Polar Lattices Research Articles

Une forme dual-extrême irrationnelle

This paper presents the five-dimensional lattices having an automorphism of order five from the point of view of duality. The first example of an irrational pair of polar lattices extreme with respect to the product of their Hermite constants arises in this set. The complete classification of the subset of isodual lattices enables us to give a partial answer to a question of Conway and Sloane.

FCA-ESPRIT: a closed-form 2-D angle estimation algorithm for filled circular arrays with arbitrary sampling lattices

This article addresses the simultaneous estimation of the azimuth and elevation angles of multiple co-channel signals incident upon an array of antennas. A new closed-form algorithm that is applicable with filled circular arrays (FCAs) is presented. The algorithm, which is called FCA-ESPRIT, is applicable with any FCA geometry, e.g., FCAs constructed using rectangular, hexagonal, polar, or random sampling lattices. The only constraint imposed on the FCA geometry is that the spacing between adjacent sensors should be half a wavelength or less on average across the entire aperture. The algorithm provides automatically paired azimuth and elevation angle estimates by solving a set of ESPRIT-like equations wherein the unknowns have the form sin(/spl theta//sub i/)e/sup j/spl phi/(i)/ (where /spl theta//sub i/ and /spl phi//sub i/ are the elevation and azimuth angles of the ith source, respectively). The algorithm provides arrival angle estimates via a closed-form procedure-it does not require computationally expensive search or optimization procedures.

Multipolar Symmetry Patterns in Molecular Nonlinear Optics

Organic materials for quadratic nonlinear optics generally follow the basic pattern of strongly dipolar quasi one-dimensional intramolecular charge transfer molecules organized in macroscopic crystalline or statistical polar lattices. This restriction has been lifted by the introduction of the much broader class of multipolar materials whereby efficient two- and three-dimensional molecules can be fruitfully exploited in self assembled or externally engineered multipolar macroscopic structures. At the molecular level, polarized harmonic scattering permits to evaluate the invariant irreducible components of the molecular quadratic tensor. Its anisotropy and dispersion can be accounted for by a three-quantum model in agreement with linear spectroscopy on poled samples, whereas the validity of the two-level model is restricted to one-dimensional systems. Permanent macroscopic multipolar organization can be implemented by purely optical photoinduced processes. Adequate choice of the polarization of "write" beams permits to imprint any desired symmetry pattern onto the (non)linear material. Photonic engineering thus complements and considerably broadens the more traditional scope of molecular engineering.

Inequalities for convex bodies and polar reciprocal lattices inR n II: Application ofK-convexity

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inRn. The Minkowski functionalźn ofU, the polar bodyU0, the dual latticeL*, the covering radius μ(L, U), and the successive minima źi,i=1, ź,n, are defined in the usual way. Let $$\mathcal{L}_n $$ be the family of all lattices inRn. Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill {\text{ }}L \in \mathcal{L}_n \hfill \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $$L \in \mathcal{L}_n $$ andxźRn/(L+U), someyźL* with źyźU0źsd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC1,C2,C3 are some numerical constants andK(RUn) is theK-convexity constant of the normed space (Rn, źźU). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovasz [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.

A deviation from the Coulomb law for interacting microscopic impurities in a perovskite-like lattice

Formulae for calculating the electric field produced by microscopic impurities in perovskite-type oxides have been derived. By way of illustration, dipole and monopole impurities in , and were considered. It was found that the lattice polarization leads to a gigantic enhancement of the interaction between dipole impurities that can explain many phenomena in polar lattices.

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR n . The Minkowski functional ∥ ∥ U ofU, the polar bodyU 0, the dual latticeL *, the covering radius μ(L, U), and the successive minima λ i (L,U)i=1,...,n, are defined in the usual way. Let ℒ n be the family of all lattices inR n . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ n andu∈R n /(L+U), somev∈L * with ∥v∥ V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U 0), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U 0) are less thanCn logn for some numerical constantC.

