Issue 4, Volume 40, December 2015

2015

1. Eupolars and their Bialternality Grid Jean Ecalle PDF

   ▸ Abstract

   This monograph is almost entirely devoted to the flexion structure generated by a flexion unit $\mathfrak{E}$ or the conjugate unit $\mathfrak{O}$, with special emphasis on the polar specialization of the units ({``eupolar structure''}).

    (i) We first state and prove the main facts (some of them new) about the central pairs of {bisymmetrals} $\mathrm{pal^\bullet/pil^\bullet}$ and $\mathrm{par^\bullet/pir^\bullet}$ and their even/odd factors, by relating these to four remarkable series of {alternals} $\{\mathfrak{re}_r^\bullet\}$, $\{\mathfrak{le}_r^\bullet\}$, $\{\mathfrak{he}_r^\bullet\}$, $\{\mathfrak{ke}_{2r}^\bullet\}$, and that too in a way that treats the swappees $\mathrm{pal}^\bullet$ and $\mathrm{pil}^\bullet$ (resp. $\mathrm{par}^\bullet$ and $\mathrm{pir}^\bullet$) as they should be treated, i.e. on a strictly equal footing.

    (ii) Next, we derive from the central bisymmetrals two series of {bialternals}, distinct yet partially (and rather mysteriously) related.

    (iii) Then, as a first step towards a complete description of the eupolar structure, we introduce the notion of {bialternality grid} and present some facts and conjectures suggested by our (still ongoing) computations.

   (iv) Lastly, two complementary sections have been added, to show which features of the eupolar structure survive, change form or altogether diappear when one moves on to the next two cases in order of importance: eutrigonometric and polynomial.

   545-636
2. Pullback attractors for strong solutions of 2d non-autonomous $g$-Navier-Stokes equations Dao Trong Quyet PDF

   ▸ Abstract

   Considered here is the first initial boundary value problem for the 2D non-autonomous $g$-Navier-Stokes equations in bounded domains. We prove the existence of a pullback attractor in $V_g$ for the continuous process generated by strong solutions to the problem. We also prove the exponential growth in $V_g$ and in $H^2(\Omega ,g)$ for the pullback attractor, when time goes to $-\infty $.

   637-651
3. Solvability of Some Classes of Systems of Dual Integral Equations Involving Fourier Transforms Nguyen Van Ngoc , Ha Tien Ngoan , Nguyen Thi Ngan PDF

   ▸ Abstract

   The aim of the present work is to consider the existence and uniqueness problems for some classes of general systems of dual integral equations involving Fourier transforms of generalized functions, which are a generalization of some systems of dual equations encountered in the mixed boundary value problems of mathematical physics and contact problems of elasticity.

   653-669
4. Disjunctive Ideals of Distributive Lattices M. Sambasiva Rao PDF

   ▸ Abstract

   Disjunctive ideals and strongly disjunctive ideals are introduced in lattices and normal lattices are also characterized in terms of annihilators. Some properties of disjunctive ideals are studied. The concept of normal prime ideals is introduced and their properties are studied. Some equivalent conditions are derived for the class of all strongly disjunctive ideals to become a sublattice of the ideal lattice.

   671-682
5. The Endomorphism Ring of a Square-free Injective Module Mai Hoang Bien PDF

   ▸ Abstract

   Let $M_R$ be an injective right module over a ring $R$. The goal of this paper is to prove that the endomorphism ring $S=\mathrm{End}(M_R)$ of $M_R$ is quasi-duo if and only if $M_R$ is square-free.

   683-687
6. Fractional Functional with two Occurrences of Integrals and Asymptotic Optimal Change of Drift in the Black-Scholes Model R. A. El-Nabulsi PDF

   ▸ Abstract

   The fractional action-like variational approach for the case of functional with two occurrences of fractional action integrals is constructed. Our motivation is based on the fact that a functional with two occurrences of integrals has many advantages in optimization problems and financial engineering issues. After deriving the corresponding fractional Euler-Lagrange equation and discussing some examples, we addressed the problem of Asian call option. We derived the asymptotically optimal change of fractional drift for the geometric average Asian call option and gave some new consequences.

   689-703
7. A Maschke Type Theorem forWeak Relative Hopf $\pi$-Modules Shuangjian Guo PDF

   ▸ Abstract

   Let $H$ be a weak Hopf $\pi$-coalgebra and let $A$ be a weak right $\pi$-$H$-comodule algebra with a total integral $\phi$. This paper is mainly devoted to a Maschke-type theorem for $(H, A)$-weak relative Hopf $\pi$-modules. As an application, we obtain a characterization for an $(H, A)$-weak Hopf $\pi$-module to be projective as an $A$-module.

   705-713
8. Jain-Baskakov Operators and its Different Generalization Prashantkumar Patel , Vishnu Narayan Mishra PDF

   ▸ Abstract

   The aim of this work is to introduce a new sequence of positive linear operators, which is an integral version of Jain operators. We studied some direct results in ordinary approximation. Weighted approximation, rate of convergence, and $A$-statistical convergence are investigated. Also, we discussed approximation properties of King-type and Stancu-type generalization of these new sequences of linear positive operators and investigated a modification of Jain-Baskakov operators with parameter $c$. At the end, we propose a $q$-analogue of Jain operators based on a $q$-integer.

   715-733
9. The Viscosity Subdifferential of the Rank Function via the Corresponding Subdifferential of its Moreau Envelopes Jean-Baptiste Hiriart-Urruty , Hai Yen Le PDF

   ▸ Abstract

   We derive the so-called viscosity subdifferential of the rank function via a limiting process applied to the Moreau envelopes of the rank function. Before that, we obtain the explicit expressions of all the generalized subdifferentials of the Moreau envelopes of the rank function.

   735-746
10. Erratum to: The Backward Problem for Ginzburg-Landau-Type Equation Dang Duc Trong , Bui Thanh Duy , Nguyen Dang Minh PDF

    747-747