On the linearized relativistic Boltzmann equation

The linearized relativistic Boltzmann equation inL ²(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL ²(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.     

e -μ+→e +μ- process in a model with theSU(2)L×SU(2)R×U(1)B−L gauge group

A model based on theSU(2)_L×SU(2)_R×U(1)_B−L gauge group is used to study the lepton recharging processe ^-μ⁺→e ⁺μ^-. It should be possible to observe this process on a muon collider in the fixed-electron-target regime or the electron-beam regime. It is shown that the given process can be used to measure not only the characteristics of physical Higgs bosons, but also the parameters of neutrino oscillations. Grodno University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 83–88, November, 1998.     

Supersymmetric SU C (4) × SU L (2) × SU R (2) and spontaneous breakdown of symmetries

A supersymmetric version of the left right symmetric partial unification group SU _C (4) × SU _L (2) × SU _R (2) is presented. The spontaneous breakdown of gauge symmetry in a favourable chain of descent has been studied in detail. The mass spectra have been calculated. The method of O’Raifeartaigh has been used to break supersymmetry. The lifting of degeneracy of mass levels between physical multiplets has been shown to occur due to radiative corrections.     

Transition from SU(2)L × SU(2)R × U(1)B-L representation to SU(2)L × U(1)Y by q-deformation and the corresponding classical breaking term of chiral U(2)

The minimal Standard Model exhibits a nontrivial chiral U(2) symmetry if the VEV and the hypercharge splitting Δ = (y-y)/2 of right-handed leptons (quarks) in a family vanish and Q = T₀ + Y independently in each helicity sector. As a generalization, we start with SU(2)_L × SU(2)_R × U(1)_(B-L) and introduce Δ as a continuous parameter which is a measure of explicit symmetry breakdown. Values 0 ? Δ ? 1/2 take the neutral generator of the isospin ½ representation to the singlet representation, i.e. ‘deformes’ the LR representation into the minimal Standard one. The corresponding classical O(3)-breaking term is a magnetic field perpendicular to the x₃-axis. A simple mapping on the fundamental Drinfeld-Jimbo q-deformed SU(2) representation is given.     

Deformation of maximal subalgebras of SU(6)

The Cartan-Chevalley generators of a L, L being a maximal subalgebra of SU(6), are written in terms of the generators of SU(6) using a boson realization and then are deformed introducing q-bosons. A procedure to obtain a deformed SU(6) starting from L _q is presented. The deformed SU(6) is not equivalent as Hopf structure to Drinfeld-Jimbo SU _q(6). This scheme provides a way to deform the embedding chain SU(6) L.     

Skein Theory and Witten–Reshetikhin–Turaev Invariants of Links in Lens Spaces

We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of an arbitrary link in L(p,q) as a function of the level rф. They are given by \frac1?r\frac{1}{\sqrt{r}} times one of p Laurent polynomials evaluated at e^{\frac 2 pi 4pr}e^{\frac {2 \pi i} {4pr}}. The congruence class of r modulo p determines which polynomial is applicable. If p L 0 mod 4, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link from the collection of the WRT invariants at all levels if p is a prime or twice an odd prime. By a more delicate argument, this is also shown to be true for p=9.     

The longitudinal structure function F L(x, Q 2) and the gluon distribution G(x, Q 2) at low x

We obtain a relation between the longitudinal structure function F _L(x, Q ²), F ₂(x, Q ²) and G(x, Q ²) at small x, using the formalism recently reported by one of the authors [2]. We also obtain a relation between F _L(x, Q ²), F ₂(x, Q ²) and its slope (dF ₂(x, Q ²))/(dlnQ ²). This provides us with the determination of the longitudinal structure function F _L(x, Q ²) from F ₂(x, Q ²) data and hence extract the gluon distribution G(x, Q ²).     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

SU(2)L×SU(2)R×U(1) gauge model

We determine here the most general electroweak interaction based on the groupSU(2)_L×SU(2)_R×U(1). When we rotate theZ ₁,Z ₂ basis to theZ,D basis such that the total interaction ofZ with the right-handed current is zero, we obtain an interaction that is free of triangle anomalies. This condition enables us to know the angle through whichZ ₁,Z ₂ basis is to be rotated. We show that the triangle anomaly free interaction obtained by others is contained here as a special case. We also determine the triangle anomaly free weak interaction whenever the neutral (Z,D) bosons are mass eigenstates and show that it reduces to the neutral sector of the standard model whenever g _R ² goes to infinity. The charged sector is also developed here. The most general elements of the masssquared matrix of theZ,D bosons are evaluated. The masses of the left- and right-handed charged bosons are also determined.     

A theorem on the support of functions in L2(R) and of their Fourier transforms

This paper is a study of the dimension of certain subspaces M of L²(R) defined by prescribing the support of the functions in M and of their Fourier transforms.     

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: <Emphasis Type="Italic">SU</Emphasis>(3)

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE{SU(3) \mathcal{ADE}} graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus \mathbbT², SU(3){\mathbb{T}^2,\,SU(3)}, and some SU(3) graphs.     

Non-abelian electrodynamics and the vacuumB (3)

By using an 0(3) gauge group, a non-Abelian theory of vacuum electrodynamics is developed in which the newly discovered longitudinal vacuum fieldsB ⁽³⁾ andi E ⁽³⁾ appear self-consistently with the usual plane wavesB ⁽¹⁾,B ⁽²⁾,E ⁽¹⁾, andE ⁽²⁾ in the circular basis (1), (2), (3), a complex representation of space. Using the charge quantization condition the vacuum Maxwell equations are given in the non-Abelian representation.     

