One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

Field equations onSE(k)-manifoldX n

An Einstein connection which is both a special connection and a (k)-connection is called anSE(k)-connection. And a generalized even-dimensional Riemannian manifoldX _n with the so-called SE(k)-condition defined by theSE(k)-connection is called theSE(k)-manifold. We obtain the necessary and sufficient condition that there is a uniqueSE(k)-connection inX _n . Next, using these results, we define theSE(k)-manifold and study the properties of the curvature tensors and the field equations in theSE(k)-manifoldX _n .     

Geometry of the submanifoldsESX n . III. Parallelism inESX n and its submanifolds

A connection which is both Einstein and semisymmetric is called anES connection. And a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through anES connection, is called ann-dimensionalES manifold and denoted byESX _n . This paper is the third part of a systematic study of the submanifoldsX _m ofESX _n . In the first part, we introduced a new concept of theC-nonholonomic frame of reference inESX _n at points ofX _m and dealt with its consequences. In the second part, the generalized fundamental equations on a hypersubmanifold ofESX _n were derived as an application of theC-nonholonomic frame of reference. The purpose of the present paper is to study parallelism inESX _n and in its submanifoldX _m , using theC-nonholonomic frame of reference and the new concept ofES _i curves.     

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX _n ⁽¹⁾ =A _n ⁽¹⁾ ,B _n ⁽¹⁾ ,C _n ⁽¹⁾ ,D _n ⁽¹⁾ . The local states taken on by the fluctuation variables are the dominant integral weights ofX _n ⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX _n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.     

Equivariant <Emphasis Type="Italic">K</Emphasis>-Theory,Generalized Symmetric Products,and Twisted Heisenberg Algebra

 For a space X acted on by a finite group Γ, the product space X ⁿ affords a natural action of the wreath product Γ _n =Γ ⁿ ⋊S _n . The direct sum of equivariant K-groups were shown earlier by the author to carry several interesting algebraic structures. In this paper we study the K-groups of Γ _n -equivariant Clifford supermodules on X ⁿ . We show that is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on ℱ⁻ _Γ(X), when Γ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups. Received: 3 February 2001 / Accepted: 17 August 2002 Published online: 10 January 2003     

Hierarchy of mean multiplicities in pn and π+n interactions at 195 GeV/c

The dependence of the average charged multiplicity, 〈n_X〉, of the system X on |t| and M²_X is studied for the reactions p(π⁺)n→p_fast(p⁺_fast) + X and p(π⁺)n→p_slow + X at 195 GeV/c. For a fixed M²_X, 〈n_X〉 changes significantly among these reactions, yielding a systematic hierarchy of mean multiplicities. The results indicate that a proton emits on the average less charged particles than a pion and more than a neutron.     

New data for the 197Au(γ, nX) and 197Au(γ, 2nX) reaction cross sections

The results from experimental and theoretical studies of the total and partial cross sections of photoneutron reactions on the ¹⁹⁷Au isotope were analyzed. The cross sections for reactions σ(γ, nX) = σ(γ, n) + σ(γ, np) + … + σ(γ, 2nX) = σ(γ, 2n) + σ(γ, 2np) + … were evaluated in the energy range 7 ≤ E _γ ≤ 30 MeV using an approach free of the shortcomings of experimental photoneutron multiplicity sorting methods. The total photoneutron reaction cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) + … = σ^exp(γ, n) + σ^exp(γ, np) + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + … were used as the initial experimental data. The contributions from the cross sections σ(γ, nX) and σ(γ, 2nX) to the cross sections σ^exp(γ, xn) were separated using the multiplicity transition functions F ₁ ^theor = σ^theor(γ, 1nX)/σ^theor(γ, xn) and F ₂ ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn), calculated within an updated version of the pre-equilibrium model of photonuclear reactions. New evaluated data for both partial reaction cross sections, i.e., σ^eval (γ, 1nX) = F ₁ ^theorσ^exp(γ, xn) and σ^eval(γ, 2nX) = F ₂ ^theorσ^exp(γ, xn), were obtained. The cross sections σeval(γ, nX) and σ^eval.(γ, 2nX) evaluated using the theoretically calculated functions F _1,2^theor are consistent with the Livermore data, but substantially contradict the Saclay data.     

