Abelian functions for cyclic trigonal curves of genus 4

We discuss the theory of generalized Weierstrass σ$\sigma$ and ℘$\wp$-functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve y³=x⁵+_λ4x⁴+_λ3x³+_λ2x²+_λ1x+_λ0$y^3=x^5+{\lambda }_4{x}^4+{\lambda }_3{x}^3+{\lambda }_2{x}^2+{\lambda }_{1}x+{\lambda }_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘$\wp$-functions, and the derivation of addition formulae.     

The topological sigma model

We obtain the invariants of Witten's topologicalσ-model by gauge fixing a topological action and using BRST symmetry. The fields and the BRST formalism are interpreted geometrically.     

The Martinelli-Parisi expansion for the non-linear sigma model

The Martinelli-Parisi expansion around the Migdal-Kadanoff transformation is computed to second order in the shift parameter for the non-linear sigma model on a triangular lattice. The computation is performed in a completely analytical way in the region of weak coupling which is the most crucial region for checking the goodness of the expansion. The value of the beta function improves definitely with respect to the zero- and first-order results, reducing to 13% the error with regard to the exact value of continuum perturbation theory.     

On phases and length of curves in a cyclic quantum evolution

The concept of a curve traced by a state vector in the Hilbert space is introduced into the general context of quantum evolutions and its length defined. Three important curves are identified and their relation to the dynamical phase, the geometric phase and the total phase are studied. These phases are reformulated in terms of the dynamical curve, the geometric curve and the natural curve. For any arbitrary cyclic evolution of a quantum system, it is shown that the dynamical phase, the geometric phase and their sums and/or differences can be expressed as the integral of the contracted length of some suitably-defined curves. With this, the phases of the quantum mechanical wave function attain new meaning. Also, new inequalities concerning the phases are presented.     

The aetiology of sigma model anomalies

Certain nonlinear sigma models with fermions are ill-defined due to an anomaly which exhibits characteristics of both the nonabelian gauge theory anomaly and the SU(2) anomaly. The simplest way to diagnose the anomaly involves consideration of the global topology of the theory. We review the mathematical methods needed for this analysis and apply them to several supersymmetric sigma models. Some of these are found to be anomalous.     

Equal sensation curves for whole-body vibration expressed as a function of driving force

Previous studies have shown that the seated human is most sensitive to whole-body vertical vibration at about 5 Hz. Similarly, the body shows an apparent mass resonance at about 5 Hz. Considering these similarities between the biomechanical and subjective responses, it was hypothesized that, at low frequencies, subjective ratings of whole-body vibration might be directly proportional to the driving force. Twelve male subjects participated in a laboratory experiment where subjects sat on a rigid seat mounted on a shaker. The magnitude of a test stimulus was adjusted such that the subjective intensity could be matched to a reference stimulus, using a modified Bruceton test protocol. The sinusoidal reference stimulus was 8-Hz vibration with a magnitude of 0.5 m/s2 rms (or 0.25 m/s2 rms for the 1-Hz test); the sinusoidal test stimuli had frequencies of 1, 2, 4, 16, and 32 Hz. Equal sensation contours in terms of seat acceleration showed data similar to those in the literature. Equal sensation contours in terms of force showed a nominally linear response at 1, 2, and 4 Hz, but an increasing sensitivity at higher frequencies. This is in agreement with a model derived from published subjective and objective fitted data.     

Tuning the electronic transport properties for a trigonal graphene flake

By applying nonequilibrium Green?s functions in combination with density-functional theory, we have investigated the effects of two side groups, NH₂ and NO₂, on the electronic transport properties of the trigonal graphene flake. It has been found that the rectifying ratios (RR) and direction can be significantly tuned by the type and the attached positions of side groups. The NH₂ group shows an obvious electron-donating characteristic, whereas NO₂ group demonstrates a poorly electron-accepting behavior in these systems. The analysis on the spatial distribution and the energy level of frontier orbitals, transmission spectra, and electrostatic potential distribution give an inside view of the observed results.     

Formulae for null curves deriving from elliptic curves

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C³${\mathbb{C}}^3$ and thus an algebraic minimal surface in R³${\mathbb{R}}^3$. We derive simple formulae for the coordinate functions of such a null curve.     

On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ⁴ = ( – E ₁) ( – E ₂) ( – E ₃) ( – E ₄). It is shown that the associated Riemann theta function may be decomposed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way.On leave from Leningrad State University and Leningrad Institute of Aviation Instrumentation.     

Spectroscopic evidence for the tricapped trigonal prism structure of semiconductor clusters

We have obtained photoelectron spectra (PES) for silicon cluster anions with up to 20 atoms. Efficient cooling of species in the source has allowed us to resolve multiple features in the PES for all sizes studied. Spectra for an extensive set of low-energy Si(-)(n) isomers found by a global search have been simulated using density functional theory and pseudopotentials. Except for n = 12, calculations for Si(-)(n) ground states agree with the measurements. This does not hold for other plausible geometries. Hence PES data validate the tricapped trigonal prism morphologies for medium-sized Si clusters.     

Symmetry of coupling coefficients for cubic and trigonal group under association

Relations between Racah matrices for products of irreducible representations of the double cubic and trigonal group are derived using the concept of association, i.e. multiplication by the representation A₂. Relations for the associated decomposition coefficients for O′ → D′₃ and the appropriate reduction coefficients are established.     

Bogomolny-type equations for gravitating gauged sigma models

It is shown that the complete set of field equations of a gauged sigma model coupled to gravity is satisfied by a set of first-order equations when the metric is static.     

The relativistic valence and conduction band edges of trigonal Te and Se

Relativistic ab initio level bands are presented for trigonal Te and Se in the vicinity of the valence band maxima and conduction band minima. The agreement with the experimental k · p band models of Te is close. The differences are associated with the missing Darwin and mass-velocity terms in the k · p models. There are distinct differences between the calculated band edges for Se and the model proposed by Moreth.