RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang - Baxter constraint for $R$ or the braid equation for $\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2), GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, one has the braid equation. Arbitray $K$ corresponds to a class (conserving the group relations independent of $K$) of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $\hat{R}(K;p,q)$ can satisfy $\hat{R}^2 = 1$ outside braid equation (BE) subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each 3-parameter $\hat{R}$. The role of $K$ in noncommutative geometries of the $(K;p,q)$,$(K;g,h)$ and $(K;q,h)$ deformed planes is studied. K is found to introduce a "soft symmetry breaking", preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.Comment: Latex, 17 pages, minor change

Higher Dimensional Multiparameter Unitary and Nonunitary Braid Matrices: Even Dimensions

A class of $(2n)^2\times(2n)^2$ multiparameter braid matrices are presented for all $n$ $(n\geq 1)$. Apart from the spectral parameter $\theta$, they depend on $2n^2$ free parameters $m_{ij}^{(\pm)}$, $i,j=1,...,n$. For real parameters the matrices $R(\theta)$ are nonunitary. For purely imaginary parameters they became unitary. Thus a unification is achieved with odd dimensional multiparameter solutions presented before.Comment: 07 page

The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra

Using a contraction procedure, we construct a twist operator that satisfies a shifted cocycle condition, and leads to the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra. The corresponding universal ${\cal R}_{h}(y)$ matrix obeys a Gervais-Neveu-Felder equation associated with the U_{h;y}(sl(2)) algebra. For a class of representations, the dynamical Yang-Baxter equation may be expressed as a compatibility condition for the algebra of the Lax operators.Comment: Latex, 9 pages, no figure

Design of a "Digital Atlas Vme Electronics" (DAVE) Module

ATLAS-SCT has developed a new ATLAS trigger card, 'Digital Atlas Vme Electronics' ("DAVE"). The unit is designed to provide a versatile array of interface and logic resources, including a large FPGA. It interfaces to both VME bus and USB hosts. DAVE aims to provide exact ATLAS CTP (ATLAS Central Trigger Processor) functionality, with random trigger, simple and complex deadtime, ECR (Event Counter Reset), BCR (Bunch Counter Reset) etc. being generated to give exactly the same conditions in standalone running as experienced in combined runs. DAVE provides additional hardware and a large amount of free firmware resource to allow users to add or change functionality. The combination of the large number of individually programmable inputs and outputs in various formats, with very large external RAM and other components all connected to the FPGA, also makes DAVE a powerful and versatile FPGA utility cardComment: 8 pages, 4 figures, TWEPP-2011; E-mail: [email protected]

Twist maps for non-standard quantum algebras and discrete Schrodinger symmetries

The minimal twist map introduced by B. Abdesselam, A. Chakrabarti, R. Chakrabarti and J. Segar (Mod. Phys. Lett. A 14 (1999) 765) for the non-standard (Jordanian) quantum sl(2,R) algebra is used to construct the twist maps for two different non-standard quantum deformations of the (1+1) Schrodinger algebra. Such deformations are, respectively, the symmetry algebras of a space and a time uniform lattice discretization of the (1+1) free Schrodinger equation. It is shown that the corresponding twist maps connect the usual Lie symmetry approach to these discrete equations with non-standard quantum deformations. This relationship leads to a clear interpretation of the deformation parameter as the step of the uniform (space or time) lattice.Comment: 16 pages, LaTe

A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula

We provide yet another proof of the classical Lagrange-Good multivariable inversion formula using techniques of quantum field theory.Comment: 9 pages, 3 diagram

Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems

It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many--fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Omega (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many--fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short--range interaction which can be arbitrarily strong, provided Omega is chosen large enough. Moreover, we give - for the first time - nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.Comment: 29 pages LaTe

Tensor Operators for Uh(sl(2))

Tensor operators for the Jordanian quantum algebra Uh(sl(2)) are considered. Some explicit examples of them, which are obtained in the boson or fermion realization, are given and their properties are studied. It is also shown that the Wigner-Eckart's theorem can be extended to Uh(sl(2)).Comment: 11pages, LaTeX, to be published in J. Phys.

Boson representations, non-standard quantum algebras and contractions

A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of $sl(2,\R)$ which directly provides its infinite and finite dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of $sl(2,\R)$ is introduced and contracted to the non-standard quantum (1+1) Poincar\'e representations. Likewise, a quantum extended Hopf $sl(2,\R)$ algebra is constructed and the Jordanian $q$-oscillator algebra representations are obtained from it by means of another contraction procedure.Comment: 21 pages, LaTeX; two new references adde

Surface site density, silicic acid retention and transport properties of compacted magnetite powder

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