Terms Of First Derivatives Research Articles

Analytic and Numerical Solutions of a Nonlinear Boundary‐Layer Problem

Solutions of a viscoinertial western‐boundary‐layer problem arising in physical oceanography are discussed for both no‐slip and stress‐free boundary conditions. A fairly complete characterization of the behavior of the solutions in a number of regimes is presented using the techniques of regular perturbation expansion, matched asymptotic expansion, and rescaling of both regular and singular solutions. A comparison with the actual solutions found by extensive numerical computations is given. Parametrization of solutions in terms of the unknown first (or second) derivative at the origin, for which a useful analytic approximation is developed, provides a convenient characterization of the entire family.

The geometry of minimal replacement for the Poincaré group

The constructs of this paper rest on two elementary facts: (1) the Poincaré group P 10 is the maximal group of isometries of Minkowski space-time M 4; (2) P 10 has a faithful matrix representation as a subgroup of GL(5, R) that maps an affine set into itself. Local action of P 10 and Yang-Mills minimal replacement are shown to induce a well-defined minimal replacement operator that maps the tensor algebra over M 4 onto the tensor algebra over a new space-time U 4. The natural frame and coframe fields of M 4 go over into a canonical system of frame and coframe fields of U 4 with both translation and Lorentz-rotation parts. The coframe fields define soldering 1-form fields for U 4 that give rise to the standard geometric quantities through the Cartan equations of structure. This leads to unique determinations of all relevant connection coefficients and the associated 2-forms of curvature and torsion that involve the compensating 1-forms for local action of both the translation and the Lorentz-rotation sectors. The metric tensor of U 4, that is induced by the minimal replacement operator, is shown to satisfy the Ricci lemma; U 4 is necessarily a Riemann-Cartan space. This space admits gauge covariant constant basis fields for the Lie algebra of the Lorentz group and for the Dirac algebra. The induced basis for the Dirac algebra evaluates the images of Dirac operators under minimal replacement, while the induced basis for the Lie algebra of L(4, R) serves to show that the holonomy group of U 4 is the Lorentz group. The minimal replacement operator is extended to include the case of a total gauge group that is the direct product of the Poincaré group and a Lie group of internal symmetries of matter fields. This provides a precise method of lifting any action integral of the matter fields from M 4 up to U 4 so that invariance properties are retained when the total group acts locally. The natural representations afforded by minimal replacement result in curvature being evaluated in terms of first order derivatives of the compensating fields that share many properties in common with the Dirac derivation algebra for spin fields. Direct interpretations of the compensating fields are obtained from the geodesic equations.

Dynamics of Native Globular Proteins in Terms of Dihedral Angles

Problems are discussed that are associated with extension of the time-independent method of conformational energy analysis of biopolymers, in which only dihedral angles are treated as independent variables, to time dependent one. For molecules with small amplitudes of fluctuations internal motions of atoms due to variations of dihedral angles are shown to be defined with respect to a coordinate system which is defined by the Eckart's condition and is moving with the molecule. Computationally efficient and explicit expressions are given (i) for a coefficient matrix to convert small changes in dihedral angles to small atomic displacements from the mean positions, and (ii) for a coefficient matrix in the expression of the kinetic energy of internal motions in terms of first order time derivatives of variable dihedral angles.

A semiclassical density matrix valid beyond the classically allowed region

We study semiclassical approximations to the density matrix of a system of Fermions in a one body potential. We derive an expression for the propagator in terms of first and second derivatives of the potential. Our result is thus exact for a harmonic oscillator potential and is equivalent to a partial resummation of the Wigner-Kirkwoodh-expansion. It leads to densities which are well-defined also in the classically forbidden region.

The use of first and second derivatives in optical model parameter searches

This paper compares the efficiencies of standard minimization procedures in carrying out nuclear optical-model fits. It is shown that first-order perturbation theory permits computation of the gradient of x 2 more than five times as fast as is possible by difference methods. A linear approximation to the second-derivative matrix in terms of first derivatives of residuals is found to be very accurate in the neighborhood of minima; it provides a way of introducing second-derivative information that is significantly superior to the use of variable-metric algorithms. The resulting restricted-step Gauss-Newton procedures are shown to be about five times as fast as direct-search methods. The use of the methods of pseudo-inverses to “freeze” linear combinations of parameters poorly determined by the data is discussed.

A computerized system for ECG monitoring

An on-line, real-time, interactive computerized ECG monitoring system has been developed that detects and quantifies heart rate, ventricular premature contractions, and atrial premature contractions. Over several months more than 50 patients were monitored by this computer system. Five-hundred waves of each of eight patients with varying arrhythmias were analyzed in detail by the computer as well as by a cardiologist. Of those 4000 waves analyzed, including 164 ventricular premature contractions (VPC's), no false positives or false negatives were detected. Characterization of each of those 4000 waves in terms of first derivative and second derivative indicates that those two parameters may be necessary and sufficient as VPC detection criteria as well as a useful new method of characterizing cardiac rhythms.

A generalized procedure for the constrained optimization problem

Abstract The necessary and the sufficient conditions for the solution of the equality constrained optimization problem with N variables and (N−1) constraints are first derived and generalized to N variables and m constraints. Directional derivatives are used in the approach. The necessary and sufficient conditions for the problem with N−1 constraints are shown to be equivalent to the unconstrained one-variable problem, when the ordinary derivatives are replaced by the corresponding directional derivatives of the objective function in the direction tangent to the intersection of the constraints. The general equality constrained optimization problem of N variables and m constraints is then analysed using the directional derivative approach. Feasible direction vectors are defined and obtained in terms of first partial derivatives of the constraints. Necessary and sufficient conditions in terms of directional derivatives are derived and their equivalent with results in the literature. Sufficient conditions hi...

A Note on the Integration of the Vorticity Equation for Quasi-Geostrophic Flow

By the transformation of integrals arising from the solution of the “quasi-geostrophic” equations for barotropic and simple types of baroclinic flow, it is shown that the isobaric height tendency can be expressed in terms of first derivatives of the heights of isobaric surfaces. This demonstrates that the procedure by which the equations are solved must tend to cancel or “average out” the large percentage errors in the geostrophic vorticity advection—i.e. errors in second and third order finite-differences. The height tendency may be computed by a simple graphical procedure, which is carried out directly on analyzed contour charts and which involves a minimum of multiplications. DOI: 10.1111/j.2153-3490.1954.tb01127.x

A Note on the Integration of the Vorticity Equation for Quasi-Geostrophic Flow

By the transformation of integrals arising from the solution of the “quasi-geostrophic” equations for barotropic and simple types of baroclinic flow, it is shown that the isobaric height tendency can be expressed in terms of first derivatives of the heights of isobaric surfaces. This demonstrates that the procedure by which the equations are solved must tend to cancel or “average out” the large percentage errors in the geostrophic vorticity advection–i.e. errors in second and third order finite-differences. The height tendency may be computed by a simple graphical procedure, which is carried out directly on analyzed contour charts and which involves a minimum of multiplications.