Non-zero Derivative Research Articles

Magnetohydrodynamic Stability of a Toroidal Plasma’s Separatrix

Large tokamaks capable of fusion power production such as ITER, should avoid large edge localized modes (ELMs), thought to be triggered by an ideal magnetohydrodynamic instability due to current at the plasma's separatrix boundary. Unlike analytical work in a cylindrical approximation, numerical work finds the modes are stable. The plasma's separatrix might stabilize modes, but makes analytical and numerical work difficult. We generalize a cylindrical model to toroidal separatrix geometry, finding one parameter Delta' determines stability. The conformal transformation method is generalized to allow nonzero derivatives of a function on a boundary, and calculation of the equilibrium vacuum field allows Delta' to be found analytically. As a boundary more closely approximates a separatrix, we find the energy principle indicates instability, but the growth rate asymptotes to zero.

Bifurcations of continuous dynamical systems under extreme and new transversality conditions

For a family of continuous-time dynamical systems, in [F. Balibrea, A. Martinez, J.C. Valverde, Local bifurcations of one-dimensional continuous dynamical systems under higher order conditions, in press] we gave sufficient conditions of higher order in terms of the first non-zero partial derivative with respect to the state variable to have (local) generic bifurcations of equilibria. Inspired by these results, here we study what happens when those conditions are extreme in the sense that all those partial derivatives vanish. Besides we deal with some others transversality conditions for the family to undergo the same bifurcations and give some counterexamples proving that, in absence of analyticity of the function which defines the system, the phenomenon could be unexpectedly different.

Perturbation selection and influence measures in local influence analysis

Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential--geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations.

Differentiability of quasi-conformal maps on the jungle gym

We obtain a result on the quasi-conformal self-maps of jungle gyms, a divergence-type group. If the dilatation is compactly supported, then the induced map on the boundary of the covering disc D is differentiable with non-zero derivative on a set of Hausdorff dimension 1. As one of the corollaries, we show that there are quasi-symmetric homeomorphisms over divergence-type groups such that for all sets E the Hausdorff dimension of E and f(E c ) cannot both be less than 1. This shows an important difference.between finitely generated and divergence-type groups.

Calculated octic centrifugal distortion coefficients of non-linear molecules

The equilibrium octic centrifugal distortion coefficients of a molecule are coefficients of terms of the type H ˜ 08 in the expansion of the effective Hamiltonian. They depend on the quartic potential constants k k l m n ′ for the vibrational modes that have non-zero inertial derivatives, and therefore give information on these constants additional to that given by the other terms that depend on the quartic potential, H ˜ 24 and H ˜ 40 . The present work describes the calculation of these H ˜ 08 terms from a given potential by perturbation theory, using a computer program that calculates the coefficients in the successive contact transformations. Results are presented for the SO 2, ClO 2, O 3, and H 2O molecules.

Thermal degradation of cyano containing ionic liquids

The long term thermal stability of ionic liquids containing dicyanamide or tricyanomethide anions is significantly lower than might be expected from temperature ramped TGA studies, and these anions, combined with N-based cations, yield polymeric products during thermal decomposition. Operational upper temperature limits for extended heating have been determined from isothermal decomposition measurements and extraction of t0.99 values, and a simple relationship between onset of decomposition (determined by non-zero 1st derivative) and T0.01/10 (the temperature at which 1% decomposition occurs in 10 h) is derived.

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have $n$ eigenvalues equal to 1, and investigate the first non-zero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of $L$-functions in families has been well-established. The motivation for this work is the expectation that through this connection with $L$-functions derived from families of elliptic curves, and using the Birch and Swinnerton-Dyer conjecture to relate values of the $L$-functions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks.

On positively differentiable curves with values in normed linear spaces

We prove a theorem that characterizes continuous normed linear space-valued curves allowing differentiable parameterizations with non-zero derivatives as those curves, all the points of which are regular (in Choquet's sense). We also state an equivalent geometric condition not involving any homeomorphisms. This extends a theorem due to Choquet, who proved a similar result for curves with values in Euclidean spaces.

Existence of solutions to MIMO relay feedback systems

This note studies the existence of solutions to MIMO linear systems under decentralized relay feedback containing hysterisis. A necessary and sufficient condition is presented to guarantee the extended solutions at the so-called intersecting instant. The suggested method is to examine the sign of the first non-zero derivative of each output right after the switches occur. Illustrative examples are given to show the cases of non-existence and non-uniqueness of solutions.

Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow

We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

Generation of cosmological large lepton asymmetry from a rolling scalar field

We propose a new scenario to simultaneously explain a large lepton asymmetry and a small baryon asymmetry. We consider a rolling scalar field and its derivative coupling to the lepton number current. The presence of an effective nonzero time derivative of the scalar field leads to $CPT$ violation so that the lepton asymmetry can be generated even in thermal equilibrium as pointed out by Cohen and Kaplan. In this model, the lepton asymmetry varies with time. In particular, we consider the case where it grows with time. The final lepton asymmetry is determined by the decoupling of the lepton number violating interaction and can be as large as order unity. On the other hand, if the decoupling takes place after the electroweak phase transition, a small baryon asymmetry is obtained from the small lepton asymmetry at that time through sphaleron effects. We construct a model in which a rolling scalar field is identified with a quintessence field.