Local state probabilities for solvable restricted solid-on-solid models:A n,D n,D n (1) , andA n (1)

The local state probabilities (LSPs) are exactly computed for four hierarchies of solvable lattice models. They are restricted solid-on-solid (RSOS) models whose local states and their adjacent conditions are specified by Dinkin diagrams of typesA _n,D _n,D _n ⁽¹⁾ andA _n ⁽¹⁾ . The LSPs are expressed in terms of modular functions characterized by branching identities among the theta functions. Their automorphic properties are used to study the critical behaviors. Some fine structures are found in the spectrum of the critical exponents.     

Density function study transition metal chromium-doped alkali clusters: the finding of magnetic superatom

The structures, stabilities and magnetic properties of CrX_n (X = Na, Rb and Cs; n up to 9) clusters are studied using density functional theory to search for the stable magnetic superatoms. The geometrical optimisations indicate the ground-state structures of CrX_n evolve toward a close packed structure with an interior Cr atom surrounded by X atoms as the cluster size increase. Their stabilities are analysed by the relative energy, gain in energy (ΔE(n)) and the highest unoccupied molecular orbital and lowest unoccupied molecular orbital gaps. Furthermore, the magnetic moments of CrX_n clusters show an odd–even oscillation. Here, we mainly focus on the CrX₇ (X = Na, Rb and Cs) clusters due to the same valence count as the known stable magnetic superatoms VNa₈, VCs₈ and TiNa₉. Although these clusters all have a filled electronic configuration 1S²1P⁶ and large magnetic moment 5 μ_B, our studies indicate that only CrNa₇ is highly stable compared to its nearest neighbours, while CrRb₇ and CrCs₇ clusters are less stable. This suggests that Cr-doped Na₇ is most appropriate for filled electronic configuration and CrNa₇ is shown to be a stable magnetic superatom. More interesting, we find CrRb₈ and CrCs₈ with the filled electronic configuration 1S²1P⁶ have higher stability and large magnetic moment 6 μ_B in their respective series.     

Study on charge transfer interaction and valence state of layered perovskite-type mixed valence complex [NH3(CH2) n NH3]2[(AuII2)(AuIIII4)(I3)2] (n = 7, 8), by means of 197Au Mössbauer spectroscopy

Cs₂[Au^I X ₂][Au^III X ₄](X = Cl, Br, and I) is well known for the three-dimensional perovskite-type gold mixed valence system. Recently, layered perovskite-type gold mixed valence complexes, [NH₃(CH₂) _n NH₃]₂[(Au^II₂)(Au^IIII₄)(I₃)₂] (n = 7 and 8), have been synthesized. We have investigated the relationship between the structural dimensionality and the Au^I–Au^III charge transfer interaction for Cs₂[Au^II₂][Au^IIII₄] and [NH₃(CH₂) _n NH₃]₂[(Au^II₂)(Au^IIII₄)(I₃)₂] (n = 7 and 8) by means of ¹⁹⁷Au Mössbauer spectroscopy.     

Additivity and non-additivity of dissociation energies in intermolecular interactions. Theoretical studies on (H2)n,n = 2-8, (CO2)n,n = 2-6 and (HF)n,n = 2-8

