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.     

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.     

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.     

Solution errors in finite element analysis

The development of the finite element method so far indicates that it is a discretization technique especially suited for positive definite, self-adjoint, elliptic systems, or systems with such components. The application of the method leads to the discretized equations in the form of $\text{u}\text{?}\text{=}\text{f(u)}$, where u lists the response of the discretized system at n preselected points called nodes. Instead of explicit expressions, vector function f and its Jacobian f,_u are available only numerically for a numerically given u. The solution of $\text{u}\text{?}\text{=}\text{f(u)}$ is usually a digital computer. Due to finiteness of the computer wordlength, the numerical solution $\text{u}{}_{\mathrm{c}}$ is in general different from u. Let $\text{u}\text{(}\text{x}\text{, t)}$ denote the actual response of the system in continuum at points corresponding to those of u. In the literature. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}$ is called the discretization errors, $\text{u}\text{-}\text{u}{}_{\mathrm{c}}$ the round-off errors, and the s is. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}{}_{\mathrm{c}}$ is called the solution errors. In this paper, a state-of-the-art survey is given on the identification, growth, relative magnitudes, estimation, and control of the components of the solution errors.     

On some convergence results for FDM with irregular mesh

The proof of convergence of the finite difference method with arbitrary irregular meshes for some class of elliptic problems is presented. By the use of the truncation error technique and stability analysis it was showed that $\text{max}{}_{\mathrm{i}}\text{¦u}{}^{\mathrm{i}}\text{? u}{}_{\mathrm{h}}{}^{\mathrm{i}}\text{¦? Ch}$, i.e., the solution u_h converges linearly with the size of the star. Correctness of this theorem was also confirmed by numerical tests.     

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.     

Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation $\text{δ(a}{}_{\mathrm{n}}\text{φ}{}_{\mathrm{p}}\text{(δχ}{}_{\mathrm{n}}\text{))=b}{}_{\mathrm{n}}\text{f(χ}{}_{\mathrm{n+1}}\text{,}\text{φ}{}_{\mathrm{p}}\text{(u)=|u|}{}^{\mathrm{p-2}}\text{u,}\text{p>1}$ where {a_n}, {b_n} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of a_n, b_n is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.     

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.     

Maximum principle and uniform convergence for the finite element method

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ? Rⁿ, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u_h converges uniformly to the exact solution u if u ? W_1,p (Ω), with p > n, and that $\text{∥u?u}{}_{\mathrm{h}}\text{∥}{}_{\mathrm{L}}\text{∞(Ω) = O(h)}$ if u ? W²,p(Ω), with 2p > n. In the case of the model problem ?Δu+au = f in Ω, u = u_o on δΩ, with a ? 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem.     

The combinational complexity of equivalence

Consider the Boolean functions and$\text{(n)=}\text{?}\text{i=1}\text{n}\text{x}{}_{\mathrm{i}}\text{nor}\text{(n)=}\text{?}\text{i=1}\text{n}\text{¬ x}{}_{\mathrm{i}}$ and the equivalence Eq(n)=and (n)∨nor(n). Let L (F) be the smallest number of arbitrary binary Boolean operations that are used in any Boolean computation for F. We prove L(Eq(n))=2n ?3, L (and(n), nor(n))=2n?2. There exist many structurally different optimal computations for Eq (n).     

Matrix equations over rings

The main purpose of this work is to establish necessary conditions and sufficient conditions for the existence of a solution of matrix equations whose coefficient matrices have elements belonging to the ring $\text{R=}\text{C}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}$ of polynomials in n variables with complex coefficients and the ring $\text{R=}\text{R}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}{}^{\mathrm{n}}$ of rational functions a(z₁,z₂,…z_n)b(z₁,z₂,…,z_n)^?1 with real coefficients and b(z₁,z₂,…,z_n)≠0 for all (z₁,z₂,…,z_n) in Rⁿ. Results obtained are useful in multidimensional systems theory and elsewhere.     

An accurate algorithm for computing the eigenvalues of a polygonal membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω₁, Ω₂,...; on each $\text{Ω}{}_{\mathrm{i}}$, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.     

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.     

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.