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Showing papers on "Constant (mathematics) published in 1994"


Journal Article
TL;DR: In this article, the authors consider a model where the cosmological constant varies with time such that the asymptotic solution for late times is characterized by a constant ratio λ(t)/ρ(t).
Abstract: We investigate the coupled system of gravity and a scalar with exponential potential. The energy momentum tensor of the scalar field induces a time-dependent cosmological “constant”. This adjusts itself dynamically to become in the “late” universe (including today) proportional to the energy density of matter and radiation. Possible consequences for the present cosmology are shortly discussed. We also address the question of naturalness of the cosmon model. Whenever cosmology encounters potential difficulties in the description of the present universe cosmologists revive the discussion about the cosmological constant [1]. The discrepancy between the critical energy density expected from inflationary cosmology and lower dynamical estimates of this density has been attributed to the cosmological constant [2]. The discussion also pertains to the age of the universe [2] and the formation of structure [3]. In fact, a cosmological constant λ of the order of today’s critical energy density in the universe (λ ≈ (2 · 10eV )) strongly affects the present universe without altering the successful predictions of the hot big bang model at early stages of the evolution of the universe. Despite many attempts [4] we have at present no satisfactory understanding why λ should be much smaller than typical energy scales of the standard model or even the Planck mass Mp. For a time-independent cosmological constant it seems even harder to explain why it should be of the order of the present energy density. The latter depends on the age of the universe rather than on fundamental constants. It looks then not very natural that a constant λ should have a value which equals the energy density just at a time within the present cosmological epoch. In this work we consider a model where the cosmological “constant” varies with time such that the asymptotic solution for late times is characterized by a constant ratio λ(t)/ρ(t)[5], [6]. We discuss consequences for present cosmology and various alternatives how “early cosmology” could have made a transition to this type of “late cosmology”. We also briefly address the question of naturalness of an asymptotically vanishing cosmological “constant”. We start from the field equations for a scalar field φ coupled to gravity in a homogenous and isotropic universe (with k = 0 and H the Hubble parameter) φ+ 3Hφ+ ∂V ∂φ = q (1) ρ+ 3H(ρ+ p) + qφ = 0 (2) H = 1 6M2 (

673 citations


Journal ArticleDOI
TL;DR: In this article, a 7-parameter theory with a linear varying thickness stretch as an extra variable allowing also large strain effects is presented, and the authors introduce a complete 3-D constitutive law without modification.
Abstract: Conventional shell formulations, such as 3- or 5-parameter theories or even 6-parameter theories including the thickness change as extra parameter, require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness. This means that the normal stress in thickness direction has to either vanish or be constant. In general, these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions. The main objective of the present study is to introduce directly a complete 3-D constitutive law without modification. Therefore, a 7-parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

379 citations


Journal Article
TL;DR: In this article, the autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms, and the asymptotic distributions of the test statistics and estimators are found.
Abstract: The autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms. Various models for 1(1) variables defined by restrictions on the deterministic terms are discussed, and it is shown that statistical inference can be performed by reduced rank regression. The asymptotic distributions of the test statistics and estimators are found. A similar analysis is given for models for 1(2) variables with a constant term.

333 citations


Journal ArticleDOI
TL;DR: In this article, the autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms, and it is shown that statistical inference can be performed by reduced rank regression.
Abstract: The autoregressive model for cointegrated variables is analyzed with respect to the role of the constant and linear terms Various models for 1(1) variables defined by restrictions on the deterministic terms are discussed, and it is shown that statistical inference can be performed by reduced rank regression The asymptotic distributions of the test statistics and estimators are found A similar analysis is given for models for 1(2) variables with a constant term

321 citations


Journal ArticleDOI
TL;DR: In this article, the scallop height is kept constant, which leads to a significant reduction in the size of the CL (cutter location) data accompanied by a reduction in machining time.
Abstract: A novel approach for the NC tool-path generation of free-form surfaces is presented. Traditionally, the distance between adjacent tool-paths in either the Euclidean space or in the parametric space is kept constant. Instead, in this work, the scallop-height is kept constant. This leads to a significant reduction in the size of the CL (cutter location) data accompanied by a reduction in the machining time. This work focuses on the zig-zag (meander) finishing using a ball-end milling cutter.

272 citations


Journal ArticleDOI
TL;DR: A solution with nearly linear space and preprocessing time and withO(n1−1/b+δ) query time is given, whered≤b≤2d−3 and δ>0 is an arbitrarily small constant.
Abstract: LetP be a set ofn points in ?d (whered is a small fixed positive integer), and let Γ be a collection of subsets of ?d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Γ-range searching problem: GivenP, build a data structure for efficient answering of queries of the form, "Given a ??Γ, count (or report) the points ofP lying in ?." Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and withO(n1?1/b+?) query time, whered≤b≤2d?3 and ?>0 is an arbitrarily small constant. The acutal value ofb is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of Γ-range searching problem, including improved ray shooting among triangles in ?3.

