Showing papers on "Tangent published in 1979"

Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces

Abstract: This study is devoted to constrained optimization problems in Banach spaces. We present different properties of tangent cones associated with an arbitrary subset of a Banach space and establish correlations with some of the existing results. In absence of both differentiability and convexity assumptions on the functions involved in the optimization problem, the consideration of these tangent cones and their polars leads us to introduce new concepts in nondifferentiable programming. Necessary optimality conditions are first developed in a general abstract form; then these conditions are made more precise in the presence of equality constraints by introducing the concept of normal subcone.

Clarke's tangent cones and the boundaries of closed sets in Rn

Abstract: Let $C$ be a nonempty closed subset of $\mathbb{R}^n$ For each $x \in C$, the tangent cone $T_C(x)$ in the sense of Clarke consists of all $y \in \mathbb{R}^n$ such that, whenever one has sequences $t_k\downarrow 0$ and $x_k \rightarrow x$ with $x_k \in C$, there exist $y_k \rightarrow y$ with $x_k + t_ky_k \in C$ for all $k$ This is not Clarke’s original definition but it is equivalent to it

On the use of directions of negative curvature in a modified newton method

Abstract: We present a modified Newton method for the unconstrained minimization problem. The modification occurs in non-convex regions where the information contained in the negative eigenvalues of the Hessian is taken into account by performing a line search along a path which is initially tangent to a direction of negative curvature. We give termination criteria for the line search and prove that the resulting iterates are guaranteed to converge, under reasonable conditions, to a critical point at which the Hessian is positive semidefinite. We also show how the Bunch and Parlett decomposition of a symmetric indefinite matrix can be used to give entirely adequate directions of negative curvature.

Higher Order, Planar Tangent-Line Envelope Curvature Theory

Abstract: Following the analogy of higher order point-path generation, new concepts for tangentline higher order envelope curvature theory are introduced. Mathematical relationships are derived to define, using the instantaneous invariants and the stretch rotation concepts, the characteristic numbers Xi and X2 for third and fourth-order contacts at the tangency point of a tangent-line generating an envelope. The newly developed tangentline higher order envelope curvature theory is applied to demonstrate its application in synthesis of mechanisms. Examples involving circular, elliptic and involute arc generation using a mechanism are presented.

Bilinear tangent yaw guidance

Abstract: This paper presents a parametric yaw steering law which has been used to provide closed-loop yaw guidance for the launch of the HEAO (High Energy Astronomy Observatory) satellite mission using the Atlas/Centaur launch vehicle. This bilinear tangent steering law provides near optimal yaw steering for maneuvers requiring insertion into orbits with a specified inclination and node. Bilinear tangent steering is shown to be optimal in both the pitch and yaw planes when a uniform gravitational field is assumed. The conditions under which the general bilinear tangent laws degenerate into linear tangent and constant attitude laws are presented. The flight computer implementation of these laws in a rotating coordinate system using real-time integration of the equations of motion is detailed. Explicit solution of the parametric guidance equations requires the inflight solution of (2x2) two-point boundary value problems in the pitch and yaw planes. Excellent results are obtained even for very large (greater than 50 deg) out-of-plane steering angles.

Octahedral Based Incremental Stress-Strain Matrices

Abstract: Some Investigators have recently suggested that appropriate nonlinear stress-strain relationships for concrete can be obtained by deriving expressions for bulk and shearing moduli based upon experimental data presented in terms of octahedral stresses and strains. The paper reviews octahedral stress-strain relationships and derives appropriate stiffness matrices, both secant and tangent, for plane stress, plane strain, and three-dimensional applications. It is shown that proper tangent stiffness matrices cannot be obtained simply by replacing secant moduli with tangent moduli. Incremental matrices suitable for implementation in nonlinear finite-element programs are presented and considered.

Inelastic Section Response by Tangent Stiffness

Abstract: A review of inelastic analysis of a cross section is made. Given the history of deformation, the history of axial force and moment may be obtained. Also, given the history of moment and axial force the history of deformation may be obtained. The tangent stiffness method is modified to avoid convergence difficulties due to “large” inelastic unloading. Stress-strain curves considered are generally nonlinear but symmetric, exhibiting hysteresis and the Bauschinger effect. The solution is checked against others. The method has been included in a discrete element frame analysis program and used to predict the behavior of a brace member for which experimental results are available.

