Vector Reachability Problem in SL(2,Z)\mathrm{SL}(2,\mathbb{Z})

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four. This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from $\mathrm{SL}(2,\mathbb{Z})$ and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from $\mathrm{SL}(2,\mathbb{Z})$. The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable

Decidability of the Membership Problem for 2×22\times 2 integer matrices

The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by $\{M_1,\dots,M_n\}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages

Optimal principal component Analysis of STEM XEDS spectrum images

STEM XEDS spectrum images can be drastically denoised by application of the principal component analysis (PCA). This paper looks inside the PCA workflow step by step on an example of a complex semiconductor structure consisting of a number of different phases. Typical problems distorting the principal components decomposition are highlighted and solutions for the successful PCA are described. Particular attention is paid to the optimal truncation of principal components in the course of reconstructing denoised data. A novel accurate and robust method, which overperforms the existing truncation methods is suggested for the first time and described in details.Comment: 21 pages, 14 figure

Time Response of Shape Memory Alloy Actuators

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Force/displacement actuators with the high output power and time response can be fabricated from shape memory wires or ribbons. Typically Ni-Ti shape memory alloys are used as an active material in such actuators. They are driven by Joule heating and air convection cooling. In the present work, the time response of various types of Ni-Ti actuators having different transformation temperatures and geometrical sizes, is studied systematically under conditions of free and forced air convection. The simple analytical model for calculating the time response is developed which accounts for the latent heat and thermal hysteresis of transformation. For all the types of considered actuators, the calculated time response is in a good agreement with that observed experimentally. Finally, on the base of the suggested model, we present the time response of Ni-Ti actuators calculated as a function of their transformation temperature and cross section dimensions

On the Identity and Group Problems for Complex Heisenberg Matrices

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time

Influencia de las estructuras protectoras en el desarrollo y productividad del tomate

La necesidad de obtener altos rendimientos de plantas agrícolas de manera constante en el Distrito Federal de Siberia es suficientemente alta. Por lo tanto, este trabajo tuvo como objetivo estudiar la influencia de las estructuras protectoras en el crecimiento, desarrollo y productividad de los tomates en la región de Novosibirsk. Los datos obtenidos permiten concluir que el uso de estructuras protectoras protege a las plantas de las heladas recurrentes, las ráfagas de viento del norte y una caída brusca de la temperatura. Las plantas cultivadas en estructuras protectoras estándar, en los primeros períodos después de la plantación en un lugar de crecimiento permanente, estuvieron expuestas a situaciones estresantes, lo que provocó una disminución de la inmunidad de la planta y, como resultado, daños por hongos y bacterias

Maximal Anderson Localization and Suppression of Surface Plasmons in Two-Dimensional Random Au Networks

Two-dimensional random metal networks possess unique electrical and optical properties, such as almost total optical transparency and low sheet resistance, which are closely related to their disordered structure. Here we present a detailed experimental and theoretical investigation of their plasmonic properties, revealing Anderson (disorder-driven) localized surface plasmon (LSP) resonances of very large quality factors and spatial localization close to the theoretical maximum, which couple to electromagnetic waves. Moreover, they disappear above a geometry-dependent threshold at ca. 1.7 eV in the investigated Au networks, explaining their large transparencies in the optical spectrum

Freestanding few-layer sheets of a dual topological insulator

The emergence of topological insulators (TIs) raised high expectations for their application in quantum computers and spintronics. Being bulk semiconductors, their nontrivial topology at the electronic bandgap enables dissipation-free charge and spin transport in protected metallic surface states. For application, crystalline thin films are requested in sufficient quantity. A suitable approach is the liquid phase exfoliation (LPE) of TI crystals that have layered structures. Bi2TeI is a weak 3D TI, which leads to protected edge states at the side facets of a crystal, as well as a topological crystalline insulator, which is responsible for protected states at the top and bottom faces. We developed an effective, scalable protocol for LPE of freestanding nanoflakes from Bi2TeI crystals. By heat treatment and sonication in isopropyl alcohol and poly(vinylpyrrolidone), crystalline Bi2TeI sheets with a thickness of ~50 nm were obtained and can therefore be considered for further processing toward microelectronic applications