Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups

We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” ∂(a)=m satisfies

Wiener–Ikehara theorems and the Beurling generalized primes

The Wiener–Ikehara theorem is classical. Here we prove a converse of the theorem with “if and only if” conditions. We also extend this theorem to cover upper estimates and lower estimates respectively. Significant applications of these theorems to the study of the Beurling generalized primes are introduced too.     

Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem

We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups. In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for a class of Gaussians on compact Gelfand pairs. This author was supported by the European Commission (TMR 1998–2001 Network Harmonic Analysis).     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

Robinson’s implicit function theorem and its extensions

S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and format. Here, Robinson’s theorem is viewed from the perspective of more recent developments in variational analysis as well as some lesser-known results in the implicit function literature on equations, prior to the advent of generalized equations. Extensions are presented which fully cover such results, translating them at the same time to generalized equations broader than variational inequalities. Robinson’s notion of first-order approximations in the absence of differentiability is utilized in part, but even looser forms of approximation are shown to furnish significant information about solutions.     

Characterization of tail distributions based on record values by using the Beurling’s Tauberian theorem

In this paper, we mainly investigate the converse of a well-known theorem proved by Shorrock (J. Appl. Prob. 9, 316–326 1972b), which states that the regular variation of tail distribution implies a non-degenerate limit for the ratios of the record values. Specifically, the converse is proved by using Beurling extension of Wiener’s Tauberian theorem. This equivalence is extended to the Weibull and Gumbel max-domains of attraction.     

Remarks on orthocenters,Pappus’ theorem and Butterfly theorems

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.     

Note on Besicovitch’s theorem on the possible values of upper and lower derivatives

LetB ₁, ..., B _k be Busemann-Feller and regular differential bases composed of intervals of the corresponding dimensions. It is proved that if B ₁, ...,B _k satisfy a certain condition (called the completeness condition), then, for their Cartesian product B ₁ × ... × B _k , an analog of Besicovitch’s theorem on the possible values of strong upper and lower derivatives is valid.     

Irreducibility and extensions of Ostrowski’s Theorem

In this paper an extension of Ostrowski’s Theorem for complex square irreducible matrices is presented. Also extensions of similar statements for square complex matrices are analyzed and completed. Most of the statements in this work cover also the case of reducible matrices.     

A new method of proof of Filippov’s theorem based on the viability theorem

Filippov??s theorem implies that, given an absolutely continuous function y: [t ₀; T] ?? ? ^d and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x ₀, there exists a solution x(·) to the differential inclusion x??(t) ?? F(t, x(t)) starting from x ₀ at the time t ₀ and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function ??(·) is the estimation of dist(y??(t), F(t, y(t))) ?? ??(t). Setting P(t) = {x ?? ? ⁿ : |x ?y(t)| ?? r(t)}, we may formulate the conclusion in Filippov??s theorem as x(t) ?? P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ?? DP(t, x)(1) ?? ?. It allows to obtain Filippov??s theorem from a viability result for tubes.