CCSD(T) and MP2 results using the aug-cc-pV5Z basis set are reported for chain, cyclic and other structures of the clusters (H₂)_n, n?=?2-8, (CO₂)_n, n?=?2-6 and (HF)_n, n?=?2-8. In chain-like structures of (H₂)_n and (CO₂)_n, with the bonding type of the dimer maintained, the dissociation energy D_e of the dimer doubles for the trimer, triples for the tetramer, and so on. Due to these systems being dominated by short-range forces, interactions are essentially restricted to neighbouring monomers. For other types of (H₂)_n and (CO₂)_n structures, the multipliers relative to the dimerisation energy can be much higher. Dissociation energies for the hexamers in S₆ symmetry of both H₂ (379?cm^?1) and CO₂ (4925?cm^?1) are over ten times the respective dimerisation energies. For the chain-like trimer of HF, however, D_e is in excess of double the dimer value. Mainly due to longer-range dipolar forces, the interactions reach beyond the neighbouring monomers. The interaction energy between HF monomers in chains follows an approximate R^?2 (R being the F–F distance) relationship, The calculated dissociation energies of the HF octamer are 15,985?cm^?1 (factor of 10.4) for the chain, and 21,003?cm^?1 (factor of 13.7) for the C_6h cyclic structure.     

The pp→K<Superscript>+</Superscript><Emphasis Type="Italic">Σ</Emphasis><Superscript>+</Superscript>n cross-section from missing-mass spectra

We utilize existing inclusive data on K⁺-meson momentum spectra of the reaction pp→K ⁺ X at T _p = 2.3-2.85GeV to deduce total cross-sections for pp→K ⁺ Σ ⁺ n. The method used to extract those cross-sections is explained and discussed in detail. Our result for T _p = 2.85GeV is consistent with the data point from a direct measurement at the same beam energy. The cross-section obtained for T _p = 2.3GeV is with 13.7±2.3μb considerably smaller than the value found in a recent experiment by the COSY-11 Collaboration at a somewhat lower beam energy, indicating that the pp→K ⁺ Σ ⁺ n reaction cross-section could exhibit a rather unusual energy dependence.     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

Scaling functions of susceptibilities toO(1/n) forn-vector models with axial anisotropy

For an axially anisotropicn-vector model withm = O(n) easy – andn – m = O(n) hard components of the order parameter, we derive the susceptibility r ^–1 along one of the equivalent easy axes and the perpendicular one r ^-1 toO(1/n) of the 1/n-expansion in the disordered phase. The results confirm predictions of the scaling theory, e.g.(g, t)=A t ^– X (B g/t ) and (g, t) =A t ^– X (B g/t ), wheret = T – T _c (g = 0),g is the anisotropy parameter andX, X denote the scaling functions. We evaluate the relevant diagrams toO(1/n) which yield the coefficientsA, A and the critical behaviour of the scaling functions and critical amplitudes explicitly for . The extreme anisotropic case, i.e.m = O(1), is discussed briefly in the large-n limit in comparison with the mean field solution.Parts of this paper were presented at the Frühjahrstagung der Deutschen Physikalischen Gesellschaft in Freudenstadt (May 1974).     

Understanding halogen‐substituent assistance in H‐atom abstraction‐based reactions of CHCl•− with CH4 − nXn (X = H,F, Cl; n = 0–3)

The effect of halogen‐substituent on hydrogen abstraction mechanisms was studied by applying density functional theory functional calculations to the gas‐phase reactions between CHCl^?? and CH_{4 ? n}X_n (X = H, F, Cl; n = 0–3), and it is found that a heavier X substituent in the substrate results in a greater stabilization of corresponding complex, a lower activation energy, a faster H‐abstraction reaction, and greater exothermicity. However, CH₄– reaction is more reactive than CH₃F– reaction under the same condition because of dominant π‐donation from the electronegative F atom. We also explored the reactivity difference for the seven reactions in terms of factors derived from bond order, second‐order perturbative energy, and activation strain model analysis. The rate constants are evaluated over a wide temperature range of 298–1000 K by the conventional transition state theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations

In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t ⁿ f(t, u(t)), where A is the generator of a C ₀-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C ₀-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S ^p-A defined similarly to the case of S ^p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A _u:= A ∩ BUC(ℝ, X) if n = 0 and to t ⁿ A _u ⊕ w _n C ₀ (ℝ, X) if n ∈ ℕ, where w _n(t) = (1 + |t|)ⁿ. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0. Dedicated to the memory of B. M. Levitan     

