A protective layer on As2S3 film for photo-resist patterning

We have developed an effective fabrication process for As₂S₃ planar waveguides. A bottom anti-reflection coating provides a layer that not only protects the As₂S₃ film from attack by the alkaline developer but also improves line edge roughness of photo-resist patterns which is a pre-requisite for waveguides with smooth sidewalls.     

Development of arbitrary-order multioperators-based schemes for parallel calculations.: Part 2: Families of compact approximations with two-diagonal inversions and related multioperators

One-parameter families of compact approximations to grid functionals with inverses of two-point operators and their properties are described. As particular examples, interpolation/extrapolations operators, quadratures formulas and approximations to derivatives are presented. Using operators from the families with fixed parameters values as basis operators, their linear combinations providing formally arbitrary-order approximations (multioperators) are constructed. Numerical illustrations are presented. Special emphasis is placed on first derivatives discretizations in the context of conservation laws. As an example, a highly accurate tenth-order scheme is outlined and tested against the Burgers’ equation. It is shown how extrapolation multioperators can be used to create boundary closures.     

A Double-Parameter Scaling Broyden–Fletcher–Goldfarb–Shanno Method Based on Minimizing the Measure Function of Byrd and Nocedal for Unconstrained Optimization

In this paper, the first two terms on the right-hand side of the Broyden–Fletcher–Goldfarb–Shanno update are scaled with a positive parameter, while the third one is also scaled with another positive parameter. These scaling parameters are determined by minimizing the measure function introduced by Byrd and Nocedal (SIAM J Numer Anal 26:727–739, 1989). The obtained algorithm is close to the algorithm based on clustering the eigenvalues of the Broyden–Fletcher–Goldfarb–Shanno approximation of the Hessian and on shifting its large eigenvalues to the left, but it is not superior to it. Under classical assumptions, the convergence is proved by using the trace and the determinant of the iteration matrix. By using a set of 80 unconstrained optimization test problems, it is proved that the algorithm minimizing the measure function of Byrd and Nocedal is more efficient and more robust than some other scaling Broyden–Fletcher–Goldfarb–Shanno algorithms, including the variants of Biggs (J Inst Math Appl 12:337–338, 1973), Yuan (IMA J Numer Anal 11:325–332, 1991), Oren and Luenberger (Manag Sci 20:845–862, 1974) and of Nocedal and Yuan (Math Program 61:19–37, 1993). However, it is less efficient than the algorithms based on clustering the eigenvalues of the iteration matrix and on shifting its large eigenvalues to the left, as shown by Andrei (J Comput Appl Math 332:26–44, 2018, Numer Algorithms 77:413–432, 2018).     

Algebraic multigrid preconditioning of the Hessian in optimization constrained by a partial differential equation

We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear‐quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation, thus allowing for it to be implemented using a variety of AMG methods and standard packages. Our analysis establishes a clear connection between the quality of the preconditioner and the AMG method used. The proposed strategy has a broad and robust applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and several numerical examples are presented.     

Biased random walks

How much can an imperfect source of randomness affect an algorithm? We examine several simple questions of this type concerning the long-term behavior of a random walk on a finite graph. In our setup, at each step of the random walk a “controller” can, with a certain small probability, fix the next step, thus introducing a bias. We analyze the extent to which the bias can affect the limit behavior of the walk. The controller is assumed to associate a real, nonnegative, “benefit” with each state, and to strive to maximize the long-term expected benefit. We derive tight bounds on the maximum of this objective function over all controller's strategies, and present polynomial time algorithms for computing the optimal controller strategy.     

Reflecting topological properties in continuous images

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ ⁺ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ ⁺. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ ⁺ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ ⁺ for compact spaces.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

Topological and equidistributional refinement of the André–Pink–Zannier conjecture at finitely many places

We present some applications of recent results in homogeneous dynamics to an unlikely intersections problem in Shimura varieties (the André–Pink–Zannier conjecture) and its refinements.     

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Let be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finite-dimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive -extensions. More generally, we find stably transitive -extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive.For groups of the form where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalizes a result of Niţică and Pollicott for -extensions.Communicated by Viviane Baladisubmitted 24/05/04, accepted 11/10/04