Issue 3, Volume 41, September 2016

2016

1. Cartan-Fubini Type Extension Theorems Jun-Muk Hwang PDF

   ▸ Abstract

   Liouville’s theorem in conformal geometry can be generalized to extension problems of holomorphic maps preserving certain structures on Fano manifolds. The most typical result of this type is Cartan-Fubini type extension theorem proved by Mok and myself. We give an introduction to this circle of problems and survey some recent results.

   369-377
2. An Overview of Motivic Homotopy Theory Marc Levine PDF

   ▸ Abstract

   Motivic homotopy theory was constructed by Morel and Voevodsky in the 1990s. It led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture on the Galois symbol. Since then the theory has turned to a systematic study of its basic invariants and structures, as well as providing numerous new applications. Morel, Cisinski-Déglise, and Röndigs-Østvær have described the relation of the motivic stable homotopy category to Voevodskys triangulated category of motives. Morel’s computation of the endomorphism of the sphere spectrum points out a close relationship to quadratic forms, and has given rise to the construction of interesting new oriented cycle theories. Asok and Fasel have applied computations of unstable motivic homotopy groups to stability problems for algebraic vector bundles. Isaksen and others have computed motivic versions of Adams-Novikov and Adams spectral sequences, and used this information to improve known computations of these spectral sequences in classical homotopy theory. We will discuss the basic ideas going into the construction of motivic homotopy theory and some of these results and applications.

   379-407
3. Trace Formulas for Motivic Volumes François Loeser PDF

   ▸ Abstract

   We present some recent trace formulas for varieties over valued fields which can be seen as analogues of Grothendieck’s Lefschetz trace formula for varieties over finite fields. This involves motivic integration and non-archimedean geometry.

   409-424
4. On the Continuity of the Geometric Side of the Trace Formula Tobias Finis , Erez Lapid PDF

   ▸ Abstract

   We extend the geometric side of Arthur’s non-invariant trace formula for a reductive group G defined over $\mathbb Q$ continuously to a natural space $\mathcal {C}(G(\mathbb {A})^{1})$ of test functions which are not necessarily compactly supported. The analogous result for the spectral side was obtained in [10]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb {Q}): \gamma_1 ∼ \gamma_2$ if $\gamma_1$ and $\gamma_2$ are conjugate in $G(\overline {\mathbb {Q}})$ and their semisimple parts are conjugate in $G(\mathbb {Q})$. All terms in the resulting decomposition are continuous linear forms on the space $\mathcal {C}(G(\mathbb {A})^{1})$, and can be approximated (with continuous error terms) by naively truncated integrals.

   425-455
5. On the Automorphism Groups of Models in $\mathbb C^2$ Ninh Van Thu , Mai Anh Duc PDF

   ▸ Abstract

   n this article, we first give a characterization of models in $\mathbb C^2$ by their noncompact automorphism groups. Then we give an explicit description for automorphism groups of models in $\mathbb C^2$.

   457-470
6. Generalization of a Theorem of Clifford Phạm Ngọc Ánh PDF

   ▸ Abstract

   We prove that the multiplicative monoid of principal ideals partially ordered by reverse inclusion, called the divisibility theory, of a Bezout ring $R$ with one minimal prime ideal is a factor of the positive cone of a lattice-ordered abelian group by an appropriate filter if the localization of $R$ at its minimal prime ideal is not a field. This result extends a classical result of Clifford (Am. J. Math. 76:631–646, 1954) saying that the divisibility theory of a valuation ring is a Rees factor of the positive cone of a totally ordered abelian group and suggests to modify Kaplansky’s (later disproved) conjecture (Fuchs and Salce, Mathematical Surveys and Monographs 84, 2001) as to whether a Bezout ring whose localization at every minimal prime ideal has at least three ideals is the factor of an appropriate Bezout domain.

   471-480
7. Finite-Time Stability and $H_{\infty}$ Control of Linear Discrete-Time Delay Systems with Norm-Bounded Disturbances Le A. Tuan , Vu N. Phat PDF

   ▸ Abstract

   This paper deals with the finite-time stability and $H_{\infty}$ control of linear discrete-time delay systems. The system under consideration is subject to interval time-varying delay and norm-bounded disturbances. Linear matrix inequality approach is used to solve the finite-time stability problem. First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system with norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance. The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.

   481-493
8. Postulation of General Unions of Lines and Double Points in a Higher Dimensional Projective Space Edoardo Ballico PDF

   ▸ Abstract

   We prove that a general union $X\subset \mathbb {P}^{r}$, $r \geq 6$, of lines and double points has the expected postulation with respect to the hypersurfaces of degree $\geq 4$.

   495-504
9. An Introduction to Vector Variational Inequalities and Some New Results Nguyen Dong Yen PDF

   ▸ Abstract

   This survey paper discusses some new results on vector variational inequalities. It can serve as an elementary introduction to vector variational inequalities and vector optimization problems. The focus point is made on the results about connectedness structure of the solution sets, which are obtained by a scalarization method and properties of semi-algebraic sets. The first major theorem says that both Pareto solution set and weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. The second major theorem asserts that both proper Pareto solution set and weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian-Fromovitz Constraint Qualification is satisfied at every point of the constraint set. In addition, it has been established that if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are shown.

   505-529
10. oundedness and Compactness of Weighted Composition Operators on Fock Spaces $\mathcal {F}^{p}(\mathbb {C})$ Pham Viet Hai , Le Hai Khoi PDF

    ▸ Abstract

    In this paper, we study weighted composition operators on the Fock spaces which are non-Hilbert spaces. Criteria for boundedness and compactness of such operators are obtained. Our results contain the corresponding results for Fock spaces which are Hilbert ones, as particular cases.

    531-537