Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

Gauge theory of a group of diffeomorphisms. I. General principles

Any (N+M)-parameter Lie group G with an N-parameter subgroup H can be realized as a global group of diffeomorphisms on an M-dimensional base space B, with representations in terms of transformation laws of fields on B belonging to linear representations of H. The gauged generalization of the global diffeomorphisms consists of general diffeomorphisms (or coordinate transformations) on a base space together with a local action of H on the fields. The particular applications of the scheme to space-time symmetries is discussed in terms of Lagrangians, field equations, currents, and source identities. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

High-Rate, Multisymbol-Decodable STBCs From Clifford Algebras

It is well known that space-time block codes (STBCs) obtained from orthogonal designs (ODs) are single-symbol decodable (SSD) and from quasi-orthogonal designs (QODs) are double-symbol decodable (DSD). However, there are SSD codes that are not obtainable from ODs and DSD codes that are not obtainable from QODs. In this paper, a method of constructing g-symbol decodable (g-SD) STBCs using representations of Clifford algebras are presented which when specialized to g = 1, 2 gives SSD and DSD codes, respectively. For the number of transmit antennas 2(a) the rate (in complex symbols per channel use) of the g-SD codes presented in this paper is a+1-g/2(a-9). The maximum rate of the DSD STBCs from QODs reported in the literature is a/2(a-1) which is smaller than the rate a-1/2(a-2) of the DSD codes of this paper, for 2(a) transmit antennas. In particular, the reported DSD codes for 8 and 16 transmit antennas offer rates 1 and 3/4, respectively, whereas the known STBCs from QODs offer only 3/4 and 1/2, respectively. The construction of this paper is applicable for any number of transmit antennas. The diversity sum and diversity product of the new DSD codes are studied. It is shown that the diversity sum is larger than that of all known QODs and hence the new codes perform better than the comparable QODs at low signal-to-noise ratios (SNRs) for identical spectral efficiency. Simulation results for DSD codes at variousspectral efficiencies are provided.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.

Derivation of the decoupling relations for composite models from anomaly constraints and n-independence

It is shown, in the composite fermion models studied by 't Hooft and others, that the requirements of Adler-Bell-Jackiw anomaly matching and n-independence are sufficient to fix the indices of composite representations. The third requirement, namely that of decoupling relations, follows from these two constraints in such models and hence is inessential.

Determination of baryon magnetic moments from QCD sum rules

The magnetic moment μB of a baryon B with quark content (aab) is written as μB=4ea(1+δB)eħ/2cMB, where ea is the charge of the quark of flavor type a. The experimental values of δB have a simple pattern and have a natural explanation within QCD. Using the ratio method, the QCD sum rules are analyzed and the values of δB are computed. We find good agreement with data (≊10%) for the nucleons and the Σ multiplet while for the cascade the agreement is not as good. In our analysis we have incorporated additional terms in the operator-product expansion as compared to previous authors. We also clarify some points of disagreement between the previous authors. External-field-induced correlations describing the magnetic properties of the vacuum are estimated from the baryon magnetic-moment sum rules themselves as well as by independent spectral representations and the results are contrasted.

New light on the optical equivalence theorem and a new type of discrete diagonal coherent state representation

In many instances we find it advantageous to display a quantum optical density matrix as a generalized statistical ensemble of coherent wave fields. The weight functions involved in these constructions turn out to belong to a family of distributions, not always smooth functions. In this paper we investigate this question anew and show how it is related to the problem of expanding an arbitrary state in terms of an overcomplete subfamily of the overcomplete set of coherent states. This provides a relatively transparent derivation of the optical equivalence theorem. An interesting by-product is the discovery of a new class of discrete diagonal representations.

Matrices of finite Lorentz transformations in a noncompact basis. III. Completeness relation for O(2, 1)

A parametrization of the elements of the three-dimensional Lorentz group O(2, 1), suited to the use of a noncompact O(1, 1) basis in its unitary representations, is derived and used to set up the representation matrices for the entire group. The Plancherel formula for O(2, 1) is then expressed in this basis.

Algebraic distributed space-time codes with low ML decoding complexity

"Extended Clifford algebras" are introduced as a means to obtain low ML decoding complexity space-time block codes. Using left regular matrix representations of two specific classes of extended Clifford algebras, two systematic algebraic constructions of full diversity Distributed Space-Time Codes (DSTCs) are provided for any power of two number of relays. The left regular matrix representation has been shown to naturally result in space-time codes meeting the additional constraints required for DSTCs. The DSTCs so constructed have the salient feature of reduced Maximum Likelihood (ML) decoding complexity. In particular, the ML decoding of these codes can be performed by applying the lattice decoder algorithm on a lattice of four times lesser dimension than what is required in general. Moreover these codes have a uniform distribution of power among the relays and in time, thus leading to a low Peak to Average Power Ratio at the relays.

Block Convolution Using Discrete Trigonometric Transforms and Discrete Fourier Transform

We derive expressions for convolution multiplication properties of discrete cosine transform II (DCT II) starting from equivalent discrete Fourier transform (DFT) representations. Using these expressions, a method for implementing linear filtering through block convolution in the DCT II domain is presented. For the case of nonsymmetric impulse response, additional discrete sine transform II (DST II) is required for implementing the filter in DCT II domain, where as for a symmetric impulse response, the additional transform is not required. Comparison with recently proposed circular convolution technique in DCT II domain shows that the proposed new method is computationally more efficient.

Boson-fermion duality in SU(m vertical bar n) supersymmetric Haldane-Shastry spin chain

By using the Y(gl(m|n)) super Yangian symmetry of the SU(m|n) supersymmetric Haldane-Shastry spin chain, we show that the partition function of this model satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom. As a byproduct of this study of the duality relation, we find a novel combinatorial formula for the super Schur polynomials associated with some irreducible representations of the Y(gl(m|n)) Yangian algebra. Finally, we reveal an intimate connection between the global SU(m|n) symmetry of a spin chain and the boson-fermion duality relation. (C) 2007 Elsevier B.V. All rights reserved.