Weak density and cupping in the d-r.e. degrees

Consider the Turing degrees of differences of recursively enumerable sets (the d-r.e. degrees). We show that there is a properly d-r.e. degree (a d-r.e. degree that is not r.e.) between any two comparable r.e. degrees, and that given a high r.e. degreeh, every nonrecursive d-r.e. degree ≦h cups toh by a low d-r.e. degree. The second author was partially supported by NSF grant DMS-8701891.     

A computably enumerable vector space with the strong antibasis property

Downey and Remmel have completely characterized the degrees of c.e. bases for c.e. vector spaces (and c.e. fields) in terms of weak truth table degrees. In this paper we obtain a structural result concerning the interaction between the c.e. Turing degrees and the c.e. weak truth table degrees, which by Downey and Remmel's classification, establishes the existence of c.e. vector spaces (and fields) with the strong antibasis property (a question which they raised). Namely, we construct c.e. sets such that the c.e. W-degrees below that of A are disjoint from the nonzero c.e. T-degrees below that of A and comparable to that of B. Received: 2 December 1998     

Residual Properties of Free Products

Let 𝒞 be a class of groups. We give sufficient conditions ensuring that a free product of residually 𝒞 groups is again residually 𝒞, and analogous conditions are given for LE-𝒞 groups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.

Moreover, we consider the pro-𝒞 topology and we characterize special HNN extensions and amalgamated free products that are residually 𝒞, where 𝒞 is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.     

On the Evolution of an Angle in a Vortex Patch

Summary. The inviscid incompressible two-dimensional motion of some initially convex singular vortex patches is examined. The angle's evolution of a tangent-slope discontinuity on a singular contour is studied from a numerical and theoretical point of view. Different numerical examples show that the angle shrinks for initial angle less than 90^o , and the angle widens when the initial angle is greater than 90^o or is ``approximately' preserved for initial angle 90^o for small time evolution. An asymptotic expansion of the initial velocity field near a singularity for a class of singular vortex patches is performed to reinforce this result analytically. Some initially nonconvex singular patches in which the evolution does not follow this rule are shown.     

Volume of Hyperbolic 3-Manifolds with Iterated Pseudo-Anosov Amalgamations (Dedicated to Professor Mitsuyoshi Kato on his sixtieth birthday)

Let : be a pseudo-Anosov homeomorphism. We will study an asymptotic behaviour of the volume of closed hyperbolic 3-manifolds N _n obtained from certain 3-manifolds M, M by attaching their boundaries by the n-th iteration ⁿ of .     

Characterizing matchings as the intersection of matroids

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function (G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, (n), for graphs with n vertices. We describe an integer programming formulation for deciding whether (G)k. Using combinatorial arguments, we prove that (n)(log logn). On the other hand, we establish that (n)O(logn/ log logn). Finally, we prove that (n)=4 for n=5,,12, and sketch a proof of (n)=5 for n=13,14,15.An earlier version appears as an extended abstract in the Proceedings of COMB01 [5]. Supported by the Gerhard-Hess-Forschungs-Förderpreis (WE 1462) of the German Science Foundation (DFG) awarded to R. Weismantel.     

Infima of d.r.e. degrees

Lachlan observed that the infimum of two r.e. degrees considered in the r.e. degrees coincides with the one considered in the D₂⁰{\Delta_2^0} degrees. It is not true anymore for the d.r.e. degrees. Kaddah proved in (Ann Pure Appl Log 62(3):207–263, 1993) that there are d.r.e. degrees a, b, c and a 3-r.e. degree x such that a is the infimum of b, c in the d.r.e. degrees, but not in the 3-r.e. degrees, as a < x < b, c. In this paper, we extend Kaddah’s result by showing that such a structural difference occurs densely in the r.e. degrees. Our result immediately implies that the isolated 3-r.e. degrees are dense in the r.e. degrees, which was first proved by LaForte.     

Remark on the Unital Quantale Q[e]

In this paper, we investigate some properties of the unital quantale Q[e], in terms of the quantic quotients of Q[e] we study the extensions of quantic nuclei of Q to Q[e] and prove that for any non-trivial quantale Q\textbf{Q}, Q[e] is not simple. Also, we give some applications for the unital quantale Q[e].     

Boundary value problems for second order nonlinear differential equations on infinite intervals

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L²(0,∞) and L²(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established.     

Splitting and cone avoidance in the D.C.E. degrees

It is shown that the Cooper splitting theorem for the n-c.e. degrees is not compatible with cone avoidance: For any n > 1, there exist n-c.e. degree a, c.e. degree b such that 0 < b < a and such that for any n-c.e. degrees x,y, if x ∨ y = a, then either b ≤ x or b ≤ y. This provides a new type of elementary difference between the classes of c.e. and d.c.e. degrees, implementable at lower levels of the high/low hierarchy.     

A study of nonlinear elliptic problems involving supercritical and exponential growth in

In this paper, we consider the multiplicity of solutions of the p‐Laplacian problems involving supercritical Sobolev growth and exponential growth in via Ricceri principle. By means of the truncation combining with Moser iteration, we can extend the result about the subcritical growth to the supercritical and exponential growth.     

