Some remarks on a conjecture in parameter adaptive control

A.S. Morse has raised the following question: Do there exist differentiable functions

$\text{f:R}{}^2\text{→ R}\text{and}\text{g:R}{}^2\text{→ R}$

with the property that for every nonzero real number λ and every (x₀, y₀) ∈ $\text{R}$² the solution (x(t),y(t)) of

$\text{x}\text{?}\text{(t) = x(t) + λf(x(t),y(t))}$

,

$\text{y}\text{?}\text{(t) = g(x(t),y(t))}$

,

$\text{x(0) = x}{}_0\text{, y(0) = y}{}_0$

, is defined for all t ? 0 and satisfies

$\text{lim}\text{t → + ∞}$

and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.     

Crosscorrelation between linearly and nonlinearly distorted versions of a given signal

A given deterministic signal x(.) is distorted by passing it through a linear time-invariant filter and also by subjecting it to the action of an instantaneous nonlinearity. The resulting time crosscorrelation of the two distorted versions of the original signal is expressed by the function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]k(t?t′)x(t?s)dt dt′}$

, where x(.) is the given signal, k(.) is the nonnegative definite impulse response of the linear filter, and g(.) is the output-input characteristic of the zero-memory nonlinear device. The problem considered is that of determining conditions on the pair (x,g) such that R₂(s) ? R₂(0) for all s and any choice of nonnegative definite filter function k; the principal result is the formulation of a necessary and sufficient condition for R₂ to have a global maximum at the origin. In particular, the peak value occurs at the origin if and only if $\text{G}{}_{\mathrm{x}}{}^1\text{(ω)X(ω)}$ is real and nonnegative for all ω ? 0, where G_x(.) and X(.) are the Fourier transforms of g[x(.)] and x(.), respectively. An equivalent condition is that the correlation function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]x(t?s)dt}$

, previously studied by Richardson, be nonnegative definite.Several examples are given, and it is shown that, unlike the case for R₁(.), monotonicity of g(.) is not a sufficient condition for R₂(.) to have a global maximum at s = 0 independently of the choice of filter characteristic k. Certain extensions of these results are given for the case when x(.) is a Gaussian random input.     

Mathematical foundations of Bézier's technique

Bezier's method is one of the most famous in computational geometry. In his book Numerical control Bezier gives excellent expositions of the mathematical foundations of this method. In this paper a new expression of the functions {f_n,i(u)}

$\text{f}{}_{\mathrm{n,i}}\text{(u)=1?}\text{Σ}\text{p=0}\text{i?1}\text{C}{}^{\mathrm{p}}{}_{\mathrm{n}}\text{u}{}^{\mathrm{p}}\text{(1?u)}{}^{\mathrm{n?p}}\text{(i=1,2,…,n)}$

is obtained.Using this formula, we have not only derived some properties of the functions {f_n,i(u)} (for instance $\text{f}{}_{\mathrm{n,n}}\text{(u) < f}{}_{\mathrm{n,n?1}}\text{(u)<...<f}{}_{\mathrm{n,1}}\text{(u)}\text{u}\text{? [0, 1]}$ and functions {f_n,i(u)} increase strictly at [0, 1] etc) but also simplified systematically all the mathematical discussions about Bezier's method.Finally we have proved the plotting theorem completely by matrix calculation.     

Lower bounds for polynomials with algebraic coefficients

We give a method, based on algebraic geometry, to show lower bounds for the complexity of polynomials with algebraic coefficients. Typical examples are polynomials with coefficients which are roots of unity, such as

$\text{Σ}\text{j=1}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>i</mtext>}}\text{X}{}^{\mathrm{i}}$

and

$\text{Σ}\text{j=i}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>pi</mtext>}}\text{X}{}^{\mathrm{j}}$

where p_j is the jth prime number.We apply the method also to systems of linear equations.     

Asymptotic expansions for the Landau density and distribution functions

Asymptotic expansions are given for the Landau function

$\text{φ(λ)=}\text{1}\text{2πi}\text{∫}\text{σ+i∞}\text{σ?i∞}\text{e}{}^{\mathrm{\mu ln\mu +\lambda \mu }}\text{dμ}$

and for its integral, as λ → ± ∞. Further, it is shown that φ(λ) is indeed a probability density function, as is always assumed in applications.     

The relation between information theory and the differential geometry approach to statistics

?encov has shown that the Riemannian metric on the probability simplex ∑x_i = 1 defined by $\text{(ds)}{}^2\text{=}\text{∑(dx}{}_{\mathrm{i}}\text{)}{}^2\text{x}{}_{\mathrm{i}}$ has an invariance property under certain probabilistically natural mappings. No other Riemannian metric has the same property. The geometry associated with this metric is shown to lead almost automatically to measures of divergence between probability distributions which are associated with Kullback, Bhattacharyya, and Matusita. Certain vector fields are associated in a natural way with random variables. The integral curves of these vector fields yield the maximum entropy or minimum divergence estimates of probabilities. Some other consequences of this geometric view are also explored.     

