A Decomposition of a Weaker Form of Continuity

The aim of this paper is to give a decomposition of a weaker form of continuity, namely gα^**-continuity by providing the concepts of K-sets and K-continuity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions

In this paper, the η-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function η is extended. In this method, a so-called η-approximated optimization problem associated with the original mathematical programming problems is constructed; moreover, an η-saddle point and an η-Lagrange function are defined. By the help of the constructed η-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

A note on δ-I-sets

We characterize δ-I-sets, discuss the relation between δ-sets, δ-I-sets and I _δ-sets and generalize some of the results.     

Decompositions of continuity and complete continuity

We introduce the notions of δ-t-sets, δ_β-t-sets, δ-B-continuity and δ_β -B-continuity and obtain decompositions of continuity and complete continuity.     

On weak neighborhood systems and spaces

We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s. This work was supported by a grant from Research Institute for Basic Science at Kangwon National University.     

Generalized (η,ρ)-Invex Functions and Semiparametric Duality Models for Multiobjective Fractional Programming Problems Containing Arbitrary Norms

In this paper, we construct several semiparametric duality models and prove appropriate duality theorems under various generalized (η,ρ)-invexity assumptions for a multiobjective fractional programming problem involving arbitrary norms.     

Generalized (η,ρ)-Invex Functions and Global Semiparametric Sufficient Efficiency Conditions for Multiobjective Fractional Programming Problems Containing Arbitrary Norms

The purpose of this paper is to develop a fairly large number of sets of global semiparametric sufficient efficiency conditions under various generalized (η, ρ)-invexity assumptions for a multiobjective fractional programming problem involving arbitrary norms.     

ζ μ -sets in generalized topological spaces

We define ζ _μ -sets, (ζ,μ)-closed sets and generalized ζ _μ -sets in a generalized topological space and investigate properties of several low separation axioms of generalized topologies constructed by the families of these sets. Characterizations of some properties of (ζ,μ)-R ₀ and (ζ,μ)-R ₁ generalized topological spaces will be given.     

The incidence class and the hierarchy of orbits

R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.      

Second order duality for minmax fractional programming

In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems. The research of Z. Husain is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral Fellowship Program No. 40/9/2005-R&D II/1739.     

Weakly Θ-precontinuous functions

A weak form of strong Θ-precontinuity, called weak Θ-precontinuity, is introduced. It is shown that weak Θ-precontinuity is strictly weaker than strong Θ-precontinuity and that it is strictly between continuity and almost weak continuity. Weak Θ-continuity is used to improve a result in the literature concerning the graph of a strongly Θ-precontinuous function. Also additional properties of these functions are investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Projective version of Malcolmson’s criterion

It is proven that the extension of Malcolmson’s methods to reppresent all elements ofR _ζ is not purely straightforward. A detailed proof of such extension is here furnished.

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Sunto Si prova che l’estensione dei metodi di Malcolmson per la rappresentazione degli elementi diR _ζ a tutte le applicazioni tra moduli proiettivi indotti non è puramente meccanica. è fornita una dimostrazione dettagliata di detta estensione.

     

Metacyclic groups and modular forms

In the paper we find the metacyclic groups of the form 〈a, b:a ^m=e, b ^s=e, b ⁻¹ ab=a ^r〉, wherem=3, 4, 5, 7, 11, 23, such that the modular forms associated with all elements of these groups by some faithful representation are multiplicative η-products. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 163–173, February, 2000.     

η-Invariant and Flat Vector Bundles

Abstract We present an alternate definition of the mod Z component of the Atiyah-Patodi-Singer η invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer. * Project supported by the Cheung-Kong Scholarship of the Ministry of Education of China and the 973 Project of the Ministry of Science and Technology of China. (Dedicated to the memory of Shiing-Shen Chern)     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Asymptotic expansions in limit theorems for stochastic processes. I

Summary. For some families of locally infinitely divisible Markov processes η ^ɛ (t), 0≦ t≦ T, with frequent small jumps, limit theorems for expectations of functionals F(η ^ɛ [0,T]) are proved of the form | E ^ɛ F(η ^ɛ [0,T])−E ⁰ F(η ⁰ [0,T])|≦ const ⋅ k(ɛ) , E ^ɛ F(η ^ɛ [0,T])=E ⁰ [F(η ⁰ [0,T])+ k(ɛ) ⋅ A ₁ F(η ⁰ [0,T])]+o(k(ɛ))  (ɛ↓ 0) , where A ₁ is a linear differential operator acting on functionals, and the constant is expressed in terms of the local characteristics of the processes η ^ɛ (t) and the norms of the derivatives of the functional F. Received: 1 April 1994 / In revised form: 30 September 1995     

The <Emphasis Type="Italic">p</Emphasis>-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with (<Emphasis Type="Italic">H</Emphasis>,η)-monotone operators

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (H,η)-monotone operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (H,η)-monotone operators, we prove the existence of solutions and the convergence of some new p-step iterative algorithms for this system of generalized mixed quasi-variational inclusions and its special cases. The results in this paper unifies, extends and improves some known results in the literature.     

Contact metric manifolds with <Emphasis Type="Italic">η</Emphasis>-parallel torsion tensor

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E ⁿ⁺¹ × S ⁿ (4) in higher dimensions.