James Angelos

On the periodic logistic equation

We show that the p-periodic logistic equation xn+1 = „n mod pxn(1 i xn) has cycles (periodic solutions) of minimal periods 1;p;2p;3p;:::: Then we extend Singer’s theorem to periodic dierence equations, and use it to show the p-periodic logistic equation has at most p stable cycles. Also, we present computational methods investigating the stable cycles in case p = 2 and 3:

Triangular truncation and finding the norm of a Hadamard multiplier

For B a fixed matrix, we study the problem of using a criterion of Haagerup to find the norm of the map A ↦ A · B, where · is the Hadamard or entrywise product of matrices. The techniques developed are applied to triangular truncation, and it is proved that if Kn is the norm of triangular truncation of n×n matrices, then Knlogn→π−1.

Strong uniqueness in Lp spaces

Abstract Let V be a finite dimensional subspace of L p , 1 p f ϵ L p / V , it is shown that the best approximation to f from V is strongly unique of order α = 2 or p . Let V be an n -dimensional Haar subspace of L 1 [ a , b ], the continuous functions on [ a , b ] with the L 1 norm. Let f ϵ L 1 [ a , b ]/ V , that is Lipschitz and so that V 1 = span{ V , f } is a Haar subspace. Then it is shown that the best approximation to f from V is strongly unique of order 2.

Local and global Lipschitz constants

Abstract Let X be a closed subset of I = [−1, 1], and let B n ( f ) be the best uniform approximation to f ϵ C [ X ] from the set of polynomials of degree at most n . An extended global Lipschitz constant is defined for f, and it is shown that this constant is asymptotically equivalent to the strong unicity constant. Estimates of the size of the local Lipschitz constant for f are given when the cardinality of the set of extremal points of f − B n ( f )is n + 2. Examples which illustrate that the local and extended global Lipschitz constants may have very different asymptotic behavior are constructed.

Local Lipschitz constants

Abstract Let X be a closed subset of I= [− 1, 1], For f ϵ C[X], the local Lipschitz constant is defined to be λ nδ (f) = sup { ∥B n (f) − B n (g)∥ ∥f − g∥: 0 , where Bn(g) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as δ → 0 of the local Lipschitz constant of f, and an explicit expression is given for this common value. The, possibly very different, characterizations of local and global Lipschitz constants are also considered.

EXISTENCE AND STABILITY OF PERIODIC ORBITS OF PERIODIC DIFFERENCE EQUATIONS WITH DELAYS

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f (n − 1,xn−k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p,k) nonautonomous p gcd(p,k) periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call the pk-Sharkovsky’s ordering of the positive integers, and extend Sharkovsky’s theorem to periodic difference equations with delays. Finally, we characterize global stability and show that the period of a globally asymptotically stable orbit must divide p.

Limit cycles for successive projections onto hyperplanes in Rn

Abstract In this paper we consider successive orthogonal projections onto m hyperplanes in R n, where m ⩾ 2 and n ⩾ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in R 2, are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in R n.

Local Lipschitz and strong unicity constants for certain nonlinear families

Abstract Let X be a compact metric space, and let V = { F ( a , x ): a ϵ A } where A is an open subset of R n , and F ( a , x ) and ∂F ∂a i , 1 ⩽ i ⩽ n , are continuous on A × X . Suppose f ϵ C ( X ) is weakly normal; that is (i) f has a best approximation F(a ∗ , ·) = B v (f) such that N = dim W(a ∗ ) ≡ dim span {( ∂F ∂a i )(a ∗ , ·): 1 ⩽ i

Oldroyd's viscosity result extended to circular disk particles dispersed in Newtonian fluids

n} is maximal, and (ii) certain weakened versions of the local Haar condition, a sign property equivalent to a form of asymptotic convexity, and Property Z hold. For those weakly normal functions f for which { x ϵ X: ¦f(x) − F(a ∗ , x)¦ = ∥f − F(a ∗ , ·)∥ } has exactly N + 1 points, we give constructions of the local Lipschitz and strong unicity constants, as well as show that B v ( f ) is differentiable.

An Extension of Sharkovsky’s Theorem to Periodic Difference Equations

A viscosity equation is formulated on the basis of Oldroyds theory for elastic and viscous properties of emulsions and suspensions by considering the drops as cylindrical rather than spherical in shape. The problem is formulated in three dimensions using cylindrical coordinates. The result can be considered as applicable to liquid, circular disk particles with negligible thickness, such as platelets, in dilute suspensions. In the present analysis, initially, stress effects are assumed uniform along the length of the cylinder, the z coordinate of velocity decays exponentially with time, and the interactive effects of the particles are assumed negligible