Lance Drager - Academia.edu (original) (raw)

Papers by Lance Drager

IEEE Transactions on Information Theory, 2003

We study the observability of a permutation on a finite set by a complex-valued function. The ana... more We study the observability of a permutation on a finite set by a complex-valued function. The analysis is done in terms of the spectral theory of the unitary operator on functions defined by the permutation. Any function can be written uniquely as a sum of eigenfunctions of this operator; we call these eigenfunctions the eigencomponents of. It is shown that a function observes the permutation if and only if its eigencomponents separate points and if and only if the function has no nontrivial symmetry that preserves the dynamics. Some more technical conditions are discussed. An application to the security of stream ciphers is discussed.

26th IEEE Conference on Decision and Control, 1987

Doug McMahon was killed in a climbing accident in late 1986. He was a member of the Department of... more Doug McMahon was killed in a climbing accident in late 1986. He was a member of the Department of Mathematics at Arizona State University. The first two authors feel fortunate to have known Doug and to have worked with him. We would also like to thank Cris Brynes and Clyde Martin for getting us together with Doug.

Trends in The Theory and Practice of Non-Linear Analysis, Proceedings of the VIth International Conference on Trends in the Theory and Practice of Non-Linear Analysis, 1985

We study the non-linear delay differential equation x′(t) + g(x(t), x (t−τ))= f(t) under a non-re... more We study the non-linear delay differential equation x′(t) + g(x(t), x (t−τ))= f(t) under a non-resonance condition which assures the existence of a unique bounded solution. Using the algebra structure of the space of bounded continuous functions we investigate the properties of this solution. We discuss some generalizations and the initial value problem.

where the aij ’s are constants. This is a system of n differential equations for the n unknown fu... more where the aij ’s are constants. This is a system of n differential equations for the n unknown functions x1(t), . . . , xn(t). It’s important to note that the equations are coupled, meaning the the expression for the derivative xi(t) contains not only xi(t), but (possibly) all the rest of the unknown functions. It’s unclear how to proceed using methods we’ve learned for scalar differential equation. Of course, to find a specific solution of (1.1), we need to specify initial conditions for the unknown functions at some value t0 of t,

I’ll use the symbol R to denote the set of real numbers. Let A be a nonempty subset of R. A numbe... more I’ll use the symbol R to denote the set of real numbers. Let A be a nonempty subset of R. A number u is an upper bound for A if a ≤ u for all a ∈ A. We say that A is bounded above if it has an upper bound. Note that saying that a number t is not an upper bound for A is equivalent to the statement “There is some a ∈ A such that t < a.” A number s is called the supremum of A (sup) if it is the least upper bound of A, i.e., s is an upper bound for A and if u is an upper bound for A then s ≤ u. We denote the supremum of A by sup(A). (A little thought shows there can be at most one number that satisfies the definition of sup.) As you recall, the real numbers are constructed by “filling in the holes” in the rational numbers. One way of saying that the holes have all been filled is the Completeness Axiom for the Real Numbers, which is stated as follows. Completeness Axiom for the Real Numbers. If A is a nonempty subset of R that is bounded above then A has a supremum. We have similar co...

These notes are meant to accompany talks in the Geometry Seminar at Texas Tech during the Spring ... more These notes are meant to accompany talks in the Geometry Seminar at Texas Tech during the Spring Semester of 2009. These notes will be updated frequently, so it's best to keep track of them on the web. Note the version time stamp at the bottom of the rst page. At this point, the seminar participants should be comfortable with the basic di erential geometry apparatus done from the point of view of principal bundles. I hope to come back and add this material to these notes at some point in the future. Thanks to the seminar participants for listening to me and straightening me out. Any errors are, of course, my own.

1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathemati... more 1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathematical device for doing this in called a norm. Here is the definition. Definition 1.1. Let V be a vector space over K. A norm on V is a function ‖·‖ : V → R : v 7→ ‖v‖ that has the following properties. (1) ‖v‖ ≥ 0 for all v ∈ V , and ‖v‖ = 0 if and only if v = 0. (2) If λ ∈ K and v ∈ V , ‖λv‖ = |λ| ‖v‖. (3) For all v1, v2 ∈ V , ‖v1 + v2‖ ≤ ‖v1‖ + ‖v2‖. This is called the triangle inequality. Exercise 1.2. Show that ‖−v‖ = ‖v‖ and ‖v1 − v2‖ = ‖v2 − v1‖. The following is a simple consequence of the triangle inequality, which we will include under that name. Proposition 1.3. For all v1, v2 ∈ V , ∣∣ ‖v1‖ − ‖v2‖ ∣∣ ≤ ‖v1 ± v2‖. Proof. We have ‖v1‖ = ‖v1 − v2 + v2‖ ≤ ‖v1 − v2‖+ ‖v2‖, by the triangle inequality, so ‖v1‖ − ‖v2‖ ≤ ‖v1 − v2‖. Switching the roles of v1 and v2 gives ‖v2‖ − ‖v1‖ ≤ ‖v2 − v1‖ = ‖v1 − v2‖. Thus, we have ± [ ‖v1‖ − ‖v2‖ ] ≤ ‖v1 − v2‖,

