An algebraic theory of functional and multivalued dependencies in relational databases

Computation of the dependency basis is the fundamental step in solving the membership problem for functional dependencies (FDs) and multivalued dependencies (MVDs) in relational database theory. We examine this problem from an algebraic perspective. We introduce the notion of the inference basis of a set M of MVDs and show that it contains the maximum information about the logical consequences of M. We propose the notion of a dependency-lattice and develop an algebraic characterization of inference basis using simple notions from lattice theory. We also establish several interesting properties of dependency-lattices related to the implication problem. Founded on our characterization, we synthesize efficient algorithms for (a): computing the inference basis of a given set M of MVDs; (b): computing the dependency basis of a given attribute set w.r.t. M; and (c): solving the membership problem for MVDs. We also show that our results naturally extend to incorporate FDs also in a way that enables the solution of the membership problem for both FDs and MVDs put together. We finally show that our algorithms are more efficient than existing ones, when used to solve what we term the ‘generalized membership problem’.

Information theory and one-dimensional disordered conductors

A novel method is proposed to treat the problem of the random resistance of a strictly one-dimensional conductor with static disorder. It is suggested, for the probability distribution of the transfer matrix of the conductor, the distribution of maximum information-entropy, constrained by the following physical requirements: 1) flux conservation, 2) time-reversal invariance and 3) scaling, with the length of the conductor, of the two lowest cumulants of ζ, where = sh2ζ. The preliminary results discussed in the text are in qualitative agreement with those obtained by sophisticated microscopic theories.

On the shear deformation theory for dynamic analysis of beams

Timoshenko's shear deformation theory is widely used for the dynamical analysis of shear-flexible beams. This paper presents a comparative study of the shear deformation theory with a higher order model, of which Timoshenko's shear deformation model is a special case. Results indicate that while Timoshenko's shear deformation theory gives reasonably accurate information regarding the set of bending natural frequencies, there are considerable discrepancies in the information it gives regarding the mode shapes and dynamic response, and so there is a need to consider higher order models for the dynamical analysis of flexure of beams.

Paraxial-wave optics and relativistic front description .1. The scalar theory

With the extension of the work of the preceding paper, the relativistic front form for Maxwell's equations for electromagnetism is developed and shown to be particularly suited to the description of paraxial waves. The generators of the Poincaré group in a form applicable directly to the electric and magnetic field vectors are derived. It is shown that the effect of a thin lens on a paraxial electromagnetic wave is given by a six-dimensional transformation matrix, constructed out of certain special generators of the Poincaré group. The method of construction guarantees that the free propagation of such waves as well as their transmission through ideal optical systems can be described in terms of the metaplectic group, exactly as found for scalar waves by Bacry and Cadilhac. An alternative formulation in terms of a vector potential is also constructed. It is chosen in a gauge suggested by the front form and by the requirement that the lens transformation matrix act locally in space. Pencils of light with accompanying polarization are defined for statistical states in terms of the two-point correlation function of the vector potential. Their propagation and transmission through lenses are briefly considered in the paraxial limit. This paper extends Fourier optics and completes it by formulating it for the Maxwell field. We stress that the derivations depend explicitly on the "henochromatic" idealization as well as the identification of the ideal lens with a quadratic phase shift and are heuristic to this extent.

Gauge theory of a group of diffeomorphisms. II. The conformal and de Sitter groups

The extension of Hehl's Poincaré gauge theory to more general groups that include space-time diffeomorphisms is worked out for two particular examples, one corresponding to the action of the conformal group on Minkowski space, and the other to the action of the de Sitter group on de Sitter space, and the effect of these groups on physical fields.

Simple Coding Scheme For Modular Arithmetic

A dual representation scheme for performing arithmetic modulo an arbitrary integer M is presented. The coding scheme maps each integer N in the range 0 <= N < M into one of two representations, each being identified by its most significant bit. The encoding of numbers is straightforward and the problem of checking for unused combinations is eliminated.

Is one-parameter scaling theory of localisation consistent? Comment on Schuh's comment

It is maintained that the one-parameter scaling theory is inconsistent with the physics of Anderson localisation.

A simplified approach to a three-part Wiener-Hopf problem arising in diffraction theory

A direct and simple approach, utilizing Watson's lemma, is presented for obtaining an approximate solution of a three-part Wiener-Hopf problem associated with the problem of diffraction of a plane wave by a soft strip.

Quantum-Ohmic Resistance Fluctuation In Disordered Conductors - An Invariant Imbedding Approach

It is now well known that in extreme quantum limit, dominated by the elastic impurity scattering and the concomitant quantum interference, the zero-temperature d.c. resistance of a strictly one-dimensional disordered system is non-additive and non-self-averaging. While these statistical fluctuations may persist in the case of a physically thin wire, they are implicitly and questionably ignored in higher dimensions. In this work, we have re-examined this question. Following an invariant imbedding formulation, we first derive a stochastic differential equation for the complex amplitude reflection coefficient and hence obtain a Fokker-Planck equation for the full probability distribution of resistance for a one-dimensional continuum with a Gaussian white-noise random potential. We then employ the Migdal-Kadanoff type bond moving procedure and derive the d-dimensional generalization of the above probability distribution, or rather the associated cumulant function –‘the free energy’. For d=3, our analysis shows that the dispersion dominates the mobilitly edge phenomena in that (i) a one-parameter B-function depending on the mean conductance only does not exist, (ii) an approximate treatment gives a diffusion-correction involving the second cumulant. It is, however, not clear whether the fluctuations can render the transition at the mobility edge ‘first-order’. We also report some analytical results for the case of the one dimensional system in the presence of a finite electric fiekl. We find a cross-over from the exponential to the power-low length dependence of resistance as the field increases from zero. Also, the distribution of resistance saturates asymptotically to a poissonian form. Most of our analytical results are supported by the recent numerical simulation work reported by some authors.

Theory For The Anomalous Hall Constant Of Mixed-Valence Systems

We show that the large anomalous Hall constants of mixed-valence and Kondo-lattice systems can be understood in terms of a simple resonant-level Fermi-liquid model. Splitting of a narrow, orbitally unquenched, spin-orbit split, f resonance in a magnetic field leads to strong skew scattering of band electrons. We interpret both the anomalous signs and the strong temperature dependence of Hall mobilities in CeCu2Si2, SmB6, and CePd3 in terms of this theory.

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.

Icosahedral DNA Nanocapsules by Modular Assembly

The construction of well-defined 3D architectures is one of the greatest challenges of self-assembly. Nanofabrication through molecular self-assembly has resulted in the formation of DNA polyhedra with the connectivities of cubes,[1] tetrahedra,[2,3] octahedra,[4, 5] dodecahedra,[3] and buckminsterfullerene.[ 3] DNA polyhedra could also function as nanocapsules and thereby enable the targeted delivery of entities encapsulated from solution. Key to realizing this envisaged function is the construction of complex polyhedra that maximize encapsulation volumes while preserving small. pore size.

Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Analysis of drop formation at conical tips : 1. Theory

The phenomenon of drop formation at conical tips under near zero flow conditions has been investigated using a theoretical approach. The analysis permits the prediction of drop profile and drop volume, until the onset of instability. A semiempirical approach based on the similarity of drop shapes has been adopted to predict the detaching drop volumes at conical tips. The effects of base diameter of the cone, cone angle, interfacial tension, and the densities of the drop and the surrounding fluid on the maximum and detached drop volumes are predicted.