A note on energy-momentum conservation in Palatini formulation of L(R) gravity

By establishing that Palatini formulation of L(R) gravity is equivalent to ω=−3/2 Brans-Dicke theory, we show that energy-momentum tensor is covariantly conserved in this type of modified gravity theory.     

Intervalley electron scattering by phonons in (AlAs)1(GaAs)3(001) superlattices

Intervalley electron scattering by phonons in (AlAs)₁(GaAs)₃(001) superlattices is studied using the pseudopotential method and a phenomenological model of the bonding forces. The deformation potentials between the conduction band extrema of the superlattice involving short-and long-wavelength phonons are calculated. It is shown that the mixing of states from the zinc-blende L valleys plays a greater role in intervalley scattering in a superlattice than the Γ-X mixing. In particular, due to L-L mixing, the Γ-X ₃ transitions, analogous to Γ-L transitions in zinc blende, have higher intensities than the analogues of Γ-X transitions (Γ₁-M ₅ and (Γ₁-Γ₃ transitions). The deformation potentials averaged over the scattering channels in the superlattice agree with the corresponding potentials in a solid solution, but all transitions in the superlattice have higher intensities for the lower states.     

Synthesis,structural study and thermal behaviour of a new compound: trirubidium bis(hydrogen sulphate) dihydrogen arsenate Rb3(HSO4)2.5(H2AsO4)0.5

Mixed crystals Rb₃(HSO₄)_2.5(H₂AsO₄)_0.5 have been prepared by slow evaporation from aqueous solution at room temperature. The crystals were characterized by X-ray single analysis, which revealed that Rb₃(HSO₄)_2.5(H₂AsO₄)_0.5 crystallizes in the space group P with lattice parameters: a = 7.471(3) Å; b = 7.636(1) Å; c = 12.193(2) Å; α = 71.91(1)°; β = 73.04(6)° and γ = 88.77(2)°. In this structure, the ordered S(1)O₄ and the disordered S(3)/AsO₄ tetrahedra are connected by O–H..O hydrogen bonds, to a zigzag chains running in the b-direction. These chains are, in turn, bonded to one another by disordered hydrogen bridges O–H..H–O, to give a planar structure, with hydrogen-bonded sheets, laying parallel to (1 0 0). Each disordered tetrahedron is linked to a tetrahedron neighbouring S(2)O₄ by ordered hydrogen bonds. Broader peaks in IR spectrum of the title material support the assumption of disordered structure. Thermal analysis of the superprotonic transition in Rb₃(HSO₄)_2.5(H₂AsO₄)_0.5 showed that the transformation to the high-temperature phase occurs by one-step process at 404 K. Thermal decomposition of this compound takes place at much higher temperatures, with an onset of approximately 473 K.     

81Br NQR Study of [NH3(CH2) n NH3]CdBr4 (n = 4 and 5) and [NH3(CH2) n NH3]ZnBr4 (n = 5 and 6)

⁸¹Br NQR frequencies and differential scanning calorimetry (DSC) were measured as a function of temperature. [NH₃(CH₂)₄ NH₃]CdBr₄ (1) and [NH₃(CH₂)₅NH₃]CdBr₄ (2) showed a doublet and quartet ⁸¹Br NQR spectrum, respectively. [NH₃(CH₂)₅NH₃]ZnBr₄ (3) and [NH₃(CH₂)₆NH₃]ZnBr₄ (4) exhibited a four-line ⁸¹Br NQR spectrum. From the NQR results, it is inferred that (1) and (2) consist of infinite two-dimensional sheets of corner-sharing CdBr₆ octahedra, whereas (3) and (4) have isolated [ZnBr₄]²⁻ tetrahedra. All of the crystals except (1) showed at least one structural phase transition above 380 K.     

<Emphasis Type="Italic">SU</Emphasis>(2) invariants of symmetric qubit states

We express the density matrix for the N-qubit symmetric state or spin-j state (j = N/2) in terms of the well-known Fano statistical tensor parameters. Employing the multi-axial representation, where the spin-j density matrix is shown to be characterized by j(2j + 1) axes and 2j real scalars, we enumerate the number of invariants constructed out of these axes and scalars. We calculate these invariants explicitly in the particular case of the pure and mixed spin-1 state.     

Proton decay in theSU(4) c ×SU(2) L ×SU(2) R model

Possible proton decays in theSU(4) _C ×SU(2) _L ×SU(2) _R unification model are discussed. There are some characteristics in the decay products, which are different from those in the standardSU(5) orSO (10) model, in certain cases.     

Two Mn(II) and Zn(II) Coordination Polymers: Aqueous-Phase Detection of Nitroaromatic Explosives and Protective Effect on Atherosclerosis

Via the solvothermal reaction between Zn(II) or Mn(II) salts and 5-(3,4-dicarboxylphenoxy)nicotinic acid (H₃L) ligand, a trifunctional N,O-building block having three diverse kinds of functional groups (O-ether, N-pyridyl and COOH), two new coordination polymers (CPs) could be generated, and their chemical formulae respectively are {[Mn₃(L)₂(H₂O)₂]·4H₂O} (1) and {[Zn(HL)]·NMP} (2). The complex 2 based on Zn(II) possesses high efficiency of fluorescence quenching for the nitrophenol (2,4,6-trinitrophenol, TNP; 4-nitrophenol, 4-NP; 3-nitrophenol, 3-NP; 2-nitrophenol, 2-NP) in the aqueous solution. Furthermore, the treatment activity of compounds on the atherosclerosis was assessed, and relevant mechanism was investigated. First of all, the ELISA assay was used to measure the content of the inflammatory cytokines released into the plasma. Besides, the levels of the NF-κb signaling pathway in the vascular endothelial cells were measured with real time RT-PCR. The hemolysis test was conducted in this research to measure the biocompatibility of the new compound.

     

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Z _p ,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZ _p factors.