Tensor product of difference posets and effect algebras

A tensor product of difference posets and/or, equivalently, of effect algebras, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. The proof uses the notion of D-test spaces generalizing test spaces of Randall and Foulis. In particular, we show that a tensor product for difference posets with a nonempty system of probability measures exists.     

A physically important class of integrals expressed as a parameter derivative

The paper deals with a class of integrals, the integrands of which contain the square of a solution of a second-order linear, ordinary differential equation. Such integrals often arise in quantum mechanics as normalization integrals or expectation values. A generalized, unified procedure for rewriting such an integral, associated with a differential equation of the Sturm-Liouville type with unspecified boundary conditions, as a parameter derivative is presented. The formula thus obtained can be used for the evaluation of various integrals of physical interest. As an application we present a simplified derivation of a formula given by de Alfaro and Regge, in which the quantal normalization integral is expressed in terms of the Jost function. Other applications to integrals involving special functions and to integrals associated with the one-dimensional Schrödinger equation are also presented. Furthermore, it is explained why an approximate formula for expectation values is much more accurate than one can expect from the usual, crude derivation of it, and why certain attempts to improve that derivation have failed.     

On extending and optimising the direct product decomposition

Abstract

The concept of the direct product decomposition (DPD) is extended to arbitrary tensors while maintaining the same theoretical reduction in storage and computation. Additionally, the structure of the DPD as introduced by Gauss and Stanton is shown to be but one of a family of direct product decompositions which may be visualised using graphs. One particular member of this family is also shown to be critically important in relating the DPD and symmetry blocking approaches. Lastly, an implementation of tensor contraction using this extended DPD based on recent work in dense tensor contraction is presented, showing how the particular DPD used to represent the tensors in memory or on disk may be divorced from the optimal DPD used for a particular tensor contraction. The performance of the new algorithm is benchmarked by interfacing with the CFOUR programme suite, where significant speedups for CCSD calculations are observed.     

Tensor product composition of algebras in Bose and Fermi canonical formalism

We consider a graded algebra with two products (σ, α) over anε-factor commutation. One of the products (σ) isε-commutative, but, in general non-associative; and the other (α) is a graded Lieε-product and a gradedε-derivative with respect to the first (σ). Using the obvious mathematical condition, namely—the tensor product of two graded algebras with the sameε-factors is another with the sameε-factor, we determine the complete structure of a two-product (σ, α) graded algebra. When theε-factors are taken to be unity and the gradation structure is ignored, we recover the algebras of the physical variables of classical and quantum systems, considered by Grgin and Petersen. With the retention of the gradation structure and the possible choice of two ε-factors we recover the algebras of the canonical formalism of boson and fermion systems for the above classical and quantum theories. We also recover in this case the algebra of anticommutative classical systems considered by Martin along with its quantum analogue.     

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and ? function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.     

A remark on matrix product operator algebras,anyons and subfactors

Letters in Mathematical Physics - We show that the mathematical structures in a recent work of...     

Sonolysis induced decomposition of metal carbonyls: kinetics and product characterization

The decomposition kinetics of Fe(CO)₅ and Mo(CO)₆ induced by sonolysis in hexadecane solvent was studied as a function of temperature (303–343 K) under an inert atmosphere. The decomposition data, obtained over at least two half lives in most of the runs, yielded first-order rate constant (k) values with correlation co-efficient (R²)>0.95. The products were characterized by various spectroscopic techniques. The transmission electron microscopy (TEM) yielded images from which the mean particle diameter (MPD) of 10 nm for Fe and <3 nm for Mo were estimated. The generation of amorphous Fe and semi-crystalline Mo particles was determined from line broadening and corresponding d-spacing values in the X-ray diffraction (XRD) spectra. The XAFS/XANES data were consistent with the production of Fe(0) metal but carbided Mo (Mo₂C). The one-step production of high-yield pyrophoric products demonstrated the applicability of sonolysis to effectively produce gram-quantity of zero-valent metals.     

On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

Letters in Mathematical Physics - In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple...     

The elliptic genus of Calabi-Yau 3- and 4-folds,product formulae and generalized Kac-Moody algebras

In this paper the elliptic genus for a general Calabi-Yau 4-fold is derived. The recent work of Kawai calculating N = 2 heterotic string one-loop threshold corrections with a Wilson line turned on is extended to a similar computation where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all cases there seems to be a generalized Kac-Moody algebra involved, whose denominator formula appears in the result.     

Intermediate product identification for ammonia decomposition at Ni (110)

Ammonia decomposition at Ni(110) has been identified to proceed via NH₃(ad) → NH₂(ad) → NH(ad) → N + H. The decomposition activation of NH is determined to be 47 kcal/mol, suggesting an amazing stability of the NiNH bond. Decomposition of NH₂ occurs up from about 350 K; no kinetic data can be given yet. NH₃ decomposition is found to proceed slower than NH₃ desorption at least below 300 K.     

Covering and gluing of algebras and differential algebras

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C^*-algebra isomorphic to a certain Podleś sphere, as well as the gluing of U_q^1/2(sl₂)-covariant differential calculi on the discs.     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

The decomposition of the product of a momentum-zero- and a mass-zero-representation of the Poincaré group

The irreducible unitary representations of a twofold covering groupE(2) of the euclidean group of the plane are set up. The product of two irreducible unitary representations ofE(2) and the irreducible unitary representations of SL(2,C) restricted to a subgroupE～(2) isomorphic toE(2) are reduced to irreducible unitary components. The results are applied to the decomposition of the product of a momentum-zero- and a mass-zero-representation of the proper orthochronous quantummechanical Poincaré group.     

Some tips on the decomposition of tensor product representations of compact connected lie groups

Let ?₁, ?₂, ?₃, ?₄ be irreducible representations of a compact connected semisimple Lie group G with highest weights Λ₁, Λ₂, Λ₃, Λ₄, respectively. Let ?₁?₂ (resp. ?₃?₄) be irreducible representations of G with highest weights Λ₁·Λ₂ (resp. Λ₃·Λ₄). It is assumed that one knows the Clebsch-Gordan series of ?₁??₃ and ?₂??₄ (resp. of S²?₁, S²?₂, A²?₁ and A²?₂). Then we formulate a result (Theorem 2) which gives information on the decomposition of ?₁?₂??₃?₄ (resp. of S²(?₁?₂) and A²(?₁?₂)). Though this result is not complete, it is useful because it delivers the information very quickly and in a very simple manner.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

The influence of product decomposition on the probability of nonthermal transitions in charge transfer reactions

The paper considers charge transfer under substantially nonequilibrium conditions. A model taking into account the excitation of the high-frequency intramolecular vibrational mode of the products along with medium reorganization is suggested. A simple analytic equation for the probability of a nonthermal transition taking into account product vibrational relaxation was obtained within the framework of the stochastic approach. The fast relaxation of high-frequency vibrational modes was shown to be capable of substantially accelerating the reaction if the slope of the term of the products was smaller than or comparable with the slope of the term of the reagents in the vicinity of the intersection point.     

KAC-moody algebras,Yang-Baxter-Zamolodchikov-Faddeev algebras and integrable field theories

A large class of integrable two-dimensional field theories exhibit Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and Kac-Moody (KM) algebras. Examples of them are chiral fermionic models, sigma models and Wess-Zumino-Witten sigma models. With their help an explicit link is found between representations of YBZF and KM algebras.     

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.