On the Consistency of a Positive Theory

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK_∞⁺. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK_∞⁺ interprets the Kelley Morse class theory. Here we prove that GPK_∞⁺ + AC_WF (AC_WF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK_∞⁺ + AC_WF is a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals.     

Special Model Axiom in Nonstandard Set Theory

We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawai's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency (with ZFC) of the existence of a k⁺ like k-saturated model of PA for a given cardinal k.     

Inconsistency of GPK + AFA

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X₁ introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X₁ in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. Mathematics Subject Classification: 03E70, 03E10.     

RADICAL THEORY FOR SEMIRINGS

Abstract

In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ?_∞(S) and ?^ε(S) and two similar ideals β_∞ (S) and β^ε(S) is widely solved. We prove ?_∞(S) ? ?^ε(S) = β^∞(S) = β^ε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M_ε(U) is always semisimple, a result which is not true for the special class M_∞(U).     

One Initial Boundary-Value Problem for Integro-Differential Equation of the Second Order With Power Nonlinearity

In the Sobolev space W _∞ ² (?⁺) we investigate one initial boundary-value problem for integro-differential equation of the second order with power nonlinearity on a semi-axis. Assuming that summary-difference even kernel serves for the considered kernel as minorant in the sense of M. A. Krasnosel’skii, we prove the existence of a nonnegative (nontrivial) solution in the Sobolev spaceW _∞ ² (?⁺). We also calculate the limits of constructed solution at infinity.     

The Set of Better Quasi Orderings is ∏

In this paper we give a proof of the II¹₂-completeness of the set of countable better quasi orderings (viewed as a subset of the Cantor space). This result was conjectured by Clote in [2] and proved by the author in his Ph.d. thesis [6] (see also [7]). Here we prove it using Simpson's definition of better quasi ordering ([15]) and as little bqo theory as possible.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ε-approximations

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (?⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω??³. Bibliography: 25 titles.     

Perturbation Theory for the Two-Particle Schrodinger Operator on a One-Dimensional Lattice

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues z_m(π) = v(m), m ∈ ℤ₊. If the potential v increases on ℤ₊, then only the eigenvalue z₀(π) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues z_m(π), m ∈ ℕ, the operator H(π) splits into two nondegenerate eigenvalues z _m ⁻ (k) and z _m ⁺ (k) under small variations of k ∈ (π − δ, π). We show that z _m ⁻ (k) < z _m ⁺ (k) and obtain an estimate for z _m ⁺ (k) − z _m ⁻ (k) for k ∈ (π − δ, π). The eigenvalues z₀(k) and z ₁ ⁻ (k) increase on [π − δ, π]. If (Δv)(m) > 0, then z _m ^± (k) for m ≥ 2 also has this property. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 212–220, November, 2005.     

Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition. For a Young function A we introduce the notion of L^A-Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L^p(w) norm of the operator by the L^p(w) norm of a maximal function associated to the complementary function of A, for any weight w in the A_∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L^p(w) norm of the operator by the L^p(w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A⁺ _∞ class.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

The Liouville Field Theory Zero-Mode Problem

We quantize the canonical free-field zero modes p, q on the half-plane p > 0 for both Liouville field theory and its reduced Liouville particle dynamics. We describe the particle dynamics in detail, calculate one-point functions of particle vertex operators, deduce their zero-mode realization on the half-plane, and prove that the particle vertex operators act self-adjointly on the Hilbert space L ²(₊) because of symmetries generated by the S-matrix. Similarly, we obtain the self-adjointness of the corresponding Liouville field theory vertex operator in the zero-mode sector by applying the Liouville reflection amplitude, which is derived by the operator method.     

On the basis property of a trigonometric system arising in the Frankl problem

We consider the function system {cos4nθ} _n=0 ^∞ , {sin(4n ? 1)θ} _n=1 ^∞ , which arises in the Frankl problem in the theory of elliptic-hyperbolic equations. We show that this system is a Riesz basis in the space L ₂(0, π/2) and construct the biorthogonal system.     

Existence and uniqueness of C0-semigroup in L∞: a new topological approachExistence et unicité de C0-semigroupe sur L∞

A sub-Markov semigroup in L^∞ is in general not strongly continuous with respect to the norm topology. We introduce a new topology on L^∞ for which the usual sub-Markov semigroups in the literature become C₀-semigroups. This is realized by a natural extension of the Phillips theorem about dual semigroup. A simplified Hille–Yosida theorem is furnished. Moreover this new topological approach will allow us to introduce the notion of L^∞-uniqueness of pre-generator. We present several important pre-generators for which we can prove their L^∞-uniqueness. To cite this article: L. Wu, Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 699–704.     

On the degree theory for general mappings of monotone type

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type f+T, where f is a mapping of class ₊(S) from a bounded open set Ω in a reflexive Banach space X into its dual X^∗, and T is a maximal monotone mapping from X into X^∗. This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type f+T. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings.     

On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

Cosimplicial DGLAs in Deformation Theory

We identify ?ech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L _∞-algebra. Applications to deformation theory are described.     

On levi–typ conditions for hypoellipticity of certain diffrential operators

The hypoelliticity is discussed for operators of the form P=D² _x+a(x)D² _y+b(x)D_ywhere a (x) and b (x) are real–valued C^∞ functions satisfying a(0)=0 and a(x) >0 for x≠0.We seek the conditions for P to be hypoelliptic, especially in the case where both a (x) and b(x) vanish to infinite order on x=0.