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.     

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.     

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.     

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.     

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.     

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.     

Gibbs energies of formation of Mn3C,M(Fe,Mn)3C and Mn23C6 from the ternary phase equilibria in the FeMnC system

The high temperature phase relations in the FeMnC system have been analyzed in light of the recently developed thermodynamic method by the authors to obtain the Gibbs energies of formation of Mn₂₃C₆ and Mn,C. A new thermodyn/amic treatment is outlined and applied to obtain the stability of the ternary carbide M(Fe,Mn)₃C without any a priori assumption of a solution model for the M₃C phase. The recommended Gibbs energies of formation for the Mn carbides, Mn₃C and Mn₂₃C₆ With γ-Mn (graphite) as the Standard states are:

The present method can be extended to obtain a consistent set of thermodynamic data for binary and ternary carbides from various ternary metal-metal-carbon phase relations.     

An explicit construction of short monotone formulae for the monotone symmetric functions

We construct formulae that assume the value 1 when and only when at least k of their n variables assume the value 1, using only conjunction and disconjunction, and having (for any fixed k) only

$\text{O}\text{(n}\text{log}\text{n)}\text{k}\text{2}{}^{\mathrm{<mtext>log</mtext><mtext>1n</mtext>}}$

occurences of variables.     

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.     

Uniform computational complexity of the derivatives of C∞-functions

We discuss the uniform computational complexity of the derivatives of C^∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C^∞[−1,1] such that the sequence $\text{{|g}{}^{\mathrm{(n)}}\text{(0)|}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is not bounded by any recursive function. On the other hand, we show that if f∈C^∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f⁽ⁿ⁾(x)| is bounded by 2^p(n) for all x∈[−1,1] for some polynomial p, then the sequence $\text{{f}{}^{\mathrm{(n)}}\text{}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is uniformly polynomial time computable.