Half Dimension Research Articles

Flat tori, lattices and bounds for commutative group codes

We show that commutative group spherical codes in R n , as introduced by D. Slepian, are directly related to flat tori and quotients of lattices. As consequence of this view, we derive new results on the geometry of these codes and an upper bound for their cardinality in terms of minimum distance and the maximum center density of lattices and general spherical packings in the half dimension of the code. This bound is tight in the sense it can be arbitrarily approached in any dimension. Examples of this approach and a comparison of this bound with Union and Rankin bounds for general spherical codes is also presented.

Extraction of higher-order coupling feature using three and one half dimension spectrum

In this paper, three and a half dimension spectrum is defined. Also, the frequency coupling properties of the spectrum are investigated by using different definitions of fifth-order cumulants of real or complex sinusoidal signals. It is shown that the three and a half dimension spectrum has ability to entirely suppress symmetrical distributed noise and no frequency coupling components, that employing three and a half dimension spectrum of complex signals can extract the attended quartic or quadratic-to-cubic frequency coupling components or procreant frequency components for coupling, and that the original frequency components and the procreant frequency components for coupling can be synchronously extracted using the three and a half dimension spectrum of real signals. Computer simulations with the underwater moving target-radiated data are used to demonstrate the performance of three and a half dimension spectrum of real or complex signals under a wide range of conditions.

Canonical bases for real representations of Clifford algebras

The well-known classification of the Clifford algebras Cl( r, s) leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For s ⩾ 1 representations of Cl( r, s) on R 2 N are obtained from representations on R N by adding two new generators while in passing from a representation of Cl( p, 0) on R N to a representation of Cl( r, 0) on R 2 N the number of generators that can be added is either 1, 2 or 4, according as the Clifford algebra represented on R N is of real, complex or quaternionic type. We have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of Cl( r, s) to a canonical form. Our algorithm for the transformation of the representations of Cl(8 d + c, 0), c ⩽ 7 to canonical forms is based on finding an abelian subalgebra of Cl(8 d + c, 0) and its invariant subspace. Computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions. The construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics.

Some New Results on L2 Cohomology of Negatively Curved Riemannian Manifolds

AbstractThe present paper is concerned with the study of the L2 cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial L2 cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

Visual Understanding of Metabolic Pathways Across Organisms Using Layout in Two and a Half Dimensions

Summary We propose a method for visualizing a set of related metabolic pathways across organisms using 2 1/2 dimensional graph visualization. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant differences among the pathways. The (dis)similarities between pathways are expressed by the Hamming distances of the underlying graphs which are used to compute a stacking order for the pathways. Layouts are determined by a global layout of the union of all pathway graphs using a variant of the proven Sugiyama approach for layered graph drawing. Our variant layout approach allows edges to cross if they appear in different graphs.

Simulation of the filling process in micro channels for polymeric materials

There is some evidence indicating that polymeric flows in micro channels differ significantly from those in macro geometries. As micro molding is attracting more attention these days, efforts need to be made to identify the significant factors that influence microscale polymeric flow behaviors and to develop new simulation schemes for micro molding. In this study, we have investigated the consequences of microscale phenomena, particularly size-dependent viscosity, wall slip and surface tension, on the filling process of polymeric materials into micro channels. The standard scheme of two and half dimensions for injection molding simulation was modified to include these microscale effects. With data currently available for polystyrene, the simulation results indicate the importance of employing size-dependent viscosity and wall slip to predict micro filling behaviors. It appears that wall slip should always occur in channels downsized to several micrometers or less, because the wall stress would otherwise be enormous. The surface tension effects turn out to be less important and can be neglected in micro injection molding in which high injection pressure is employed.

Ground penetrating radar study of iron age tombs in Southeastern Kazakhstan

AbstractAn international archaeological and geophysical expedition was carried out to study Iron Age tombs of the Saka culture in southeastern Kazakhstan. Single‐fold and multifold ground‐penetrating radar (GPR) surveys, in two and two and a half dimensions, focused on the identification of burial chambers beneath the mound‐shaped Saka tumuli, named kourgans. Primary objectives of the geophysical study were the testing and optimization of GPR techniques for studying this type of tomb and the reconstruction of subsurface images of use for the design of archaeological excavations. Single‐fold GPR grids with line spacing ranging between 50 and 200 cm were obtained at each of the 13 sites selected. After identification of the sectors of major potential interest from data processing performed directly in the field and interpretation of the single‐fold grids, multifold profiles with average 1200 per cent fold were designed and performed. Processing of the data focused on velocity analyses, pre‐stack coherent noise attenuation by means of FK and KL filters, stacking and analysis of the instantaneous attributes of the complex radar trace. Multifold GPR data processing techniques achieved a substantial increment in the signal to noise ratio at most sites, allowed accurate depth conversions of the stacked profiles (average deviation of estimated target location from the actual target was less than 5% of total depth) and provided indications to identify the characteristic response from the target of interest through the joint interpretation of pre‐stack gathers, velocity analysis and amplitude versus offset analysis. Interpretation of stacked sections and three‐dimensional models obtained from the single‐fold/multifold grid allowed the identification of a burial chamber located at an estimated depth of 140–150 cm, which was excavated successfully by the archaeological team. Further reflectors with similar characteristics are shown from other sites, which will be excavated by the next archaeological expeditions to the area. Copyright © 2001 John Wiley & Sons, Ltd.

