Nondegenerate Point Research Articles

On the nature of critical points in leakage regimes of a conductor-backed coplanar strip line

Leaky dominant mode propagation regimes of a conductor-backed coplanar strip line are rigorously analyzed using the concept of critical or equilibrium points from catastrophe and bifurcation theories, in conjunction with a full-wave integral equation solution. The existence of nondegenerate or Morse critical points (MCPs) and degenerate or fold (turning) critical points (FPs) in coupling and leakage regions are associated with the occurrence of improper real and complex (leaky) solutions. The locations and types of critical points determine the stability and instability of the transmission line system with respect to small changes in geometrical parameters. The dispersion behavior of improper real and complex (leaky) solutions are efficiently reproduced in the local neighborhood of MCPs and FPs using a Taylor series expansion about those points. The qualitative and quantitative dynamic behavior of the transmission line modes can be investigated by examining the evolution of nondegenerate and degenerate points versus some structural parameter, such as strip width. The proposed analysis enables the prediction of bifurcation situations and the existence of improper real and complex solutions and gives a complete description of the system's structural behavior.

Hamiltonian triangulations for fast rendering

High-performance rendering engines are often pipelined; their speed is bounded by the rate at which triangulation data can be sent into the machine. An ordering such that consecutive triangles share a face, which reduces the data rate, exists if and only if the dual graph of the triangulation contains a Hamiltonian path. We (1) show thatany set ofn points in the plane has a Hamiltonian triangulation; (2) prove that certain nondegenerate point sets do not admit asequential triangulation; (3) test whether a polygonP has a Hamiltonian triangulation in time linear in the size of its visibility graph; and (4) show how to add Steiner points to a triangulation to create Hamiltonian triangulations that avoid narrow angles.

Bifurcation phenomena in confined thermosolutal convection with lateral heating: Commencement of the double-diffusive region

As has recently been reported by Tsitverblit and Kit [Phys. Fluids A 5, 1062 (1993)], a vertical rectangular enclosure containing stably stratified brine and differentially heated from its side walls is characterized by complex steady bifurcation phenomena. In the present work, the structure of steady solutions in the enclosure has been studied in detail for several values of the salinity Rayleigh number, RaS, fixed near the commencement of the double-diffusive region. It was found that when the thermal Rayleigh number, RaT, is either very small or sufficiently large, the steady solution is unique while in an intermediate region of this parameter, there exists a great variety of the multiple steady flows, being the result of nondegenerate hysteresis points and isolas of asymmetric solutions forming as RaS is increased. In particular, at the maximal value of RaS considered there have been observed symmetric and asymmetric one-, two-, three-, four-, and five-cell flows. Despite the multiplicity of the flow patterns, a critical interval of the buoyancy ratio has been distinguished, above and below which the generic characteristics of the steady solutions were found to resemble the respective features of the ‘‘successive’’ and ‘‘simultaneous’’ regimes of layer formation whose existence was established in previous studies. Although the set of the steady solutions has been found to contain no linearly stable multicell flows, the perturbation was so long retained in the close proximity of the unstable steady solutions that such flows could be easily observable in the experiment. In spite of the appreciably different range of the Rayleigh numbers, the physically meaningful parameters suggested in previous studies were found to be represented in the present results.

Solutions of the variational problem in the Curie—Weiss—Potts model

The variational problem for the Curie—Weiss—Potts model is solved completely. The results extend those of Ellis and Wang (1990, 1992), in which we study limit theorems and parameter estimations for the model and consider only the case of zero external field. In contrast to the Curie—Weiss model, this model has phase transitions in non-zero external field. All the solutions of the variational problem are non-degenerate points, so all the results in Ellis and Wang (1990, 1992) can be easily extended to the case considered here. We will also point out that simultaneous parameter estimation is impossible.

New descent rules for solving the linear semi-infinite programming problem

The algorithm described in this paper approaches the optimal solution of a continuous semi-infinite linear programming problem through a sequence of basic feasible solutions. The descent rules that we present for the improvement step are quite different when one deals with non-degenerate or degenerate extreme points. For the non-degenerate case we use a simplex-type approach, and for the other case a search direction scheme is applied. Some numerical examples illustrating the method are given.

Multi‐valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

AbstractMulti‐valued solutions are constructed for 2 × 2 first‐order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non‐zero 3‐jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non‐tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n‐th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.

An Algorithm for Computing a Nondegenerate Hysteresis Point

We propose an algorithm for computing nondegenerate hysteresis points arising in bifurcation problems with two parameters such as G:\boldsymbol R^2 \times \boldsymbol R^N → \boldsymbol R^N . A combination of methods in [5] and [7] requires finding zeros of an extended system of 4N+3 variables. Shintani and Kanda [6] proposes another algorithm, in which computation of extended system of 3N+3 variables is sufficient. On the contrary, in our algorithm, we should find zeros of another extended system of 3N+2 variables.

Homoclinic orbits on compact manifolds

Under a suitable assumption on the potential energy V, we prove the existence of a homoclinic orbit of the equation D t ( x′( t)) + grad V( x( t)) = 0, x ϵ M, where M is a compact Riemannian manifold. Our assumption is satisfied, for instance, if the function V has a unique nondegenerate maximum point.

Convergence properties of trust region methods for linear and convex constraints

We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions.

