Random walks on Cayley graphs of complex reflection groups

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

Quadrature method for broadband measurement of microwave path parameters

A mathematical model has been developed for the procedure for broadband automatic measurement of the complex reflection coefficient and power level in the microwave path based on the quadrature measurement method. Analytical expressions are obtained for the quadrature components at the outputs of a quadrature demodulator connected with a switch to two non-directional measuring probes for the case of an unmatched microwave path. A modified system of nonlinear equations with respect to measured values was formed and solved, which allows setting zero assumptions. A block diagram of a broadband device for built-in automatic control of microwave channel parameters has been developed, which implements the developed measurement procedure and the solution of a system of measurement equations to determine the current values of the power level factor, as well as the modulus and argument of the complex reflection coefficient.

Completely positive maps for imprimitive complex reflection groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

McKay Quivers and Lusztig Algebras of Some Finite Groups

We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde {A}(\mathsf {G})$ A ~ ( G ) of a finite group $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) , which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.

“Shakespeare’s Pioneer Migrant Othello”: Desire, Migration, Pathology and the Politics of Citizenship in The Nature of Blood

In a historical period characterised by political and social occurrences of immigration crises, Shakespeare’s and early modern dramatic attention to aliens and foreigners invite us to re-read the anguish of Shakespeare’s alien and to investigate the ways in which, in the specific case of Othello, the Moor’s experience prefigures subsequent migrations and the contemporary immigrants’ struggle for integration. Caryl Phillips’ creative re-appropriation of Shakespeare’s Othello in The Nature of Blood (1998) gives voice to the psychological anguish of a migrant in the guise of an unnamed Othello-like black general newly arrived in Venice. In a never-ending dialogue between present and past, Phillips articulates a complex reflection on the immigrants’ desire to be granted recognition as a legitimised individual with a social identity, while problematising the idea of home and the sense of belonging, here understood as the subjective feeling of identification with a city/nation. With such a complex redefinition in mind, my reading conceptualises Othello (alias the migrant) as an ethnopsychiatric “symptom”, in a psychopathology of migration and exile which exposes the meanings and practices of belonging produced and supported by host societies.

NORMAL REFLECTION SUBGROUPS OF COMPLEX REFLECTION GROUPS

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.

ДОСЛІДЖЕННЯ ВІДБИТТЯ УЛЬТРАЗВУКОВИХ ХВИЛЬ ВІД ОДНОШАРОВИХ ТЕКСТИЛЬНИХ ПОЛОТЕН ТА ДВОШАРОВИХ ТЕКСТИЛЬНИХ ПАКЕТІВ ІЗ РІЗНИМ РОЗМІРОМ ПОР

Investigate the influence of single-layer textile fabrics and two-layer textile bags on the parameters of ultrasonic waves that interact with them. In particular, to investigate the dependence of the complex reflection coefficient of ultrasonic waves on the total thickness and basis weight of textile material. Methodology. The analytical part of the study consisted in obtaining an expression for the complex reflection coefficient of ultrasonic waves and in modeling its dependence on the thickness and basis weight of different textile materials. The proposed method of control of these technological parameters consists in irradiation of textile materials with ultrasonic waves with the subsequent reception of the waves reflected from textile material, their digitization and carrying out the computer analysis of the received results. Findings. Analytical expressions are obtained that relate the thickness and basis weight of single-layer canvases, two-layer textile bags with a complex spatial structure, on the one hand, and the reflection coefficient of ultrasonic waves from such materials, on the other hand. Analytical calculations were made and mathematical modeling was performed based on the results of theoretical research. Originality. As a result of theoretical studies, it is determined how the thickness and properties of each of the two layers of porous textile materials affect the reflection of ultrasonic waves. This makes it possible on the basis of ultrasonic measurements to determine both the total thickness of textile materials and their basis weight with the accuracy required for their production. Practical value. The obtained analytical dependences are another step towards the creation of control and measuring equipment to determine the properties of single-layer textile fabrics and two-layer porous textile bags. This will help determine the overall thickness and basis weight of materials with a complex spatial structure.

Complex reflection groups and K3 surfaces I

We construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$--singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author. Comment: 26 pages

Oblique Incidence Ultrasonic Reflectometry Device Based on c-axis Tilted ScAlN Films for Evaluating Viscoelastic Properties of Liquids Above 100 MHz

Ultrasound-based evaluation of fluid properties allows for real-time measurement of small amounts of liquid samples. The ultrasonic reflection method is used to obtain the complex reflection coefficient, which can be used to evaluate the viscoelastic properties of liquids. However, this method has not been used with shear waves at frequencies above 100 MHz because shear-mode piezoelectric films are difficult to obtain at such frequencies. We propose using the oblique incidence reflectometry with quasi-shear waves excited by c-axis tilted scandium aluminum nitride (ScAlN) thin films to realize high-sensitivity evaluation of the viscoelastic properties of liquids in the above 100 MHz. In experiments, the shear elasticity and shear viscosity of glycerin solutions were estimated from their complex reflection coefficients.