On some additive mappings in rings with involution

Summary LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax ^* – xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx ^* =x ^* x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx R, forcesR to be commutative.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

The algebraic structure of the isomorphic types of tally,polynomial time computable sets

 We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P _T into six parts: D ⁻, D^ ⁻ , C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets. 1. is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in can be distinguished by first order formulas. 3. There exist infinitely many nontrivial automorphisms for . 4. is not distributive, but any interval in is a countable distributive lattice. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID="ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID="ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies. Received: 23 July 1999 / Published online: 27 March 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=" ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID=" ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.     

Monotonicity and the Expressibility of NP Operators

We investigate why similar extensions of first-order logic using operators (that is, generalized quantifiers) corresponding to NP-complete decision problems apparently differ in expressibility: the logics capture either NP or L^NP. It had been conjectured that the complexity class captured is NP if and only if the operator is monotone. We show that this conjecture is false. However, we provide evidence supporting a revised conjecture involving finite variations of monotone problems. Mathematics Subject Classification: 68Q15, 03D15, 03C13.     

The metamathematics of scattered linear orderings

Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II ₁ ¹ -CA₀ is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR₀ is equivalent to a version of the Continuum Hypothesis for linear orderings, which states that every countable linear ordering either has countably many or continuum many Dedekind cuts.Research partially supported by NSF grant # DCR-8606165. AMS Subject Classification 03F35, 03F15, 03D55     

Cupping the Recursively Enumerable Degrees by D.R.E. Degrees

We prove that there are two incomplete d.r.e. degrees (the Turingdegrees of differences of two recursively enumerable sets) suchthat every non-zero recursively enumerable degree cups at leastone of them to 0', the greatest recursively enumerable (Turing)degree. 1991 Mathematics Subject Classification: primary 03D25,03D30; secondary 03D35.     

Specialising Aronszajn trees by countable approximations

We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.The first author was partially supported by a Minerva fellowship. The second author's research was partially supported by the ``Israel Science Foundation', founded by the Israel Academy of Science and Humanities. This is the second author's work number 778. Mathematics Subject Classification (2000): 03E15, 03E17, 03E35, 03D65     

A Conjecture Concerning the Spectrum of a Sentence

We give a plausible-sounding conjecture involving the number of n-equivalence classes of structures of size m which would imply that the complement of a spectrum is also a spectrum. Mathematics Subject Classification : 03B10, 03D15, 68Q15.     

Optimal Domains for <Emphasis Type="Italic">L</Emphasis><Superscript>0</Superscript>-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L⁰([0, 1]), where E is an F–function space and L⁰([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions. Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).     

The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic

By RCA₀, we denote a subsystem of second order arithmetic based on ⁰₁ comprehension and ⁰₁ induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilberts Nullstellensatz in RCA₀.Mathematics Subject Classification (2000): Primary 03F35; Secondary 03B30, 12D10     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order

Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iu_x  +  u_tt + (|u|^p + |u|^2p)u + u_xt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.     

A transference theorem for hermite expansions

A transference theorem for multipliers of Hermite expansions is proved. The result allows to transfer weightedL ²(ℝ ⁿ ) estimates from lower to higher dimensions. Research of the author supported by grant BFM2003-06335-603-03 of the D.G.I..     

On multilinear operators commuting with Lie derivatives

LetE ₁, ...,E _k andE be natural vector bundles defined over the categoryMf _m ⁺ of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm2. LetM be an object ofMf _m ⁺ which is connected. We give a complete classification of all separately continuousk-linear operatorsD : _c(E ₁ M) × ... × _c(E _k M) (EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy

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.     

Polynomial Representations via Spectral Decompositions

We show using functional analysis methods, that under some conditions, such as the positivity of a homogeneous polynomial, one can give a representation of polynomials which are positive on semi-algebraic sets, allowing irrational fractions with denominators of the form (1+t² ₁) _1/2 ...(1+t ² _n) _n/2, _iZ₊, i = 1, ..., n     

A cohesive set which is not high

We study the degrees of unsolvability of sets which are cohesive (or have weaker recursion-theoretic “smallness” properties). We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55.     

On End-Extensions of Models of ¬exp

Every model of IΔ₀ is the tally part of a model of the stringlanguage theory Th-FO (a main feature of which consists in having induction on notation restricted to certain AC⁰. sets). We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ₀ + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models. Mathematics Subject Classification: 03F30, 03C62, 68Q15.     

A Tauberian Theorem for Ergodic Averages,Spectral Decomposability,and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (,) is a -finite measure space, and 1 < p < . Let T:L^p( L ^p() be a bounded invertible linear operator such that T and T ^–1 are positive. Denote by _n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {T^j+T^–j}_{j =1} . Martín-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of E_n(T)_n=1 ; (b) a uniform A _p p estimate in terms of discrete weights generated by the pointwise action on of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {T^j+T^–j}_j=1 already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform A _p weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages E_n(T)_{n = 1} .