199 citations


Book ChapterDOI
21 Jun 1994
TL;DR: With this approach, systems with unsynchronized and drifting clocks can be modeled, a general differential equation can be abstracted by breaking the state space into regions with constant differential inclusions, and many previously presented hybrid system examples can be verified.
Abstract: A hybrid system is modeled with a finite set of locations and a differential inclusion associated with each location. We discuss a subclass of hybrid systems with constant rectangular differential inclusions. The continuous state of the system is x ∃ IRn with xi evolving with differential inclusion x i [L i , U i ] where L i , U i are integers (i.e., the slope of trajectory of xi could be changing, but is restricted to remain within [L i , U i ]). A transition from one location to another can be made provided the state satisfies the enabling condition for the transition. The state can also be initialized to a new value during the transition. The differential inclusion for x i can be changed when x i is an integer or when x i is initialized to a new value. We show that the verification problem for this class of hybrid systems is decidable. With this approach, systems with unsynchronized and drifting clocks can be modeled, a general differential equation can be abstracted by breaking the state space into regions with constant differential inclusions, and many previously presented hybrid system examples can be verified.

170 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the breakdown at short times of the much used formula for the Hahn echo amplitude in a constant gradient in unbounded space: M(2τ)/M(0)=exp(−2D0g2τ3/3).
Abstract: Transverse magnetization of spins diffusing in a bounded region in the presence of a constant field gradient is studied. We investigate the breakdown at short times of the much used formula for the Hahn echo amplitude in a constant gradient in unbounded space: M(2τ)/M(0)=exp(−2D0g2τ3/3). Here D0 is the diffusion constant in unbounded space and g is the field gradient multiplied by the gyromagnetic ratio. We find that this formula is replaced by M(2τ)/M(0)=exp[−2Deffg2τ3/3 +O(D5/20g4τ13/2S/V)] with an effective diffusion coefficient Deff(2τ) =D0[1−α√D0τ(S/V) +...], where α is a constant and S/V is the surface to volume ratio of the bounded region. Breakdown is complex but we find that the interplay between a natural length scale lc=(g/D0)−1/3 and the geometry of the region governs the problem. The long‐time behavior of the free induction decay and echo amplitude are then considered where pure exp[−const t] decay is expected. We consider some simple geometries and find in addition to the well‐known result, ...

169 citations


Book ChapterDOI
01 Jan 1994
TL;DR: The integrability of the sine-Gordon equation and other integrable equations has been studied for a long time as discussed by the authors, including the Backlund transform, which has clear geometrical interpretation.
Abstract: Many of the equations which now are called integrable have been known in differential geometry for a long time. Probably the first was the famous sine-Gordon equation, which was derived to describe surfaces with constant negative Gaussian curvature. At that time many features of integrability of the sine-Gordon and other integrable equations were discovered 1, namely those which have clear geometrical interpretation (for example, the Backlund transform).

164 citations


Journal ArticleDOI
TL;DR: The first nontrivial general upper bound for this problem is shown, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2λq(n) for some constantq depending on the shape and degree of the surfaces.
Abstract: We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d?3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(nd?1+?), for any ?>0; the constant of proportionality depends on ?, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2?q(n)) for some constantq depending on the shape and degree of the surfaces (where ?q(n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n2+?), and give several applications of the new bounds.

152 citations


Journal ArticleDOI
TL;DR: In this paper, the exact exchange correlation potential and energy are compared with the corresponding quantities, obtained for the same densities, using approximate density functionals, namely the local density approximation and several generalized gradient approximations.
Abstract: We consider a model, given by two interacting electrons in an external harmonic potential, that can be solved analytically for a discrete and infinite set of values of the spring constant. The knowledge of the exact electronic density allows us to construct the exact exchange–correlation potential and exchange–correlation energy by inverting the Kohn–Sham equation. The exact exchange–correlation potential and energy are compared with the corresponding quantities, obtained for the same densities, using approximate density functionals, namely the local density approximation and several generalized gradient approximations. We consider two values of the spring constant in order to study the system in the low correlation case (high value of the spring constant) and in the high correlation case (low value of the spring constant). In both cases, the exchange–correlation potentials corresponding to approximate density functionals differ from the exact one over the entire spatial range. The approximate correlation potentials bear no resemblance to the exact ones. The exchange energy for generalized gradient approximation functionals is much improved compared to the result obtained within the local density approximation but the correlation energy is only a little improved.

Journal ArticleDOI
TL;DR: In this paper, the authors examine the reasonableness for public policy analysis of non-constant discounting methods that, unlike constant discounting, can accord considerable importance to outcomes in the distant future.

