Acta Mathematica Vietnamica
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Issue 2, Volume 44, June 2019
2019
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Negative Index Materials: Some Mathematical Perspectives Hoai-Minh Nguyen PDF325-349
▸ Abstract
Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones. -
Relative Functoriality and Functional Equations via Trace Formulas Yiannis Sakellaridis PDF351-389
▸ Abstract
Langlands’ functoriality principle predicts deep relations between the local and automorphic spectra of different reductive groups. This has been generalized by the relative Langlands program to include spherical varieties, among which reductive groups are special cases. In the philosophy of Langlands’ “beyond endoscopy” program, these relations should be expressed as comparisons between different trace formulas, with the insertion of appropriate $L$-functions. The insertion of $L$-functions calls for one more goal to be achieved: the study of their functional equations via trace formulas.
The goal of this article is to demonstrate this program through examples, indicating a local-to-global approach as in the project of endoscopy. Here, scalar transfer factors are replaced by “transfer operators” or “Hankel transforms” which are nice enough (typically, expressible in terms of usual Fourier transforms) that they can be used, in principle, to prove global comparisons (in the form of Poisson summation formulas). Some of these examples have already appeared in the literature; for others, the proofs will appear elsewhere.
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On Relative Trace Formulae: the Case of Jacquet-Rallis Pierre-Henri Chaudouard PDF391-430
▸ Abstract
We give an account of recent works on Jacquet-Rallis’ approach to the Gan-Gross-Prasad conjecture for unitary groups. We report on the present state of the Jacquet-Rallis relative trace formulae and on some current applications of it. We give also a precise computation of the constant that appears in the statement “Fourier transform and transfer commute up to a constant”.
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Second-Order Optimality Conditions and Solution Stability to Optimal Control Problems Governed by Stationary Navier-Stokes Equations Bui Trong Kien PDF431-448
▸ Abstract
In this paper, a class of parametric optimal control problems governed by stationary Navier-Stokes equations with mixed pointwise constraints is considered. We give no-gap second-order necessary and sufficient conditions for unperturbed problem. We show that if the strictly second-order sufficient condition for unperturbed problem is valid and the objective function is locally Lipschitz continuous, then the solution map is locally upper Hölder continuous at the reference parameter. -
Zeros Sets of $H^p$ Functions in Lineally Convex Domains of Finite Type in $\mathbb C^n$ Philippe Charpentier , Y. Dupain PDF449-467
▸ Abstract
In this note, we extend N. Th. Varopoulos result on zero sets of $H^p$ functions of strictly pseudo-convex domains in $\mathbb C^n$ to lineally convex domains of finite type. -
Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations I: the $d$-Concavity and the Comparison Principle Ha Tien Ngoan , Thai Thi Kim Chung PDF469-491
▸ Abstract
We introduce the notion of $d$-concavity, $d \geq 0$, and prove that the nonsymmetric Monge-Ampère type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Ampère type equations in the case when they are so-called $\delta$-elliptic with respect to compared functions with $0 \leq \delta < 1$. -
On Chromatic Numbers of Two Extensions of Planar Graphs Khosro Tajbakhsh PDF493-500
▸ Abstract
In this paper, the acyclic chromatic and the circular list chromatic numbers of a simple $H$-minor free graph $G$ is considered, where $H \in\{K_5,K_{3,3}\}$. It is proved that the acyclic chromatic number (resp. the circular list chromatic number) of a simple $H$-minor free graph $G$ where $H \in \{K_5,K_{3,3}\}$ is at most 5 (resp. at most 8) and we conclude that $G$ is star 20-colorable. These results generalize the same known results on planar graphs. Moreover, some upper bounds for the coloring numbers of $H$-minor free graphs for $H\in \{K_5,K_{3,3},K_{r,s}\}$ and $r \leq 2$ are obtained. These results generalize some known results and give some new results on group choice number, group chromatic number, and the choice number of the mentioned graphs with much shorter proofs.
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The Einstein-Vlasov-Scalar Field System with Gowdy or $T^2$ Symmetry in Contracting Direction Alex Lassiye Tchuani , David Tegankong , Norbert Noutchegueme PDF501-517
▸ Abstract
We prove in the case of cosmological models for the Einstein-Vlasov-scalar field system with Gowdy symmetry, that the solutions exist globally in the past. The sources of the equations are generated by a distribution function and a scalar field, subject to the Vlasov and the wave equations respectively. The result is generalized for the case of $T^2$ symmetry. Using previous results, we deduce geodesic completeness. -
On Hölder Estimates with Loss of Order One for the $\bar {\partial }$ Equation on a Class of Convex Domains of Infinite Type in $\mathbb {C}^{3}$ Ly Kim Ha PDF519-530
▸ Abstract
In this paper, we establish a Hölder continuity with loss of order one for the Cauchy-Riemann equation on a class of smoothly bounded, convex domains of infinite type in the sense of Range in $\mathbb {C}^{3}$. Let $\Omega$ be such a domain and let $\varphi$ be a $(0,1)$-form defined continuously on $\bar {\Omega }$. Then, if $\varphi$ is Lipschitz continuity on $b\Omega$, in the sense of distributions, there exists a function $u$ belonging to a “suitable” Hölder class such that $$\bar{\partial} u=\varphi \quad \text{ in } {\Omega}.$$
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A Two-Step Extragradient-Viscosity Method for Variational Inequalities and Fixed Point Problems Dang Van Hieu , Dang Xuan Son , Pham Ky Anh , Le Dung Muu PDF531-552
▸ Abstract
This paper develops a two-step extragradient-viscosity method for finding a common solution to a variational inequality problem and a fixed point problem in a Hilbert space. Combining the two-step extragradient algorithm and the viscosity one, we propose a strongly convergent method, which contains as special cases some existing algorithms. Several numerical experiments are implemented to demonstrate the performance of the proposed method in comparison with classical hybrid extragradient-like algorithms. -
On Weakly Locally Finite Division Rings Trinh Thanh Deo , Mai Hoang Bien , Bui Xuan Hai PDF553-569
▸ Abstract
Weakly locally finite division rings were considered in Deo et al. (J. Algebra 365, 42–49, 2012), where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer $n \geq 0$ or $n = \infty$, we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is $n$. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well-known Kurosh Problem and with one of Herstein’s conjectures.