro-differential equations is proposed. It resides in reducing an integro-differential equation to a system of boundary differential equations and in computing symmetries and conservation laws of this system. A geometry of boundary differential equations is constructed like the differential case. Results of the computation for the Smoluchowski's coagulation equation are given.     

Phase-locking in the multidimensional Frenkel-Kontorova model

We consider the multidimensional Frenkel-Kontorova model with one degree of freedom which is a variational problem for real functions on the lattice . For every vector there is a special class of minimal solutions lying in finite distance to the linear function with . Due to periodicity properties of the variational problem is called the rotation vector of these solutions. The average action of a minimal solution is obtained by averaging the variational sum over . One shows that this average action is the same for any minimal solution with finite distance to the linear function with rotation vector . Our main results concern the differentiability properties of as a function of the rotation vector: Typically, is not differentiable at . This will be interpreted in a dual form as phase-locking. The phase of is defined by the unique vector in for which is minimal. If one perturbs the variational principle by changing the parameter , the non-differentiability of at forces the phase to be locked onto the rational value . Received: 28 April 1994; in final form 11 April 1996     

HOLOMORPHIC MAPS ON DOMAINS WITH EMPTY INTERIOR

Abstract

We bring to light further infinite dimensional domains with empty interior on which holomorphic maps can be constructed.     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

On the geometry of the set of square roots of elements inC *-algebras

This research was partially supported by the CONICET and the University of Buenos Aires.     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

On the Courant-Fischer theory for Krein spaces

Let J=I_r⊕-I_n-r,0<r<n. An n×n complex matrix A is said to be J-Hermitian if JA=A^∗J. An extension of the classical theory of Courant and Fischer on the Rayleigh ratio of Hermitian matrices is stated for J-Hermitian matrices. Applications to the theory of small oscilations of a mechanical system are presented.