The “Essential Tension” at Work in Qualitative Analysis: A Case Study of the Opposite Points of View of Poincaré and Enriques on the Relationships between Analysis and Geometry

An analysis of the different philosophic and scientific visions of Henri Poincaré and Federigo Enriques relative to qualitative analysis provides us with a complex and interesting image of the “essential tension” between “tradition” and “innovation” within the history of science. In accordance with his scientific paradigm, Poincaré viewed qualitative analysis as a means for preserving the nucleus of the classical reductionist program, even though it meant “bending the rules” somewhat. To Enriques's mind, qualitative analysis represented the affirmation of a synthetic, geometrical vision that would supplant the analytical/quantitative conception characteristic of 19th-century mathematics and mathematical physics. Here, we examine the two different answers given at the turn of the century to the question of the relationship between geometry and analysis and between mathematics, on the one hand, and mechanics and physics, on the other.Copyright 1998 Academic Press.Un'analisi delle diverse posizioni filosofiche e scientifiche di Henri Poincaré e Federigo Enriques nei riguardi dell'analisi qualitativa fornisce un'immagine complessa e interessante della “tensione essenziale” tra “tradizione” e “innovazione” nell'ambito della storia della scienza. In linea con il proprio paradigma scientifico, Poincaré vedeva nell'analisi qualitativa un mezzo per preservare il nucleo del programma riduzionista calssico, anche se cio comportava una lieve “distorsione delle regole”. Nella mente di Enriques, l'analisi qualitativa rappresentava l'affermazione di un punto di vista sintetico e geometrico che avrebbe soppiantato la concezione analitico-quantitativa caratteristica della matematica e della fisica matematica del 19° secolo. Il nostro scopo principale è di esaminare due diverse risposte date a cavallo del secolo alla questione dei rapporti tra geometria e analisi e tra matematica da un lato e meccanica e fisica dall'altro.Copyright 1998 Academic Press.AMS subject classification: 01A55     

Splitting into degrees with low computational strength

We investigate the extent to which a c.e. degree can be split into two smaller c.e. degrees which are computationally weak. In contrast to a result of Bickford and Mills that ${\mathbf{0}}^{\mathbf{\prime }}$ can be split into two superlow c.e. degrees, we construct a SJT-hard c.e. degree which is not the join of two superlow c.e. degrees. We also prove that every high c.e. degree is the join of two array computable c.e. degrees, and that not every high₂ c.e. degree can be split in this way. Finally we extend a result of Downey, Jockusch and Stob by showing that no totally ω-c.a. wtt-degree can be cupped to the complete wtt-degree.     

On the Density of the Pseudo-Isolated Degrees

A d.c.e. (2-computably enumerable) degree d is pseudo-isolatedif d itself is non-isolated (in the sense that no computablyenumerable (c.e.) degree below d can bound the c.e. degreesbelow d) and there is a d.c.e. degree b < d bounding allc.e. degrees below d. We prove in this paper that the pseudo-isolateddegrees are densely distributed in the c.e. degrees. 2000 MathematicsSubject Classification 03D25, 03D28.     

On the r.e. predecessors of d.r.e. degrees

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets) and R[d] be the class of less than d r.e. degrees in whichd is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e. degree defined by Lachlan for a d.r.e.set d is just the minimum degree in which D is r.e. Then we study for a given d.r.e. degree d class R[d] and show that there exists a d.r.e.d such that R d] has a minimum element 0. The most striking result of the paper is the existence of d.r.e. degrees for which R[d] consists of one element. Finally we prove that for some d.r.e. d R[d] can be the interval [a,b] for some r.e. degrees a,b, a b d. Received: 17 January 1996     

On a problem of Ishmukhametov

Given a d.c.e. degree d, consider the d.c.e. sets in d and the corresponding degrees of their Lachlan sets. Ishmukhametov provided a systematic investigation of such degrees, and proved that for a given d.c.e. degree d > 0, the class of its c.e. predecessors in which d is c.e., denoted as R[d], can consist of either just one element, or an interval of c.e. degrees. After this, Ishmukhametov asked whether there exists a d.c.e. degree d for which the class R[d] has no minimal element. We give a positive answer to this question.     

Sopra l'iperbolicità dei sistemi con vincoli e considerazioni sul superfluido e la magnetoidrodinamica

The symmetrization of quasi-linear systems with constraints and convex supplementary conservation law allows to solve the difficulties encountered in the search of the hyperbolicity of the superfluid and magnetohydrodynamics equations.

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Sopra l'iperbolicità dei sistemi con vincoli e considerazioni sul superfluido e la magnetoidrodinamica

Sunto Si nota come siano superate dalla simmetrizzazione dei sistemi con vincoli e legge supplementare convessa, le difficoltà sorte a proposito dell'iperbolicità delle equazioni del superfluido e della magnetoidrodinamica.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R. e M.P.I, contratto (40%) 20120201/83 A 218142055.     

Recursively enumerable bw-degrees

For every nonrecursive recursively enumerable (r.e.) set A are constructed bw-incomparable r.e. sets Bi, i ε N, such that B_i<_bwA and B_i ≡_wA. The existence of an infinite antichain of r.e. m-degrees in every nonrecursive r.e. bw-degree, and also that of an r.e. set A with the property aⁿn+1, n ε n, is proved.     

Joining to high degrees via noncuppables

Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low₂ c.e. degree to a nonlow₂ degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low₂ c.e. degree b. It is natural to ask in which subclass of low₂ c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results.     

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable functions f ₁(e, i), f ₂(e, i), g ₁(e, i), g ₂(e, i) such that for all e, i ∈ ω, (1) $ {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (2) $ {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (3) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \oplus {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}; $ (4) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset};{\text{and}} $ (5) $ {\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset}. $ It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.