Journal ArticleDOI
TL;DR: In this article, the Steepest Gradient (SG) algorithm was developed and its performance was shown to be superior to that of the Clipped Optimal Solution (CLOP) solution.
Abstract: SUMMARY This paper attempts to clarify the question of what the optimal semi-active suspension is that minimizes a deterministic quadratic performance index. The optimal control law is a time-varying solution that involves three related Riccati equations. The constant Riccati solution (the so-called “clipped optimal” solution) is not optimal, although its performance is generally quite close to that of the time-varying solution. As the time-varying solution cannot be practically implemented, several constant gain sub-optimal solutions are investigated. A new semi-active algorithm, called the “steepest gradient” algorithm, is developed and its performance is shown to be superior to that of the “clipped optimal” solution.


Journal ArticleDOI
TL;DR: It is found that the optimal dimensionality of pipelined-channel networks is higher than that of nonpipelined- channel networks, with the difference being greater under looser wiring constraints.
Abstract: In a pipelined-channel interconnection network, multiple bits may be simultaneously in flight on a single wire. This allows the cycle time of the network to be independent of the wire lengths, significantly affecting the network design trade-offs. This paper investigates the design and performance of pipelined channel k-ary n-cube networks, with particular emphasis on the choice of dimensionality and radix. Networks are investigated under the constant link width, constant node size and constant bisection constraints. We find that the optimal dimensionality of pipelined-channel networks is higher than that of nonpipelined-channel networks, with the difference being greater under looser wiring constraints. Their radix should remain roughly constant as network size is grown, decreasing slightly for some unidirectional tori and increasing slightly for some bidirectional meshes. Pipelined-channel networks are shown to provide lower latency and higher bandwidth than their nonpipelined-channel counterparts, especially for high-dimensional networks. The paper also investigates the effects of switching overhead and message lengths, indicating where results agree with and differ from previous results obtained for nonpipelined-channel networks. >

Journal ArticleDOI
TL;DR: In this paper, the authors studied all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg-Landau equation on the complex plane.
Abstract: In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.

Book
01 Oct 1994
TL;DR: Real Finsler geometry as discussed by the authors and complex complex FINF geometry with constant holomorphic curvature, with a constant curvature and a constant number of vertices, respectively.
Abstract: Real Finsler geometry.- Complex Finsler geometry.- Manifolds with constant holomorphic curvature.


Journal ArticleDOI
TL;DR: In this article, the authors reported new results of nucleation measurements carried out in their laboratory at a variety of constant total pressures for 1-propanol vapor with nitrogen, helium, and hydrogen as background gases.
Abstract: We report new results of nucleation measurements carried out in our laboratory at a variety of constant total pressures for 1-propanol vapor with nitrogen, helium, and hydrogen as background gases and for 1-butanol vapor with nitrogen and hydrogen as background gases. In addition, we report new results of nucleation experiments carried out at a variety of constant temperatures for 1-butanol with helium and hydrogen as background gases. The 1-butanol constant temperature results compare favorably with the results of the 1-butanol constant total pressure experiments, and the constant total pressure data for 1-propanol obtained in this investigaton compare favorably with constant temperature results obtained from 1-propanol experiments described in the literature. The results of this investigation (continue to) suggest a significant role of the background gas in the nucleation process. Our constant pressure and constant temperature experiments reported here give new results that reinforce our earlier observa...

Journal ArticleDOI
01 Mar 1994
TL;DR: For Schrodinger operators on an interval with convex potentials, the gap between the two lowest eigenvalues is minimized when the potential is constant as discussed by the authors, and the gap is bounded by a constant number of potentials.
Abstract: For Schrodinger operators on an interval with convex potentials, the gap between the two lowest eigenvalues is minimized when the potential is constant.


Journal ArticleDOI
TL;DR: In this article, the authors investigated the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved, and they obtained results for autonomous and non-autonomous equations with one or many delays.
Abstract: The idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments (EPCA) has been suggested by Gyijri [l], who proved convergence of the method for linear and nonlinear delay equations on compact intervals, and under certain conditions also on the half line. The approximating equations with piecewise constant argument can, in turn, be solved by use of difference equations. The latter then also provide an approximation of the original delay equation. (The theory of EPCA was initiated and studied by Cooke and Wiener in [2] and [3].) In this paper, we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved. A number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features. Readers may refer to the papers of Kloeden and Lorenz [4] and Beyn [5] for some of this work. For delay differential equations, questions of this sort have been studied by Cryer [6], Barwell [7], Zennaro [8], and later authors. The paper of Strehmel et al. [9] contains an up-to-date list of references. In our method of approximation, the delay equation is first replaced by an EPCA, and then by a difference equation, and our objective is to relate the qualitative dynamics of these three equations. We obtain results for autonomous and nonautonomous equations with one or many delays. For autonomous equations, the resulting difference algorithm is just a simple Euler scheme, but for nonautonomous equations, it includes other possibilities. In order to make the method and the nature of results as clear as possible, we begin with a particular test equation, k(t) = -pz(t 7), p > 0, 7 > 0.