Permutations related to secant, tangent and Eulerian numbers

Abstract: It is well known that 1 where An denotes the number of "up-down" or alternating permutations 2 of 1, 2, …, n. The numbers A2n and A2n+1 are known as the secant and tangent numbers respectively and A2n = (—l)"E2n , where En is the Euler number.

Elliptic polarization represented by the Carter and Smith charts

Abstract: The graphical method using the Poincare sphere for representing elliptically polarized light is based on the fact that the latitude and longitude of a point on the unit-radius sphere represent the ellipticity and the azimuth of the corresponding light ellipse. Stereographic projections of the unit-diameter Poincare sphere either on a western plane tangent at the equator of the sphere or on an equatorial plane tangent to either of the poles lead to a representation of any elliptical polarization state on the so-called Carter charts, while charts derived from orthographic projections in the Poincare sphere and consisting again of families of orthogonal circles traced inside either the equator or the principal meridian of the unit-radius Poincare sphere yielding either ellipticity–azimuth or amplitude ratio–phase angle niveau lines constitute the Smith charts. Both types of charts consist of families of orthogonal circles. These families of circles, traced on either chart and corresponding to parametric families with different properties for the corresponding elliptically polarized light, yield an easy and accurate solution to the problem of graphical evaluation of the optical properties of the resulting polarization state when an elliptically polarized plane wave is passing through different optically active elements.

Converter for changing Cartesian vector variables into polar vector variables

Abstract: A coordinate converter useful for field-oriented control of a rotating-field machine includes a divider, a first adder, a multiplier and a second adder by which the magnitude of the vector and the tangent of the half-angle relative to one axis can be determined. An ancillary unit forms an angle-proportional variable from the half-angle tangent. A rotating vector can also be processed.

Analysis of Six-Port Measurement Systems

Abstract: A comparative scattering parameter analysis of various microwave six-port circuit configurations has been performed in relation to applications in single and dual six-port automated microwave measurement systems. The aim of this analysis was a confirmation of the predictions of the existing general six-port theories and a search for possible simplifications of these theories to be achieved through selective restrictions of their generality. Two circuit configurations have been, so far, analyzed. The first is the proposed preferred configuration having three q-points close to the 120/spl degree/ optimum locations and a fourth q-point at a great distance from the origin. The second is a "pseudo-symmetric" circuit configuration having four q-points nominally at the cardinal points of the gamma-plane unit circle. Microwave measurement systems based on the six-port principle fully reconstruct complex wave-vector ratios from sets of redundant magnitude-only readings. This leads to determining points in a complex plane as intersections of three or more circles. A simple conformal mapping may be used to visualize the resolution of this method and its sensitivity to errors in the magnitude readings. This mapping transforms families of constant-magnitude-ratio circles in straight parallel lines. The coordinates in the transformed plane are the measured magnitude ratio in dB and the angle between the tangents to the intersecting circles at the intersection points.

Some special solutions of the Helmholtz equation

Abstract: Functions are constructed which describe a cylindrical wave diverging from the origin and tangent to a semi-infinite plane wave. The use of such functions is illustrated by the example of the problem of diffraction of a plane wave by a semi-infinite screen.

Phase extension and refinement

Abstract: Two procedures are described to extend and refine phases starting from a medium-sized set of known phases. From tests with 376 and 400 atom structures it was found that for extension purposes the tangent formula is suitable; for refinement purposes the tangent formula was adapted in order to maintain the enantiomorph.

Entirely optimised model delta wing - has pressure drag minimised at mach 2 and is defined by equations for neutral plane and thickness distribution

Abstract: The entirely optimised model data wing (1) is intended for a cruising speed of Mach 2. Both its plan form and its surface configuration have been chosen to minimise pressure drag at that speed. The shape of the delta wing is specified using an orthogonal system of coordinates with the origin at the nose or sharp tip of the wing. Using this system of coordinates, the equation defines the neutral plane Z(x1, x2) of the wing where x1 is the distance in the approach flow direction and (x2) is the distance along the span. For shockfree entry at cruising speed the centreline of the cross-sectional profiles is tangent to the axis x1. The equation for the thickness distribution about the neutral plane is given.

