Phase conventions consistent with duality between SUn and S L

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.

Automorphic Forms with Anisotropic Periods on a Unitary Group

This thesis consists of four independent papers. In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth. In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1. In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone. In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.

Complex Conjugation of Group Representations by Inner Automorphisms

Representations of the unitary group U (n) symmetry adapted to the subgroup sequences U (n) U(nj)U(in) U (inj) are considered using double coset decomposition. The matrix elements of the double coset representatives are related to identical coefficients developed

Parametrization of $4\times 4$-matrices $G$ of the complex linear group $GL(4,C)$ in terms of four complex 4-vector parameters $(k,m,n,l)$ is investigated. Additional restrictions separating some subgroups of $GL(4,C)$ are given explicitly. In the given parametrization, the problem of inverting any $4\times 4$ matrix $G$ is solved. Expression for determinant of any matrix $G$ is found: $\det G = F(k,m,n,l)$. Unitarity conditions $G^{+} = G^{-1}$ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups $G_{1}$, $G_{2}$, $G_{3}$ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators $\Lambda_{k}$, being of Gell-Mann type, substantially differs from the basis $\lambda_{i}$ used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of $GL(4,C)$ can be used $\{\Lambda_k\} = \{\alpha_i\oplus\beta_j\oplus(\alpha_iV\beta_j = {\boldsymbol K} \oplus {\boldsymbol L}\oplus{\boldsymbol M})\}$, which permit to factorize SU(4) transformations according to $S = e^{i\vec{a}\vec{\alpha}} e^{i\vec{b}\vec\beta}} e^{i{\boldsymbol k}{\boldsymbol K}} e^{i{\boldsymbol l}{\boldsymbol L}} e^{i\boldsymbol m}{\boldsymbol M}}$, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups.

Double Coset Decompositions of Reductive Lie Groups Arising from Two Involutions

Influence of the structural state and crystallographic texture of Zr-2.5% Nb alloy samples on the anisotropy of their thermal expansion

The nature, properties, and behavior of physical systems depend upon the identity of the constituents and the nature of interactions between them and upon the degree of aggregation and the ambient conditions. Investigations of the energy level structure, elementary excitations, morphology (shape or crystallographical structure), phase transformations and dynamics of finite systems, and their dependence upon the degree and form of aggregation, are common endeavors in the physical sciences spanning a wide spectrum of interaction forms and strengths, spatial dimensions, and temporal scales. Thus the evolution of energetic, structural, dynamical and thermodynamic properties of matter as a function of the degree of aggregation (i.e., size of the system, or number of particles composing it) is of fundamental interest in diverse fields cutting across the disciplines of atomic, molecular, nuclear, intermediate and high-energy particle physics, astrophysics, chemical dynamics, materials science and condensed matter physics. In this context it is of interest to note close analogies between some of the properties and phenomena exhibited by atomic and molecular clusters and those found in atomic nuclei, despite gross differences in the nature of binding in these systems and their spatial extent.

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open‐shell, and virtual orbitals. When acting on the closed‐shell reference state with nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand‐Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). The tensor operators generating the M‐tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group SM. This provides an alternative to the Nagel‐Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) represent a convenient basis for a spin‐adapted full coupled cluster calculation for closed‐shell systems. For a high‐spin reference determinant with n, singly occupied open‐shell orbitals, the corresponding representation of U(n), n=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration‐state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first‐, second‐, and third‐order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin‐adapted open‐shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin‐restricted open‐shell many‐body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.

Dion-Jacobson (DJ) quasi-two-dimensional (2D) halide perovskites (HPs) have emerged as a robust platform for optoelectronic applications, yet their crystallization pathways, encompassing nucleation, growth and phase transformation, under non-equilibrium synthesis conditions, remains insufficiently understood. Herein, we employ a robot-based high-throughput experimental platform to systematically probe the multidimensional synthesis and compositional space of 1,4-butanediammonium (BDA) and methylammonium (MA) and formamidinium (FA)-based DJ quasi-2D HP system. By integrating an automated spectral diagnosis-based data analysis toolkit into the experimental workflow, we obtain an intuitive and quantitative understanding of the phase growth dynamics within this complex materials system. Our studies reveal two competing phase transformation behaviors—lattice disintegration and lattice coarsening—where the directionality of each is governed by delicate solid–liquid chemical equilibria. The experimental workflow identifies an optimized temperature-benign compositional window that promotes the formation of a robust 3D lattice in DJ quasi-2D HPs with homogenized phase distribution, enabling simultaneous improvements in photovoltaic performance of HPs and thermal stability. Overall, the integrated automated experiment-analysis workflow provides a powerful approach for deciphering the complex and dynamic processes in quasi-2D HPs, and offers a versatile platform for guiding materials design and application strategies.

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We

The kinetic properties and high-speed processes during phase transformations and related effects of giant deformations in micro- and nanosamples of functional nanomaterials in alternating electric and thermal fields have been studied. Theoretically studied the processes of controlled deformation (activation) and heat distribution at small sample sizes, in which the manifestation of such phenomena as thermoelastic martensitic phase transition and associated shape memory effect (SME) is possible. A computing model of the speedwork of a composite actuator has been constructed and the dependence of the maximum activation frequency on the linear dimensions of the micro-actuator has been determined.

The article addresses Central European historical experiences of the twentieth century manifesting in the fates of Sándor Márai, Joseph Roth and Stefan Zweig. Entangled in the speeding wheel of the modern history, the three writers experienced excessive historical discontinuities (wars, revolutions, dictatorships) which they conceptualized in terms of Epicureanism and Stoicism. To a great extent mythicized Epicurean ‘lightness of being,’ carefree travelling, journalistic openness coexist with the Stoic inward diaristic safeguarding of the self from the historical burden in their texts. While in the Epicurean approach to life, individual is a master of his own fate realizing positive freedom, the centripetal Stoic worldview entails a search of negative freedom from the overwhelming historical fate and a withdrawal to inner (diaristic) self as the only anchor in volatile times. Moreover, the three writers’ historical experiences shaped their double displacement. Whereas its spatial dimension (exilic nomadism) made their self-identifications oscillate between homo politicus and homo poeticus, its temporal aspect – in the article’s foreground – implied the need to narratively inscribe one’s self within a meaningful order of time reconfigured in personal writing.