North-Holland Mathematics Studies, 1985

... Comp. and Math. w. Appls., to appear. [12] W. Layton and R. Mattheij, Estimates over Infinite... more ... Comp. and Math. w. Appls., to appear. [12] W. Layton and R. Mattheij, Estimates over Infinite Intervals of Approximations to Initial Value Problems, Report 8338, Math. Dept.,Cath. Univ. Nijmegen, The Netherlands, 1983. [13] WS ...

Several authors have considered observability problems for the heat equation and relatedpartial d... more Several authors have considered observability problems for the heat equation and relatedpartial differential equations.A basic problem is to determine what kinds of sampling providesufficient information to uniquely determine the initial heat distribntion.We address the case wherethe temperature is measured while travelling along a curve.We consider the special case where the space is a flat torus(of arbitrary dimension)and thecurve is a geodesic.It is shown that,in this case,the observed temperature is sufficient informationto uniquely determine the initial heat distribution if and only if the geodesic is dense in the torus.In the case of a torus,Fourier analysis techniques can be used to write down the solution of theheat equation.This allows us to derive an explicit representation of the observed temperature interms of the initial distribution.We use this representation and some ideas from the theory ofalmost periodic functions to show that the Fourier coefficients of the initial distribution can berecovered from the observation.

North-Holland Mathematics Studies, 1984

Publisher Summary This chapter describes some results concerning the qualitative properties of bo... more Publisher Summary This chapter describes some results concerning the qualitative properties of bounded solutions of the nonlinear (scalar) delay differential equation. The chapter extends these results and places them in a general framework by considering certain subalgebras of the algebra of bounded continuous functions on the line.

Thesis (Ph. D.)--Brandeis University, 1978. "UMI: 7819936." Includes bibliographical re... more Thesis (Ph. D.)--Brandeis University, 1978. "UMI: 7819936." Includes bibliographical references (leaves 203-204). Photocopy.

Differential Geometry: The Interface between Pure and Applied Mathematics, 1987

The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, 1986

Harmonic analysis is used to show that an ergodic translation on a compact abelian group is obser... more Harmonic analysis is used to show that an ergodic translation on a compact abelian group is observed by almost all continuous scalar functions.

Communications in Information and Systems, 2012

The authors consider a natural reachability problem on a Riemannian manifold. Given an initial po... more The authors consider a natural reachability problem on a Riemannian manifold. Given an initial point on a manifold together with an initial subspace of the tangent space at that point, consider piecewise smooth curves such that the velocity at each point along the curve is tangent to the parallel translation of the given initial subspace. The problem is to determine or characterize the set of points reachable by such curves. The authors show that the problem can be formulated in terms of the standard control theory machinery of singular distributions and vector fields by lifting to the frame bundle. It is shown that if the initial velocity subspace is tangent to a totally geodesic submanifold, then the reachable set is contained in that submanifold. Thus our problem makes contact with the existence and uniqueness problem for totally geodesic submanifolds. In the absence of a general result along these lines, it is natural to consider special cases. The authors consider the case where M is the three dimensional Heisenberg group. We show that in this example, all points are reachable and further, the final configuration of the subspace carried along can be specified as well. This stronger result will be expressed in terms of the orthonormal frame bundle.

SIAM Review, 1983

At the Georgia Institute of Technology, a computer program is used in freshman calculus which gra... more At the Georgia Institute of Technology, a computer program is used in freshman calculus which graphically illustrates upper and lower Riemann sums and generates values of their differences. The students often observe that the differences Deltan\Delta _n Deltan seem to be proportional to 1/n{1 / n}1/n, where n is the number of subdivisions; but this is only approximate.We make this rigorous by showing that \[ \Delta _n = \frac{V} {n} + O\left( {\frac{1} {{n^3 }}} \right)\quad {\text{ as }}n \to \infty ,\] for nice functions, where V is the total variation. The proof is simple, and is a nice illustration of the ideas of asymptotic analysis, and several other techniques of analysis.