Elastic energy and metastable phase equilibria for coherent mixtures in cubic systems

Abstract: Expressions were derived for the elastic energy due to coherency for cubic systems for an isotropic structure and for (100) or (111) habit planes for a lamellar structure. For the metastable equilibria the usual tangent compositions are replaced by compositions that are tangent to the elastic energy curve. For a loss of coherency there is an energy decrease due to the elastic effects and a further decrease associated with compositional changes. Information contained within this treatment permits calculation of the x-ray diffraction effects for such structures.

A new fast direct solution to the problem of the sphere tangent to four spheres

Abstract: A simple non-iterative process to solve in 2D, 3D and hyperspaces the problem of the tangency of spheres is presented and proved. Solutions are obtained immediately with high numerical precision.

An Efficient Geodesic Path Solution for Prolate Spheroids

Abstract: : A major task in the application of the geometrical theory of diffraction (GTD) to the problem of electromagnetic radiation from a general convex surface is to determine the unique geodesic path that starting from the source location traverses the surface and has a tangent at some point along the path which in the desired radiation direction. In this report a numerically efficient and accurate scheme has been developed for solving this problem in the case of a general convex surface of revolution. The surface of revolution is of interest in that it provides an analytic model for the aircraft fuselage structure. A computer program is also developed to solve the governing nonlinear equation using the secant method for iteration. This method is proved to be simple and converges to the accurate results in just a few steps. Numerical results for the family of geodesic curves are also presented for the cases of sphere, prolate spheroid and cylinder.

Convergence Behaviour of Some Iteration Procedures for Exterior Point Method of Centres Algorithms

Abstract: : The convergence rate for a number of iterative procedures for the method of centres, was studied in connection with the investigation of methods for extending the applicability of flight directors. By the use of the Kuhn-Tucker conditions and the duality properties for convex programming problems, it was shown that the augmented cost function, arising in this method, has a second order zero at the optimum point. From this flows the results: that the Staha and Morrison iteration procedures are linearly convergent; the tangent iteration procedure is quadratically convergent; and two interpolation polynomial iteration procedures proposed by the author to overcome the deficiencies of the tangent method away from the optimum point are super-linearly convergent and are thus worthy of further investigation. (Author)

On the curvatures of midcurve and gable curve

Abstract: Introducing in plane affine differential geometry a gable curve to a convex arc as the set of points of intersection for pairs of tangents to the convex arc at endpoints of parallel chords it is shown, that the gable curve and the midcurve for the same chords combined form a curve with a point of inflection at their meeting point P and such that the ratio of their curvatures tends to 3 at P, independent of the convex arc.

Analysisof Six-Portmeasurement Resolution by Conformal Mapping

Abstract: Microwave measurement systems based on the six-port principle fully reconstruct complex wave-vector ratios from sets of redundant magnitude-only readings. This leads to determining the representative points in the complex plane of the vector-ratios as intersections of three or more circles. A simple conformal mapping may be used to visualize the resolution of this method and its sensitivity to errors in the magnitude readings. This mapping transforms families of constant-magnitude-ratio circles in straight parallel lines. The coordinates in the transformed plane are the measured magnitude ratio in dB and the angle between the tangents to the intersecting circles at the intersection points. An example of computer-generated mapping representing a typical location of the q-points is presented and discussed.

Early Tangent Constructions

Abstract: In modern calculus courses the treatment of differentiation and the construction of tangent lines to curves usually precede the treatment of integration and the calculation of areas under curves. This is a reversal of the historical sequence of discovery; as we have seen in the preceding chapters, the calculation of curvilinear areas dates back to ancient times. However, apart from simple constructions of tangent lines to conic sections (with the static Greek view of a tangent line as a line touching the curve in only one point), and the isolated example of Archimedes’ construction of the tangent to his spiral, tangent lines were not studied until the middle decades of the seventeenth century.