Finely locally injective finely harmonic morphisms

We consider fine topology in the complex plane C and finely harmonic morphisms. We use oriented Jordan curves in the plane to prove that for a finely locally injective finely harmonic morphism f in a fine domain in C, either f or f is a finely holomorphic function. This partially extends result by Fuglede, who considered a kind of continuity for the fine derivatives of the finely harmonic morphism. As a consequence of this we obtain a both necessary and sufficient condition for a function f to be finely holomorphic or finely antiholomorphic. We do not know if the condition of finely local injectivity (q.e.) is automatically fulfilled by any non-constant finely harmonic morphism.     

The Riesz decomposition of finely superharmonic functions

This paper answers a question of Fuglede about minimal positive harmonic functions associated with irregular boundary points. As a consequence, an old and central problem of fine potential theory, concerning the Riesz decomposition, is resolved. Namely, it is shown that, on certain fine domains, there exist positive finely superharmonic functions which do not admit any positive finely harmonic minorant and yet are not fine potentials.     

Fonctions finement harmoniques et l'axiome de domination

In a harmonic space with the domination Axiom (axiom D), B. Fuglede [5] has introduced the sheaf property of the cones of the finely hyperharmonic functions (defined in the fine opens). In [7], [8], J. Luke, J. Malý and L. Zajíek have studied a notion analogous to the finely hyperharmonic functions without supposing axiom D, and have proved ([8] theorem 12.16) that if the cones of the positive finely hyperharmonic functions make a sheaf, then axiom D is satisfied. See N. Boboc, Gh. Bucur and A. Cornea [3] for the first result of this type given within the context of theH-cones.In this paper, we prove axiom D is a consequence of the sheaf property even for the smallest class of the functions (the class of the finely harmonic functions; absolute-value bounded in a convenient sense). This result implies that of Lukeet al. cited above. Consequently, the tow fine properties of the sheaf are equivalent, which has not been evident previously.

     

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1] ^s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.     

A Remark on Fine Differentiability

In [7], B. Fuglede has proved that finely holomorphic functions on a finely open subset U of the complex plane C are finely locally extendable to usual continuously differentiable functions. We shall adopt B. Fuglede’s approach to show that the same remains true even for functions which have only finely continuous fine differential on U. In higher dimensions, an analogous result may be obtained and the result can be applied to finely monogenic functions which were introduced recently as a higher dimensional analogue of finely holomorphic functions. I acknowledge the financial support from the grant GA 201/05/2117. This work is also a part of the research plan MSM 0021620839, which is financed by the Ministry of Education of the Czech Republic.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

On the riesz charge of the lower envelope of δ-subharmonic functions

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more -subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.     

A theorem in fine potential theory and applications to finely holomorphic functions

Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

Controlled algebraic K-theory of integral group ring of SL(3, ?)

We calculate the lower Controlled Algebraic K-theory of any finitely generated infinite subgroup of SL(3,Z), the group of 3×3 integral matrices of determinant 1.     

Continuity of eigenvalues for Schrödinger operators, $L^p$-properties of Kato type integral operators

Given an arbitrary relatively compact (finely) open subset of -eigenvalues of are studied where is the Dirichlet Laplacian on D and are measures on such that is continuous and is bounded for every ball X in being Green's function for X). Moreover, it is shown that these eigenvalues depend continuously on D and . The results are based on very general compactness and convergence properties of integral operators of Kato type which are developed before. Received: 9 November 2000 / Published online: 24 September 2001     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps

This paper is concerned with regularity results for starting points of continuous manifold-valued martingales with fixed terminal value under a possibly singular change of probability. In particular, if the martingales live in a small neighbourhood of a point and if the stochastic logarithm M of the change of probability varies in some Hardy space H_r for sufficiently large r<2, then the starting point is differentiable at M=0. As an application, our results imply that continuous finely harmonic maps between manifolds are smooth, and the differentials have stochastic representations not involving derivatives. This gives a probabilistic alternative to the coupling technique used by Kendall (1994).     

V -Modell der isotropen Ebene

A new model of isotropic plane called V-model is built, where V-points are points in the usual sense and V-straight lines are conics. A relationship with the affine model of isotropic plane is established. Furthermore, conics in the V-model are constructed.     

Finite invariant simplices in infinite graphs

It is shown that ifG is a graph which is contractible or dismantlable or finitely ball-Helly, and without infinite paths; or which is bounded, finitely ball-Helly and without infinite simplices then: (i) any contraction ofG stabilizes a finite simplex; and (ii)G contains a finite simplex which is invariant under any automorphism.     

THE REGULARITY DEGREE AND EPIMORPHISMS IN THE CATEGORY OF COMMUTATIVE RINGS

Elements of the universal (von Neumann) regular ring T(R) of a commutative semiprime ring R can be expressed as a sum of products of elements of R and quasi-inverses of elements of R. The maximum number of terms required is called the regularity degree, an invariant for R measuring how R sits in T(R). It is bounded below by 1 plus the Krull dimension of R. For rings with finitely many primes and integral extensions of noetherian rings of dimension 1, this number is precisely the regularity degree.

For each n ≥ 1, one can find a ring of regularity degree n + 1. This shows that an infinite product of epimorphisms in the category of commutative rings need not be an epimorphism.

Finite upper bounds for the regularity degree are found for noetherian rings R of finite dimension using the Wiegand dimension theory for Patch R. These bounds apply to integral extensions of such rings as well.     

The algebraic surfaces on which the classical Phragmén-Lindelöf theorem holds

Let V be an algebraic variety in . We say that V satisfies the strong Phragmén-Lindelöf property (SPL) or that the classical Phragmén-Lindelöf Theorem holds on V if the following is true: There exists a positive constant A such that each plurisubharmonic function u on V which is bounded above by |z|+o(|z|) on V and by 0 on the real points in V already is bounded by A| Im z|. For algebraic varieties V of pure dimension k we derive necessary conditions on V to satisfy (SPL) and we characterize the curves and surfaces in which satisfy (SPL). Several examples illustrate how these results can be applied.     

Families of cuts with the MFMC-property

A family ℱ of cuts of an undirected graphG=(V, E) is known to have the weak MFMC-property if (i) ℱ is the set ofT-cuts for someT⊆V with |T| even, or (ii) ℱ is the set of two-commodity cuts ofG, i.e. cuts separating any two distinguished pairs of vertices ofG, or (iii) ℱ is the set of cuts induced (in a sense) by a ring of subsets of a setT⊆V. In the present work we consider a large class of families of cuts of complete graphs and prove that a family from this class has the MFMC-property if and only if it is one of (i), (ii), (iii).     

Spectra for Factorable Matrices on ℓ p

We obtain the spectra and fine spectra for factorable matrices, considered as bounded linear operators over ℓ^p , 1 < p < ∞.     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.