Angle distortion and conformality

We offer an alternative to the usual proofs for the equivalence between conformality and analyticity with nonzero derivative. Our proof uses the z̄-derivative to quantify angle distortion, and thereby foreshadows some important ideas in the study of quasiconformal mappings.

Center conditions for a lopsided quintic polynomial vector field

We consider in this article, a lopsided quintic polynomial vector field\(X = - x\frac{\partial }{{\partial y}} + y\frac{\partial }{{\partial x}} + \sum\nolimits_{i + j = 5} {(a_{ij} x^i y^j \frac{\partial }{{\partial x}})} \). We first compute the first nonzero derivative of the return mapr→L(r,e). We study then the necessary and sufficient conditions for the existence of a center.

Center conditions for a lopsided quartic polynomial vector field

We consider in this article 1 1 We wish to express our thanks to Pr. J.P. Françoise for his advice during the period of this work. a lopsided quartic polynomial vector field X=−x ∂ ∂y +y ∂ ∂x + ∑ i+j=4 (b ijx iy j ∂ ∂y ). We first compute the first non-zero derivative of the return map r↦ L( r, ε). We study then the necessary and sufficient conditions for the existence of a center.

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(x,t) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x0, t) with non-zero derivative at a single sensor location x = x0 is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x,t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.

Limit distributions of unbiased estimators in natural exponential families

We obtain the possible limit distributions of unbiased estimators of functions of the parameter of a natural exponential family. The limit distribution depends on <$>j<$>, the order of the first non-zero derivative at the true (but usually unknown) value of the parameter. We show that if <$>j \\geq 2<$> then the umvu and the maximum likelihood estimators are not asymptotically equivalent.

A note on quantum thermodynamic rate theories

The relationship between quantum transition state theory, the mixed quantum classical rate theory and the Hansen–Andersen analytic continuation methods is analyzed. We prove that the first three time derivatives of a coordinate dependent operator are the same in quantum and classical mechanics. As a result, the mixed quantum classical theory, in which the quantum projection operator is replaced by the classical, may be considered as a specific case of the Hansen–Andersen methodology. The same holds true for quantum transition state theory for one dimensional systems, where the exact quantum propagator is replaced by its parabolic barrier approximation. In the multidimensional case, quantum transition state theory errs somewhat in the second nonzero time derivative, however it may be reformulated to assure that it too remains exact for the first two nonzero initial time derivatives. Further systematic improvement of the mixed quantum classical theory may be obtained by including higher order terms in the ℏ2 expansion of the Wigner–Liouville equation. An iterative solution of the integral form of the Wigner–Liouville equation is suggested, which is based on propagation of classical trajectories only.

Comparison of electromagnetic field for two different lightning pulse current models

AbstractIn this paper the electromagnetic field produced by a lightning current analytically described by the so called double exponential model is compared with the field produced by the same current (same peak, rise and decay time) analytically described by the model proposed by Heidler. The exponential model has been widely used in literature for its simplicity and its relatively good accuracy. The Heidler model is more complex, but removes the main problem shown by the double exponential: the non‐zero derivative at the beginning of the lightning pulse in contrast with the physical phenomenon. In the lightning electromagnetic pulse (LEMP) simulations both models are now used. In the paper we aim to make a comparison between the two. In particular we show that the two models are fairly equivalent in the frequency range up to 2 MHz, namely the range of interest of typical lightning phenomena. In addition, the double exponential model is shown to be conservative for higher frequencies.

Aircraft Nonminimum Phase Control in Dynamic Sliding Manifolds

The approximate causal nonminimum phase output tracking problem is considered for an F-16 nonlinear sixdegree-of-freedom modelandaddressed via sliding modecontrol.Asymptoticoutputtracking-errordynamicswith desired eigenvalue placement are provided in case of tracking causal reference output proe le with a e nite number of nonzero time derivatives (piecewise polynomial spline model ) in the presence of unmatched disturbances of the same kind. A complete constructive algorithm for tracking controller design is built for a class of uncertain nonlinear multi-input/multi-output systems with known linear unstable internal dynamics. An analysis is made of the issue that the given nonlinear aircraft model yields to the approach developed.

Diffusion Waves and their Uses

The concept of waves is an integral part of our scientific culture and has nourished physicists, pure and applied alike, for centuries. Many important discoveries in physics, including quantum mechanics, have involved wave phenomena. The wave concept owes some of its scientific success to its mathematical tractability. Linear wave equations—the sort that describe the transmission of sound and radio waves through air—contain a nonzero second-order time derivative, which gives rise mathematically to the rich and familiar array of properties we associate with waves, such as wavefront propagation, reflection, and refraction.