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.     

Averaging shifted histograms

A method which consists in shifting different histograms of the same spectrum and then taking their average is presented in order to smooth the data and to increase the localization accuracy and separation of the peaks. The statistical properties of this method are investigated. The average of two histograms with shifted bin limits is studied. It is shown that for histograms with random bin limits, distributed according to

$\text{F}{}_{\mathrm{i}}\text{(x)=}\text{∫}\text{?∞}\text{x}\text{?}{}_{\mathrm{i}}\text{(ξ, μ}{}_{\mathrm{i}}\text{, σ)}\text{d}\text{ξ}$

; where the standard deviation σ is very small compared to the difference of the means (μ_i+1 ? μ_i) for ll i the zero order approximation to the variance of this histogram is given by:

$\text{var}\text{∽}\text{(H)=}\text{∑}\text{i=0}\text{m}\text{(A}{}_{\mathrm{i+1}}\text{?a}{}_{\mathrm{i}}\text{)}{}^2\text{F}{}_{\mathrm{i+1}}\text{(x)(1?F}{}_{\mathrm{i+1}}\text{(x))}$

, where

$\text{a}{}_{\mathrm{i}}\text{=}\text{1}\text{x}{}_{\mathrm{i=1}}\text{?x}{}_{\mathrm{i}}\text{∫}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i+1}}\text{g(ξ)}\text{d}\text{ξ}$

and g is an unknown function fitted by the histogram. Formula (1) gives also the relation:

$\text{v}\text{a}\text{?}\text{r}\text{(}\text{(H}{}_1\text{+ H}{}_2\text{)}\text{2}\text{) =}\text{1}\text{4}\text{(}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_1\text{(x)) +}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_2\text{(x))}$

, when H₁ and H₂ have stochastically independent bin limits.When the histogram H is considered as a spline function S of order one it is shown that for the minimization criterion with respect to the coefficient of the spline:

$\text{M}{}_1\text{=}\text{min}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{(g(x) ? S}{}_1\text{(x))}{}^2\text{d}\text{x}$

, the following result holds: $\text{M}{}_{\mathrm{a}}\text{?}{}^1{}_2\text{(M}{}_1\text{+ M}{}_2\text{)}$, where $\text{S}{}_{\mathrm{a}}\text{(x) =}{}^1{}_2\text{(S}{}_1\text{(x) + S}{}_2\text{(x))}$. If the number of shifted histograms tends to infinity, then

$\text{S}{}_{\mathrm{\infty }}\text{(x) = [Γ(x + h) + Γ(x ? h) ? 2Γ(x)]/h}{}^2$

, where $\text{Γ(x) =}\text{∫}\text{?∞}\text{x}\text{∫}\text{?∞}\text{η}\text{g(ξ)}\text{d}\text{ξ}\text{d}\text{η}$, and h is a constant bin size. Then

$\text{M}{}_{\mathrm{\infty }}\text{≈}\text{h}{}^4\text{144}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{g″}{}^2\text{(x)}\text{D}\text{x}$

. Extensions to two-dimensional histograms and to higher order (empirical distributions) are presented.     

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.     

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.     

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.     

Improved M-matrix condition for the existence of positive-real weighting

It is proved that, for a given stable transfer matrix G(s), there exists a constant diagonal matrix W which makes WG(s) positive-real if Re g_ii(jω) ≥ 0 and I?? is an M-matrix where ? = (?_jk) is defined by ?_ii = 0 and $\text{?}{}_{\mathrm{jk}}\text{=}\text{sup}{}_{\mathrm{\omega }}\text{|g}{}_{\mathrm{ik}}\text{(gω) |/(}\text{Re}\text{[g}{}_{\mathrm{ii}}\text{(jω)]·}\text{Re}\text{[g}{}_{\mathrm{kk}}\text{(jω])}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

A comparison of CPU and GPU implementations for solving the Convection Diffusion equation using the local Modified SOR method

In this paper we study a parallel form of the SOR method for the numerical solution of the Convection Diffusion equation suitable for GPUs using CUDA. To exploit the parallelism offered by GPUs we consider the fine grain parallelism model. This is achieved by considering the local relaxation version of SOR. More specifically, we use SOR with red-black ordering using two sets of parameters _ω1ij${\omega }_{1\mathit{ij}}$ and _ω2ij${\omega }_{2\mathit{ij}}$ for the 5 point stencil. The parameter _ω1ij${\omega }_{1\mathit{ij}}$ is associated with each red (i + j   even) grid point (i,j)$(i\text{,}j)$, whereas the parameter _ω2ij${\omega }_{2\mathit{ij}}$ is associated with each black (i+j$(i+j$ odd) grid point (i,j)$(i\text{,}j)$. The use of a parameter for each grid point avoids the global communication required in the adaptive determination of the best value of ω$\omega$ and also increases the convergence rate of the SOR method (Varga, 1962) [38] and (Young, 1971) [41]. We present our strategy and the results of our effort to exploit the computational capabilities of GPUs under the CUDA environment. Additionally, two parallel CPU programs utilizing manual SSE2 (Streaming SIMD Extensions 2) and AVX (Advanced Vector Extensions) vectorization were developed as performance references. The optimizations applied on the GPU version were also considered for the CPU version. Significant performance improvement was achieved with all three developed GPU kernels differentiated by the degree of recomputations thus affecting the flops per element access ratio.     

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.     

Kleene chain completeness and fixedpoint properties

Let L denote the nonscalar complexity in k(x₁,…, x_n). We prove $\text{L(?,??/?x}{}_1\text{,…,??/?x}{}_{\mathrm{n}}\text{)?3L(?)}$. Using this we determine the complexity of single power sums, single elementary symmetric functions, the resultant and the discriminant as root functions, up to order of magnitude. Also we linearly reduce matrix inversion to computing the determinant.     

Real-time estimation scheme for the spot cross volatility of jump diffusion processes

Given a finite set of observed data {_Xtk(_ω0),_Ytk(_ω0)}$\{X_{t_k}({\omega }_0)\text{,}{Y}_{t_k}({\omega }_0)\}$ of just one sample path at n$n$ regularly spaced time of the processes _Xt$X_t$ and _Yt$Y_t$ satisfying d_Xt=_a0(t)dt+_a1(t)d_W1(t)+_a2(t)d_W2(t)+d_J1(t),d_Yt=_b0(t)dt+_b1(t)d_W1(t)+_b2(t)d_W2(t)+d_J2(t),t∈[0,T]$d{X}_t=a_0(t)dt+a_1(t)d{W}_1(t)+a_2(t)d{W}_2(t)+d{J}_1(t)\text{,}d{Y}_t=b_0(t)dt+b_1(t)d{W}_1(t)+b_2(t)d{W}_2(t)+d{J}_2(t)\text{,}t\in [0\text{,}T]$, where _J1,_J2$J_1\text{,}{J}_2$ are jump process, we are to investigate a numerical scheme for the estimation of the value _νX,_Y(t)=_a1(t)_b1(t)+_a2(t)_b2(t)${\nu }_{X\text{,}Y}(t)=a_1(t)b_1(t)+a_2(t)b_2(t)$ called cross volatility. Our framework also contains the volatility estimation problem as a special case. We will show that our scheme works under mild assumptions on the activity of the jump process _Jt$J_t$.