Values of rational functions on non-hilbertian fields and a question of Weissauer

We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions onK: we construct a non-hilbertian subfieldK of such that, given anyf ₁ ,…,f _h ∈K(x), each of degree ≥2, the union ∪ _z=1 ^h f _z(K) does not containK. See e.g. [FrJ], [L1], [L2], [Sch], [Se1], or [Se2] for the classical theory of hilbertian fields.     

Topological rigidity of algebraic ℙ<Subscript>3</Subscript>-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus g≥1 is itself a ruled surface over a curve of genus g. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in the case g≥2. Received: August 30, 2001; in final form: April 12, 2002?Published online: March 12, 2003     

Combinatorial obstructions to the lifting of weaving diagrams

This paper deals with various connections of oriented matroids [3] and weaving diagrams of lines in space [9], [16], [27]. We encode the litability problem of a particular weaving diagramD onn lines by the realizability problem of a partial oriented matroid χ _D with2n elements in rank 4. We prove that the occurrence of a certain substructure inD implies that χD is noneuclidean in the sense of Edmonds, Fukuda, and Mandel [12], [14]. Using this criterion we construct an infinite class of minor-minimal noneuclidean oriented matroids in rank 4. Finally, we give an easy algebraic proof for the nonliftability of the alternating weaving diagram on a bipartite grid of 4×4 lines [16].     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

Isomorphism rigidity of irreducible algebraic ℤd-actions

An irreducible algebraic ℤ ^d -actionα on a compact abelian group X is a ℤ ^d -action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤ ^d -actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤ ^d -actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤ ^d -actions with d≥2. Oblatum 30-IX-1999 & 4-V-2000?Published online: 16 August 2000     

Hyperbolic Calculus

The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universal) Clifford algebra forIR ^1,0 [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus” analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical string motions.     

On the diophantine equation x 2 + p 2k = y n

Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23]. Received: 25 April 2008     

Retrieving random media

Benjamini asked whether the scenery reconstruction methods of Matzinger (see e.g. [21], [22], [20]) can be done in polynomial time. In this article, we give the following answer for a 2-color scenery and simple random walk with holding: We prove that a piece of the scenery of length of the order 3 ⁿ around the origin can be reconstructed – up to a reflection and a small translation – with high probability from the first 2 · 3^{10 αn} observations with a constant α > 0 independent of n. Thus, the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct. The probability that the reconstruction fails tends to 0 as n→∞. In contrast to [21], [22], and [20], the proofs in this article are all constructive. Our reconstruction algorithm is an algorithm in the sense of computer science. This is the first article which shows that the scenery reconstruction is also possible in the 2-color case with holding. The case with holding is much more difficult than [22] and requires completely different methods.     

The rationality of vector valued modular forms associated with the Weil representation

 In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds' question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen, and Zagier [Math. Ann., 278, 497–562]. Received: 27 September 2001 / Revised version: 22 July 2002 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 11F30; 11F27     

Rational values of the arccosine function

We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind. Research partially supported by grant BFM2003-06335-C03-03 of the DGI (Spain).     

Periodic harmonic functions on lattices and points count in positive characteristic

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.      

A de Montessus type convergence study for a vector-valued rational interpolation procedure

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F(z), where F: ℂ → ℂ^N, were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F(z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

How edge-reinforced random walk arises naturally

 We give a characterization of a modified edge-reinforced random walk in terms of certain partially exchangeable sequences. In particular, we obtain a characterization of an edge-reinforced random walk (introduced by Coppersmith and Diaconis) on a 2-edge-connected graph. Modifying the notion of partial exchangeability introduced by Diaconis and Freedman in [3], we characterize unique mixtures of reversible Markov chains under a recurrence assumption. Received: 22 January 2002 / Revised version: 24 September 2002 Published online: 15 April 2003 Most of this paper was written while the author was working at EURANDOM in Eindhoven, The Netherlands. Mathematics Subject Classification (2000): 60K37, 60G09     

On the topology of spherically symmetric space-times

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below.     

Four-manifolds with π<Subscript>1</Subscript>–free second homotopy

 We study the homotopy type of closed connected orientable topological 4–manifolds M with Λ–free second homotopy group, where Λ is the integral group ring of π₁(M). This is related with problem N.4.53 of [23], and extends some results proved for the class of closed 4–manifolds with free fundamental group [10][11]. Other applications on special classes of closed topological manifolds complete the paper. Received: 27 November 2001 / Revised version: 28 October 2002 Published online: 19 May 2003 Work performed under the auspices of the GNSAGA of the CNR of Italy and partially supported by Ministero per la Ricerca Scientifica e Tecnologica of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse, and by a research grant of the University of Modena and Reggio Emilia. Mathematics Subject Classification (2000): Primary: 57 N 65, 57 R 67; Secondary: 57 Q 10, 57 R 80     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

A unified theory of<Emphasis Type="Italic">θ</Emphasis>-continuity for functions

We introduce a new notion ofθ-M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory ofθ-continuity [15],θ-semi-continuity [4], quasi-irresoluteness [9],θ-preirresoluteness [29], and weakβ-irresoluteness [30].     

A gould type integral with respect to a multisubmeasure

In [GOULD, G. G.: Integration over vector-valued measures, Proc. London Math. Soc. (3) 15, (1965), 193–205], G. G. Gould introduced a type of integral of a bounded, real valued function with respect to a finite additive set function taking values in a Banach space, integral which is more general than the Lebesgue one. Recently, A. Precupanu and A. Croitoru gave the generalization, defining a Gould type integral for multimeasures with values in _kc(X), X being a Banach space ([PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, I, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 48 (2002), 165–200]). Taking as starting point this work and [PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, II, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 49 (2003), 183–207], we define here the notion of a Gould type integral with respect to a _bf(X)-valued multisubmeasure, pointing out important properties of it. We also establish that, even if we deal with multisubmeasures, the integral is still a multimeasure.      

Point-Free Version of Kakutani Duality

There are many results proved using the Axiom of Choice. Using point-free topology, we can prove some of these results without using this axiom. B. Banaschewski in [Pointfree Topology and the Spectra of f-rings, Ordered algebraic structures (Curacoa, 1995), Kluwer, Dordrecht, 123–148], studying the spectra of f-rings, describes the point-free version of the classical Gelfand duality without using the Axiom of Choice In this paper, referring to [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of ℓ-Modules, To appear in J. Pure Appl. Algebra], we describe a point-free version of the classical Kakutani duality. For this, using one of the spectra given in [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we find an adjunction between the category of compact completely regular frames with frame maps and the category of Archimedean bounded Riesz spaces with continuous Riesz maps.