Vector Ambiguity and Freeness Problems in SL (2, ℤ).

We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y=Mx, where M∈S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL(2,Z), which is the set of 2×2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups

Vector Reachability Problem in SL(2,Z)

This paper solves three open problems about the decidability of the vector and scalar reachability problems and the point to point reachability by fractional linear transformations over finitely generated semigroups of matrices from . Our approach to solving these problems is based on the characterization of reachability paths between vectors or points, which is then used to translate the numerical problems on matrices into computational problems on words and regular languages. We will also give geometric interpretations of these results

The Identity Problem for Matrix Semigroups in SL2(Z) is NP-complete

In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of $2\times 2$ matrices from the modular group $\text{PSL}_2(\mathbb Z)$ and thus the Special Linear group $\text{SL}_2(\mathbb Z)$ is solvable in $\mathbf{NP}$. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from $\text{SL}_2(\mathbb Z)$ or $\text{PSL}_2(\mathbb Z)$ generates a group or free semigroup is also decidable in $\mathbf{NP}$. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhum\"aki, was in \EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet $\{s,r\}$ and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known $\mathbf{NP}$-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in $\text{SL}_2(\mathbb Z)$ are $\mathbf{NP}$-complete

Peculiarities of angiogenesis in testicular embryonal carcinoma

Despite the relatively low incidence of testicular germ cell tumors (TGCT), which amount only 1% of all cancers in men worldwide in the epidemiological aspect, unlike tumors of other localizations, it does not become more frequent with age, but reaches its peak in young men of working age what attaches to the problem not only great medical, but also social significance.Estimation of angiogenesis in the tumor is considered as one of the markers for predicting the course of disease, presence of metastases and sensitivity to antitumor antiangiogenic therapy. At the same time, data concerning the study of the mechanisms of blood vessels formation, the work out of methods for estimation of tumorous angiogenesis as well as the use habits of IHC markers of endothelial cells (CD31 and CD34) for assessing the aggressiveness and prognosis of TGCT and, in particular, EC, are absent in the available literature sources.Investigation of peculiarities of neoangiogenesis in the dynamics of testicular embryonal carcinoma progression revealed: significant increasing of relative area of CD31 and CD34 expression as well as vascular density during transition from the initial to the late stages of tumor progression; formation of intratumoral vessels in the embryonal carcinoma occurs by angiogenesis and vasculogenesis with participation of progenitor endothelial cells; embryonal carcinoma is characterized by vasculogenic mimicry in the form of channels formation that do not have endothelial lining

Towards Uniform Online Spherical Tessellations

The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as “incremental generation”) plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is 1.618 and for at least four points is no less than 1.726

Physicochemical investigation of anodic processes involved in silver electrowinning in refining technology

During silver electrowinning in refining technology, irrespectively of the nature of electrolyte and anode material, an anode deposit is formed due to oxidation of singly charged silver ions to higher degrees of oxidation (+2 and +3) under polarization. Absorption spectra of Ag(II) have been obtained in solutions with various concentrations of silver ions and nitric acid using anodic polarization in combination with electronic absorption spectroscopy; silver ions of high oxidation degrees were found to be unstable in solutions. At anode potentials φa ≥ 1.15-1.22 V, Ag(I) ions in liquid phase are oxidized to Ag(II), which paint the anolyte; their content depends on silver and nitric acid concentrations in the electrolyte. The deposit crystallizes on anode at potential φa ≈ 1.5 V. Chemical analysis, X-ray powder diffraction, and a thermogravimetric study of the anode deposit have made it possible to identify its initial composition as Ag 7O8NO3. The deposit composition varies in time as a result of reduction of highly oxidized silver. © 2013 Pleiades Publishing, Ltd

Multidisciplinary Intention: Revealing The Consequence of Psychosocial Factors on the Adherence to Treatment of the Patients with Type 2 Diabetes Mellitus

Background: Diabetes Mellitus type 2 (T2DM) has been becoming a globalworld health problem affecting countries with different income level (Mogre,Johnson, Tzelepis, Shaw, & Paul, 2017). The low level of compliance and theincreased level of the diabetes distress has been closely associated with destituteglycemic control. Some diabetes-specific psychological variables may beimportant for psychological status of patients and glycemic control to theachievement of individual glycemic targets (Indelicato et al., 2017). Therefore,alexithymia – psychological clinical characteristic of the reduced emotionalcomponent that could have been associated with the metabolic syndrome inpatients with type 2 diabetes, for example (Lemche, Chaban, & Lemche, 2014). Itis important to consider the clinical evaluation of mental aspects in patients withT2DM, however, in order to preventing potentially unfavorable self-care behaviorleading to complications of this disease, including a decrease in the quality of lifelevel (Conti et al., 2017). Patient's level of the quality of life is the one of mainimportant components of treatment adherence and clinical decision making forimproving the effectiveness of therapy (Chaban, Khaustova, & Bezsheiko, n.d.).The low level of medication compliance and quality of life could have beenaddicted in consequence to non-adherence and respectively – inefficiency oftherapy of these patients. Therefore, multidisciplinary connection – involvingmental health care specialists can potentially improve treatment effectiveness(Kogut at al., 2018)

Non-destructive testing of multilayer medium by the method of velocity of elastic waves hodograph

The method of velocity of elastic waves hodograph, aimed at non-destructive testing of structurally heterogeneous composite materials and products based on them, as well as multilayer products and constructions, is considered. The theoretical basis for determining the propagation velocity of elastic waves in a multilayer medium by the hodograph method is given. Based on the studies, recommendations are given for determining the propagation velocity of elastic waves in each individual layer of a multilayer medium, which allows non-destructive testing of the physicomechanical characteristics of each layer of a multilayer medium. It is shown that in addition to simple multiple reflections in a homogeneous medium, in a multilayer medium with parallel interfaces consisting of two or more layers, complex types of multiple reflected waves and mixed waves (reflected-refracted and refracted-reflected) can arise. The main task of applying the low-frequency ultrasonic method is to determine the acoustic parameters of the propagation of elastic waves (velocities, amplitudes, spectra). The main methods for determining the elastic wave velocities are considered, based on the hodograph equation of the indicated reflected waves in a multilayer medium