The multidimensional stationary phase method. The second term of the asymptotic expansions

The second term in the asymptotic expansion of the contribution from a nondegenerate stationary point is determined for the multidimensional case.

Importance of the L2tide in the straits of Georgia and Juan De Fuca

Abstract The semidiurnal L2 tide, in a model simulation of mixed tides in the straits of Georgia and Juan de Fuca, is the only tidal constituent with a nondegenerate amphidromic point in the system.

Extremal properties of quadratic differentials with strip-shaped domains in the structure of the trajectories

One considers the modulus problem for a family of homotopic classes {Hi}, in the extended complex plane , of the following types. The classes Hn consist of closed Jordan curves, homotopic to appropriate nondegenerate contours or point curves, and also of arcs with endpoints in (distinct or coinciding) distinguished points in . One establishes the relation of the indicated extremal metric problem and the problem on the extremal decomposition of in the family of systems of mutually nonoverlapping domains {Di}, associated with the family of the classes {Hi}. The results of this paper complement a previous theorem of the author [Moduli of families of curves and quadratic differentials. Trudy Mat. Inst. Akad. Nauk SSSR, Vol. 139, 1980].

Projected gradient methods for linearly constrained problems

The aim of this paper is to study the convergence properties of the gradient projection method and to apply these results to algorithms for linearly constrained problems. The main convergence result is obtained by defining a projected gradient, and proving that the gradient projection method forces the sequence of projected gradients to zero. A consequence of this result is that if the gradient projection method converges to a nondegenerate point of a linearly constrained problem, then the active and binding constraints are identified in a finite number of iterations. As an application of our theory, we develop quadratic programming algorithms that iteratively explore a subspace defined by the active constraints. These algorithms are able to drop and add many constraints from the active set, and can either compute an accurate minimizer by a direct method, or an approximate minimizer by an iterative method of the conjugate gradient type. Thus, these algorithms are attractive for large scale problems. We show that it is possible to develop a finite terminating quadratic programming algorithm without non-degeneracy assumptions.

Rationality for isolated expansive sets

An expansive isolated set of a diffeomorphism with nondegenerate periodic points has a rational zeta function.

The behavior of oscillatory integrals with degenerate stationary points

(see Theorem 1.1 in §1). When the phase function {x) has only non-degenerate stationary points (in supp[/o]) or is analytic, I{a) is expanded asymptotically as |a|->oo (cf. Hormander [1], Varchenko [8], Kaneko [4], etc.),and then we can obtain the above estimate through that expansion. But it seems difficult to do so when all derivatives of $(x) vanish at some points. We take this case into consideration. In our methods we do not employ the asymptotic expansion. We can apply the above estimate to a scattering inverse problem studied by Ikawa [3]. Consider the scattering by obstacles formulated by Lax and Phillips[5]. Then the scattering matrix S(z) is meromorphic in the whole complex plane and analytic on the lower half plane {z: Imz^O} (cf. Chapter V of [5]). Ikawa [2], [3] examine the distribution of the poles of S(z) in the case where the obstacles consist of two convex bodies. In \2~] it is proved that if

The prediction of critical points for thermal explosions in a finite volume

A technique for predicting critical points for thermal ignition by solving specific extended systems of equations is presented. In particular, two extended systems for locating limit points and nondegenerate hysteresis points are discussed. These are implemented in a finite-element formulation, which allows the points of ignition, extinction, and loss of criticality to be calculated for explosions in a finite volume of arbitrary shape. This is demonstrated for the special case of thermal ignition in a finite cylinder. Continuation methods are used to follow the paths of critical points as the aspect ratio of the cylinder varies.

Oscillatory Bifurcations in Singular Perturbation Theory, II. Fast Oscillations

We construct closed orbits for a singular perturbation problem near a nondegenerate equilibrium point of its reduced system, in case the linearization of the fast part of the flow has a complex conjugate pair of pure imaginary eigenvalues. The mechanism for constructing the closed orbits is essentially different from the mechanism of a Hopf bifurcation.

Topological degree and number of Nash equilibrium points of bimatrix games

In this paper, using the topological degree, we give a new proof of a well-known result: the number of Nash equilibrium points of a nondegenerate bimatrix game is odd. The calculation of the topological degree allows the localization of the whole set of non-degenerate equilibrium points.

Asymptotic analysis of Gaussian integrals. I. Isolated minimum points

This paper derives the asymptotic expansions of a wide class of Gaussian function space integrals under the assumption that the minimum points of the action are isolated. Degenerate as well as nondegenerate minimum points are allowed. This paper also derives limit theorems for related probability measures which correspond roughly to the law of large numbers and the central limit theorem. In the degenerate case, the limits are non-Gaussian.

The central limit theorem for the spectrum of the random one-dimensional Schrödinger operator

Let HL = −d2/dt2+q(t,ω) be an one-dimensional random Schrodinger operator in ξ2(−L, L) with the classical boundary conditions. The random potential q(t,ω) has a form q(t, ω)=F(xt), where xt is a Brownian motion on the Euclidean v-dimensional torus, F∶Sv→ R1 is a smooth function with the nondegenerated critical points, minsv F = 0. Let\(N_L (\lambda ) = \sum _{\lambda _i (L) \leqslant \lambda } 1(\lambda _i (L,\omega )\) are the eigenvalues of HL) be a spectral distribution function in the “volume” [− L,L] and N(λ) = limL→∞(1/2L)NL(λ) be a corresponding limit distribution function.