Journal ArticleDOI
TL;DR: In this paper, the derivation of d = 1.702 was discussed and it was shown that the scaling could be improved by the factor (w//l3) (15/16) = (1.70044).
Abstract: Cox (1970) observed that the most apparent method of scaling ti to coincide with 4 is to standardize the logistic variable, which is done by multiplying x by Tw/In = 1.81380. Johnson and Kotz (1970) graphically showed that the scaling could be improved by the factor (w//l3) (15/16) = 1.70044. However, Haley (1952) outlined the theoretical derivation of d = 1.702. Because the use of d is widespread and Haley's (1952) unpublished work is not easily accessible, the derivation of d re-presented in this brief note provides a

Journal ArticleDOI
TL;DR: In this article, the Mori-Tanaka method is extended to linearly variable overall and local fields for statistically homogeneous composite materials with variable reinforcement density, and linear and constant field approaches provide different estimates of overall properties for small representative volumes, but nearly identical estimates for large volumes.

Journal ArticleDOI
TL;DR: In this paper, the Lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs with respect to right-hand side perturbations are given in terms of norms of pseudoinverses of submatrices of the matrices involved.
Abstract: The main purpose of this paper to give Lipschitz constants for basic optimal solutions (or vertices of solution sets) and basic feasible solutions (or vertices of feasible sets) of linear programs with respect to right-hand side perturbations. The Lipschitz constants are given in terms of norms of pseudoinverses of submatrices of the matrices involved, and are sharp under very general assumptions. There are two mathematical principles involved in deriving the Lipschitz constants: (1) the local upper Lipschitz constant of a Hausdorff lower semicontinuous mapping is equal to the Lipschitz constant of the mapping and (2) the Lipschitz constant of a finite- set-valued mapping can be inherited by its continuous submappings. Moreover, it is proved that any Lipschitz constant for basic feasible solutions can be used as an Lipschitz constant for basic optimal solutions, feasible solutions, and optimal solutions.

Journal ArticleDOI
TL;DR: In this paper, a simple method for making work function measurements in gas ambient is presented, which makes use of a Kelvin probe calibrated using a clean solution surface of an electrochemical half-cell.

Journal ArticleDOI
01 Mar 1994
TL;DR: A general method is described that makes it possible to simulate the dynamic equilibrium that exists in a chemically reacting system through the use of a recently developed grand canonical molecular dynamics method.
Abstract: The dynamic equilibrium that exists in a chemically reacting system can be simulated using classical me chanics if the appropriate statistical mechanical ensem ble is chosen. This paper describes a general method that makes it possible to simulate this equilibrium in a simple chemical reaction through the use of a recently developed grand canonical molecular dynamics method. After a brief description of the method, an example calculation is performed that simulates the acid-base equilibrium between acetic acid and water. The computational demands of this application are discussed along with a description of a new MPP algo rithmic approach to this application.

Journal ArticleDOI
TL;DR: In this article, it was shown that a Boolean function f is computed by a constant depth circuit with 2 m AND-, OR-, and NOT-gates and m majority gates.
Abstract: Suppose that a Boolean functionf is computed by a constant depth circuit with 2 m AND-, OR-, and NOT-gates—andm majority-gates. We prove thatf is computed by a constant depth circuit with\(2^{m^{O(1)} }\) AND-, OR-, and NOT-gates—and a single majority-gate, which is at the root.

Journal ArticleDOI
TL;DR: The authors derive a simple sufficient condition for the output of a discrete-time, time-invariant bilinear system to be bounded whenever the input signal to the system is bounded by a finite constant.
Abstract: The authors derive a simple sufficient condition for the output of a discrete-time, time-invariant bilinear system to be bounded whenever the input signal to the system is bounded by a finite constant. >

Book ChapterDOI
01 Jan 1994
TL;DR: In this article, it was shown that the projection needed by the CAD algorithm needs only to compute the resultants, the discriminants, leading coefficients and the constant coefficients of the input polynomials, provided they have be made square.
Abstract: It is proved that the projection needed by the CAD algorithm needs only to compute the resultants, the discriminants, the leading coefficients and the constant coefficients of the input polynomials, provided they have be made square—free and relatively prime by GCD computations. This improves [7] first improvement by removing any dimension condition and dropping out from the projection set all coefficients of the input polynomials, other than the leading and the constant ones.