Assessing and characterizing inter-and intra-die variation using a statistical metrology framework: A CMP case study

As device dimensions continue to shrink below half micron dimensions and the demand for both yield and performance rises, the assessment and control of process and device variation becomes critical. Yield loss from parametric sources will rise unless process margins are characterized to a higher degree than currently available [1]. Process and device variation in VLSI manufacturing occurs with different scope and extent. Process perturbation, drifts, and equipment factors result in lot-to-lot variation, while within-wafer variation may stem from process non-uniformities. While substantial work has focused on the control of lot and wafer-level variation [2], die-level variation has received little attention despite the potential impact on circuit performance [3]. This spatial variation has both systematic and random components. Currently, device variation is often lumped into a single large distribution as part of a “worstcase” approach to modeling. A methodology is needed to transform this large distribution into deterministic components that can be compensated or designed around. We present statistical metrology as a methodology to assess variation and parameterize any resulting circuit and process impact. Variation assessment is achieved by identifying and explicitly modeling the individual sources of variation. The resulting quantification of variation components enables better process modeling, facilitates process control, and provides realistic data for statistical circuit modeling. In this paper we use statistical metrology methods to study inter-level dielectric (ILD) thickness variation for two representative chemical-mechanical polishing (CMP) processes. We suggest that three phases of experimental design are appropriate to (1) identify important factors; (2) construct explicit variation models, and (3) quantify the impact of variation in realistic chip environments. We present a case study in which these phases as well as new modeling methods, in particular a modified form of repeated measure ANOVA (Analysis of Variance), are used to model dieand waferlevel variation for representative CMP/ILD processes. STATISTICAL METROLOGY METHODOLOGY

A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kahler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let \(\cal q\) (resp. \({\Bbb R} P^n\)) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \({\Bbb C} P^n\) of constant holomorphic sectional curvature 4\( \lambda \). We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) \(\leq \) volume (\(\cal q\))/ volume \(({\Bbb C} P^n)\) (resp. \(\leq \) volume \(({\Bbb R} P^n)\) / volume \(({\Bbb C} P^n)\)), then there is a holomorphic isometry between M and \({\Bbb C} P^n\) taking P isometrically onto \(\cal q\) (resp. \({\Bbb R} P^n\)). We also classify the Kahler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \({\Bbb R} P^n_r\) of radius r around \({\Bbb R} P^n\) in \({\Bbb C} P^n\) and have the first Dirichlet eigenvalue not lower than that of \({\Bbb R} P^n_r\).

Kinetic and Structural Evolution of Self‐gravitating, Magnetized Clouds: 2.5‐dimensional Simulations of Decaying Turbulence

The molecular component of the Galaxy is comprised of turbulent, magnetized clouds, many of which are self-gravitating and form stars. To develop an understanding of how these clouds' kinetic and structural evolution may depend on their level of turbulence, mean magnetization, and degree of self-gravity, we perform a survey of direct numerical MHD simulations in which three parameters are independently varied. Our simulations consist of solutions to the time-dependent MHD equations on a two-dimensional grid with periodic boundary conditions; an additional half dimension is also incorporated as dependent variables in the third Cartesian direction. Two of our survey parameters, the mean magnetization parameter β≡c2sound/v2Alfven and the Jeans number nJ≡Lcloud/LJeans, allow us to model clouds that either meet or fail conditions for magneto-Jeans stability and magnetic criticality. Our third survey parameter, the sonic Mach number ≡σvelocity/csound, allows us to initiate turbulence of either sub- or super-Alfvenic amplitude; we employ an isothermal equation of state throughout. We evaluate the times for each cloud model to become gravitationally bound and measure each model's kinetic energy loss over the fluid-flow crossing time. We compare the evolution of density and magnetic field structural morphology and quantify the differences in the density contrast generated by internal stresses for models of differing mean magnetization. We find that the values of β and nJ, but not the initial Mach number , determine the time for cloud gravitational binding and collapse: for mean cloud density nH2=100 cm-3, unmagnetized models collapse after ~5 Myr, and magnetically supercritical models generally collapse after 5-10 Myr (although the smallest magneto-Jeans stable clouds survive gravitational collapse until t~15 Myr), while magnetically subcritical clouds remain uncollapsed over the entire simulations; these cloud collapse times scale with the mean density as tg∝n−½H2. We find, contrary to some previous expectations, less than a factor of 2 difference between turbulent decay times for models with varying magnetic field strength; the maximum decay time, for B~14 μG and nH2=100 cm-3, is 1.4 flow crossing times tcross=L/σvelocity (or 8 Myr for typical giant molecular cloud parameters). In all models we find turbulent amplification in the magnetic field strength up to at least the level βpert≡c2sound/δv2Alfven=0.1, with the turbulent magnetic energy between 25% and 60% of the turbulent kinetic energy after one flow crossing time. We find that for non-self-gravitating stages of evolution, when clouds have =5-10, the mass-averaged density contrast magnitudes log(ρ/) are in the range 0.2-0.5, with the contrast increasing both toward low and high β. Although our conclusions about density statistics may be affected by our isothermal assumption, we note that only the more strongly magnetized models appear to be consistent with estimates of clump/interclump density contrasts inferred in Galactic giant molecular clouds.