Evidence for chemical processing of precometary icy grains in circumstellar environments of pre-main-sequence stars

We report the detection of a broad absorption feature near 2166/cm in the spectrum of the Taurus cloud cource Elias 18. This pre-main-sequence source is the second in Taurus, the third in our survey, and the fifth known in the sky to show the broad 2166/cm absorption feature. Of equal importance, this feature is not seen toward several other embedded sources in our survey, nor is it seen toward the source Elias 16, located behind the Taurus cloud. Laboratory experiments with interstellar ice analogs show that such a feature is associated with a complex C triple bonded to N containing compound (called X(C triple bonded to N)) that results from high-energy processing (ultraviolet irradiation or ion bombardment) of simple ice components into more complex, organic components. We find a nonlinear anticorrelation between the abundance of X(C triple bonded to N) and frozen CO components in nonpolar lattices. We find no correlation between the abundance of X(C triple bonded to N) and frozen CO in polar lattices. Because the abundances of frozen CO and H2O are strongly correlated with each other and with visual extinction toward sources embedded in and located behind the Taurus molecular cloud, these ice components usually are associated with intracloud material. Our results indicate that X(C triple bonded to N) molecules result from chemical processing of dust grains dominated by nonpolar icy mantles in the local environments of pre-main-sequence stars. Such processing of icy grains in the early solar system may be an important source of organic compounds observed in minor solar system bodies. The delivery of these organic compounds to the surface of the primitive Earth through comet impacts may have provided the raw materials for prebiotic chemistry.

Absolute deformation potentials in semiconductors

The absolute deformation potentials (ADP's) in dielectrics depend in general on shape and surface effects, owing to the long-range nature of Coulomb interactions. We prove that, for a given direction of uniaxial strain, he ADP's of crystalline solids are well defined in terms of bulk properties in Bravais and diamond lattices, while they are not in polar lattices. We give an explicit expression for an arbitrary strain orientation and calculate them in Si and Ge using ab initio methods.

Sur un problème de dualité lié aux sphères en géométrie des nombres

We study extremal pairs of polar lattices with respect to the product of the norms of minimal vectors. This notion of a “dual-extreme” lattice is compared with the usual notion of an extreme lattice. This study was suggested by the problem of twin-classes in number fields raised by Zimmert in an analytical context.

Polar lattices from the point of view of nuclear spaces

Polar lattices from the point of view of nuclear spaces

A result on two-dimentional polar lattices

The Journal of the National Science Foundation of Sri Lanka publishes the results of research in all aspects of Science and Technology. The journal also has a website at http://www.nsf.gov.lk/. 2021 Impact Factor: 0.682The JNSF provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.

The lattice structure of pseudo-random number generators

The pairs, triples, etc. from most congruential pseudo-random number generators are known to lie on a lattice, and the ‘uniformity’ of these lattices is reflected in the quality of the output of the generator. Various characteristics of the lattices have been proposed as summaries of the quality of a generator, including the so-called lattice and spectral tests. This paper exploits the concept of polar lattices to show that these charac­terizations are essentially equivalent, and that they can be found to an approximation sufficient for assessing the quality of the generator without extensive searches. Checking generators is important, for many of those provided on small computers are inadequate for serious work.

Positive points in polar lattices II

AbstractThe problem of positive points in polar lattices, discussed by Hossain and Worley for the distance functions Ft(x1, x2) = │x1│+│tx2 │ and , is considered for a general distance function F. Best possible results are obtained.

Locally polar lattices

Lattices are studied and characterized in which all intervals above points are polar spaces.

Positive points in polar lattices

AbstractThe authors investigate max min μF(x) F(y)for two standard distance functions F in R2, where μ denotes the area of {x ∈ R2; F(x)≤ 1}, the maximum is over all (geometric) lattices Λ in R2 and the minimum is over all positive points x ∈ Λ and y ∈ Λ* (the polar lattice of Λ). An application is given to a problem on fractional parts.

Thermodynamic and dielectric properties of polar lattices

Monte Carlo results are presented for the free energy and dielectric constant of systems of permanent dipoles disposed on the sites of simple cubic and face-centred cubic lattices as functions of the parameter ϑ = μ2ρ/kT. Alternative schemes are considered for taking account of the long-range character of the dipolar interaction, based on an Ewald-type sum or a reaction-field approximation; use of such a procedure is essential if the system is to have the correct dielectric properties. Different methods of calculating the dielectric constant are also compared. It is shown that the two methods of treating the long-range contribution to the energy lead to similar results for the dielectric constant, but use of the Ewald method apparently introduces a small but systematic error. The results on both free energy and dielectric constant are discussed in the light of predictions of a number of analytical approaches. In the case of the simple cubic lattice the general trend in the computed dielectric constant as ...

Polaronic Impurity Hopping Conduction

AbstractImpurity hopping conduction in polar lattices and amorphous substances is investigated by application of the canonical transformation of the small polaron theory. In the case of acoustic coupling only, the results of Miller and Abrahams and Gosar for one‐phonon processes are recovered and compared with the contributions of multi‐phonon processes. In the polar case the assisting roles of acoustic and polar phonons are discussed seperately at temperatures T > θ/2 and T <θ/2 (θ being the Debye temperature). The type of the temperature dependence of the conductivity is determined.