A distribution-free geometric upper bound for the probability of error of a minimum distance classifier

An upperbound to the probability of error per class in a multivariate pattern classification is derived. The bound, given by

$\text{P(E|class w}{}_{\mathrm{i}}\text{)≤}\text{N}\text{R}{}^2{}_{\mathrm{i}}$

is derived with minimal assumptions; specifically the mean vectors exist and are distinct and the covariance matrices exist and are non-singular. No other assumptions are made about the nature of the distributions of the classes. In equation (i) N is the number of features in the feature (vector) space and R_i is a measure of the “radial neighbourhood” of a class. An expression for R_i is developed. A comparison to the multivariate Gaussian hypothesis is presented.     

Normal forms for nonautonomous difference equations

We extend Henry Poincaré's normal form theory for autonomous difference equations χ_{k + 1} = f(χ_k) to nonautonomous difference equations χ_{k + 1} = f_k(χ_k). Poincaré's nonresonance condition $\text{α}{}_{\mathrm{j}}\text{-П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{=1α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for eigenvalues is generalized to the new nonresonance condition λ_j∪$\text{α}{}_{\mathrm{j}}\text{⊔П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for spectral intervals.     

Truncation error for band-limited nonstationary processes

Let x(t) be a real-valued random process band-limited to the interval [$\text{?1}\text{2T}\text{,}\text{1}\text{2T}$] for some T > 0. In this note we find an upper bound on the mean square of the truncation error involved when x(t) is approximated in the interval $\text{|t| ?}\text{T}\text{2}$ by the finite selection

$\text{∑}\text{n=?N}{}_1\text{N}{}_2\text{x(nt)}\text{sin}\text{π(t?nT)}\text{T}\text{π(t?nT)}\text{T}$

of terms from its sampling expansion representation.     

The decentralized stabilization and control of a class of unknown non-linear time-varying systems

The following problem is considered in this paper. Suppose a system S consists of a set of arbitrary interconnected subsystems S_i, i = 1, 2, …, Ω; is it possible to stabilize and satisfactorily control the whole system S by using only local controllers about the individual subsystems without a knowledge of the manner of the actual interconnections of the whole system? Sufficient conditions are obtained for such a result to hold true; in particular it is shown that a system S consisting of a number of subsystems S_i connected in an arbitrary way between themselves with finite gains: $\text{S}{}_{\mathrm{i}}\text{:}\text{x}\text{?}{}_{\mathrm{i}}\text{=}\text{A}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}\text{+}\text{b}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)u}{}_{\mathrm{i}}\text{, y}{}_{\mathrm{i}}\text{=}\text{c}\text{′}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}$ where A_i and b_i have a particular structure, may be satisfactorily controlled by applying only local controllers C_i about the individual subsystems: C_i: u_i = K′_i(?)x_i where K_i is a constant gain matrix with the scalar ? appearing as a parameter, provided ? is large enough.     

An optimal triangulation for second-order elliptic problems

Let Ω be a polygonal domain in $\text{R}{}^{\mathrm{n}}$, τ_h an associated triangulation and u_h the finite element solution of a well-posed second-order elliptic problem on (Ω, τ_h). Let M = {M_i}^{p + q}_{i = 1} be the set of nodes which defines the vertices of the triangulation τ_h: for each i,$\text{M}{}_{\mathrm{i}}\text{= {x}{}_{\mathrm{i}}{}^{\mathrm{l}}\text{¦1 ? l ?n}}$ in Rⁿ. The object of this paper is to provide a computational tool to approximate the best set of positions M? of the nodes and hence the best triangulation $\text{}\text{?}\text{gt}{}_{\mathrm{h}}$ which minimizes the solution error in the natural norm associated with the problem.The main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates x_i^l, 1 ? l ? n, of each of the variable nodes M_i, i = 1,…, p.     

computer aided intuition in abstract algebra

Two examples are given in which the computer was used to supplement intuition in abstract algebra. In the first example, the computer was used to search Cayley tables of 4 element groupoids to find those which are 5-associative but not 4-associative. (n-associative means that the product of any n elements is independent of the way the factors are grouped by parentheses.) The computer generated examples suggested the existence of n element groupoids which are (2^n?2+1)-associative but not (2^n-2)-associative, for each integer n≧4.In the second example, the computer counted the numbers g₂(m) of invertible 2×2 matrices with entries chosen from the ring Z_mof integers, for m = 2, 3, 4,…, 18. The insight gained from these results led to a proof that there are

invertible n×n matrices over Z_m.Some applications to graduate and undergraduate instruction are indicated.     

Entropy and set covering

If the set covering constraints are Ax ? 1 and x_j ∈ {0,1}, the prior probability that the jth subset participates in an optimal covering (independently of subset costs) is shown to be given by the principal row eigenvector of A¹A, where $\text{a}{}_{\mathrm{ji}}{}^1\text{= 1 ? a}{}_{\mathrm{ij}}$. These probabilities lead to new and interesting objective functions, which are shown to be equivalent to cross entropy or weighted cross-entropy. The probabilities can also be used to obtain better bounds for heuristic solutions to optimal covering and set representation problems.