幻数尺寸Li-n-1,Lin,Li+ n+1(n=20,40)团簇的几何结构、电子与光学性质的第一性原理研究

基于密度泛函理论,采用第一性原理分子动力学模拟退火方法,对Li^-_n-1,Li_n,Li⁺_n+1 (n=20,40)的最低能量结构进行了全局搜索. 发现锂团簇的生长模式是以单个或多个嵌套的正多面体为核心,其余原子以五角锥为基本单元围绕核心生长. 基于最低能量结构的第一性原理电子结构计算得到锂团簇的分子轨道能级分布与无结构凝胶模型给出的电子壳层完全一致. 在总电 关键词： 团簇 电子结构 极化率 光吸收     

Electronic transport through a C60– n X n (X=N and B) molecular bridge

By applying the non-equilibrium Green's function (NEGF) technique, the Landauer–Buttiker theory and the Fisher–Lee formula, we have investigated the transport behavior of a C_{60– n} X _n (X=N, B) molecule coupled to two semi-infinite SWCNT electrodes. In this study, the coupling through the carbon, boron and nitrogen atoms to the electrodes will be considered. We study the effects of different contact geometries, the electron–phonon interaction and the number of doped atoms on the current value and negative differential resistance (NDR) behavior of C_{60– n} X _n . Our results indicate that the transmission coefficient and the NDR behavior of C_{60– n} X _n vary on changing n and X. Moreover, NDR behavior is observed in C_{60– n} X _n with different contacts and in C₆₀ with C₅ and C₆ contacts. C_{60– n} X _n molecules are suggested for the operation of devices with a nanoscale current.     

The Krichever map,vector bundles over algebraic curves,and Heisenberg algebras

We study the GrassmannianGr _x ⁿ consisting of equivalence classes of rankn algebraic vector bundles over a Riemann surfaceX with an holomorphic trivialization at a fixed pointp. Commutative subalgebras ofgl(n, H ),H being the ring of functions holomorphic on a punctured disc aboutp, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated togl(n, H ₎. One can obtain, by the Krichever map, points ofGr _x ⁿ (and solutions of mcKP) from coveringsf: YX and other geometric data. Conversely for every point ofGr _x ⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr _x ⁿ , an algebraic curve coveringX and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.The research was partially supported by US Army grant DAA L03-87-K-0110 and NSF grant DMS 9106938     

Anomalous concentration dependence of the coordination behavior of Cl− ion to Ln3+ ion (Ln3+ = rare‐earth ion) in anhydrous LnCl3 alcohol solutions

Raman spectroscopic measurements were carried out for the anhydrous LnCl₃·20ROH·XLiCl solutions (Ln³⁺ = La³⁺− Lu³⁺, X = 0–3; ROH = MeOH, EtOH, n‐PrOH) in the liquid state. The salt concentration (X) dependence of the wavenumber for the Ln–Cl stretching Raman band (ν_Ln–Cl) is examined in conjunction with the formation of chloro‐rare‐earth complexes. We have obtained very intriguing results including the fact that the chloro complexations of the middle rare‐earth ions (e.g. gadolinium, holmium ions, etc.) in the MeOH and EtOH solutions show peculiar behavior with regard to the salt concentration dependence: the ν_Ln–Cl wavenumber increases with the increasing chloride concentration. However, the ν_Ln–Cl wavenumbers of the light and heavy rare‐earth (e.g. lanthanum, lutetium, etc.) salt solutions show normal behavior; i.e. ν_Ln–Cl decreases with the increasing chloride concentration. On the other hand, in the n‐PrOH solutions, the ν_Ln–Cl frequency in the solutions of all the rare‐earth elements exhibits a normal behavior. We now present a possible mechanism for this anomalous concentration dependence of coordination of Cl⁻ ions to Ln³⁺ ions in anhydrous LnCl₃ alcohol solutions. Copyright © 2007 John Wiley & Sons, Ltd.