On the η‐function for bisingular pseudodifferential operators

In this work we consider the η‐invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the η‐function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the K‐theory of the algebra of 0‐order (global) bisingular operators. With these preparations we establish the regularity properties of the η‐function at the origin for global bisingular operators which are self‐adjoint, elliptic and of positive orders.     

Increasing η ‐representable degrees

In this paper we prove that any Δ₃⁰ degree has an increasing η ‐representation. Therefore, there is an increasing η ‐representable set without a strong η ‐representation (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Feasible Operations and Applicative Theories Based on λη

We study a theory PTO(λη) of polynomial time computability on the type of binary strings (in the style of [6]), as embedded in full lambda calculus with total application and extensionality. We prove that the closed terms of (provable) type W → W are exactly the polynomial time operations. This answers a conjecture of Strahm [13].     

The Hankel Convolution and the Zemanian Spaces βμ and β'μ

In this paper we give structure theorems for the elements of the Zemanian spaces β'_μ and β'_μ,a' Also, bounded sets and convergent sequences in β'_μ,a' are characterized through representations as derivatives of measurable functions. Finally, we analyze the Hankel convolution on the above spaces.     

Automorphisms of η‐like computable linear orderings and Kierstead's conjecture

We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η‐like computable linear ordering , such that has no interval of order type η, and such that the order type of is determined by a ‐limitwise monotonic maximal block function, there exists computable such that has no nontrivial automorphism.     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

A Short Proof That χ Can be Bounded ε Away from Δ + 1 toward ω

In 1998 the second author proved that there is an such that every graph satisfies . The first author recently proved that any graph satisfying contains a stable set intersecting every maximum clique. In this note, we exploit the latter result to give a much shorter, simpler proof of the former. Working from first principles, we omit only some five pages of proofs of known intermediate results (which appear in an extended version of this paper), and the proofs of Hall's Theorem, Brooks' Theorem, the Lovász Local Lemma, and Talagrand's Inequality.     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

The univalence of α‐project starlike functions

We solve the univalence problem in the class of α‐project starlike function. The α‐project close‐to‐convex function, α‐project Bazilevi? and α‐project Φ‐like functions are also considered here.     

The spectrum of cyclic BSA(ν, 3, λ; α) with α = 2, 3

Balanced sampling plans excluding contiguous units (or BSEC) were first introduced by Hedayat, Rao, and Stufken in 1988. In this paper, we generalize the concept of a cyclic BSEC to a cyclic balanced sampling plan to avoid the selection of adjacent units (or CBSA for short) and use Langford and extended Langford sequences to construct a cyclic BSA(ν, 3, λ; α) with α = 2, 3. We finally establish the necessary and sufficient conditions for the existence of a cyclic BSA(ν, 3, λ; α) where α = 2, 3. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].     

On the Distributions δk and (δ')k

Powers and products of distributions have not as yet been defined to hold true in general. In this paper, we choose a fixed δ-sequence and use the concept of neutrix limit to give meaning to the distributions δ^k and (δ')^k for some k. These may be regarded as powers of Dirac delta functions.     

The Möbius function and the residue theorem

A classical conjecture of Bouniakowsky says that a non-constant irreducible polynomial in Z[T] has infinitely many prime values unless there is a local obstruction. Replacing Z[T] with κ[u][T], where κ is a finite field, the obvious analogue of Bouniakowsky's conjecture is false. All known counterexamples can be explained by a new obstruction, and this obstruction can be used to fix the conjecture. The situation is more subtle in characteristic 2 than in odd characteristic. Here, we illustrate the general theory for characteristic 2 in some examples.     

The Distributional Hankel Transform of δ(k)(m2×P

In this note we give a sense to certain kinds of n-dimensional distributional Hankel transforms of the Dirac measure δ^(k)(m² + P). The most important result is the interchange formula between the product and the convolution of the Hankel transform of δ^(k)(m² + P).