Discrete Schrodinger operators and topology

This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic Wronskian is a 1-cycle in the graph. This is a vector-valued symplectic 2-form on the space of solutions. This construction was applied to the Scattering Theory on the graphs with finite number of tails. The asymptotic values of solutions is a Lagrangian Plane of half dimension. This property determines all unitary properties of Scattering Matrix, which is also symmetric. All higher order discrete operators and operators on higher dimensional simplicial complexes are included in this scheme. Nonlinear analog of that was invented by the present author in collaboration with A.S.Schwarz.

Stress singularities in 2D orthotropic corners

A new technique for analysis of two-dimensional linear elastostatic solutions with stress singularities at orthotropic corners is developed. An explicit and general representation of the associated eigenequation given as ‘zero determinant condition’ of a matrix with half dimension in comparison with the current approach is derived by application of Stroh relations of anisotropic elasticity. The technique is directly applicable to anisotropic corners. Analytical formulae for stress singularity exponents, roots of the associated eigenequations, at orthotropic half-plane and semiinfinite crack problems for all combinations of basic homogeneous boundary conditions, including slip with friction, are derived. it is noteworthy that the singularity exponents are invariant with respect to the relative orientation of boundary edges and orthotropic material for nine of these combinations. Numerical analysis of singularity exponents for some configurations typical in the modelization of material tests of fiber-matrix composite materials is presented.

A new approach to the random phase approximation

With the aim of obtaining a practical approach to molecular photoionization continua, a new formulation is given for the random phase approximation (RPA). This development is based on the so-called algebraic-diagrammatic construction (ADC) used previously to derive higher-order approximations to the polarization and other propagators. In the ADC reformulation the RPA pseudo-eigenvalue equations split into two equivalent, generally Hermitian, secular problems of half dimension. Here the elements of the resulting ADC secular matrices are given in the form of `regular' perturbation expansions. Similar perturbation expansions result for the `effective' transition amplitudes required to calculate spectral intensities. Approximation schemes converging to the full RPA are obtained by truncating these perturbation expansions at successively higher-order n. For the lowest orders n = 1 and 2 the ADC(n)/RPA schemes are specified explicitly. In addition to the perturbation-theoretical approach, direct closed-form expressions are given for the ADC secular matrix and effective transition amplitudes relating these quantities to the RPA eigenvalues and eigenvectors. As shown by these relations, the ADC reformulation can be viewed as a specific form of quasi-degenerate perturbation theory (QDPT) applied to the RPA pseudo-eigenvalue problem. In the single-channel (SC) approximation the ADC secular equations reduce to a one-electron eigenvalue problem for an energy-independent non-local potential. A particularly useful method is the single-channel ADC(1) scheme combining the scattering solutions of the familiar frozen-core Hartree - Fock (FCHF) model with a simple first-order expression for the transition moments. Illustrative applications of the SC-ADC(1) method to the photoionization in and are reported. These calculations show that ground state correlation can quite substantially influence both the magnitude and the shape of the cross sections as a function of energy.

THE BOUNDARY STRIP METHOD IN ELASTOSTATICS AND POTENTIAL EQUATIONS

The present paper develops the idea of the boundary strip method, and presents its fundamentals, merits, applications and also some closed-form or non-element solutions based on it. The present approach combines the Boundary Integral Equation Method (BIEM) and the finite strip method, taking the advantages of both. The finite strip method is installed into the BIEM by expanding the unknown parameters of problems in terms of trigonometric series. This combination creates a new powerful numerical method with three advantages over other numerical methods, namely, a shorter computation time, a better accuracy and a reduction of one and a half dimensions in mesh generation. Applications in two-dimensional potential and field problems demonstrate the efficiency and the accuracy of the proposed method. Finally, closed-form presentations for Laplace equation and elastostatics are given, along circular segments.

On the cultivation of the compass rose

Abstract The author has put forward, in the past, a definition of cartography as of paying homage to the particular fractal surface we inhabit; of fulfilling an obligation to one of our parents (the other being Father Time) by portraying in all her wonderful oddity this goddess of (roughly speaking) two and a half dimensions. Having read further contributions to the question he suggests another definition: Cartography is the cultivation by graphic means of the compass rose.

Generalization of a theorem of Griffiths concerning algebraic cycles

We generalize a theorem of Griffiths concerning the fact that a primitive cycle of half dimension on a hypersurface in p2m+1 yields cycles algebraically not equivalent to zero but homologous to zero on hyperplane sections.