Remarks on a Paper of Leighton
Counterexamples to theorems for zeros of solutions of second order linear differential equations
DIFFERENTIAL EQUATION↗
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Liapunov's second method
Stability theorem based on Liapunov function, and using invariance property of limit sets of solutions to differential equations
THEOREM PROVING↗
Some stability theorems for ordinary difference equations
Stability theorems using invariance properties of difference equation solutions
INVARIANCE↗
On the optimal control of a system governed by a linear parabolic equation with white noise inputs
Optimum control of system governed by linear parabolic equation with white noise inputs
WHITE NOISE↗
Bounds for the zeros of polynomials Final report, 1 Apr. 1966 - 30 Sep. 1967
Theorem application to companion matrix of polynomial
POLYNOMIAL↗
On an approximation theorem of Kupka and Smale.
Proof of Kupka and Smale approximation theorem concerning differential equations defined on compact manifold
MANIFOLD↗
Kinematics of finite, rigid-body displacements.
Rigid body kinematics for finite displacements, proving several theorems analytically by matrix algebraic methods using mass density description
Thurnauer, P. G.↗
Infinite dimensional multipoint methods and the solution of two point boundary value problems
Multipoint methods for two point boundary value problems with Banach space self mapped, proving convergence theorems for iterative solutions
Bosarge, W. E., Jr.↗
Nth-order flat approximation of the signum function by a polynomial
In the interval studied, the signum function, sgn x, was demonstrated to be uniquely approximated by an odd polynomial f sub n (x) of order 2n-1, for which the approximation is nth order flat with respect to the points (1,1) and (-1,-1). A theorem was proved which states that for even integers n or = 2, the approximating polynomial has a pair of nonzero real roots + or - x sub n such that the x sub n form a monotonically decreasing sequence which converges to the root of 2 as n approaches infinity. For odd n i, f sub n (x) represents a strictly increasing monotonic function for all real x. As n tends to infinity, f sub n (x) converges to sgn x uniformly in two interval ranges.
Hosenthien, H. H.↗
On pursuit with curvature constraints.
A theorem is proved extending results of Cockayne on pursuit with curvature constraints. Let two points (pursuer and evader) move in Euclidean 3-space with constant speeds. Provided the pursuer has greater speed and greater normal acceleration, it is shown that pursuit is always successful. The methods used are similar to Cockayne's. The pursuer, by some preliminary maneuvers, sets up a condition where he is leaving the line of sight in the same direction and with the same speed as the evader. It is shown that from this instant, the pursuer can, without violating constraints, keep the line of sight parallel to the original and ultimately collide with the evader.
Rublein, G. T.↗
Applications of partial orderings to the study of positive definiteness, monotonicity, and convergence of iterative methods for linear systems.
Most of the known results concerning convergence of iterative methods for solving linear systems involve either positive definiteness or monotonicity. In this paper a new concept, called K-semipositivity, is introduced, which provides a link between convergence theory, monotonicity, and positive definiteness. By using this concept, together with partial orderings in Euclidean n-space, several new convergence theorems are proved. Application to Jacobi's methods and the theory of regular splittings shows the usefulness of these new results.
Vandergraft, J. S.↗
A result on differential inequalities and its application to higher order trajectory derivatives
A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.
Gunderson, R. W.↗
Behavior near constant solutions of functional differential equations
Techniques have been developed to determine in a systematic way the local behavior near constant solutions. Local integral manifolds play a very important role in this development, as they have also for ordinary differential equations. An attempt is made to indicate a few more applications of these methods to some problems in bifurcation in the spirit of Sotomayor (to appear) and to a growth model of Cooke and Yorke (to appear). It is also shown how to prove a theorem on stability under constantly acting disturbances using these methods.
Hale, J. K.↗
The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion
The problem of an anisotropic ellipsoidal inclusion which undergoes a stress-free transformation strain (in the sense of J. D. Eshelby) is considered, and the following theorem is proved: if an ellipsoidal region in an infinite anisotropic linear elastic medium undergoes, in the absence of its surroundings, a stress-free transformation strain which is a polynomial of degree M in given position coordinates, then the final stress and strain state in the transformed inclusion, when constrained by its surroundings, is also a polynomial of degree M in those position coordinates.
Asaro, R. J.↗
A comparison of stratified versus regression estimators
LANDSAT data acquired over an agricultural area along with ground enumeration of the same area are used to obtain crop acreage estimates which are better (as measured in terms of bias and variance) than can be obtained from either data source alone. Two basic approaches considered within the AgRISTARS program are a stratified crop acreage estimator and a regression estimator. A statement of the problem was mathematically formulated and some theorems were proved which relate to the variance of the two estimators. For a particular set of data, the regression and stratified estimators are compared in terms of certain easily computed parameters.
Takacs, H. C.↗
Construction of solutions for some nonlinear two-point boundary value problems
Constructive existence and uniqueness results for boundary value problems associated with some simple special cases of the second order equation y'' = f(x,y,y') 0 or = x or = 1, are sought. The approach considered is to convert the differential equation and boundary conditions to an integral equation via Green's functions, and then to apply fixed point and contraction map principles to a sequence of successive approximations. The approach is tested on several applied problems. Difficulties in trying to prove general theorems are discussed.
Pennline, J. A.↗
Sturm-Liouville eigenproblems with an interior pole
The eigenvalues and eigenfunctions of self-adjoint Sturm-Liouville problems with a simple pole on the interior of an interval are investigated. Three general theorems are proved, and it is shown that as n approaches infinity, the eigenfunctions more and more closely resemble those of an ordinary Sturm-Liouville problem. The low-order modes differ significantly from those of a nonsingular eigenproblem in that both eigenvalues and eigenfunctions are complex, and the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. In addition, the WKB is shown to be accurate for all n, and all eigenvalues of a normal one-dimensional Sturm-Liouville equation with nonperiodic boundary conditions are well separated.
Boyd, J. P.↗