Virtual Polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications

Critical configurations of planar robot arms

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration

Ferromagnetic Wires Composite Media with Tunable Scattering Spectra at Microwaves

We demonstrate composite media with ferromagnetic wires that exhibit a frequency region at the microwave regime with scattering spectra strongly dependent on an external magnetic field or stress. These tunable composite materials have recently been proposed theoretically; however, no direct experimental verification has been reported. We used composite materials with predominantly oriented CoFeCrSiB glass-coated amorphous wires having large magnetoimpedance at GHz frequencies. The free space measurements of reflection and transmission coefficients were conducted in the frequency range 1-8 GHz in the presence of an external static magnetic field or stress applied to the whole sample. In general, the transmission spectra show greater changes in the range of 10dB for a relatively small magnetic field of few Oe or stress of 0.1 MPa. The obtained results are quantitatively consistent with the analytical expressions predicted by the effective medium arguments. The incident electromagnetic wave induces an electrical dipole moment in each wire, the aggregate of which forms the effective dipole response of the whole composite structure in the radiative near or far field region. The field and stress dependences of the effective response arise from a field or tensile stress sensitivity of the ac surface impedance of a ferromagnetic wire. In the vicinity of the antenna resonance the variations in the magneto-impedance of the wire inclusions result in large changes of the total effective response. A number of applications of proposed materials is discussed including the field tunable microwave surfaces and the self-sensing media for the remote non-destructive evaluation of structural materials

Laser beam welding of a low density refractory high entropy alloy

In particular,the effect of different pre-heating temperatures was examined. Due to the low ductility of the material, laser beam welding at room temperature resulted in the formations of hot cracks. Sound butt joints without cracks were produced using pre-heating to T≥600 ◦C. In the initial as-cast condition, the alloy consisted of coarse bcc grains with a small amount of lens-shaped C15 Laves phase particle

Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials

We study the optical properties of metamaterials made from cut-wire pairs or plate pairs. We obtain a more pronounced optical response for arrays of plate pairs -- a geometry which also eliminates the undesired polarization anisotropy of the cut-wire pairs. The measured optical spectra agree with simulations, revealing negative magnetic permeability in the range of telecommunications wavelengths. Thus, nanoscopic plate pairs might serve as an alternative to the established split-ring resonator design.Comment: 3 pages, 4 figures, submitted to Opt. Let

Udmurt Folk Calendar and Rural Society

The present paper aims to analyze rituals and festivals of the Udmurt calendar which convey the idea of interrelationship between the producing energy of the nature and of the human being taking into consideration folk beliefs about time as well as age and gender stratification of a rural community. It is established that men played an important role in the process of praying, while “female” theme was the quintessence of spring rites the main functions of which were fertility and production. Elderly women played a special role - not only did they supervise ritual activities, but also initiated some occasional rites, and, if required, they could also establish a contact with the other world. The youth took an active part in festivals held in the beginning of half-years considered as equivalent periods ( palar ) of a calendar cycle, i.e. in spring and in autumn. Young boys and girls were main participants in spring and summer merrymaking and round dances as well as autumn and winter gatherings. The rituals of this cycle were aimed at making pairs with a view of entering into a marriage. A holiday eve could be associated with puberty. Children participated in games which corresponded to rituals. The research presents main characters of the ritual scenario which symbolize quality changes in the nature’s cycle and human life. The article focuses on the functions of those characters and peculiarities of their behavioural stereotypes

Structure and mechanical properties of near-eutectic refractory Al-Cr-Nb-Ti-Zr high entropy alloys

In this study, the structure and mechanical properties of the near-eutectic refractory Al₂₄Cr₁₁Nb₁₁Ti₃₇Zr₁₇, Al₂₀Cr₂₀Nb₂₅Ti₂₅Zr₁₁₀, and Al₂₈Cr₁₅Nb₁₅₅Ti₂₀Zr₂₂ (at.%) high entropy alloys were investigated. Both the Al₂₄Cr₁₁₁Nb₁₁Ti₃₇₇Zr₁₇ and Al₂₀Cr₂₀Nb₂₅Ti₂₂₅Zr₁₀₀ alloys had a hypoeutectic structure, while the Al₂₈Cr₁₅Nb₁₅Ti₂₀Zr₂₂ alloy had a hypereutectic one. The proeutectic phase in the Al₂₄Cr₁₁Nb₁₁Ti₃₇Zr₁₇ and Al₂₀Cr₂₀Nb₂₅Ti₂₅Zr₁₀ alloys was the ordered B2 phase and in the Al₂₈Cr₁₅Nb₁₅Ti₂₀Zr₂₂ alloy - the C14 Laves phase. In all three alloys, the eutectic regions possessed a fine lamellar structure composed of a mixture of the C14 Laves and B2 phases. The densities of the alloys were 5.20-5.89 g/cm

Crystal structure prediction of organic pigments: quinacridone as an example

The structures of the α, β and γ polymorphs of quinacridone were predicted using Polymorph Predictor software in combination with X-ray powder diffraction patterns of limited quality. The present work demonstrates a method to obtain crystal structures of industrially important pigments when only a low-quality powder pattern is available