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.     

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.     

On the complexity of simple arithmetic expressions

Let $\text{E}$ be the set of all simple arithmetic expressions of the form E(x) = xT_l…T_k where x is a nonnegative integer variable and each T_i is a multiplication or integer division by a positive integer constant. We investigate the complexity of the inequivalence and the bounded inequivalence problems for expressions in $\text{E}$. (The bounded inequivalence problem is the problem of deciding for arbitrary expressions E₁(x) and E₂(x) and a positive integer l whether or not E₁(x) ≠ E₂(x) for some nonnegative integer x<l. If l = ∞, i.e., there is no upper bound on x, the problem becomes the inequivalence problem.) We show that the inequivalence problem (or equivalently, the equivalence problem) for a large subclass of $\text{E}$ is decidable in polynomial time. Whether or not the problem is decidable in polynomial time for the full class $\text{E}$ remains open. We also show that the bounded inequivalence problem is NP-complete even if the divisors are restricted to be equal to 2. This last result can be used to sharpen some known NP-completeness results in the literature. Note that if division is rational division, all problems are trivially decidable in polynomial time.     

Simulations among multidimensional turing machines

For all d ? 1 and all e >d, every deterministic multihead e-dimensional Turing machine of time complexity T(n) can be simulated on-line by a deterministic multihead d-dimensional Turing machine in time O(T(n)^1+1?d?1?(logT(n))⁰⁽¹⁾). This simulation almost achieves the known lower bound $\text{Ω(T(n)}{}^{\mathrm{1+1?}}\text{d?1?e)}$ on the time required. The simulation is interpreted in terms of dynamic embeddings among arrays with local access.     

A method for the calculation of eigenfunction expansions

In this paper, a method is presented for the calculation of the coefficients of the series expansion of a function f(t), in the base orthonormal set made up by the eigenfunctions of the self-adjoint operator $\text{L: L(x(t)) = (}\text{d}\text{dt}\text{)( p(t)(}\text{dx(t)}\text{dt}\text{))?q(t)x(t)}$. We show that the values of the numbers $\text{t}\text{}$x_k> can be obtained by solving the differential equation L + λ) y(t) = Kf(t), in the interval of definition, for each of the eigenvalues λ of L and by using as initial conditions those which determine one of its associated orthonormal functions. This makes the method specially interesting for its implementation on a hybrid computer: One advantage of the proposed method is that the analysis of f(t) does not require the simultaneous presence of the functions of the base set and the problem signal, thus eliminating both the problems of the synchronized generation of signals and the need for storing it in memory.     

Computer-aided design via optimization: A review

Many design problems, including control design problems, involve infinite dimensional constraints of the form φ(z, α) ≤ 0 for all α ? $\text{A}$, where α denotes time or frequency or a parameter vector. In other design problems, tuning or trimming of certain parameters, after manufacture of the system, is permitted; the corresponding constraint is that for each α in $\text{A}$ there exists a value τ (of the tuning parameter) in a permissible set T such that φ(z, α, t) < 0. Recent algorithms for solving design problems having such constraints are summarized.     

Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials

We improve some lower bounds which have been obtained by Strassen and Lipton. In particular there exist polynomials of degree n with 0–1 coefficients that cannot be evaluated with less than $\text{n}\text{/(4}\text{log}\text{n)}$ nonscalar multiplications/divisions. The evaluation of p(x) $\text{∑}\text{δ=0}\text{n}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>2\delta </mtext>}}\text{x}{}^{\mathrm{\delta }}$ requires at least $\text{n}\text{(12}\text{log}\text{n)}$ multiplications/divisions and at least $\text{n/(8}\text{log}\text{n)}$ nonscalar multiplications/divisions. We specify polynomials with algebraic coefficients that require $\text{1}\text{2}\text{n}$ multiplications/divisions.     

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.     

Connective stability of competitive equilibrium

The purpose of this paper is to derive necessary and sufficient conditions for connective stability of non-linear matrix systems described by the equation $\text{x}\text{?}\text{= A(t, x) x}$, where the matrix A(t, x) has time-varying nonlinear elements. The obtained results can be used to study stability of competitive equilibrium in as diverse fields as economics and engineering, model ecosystems and arms races.     

On the algorithmic complexity of associative algebras

The complexity L(A) of a finite dimensional associative algebra A is the number of non-scalar multiplications/divisions of an optimal algorithm to compute the product of two elements of the algebra. We show

$\text{L(A) ? 2ddim A?t}$

, where t is the number of maximal two-sided ideals of A.     

Random functions with given time correlation

First, on any sequence of real numbers (x_λ), $\text{λ ? [? Λ}{}_1\text{+ Λ] ?}\text{Z}$, the pseudo probability Pr(x, x′) of the event x_λ?[x, x′[ is defined to be the limit when Λ → ∞ of the ratio of the number of x_λ?[x, x′[ to the total number of x_λ. The a.d.f. (asymptotic distribution function) of the sequence is then defined by F(x) = Pr(? ∞, x); it possesses the properties of a d.f. (distribution function). Consequently, what is said below applies equally to a sequence of r.v. (random variables) or to a sequence of p.r.v. (pseudorandom variables) consisting of a sequence (_nx_λ), $\text{n ? [? N, + N] ?}\text{Z}$ of sequences _nx_λ, $\text{λ ? [? Λ, + Λ] ?}\text{Z}$.A weyl's polynomial ?_n(λ) is a polynomial such that one of its coefficieints other than ?(0) is irrational. Then any sequence, the fractional part of ?_n(λ), $\text{λ ? [? Λ, + Λ] ?}\text{Z}$, is asymptotically equidistributed on [0, 1].A property is given which permits the construction of a sequence (_nx_λ), $\text{n ? [? N, + N] ?}\text{Z}$ of pseudostochastically independent sequences _nx_λ, $\text{λ ? [? Λ, + Λ] ?}\text{Z}$.It is known that setting Y_n = F^{(? 1)}(X_n), it is possible to transform any sequence of r.v. X_n     

Curve generation of implicit functions by incremental computers

The conventional numerical solution of an implicit function f(x, y) = 0 is substantially complicated for calculating by any computer. We propose a new method representing the argument of the implicit function as a unary function of a parameter, t, if the continuous and unique solution of f(x, y) = 0 exists. The total differential $\text{d}\text{f}\text{d}\text{t}$ constitutes simultaneous differential equations of which the solution about x and y is unique. The Newton-Raphson method must be used to calculate the values near singular points of an implicit function and then the sign of dt has to be decided according to four special cases. Incremental computers are suitable for curve generation of implicit functions by the new method, because the incremental computer can perform more complex algorithms than the analog computer and can calculate faster than the digital computer. This method is easily applicable to curve generation in three-dimensional space.     

A recursive and a grammatical characterization of the exponential-time languages

We characterize the class of all languages which are acceptable in exponential time by means of recursive and grammatical methods. (i) The class of all languages which are acceptable in exponential time is uniquely characterized by the class of all (0-1)-functions which can be generated, starting with the initial functions of the Grzegorczyk-class $\text{E}$², by means of subtitution and limited recursion of the form f(x, y + 1) = h(x, y), f(x, y), f(x, l(x, y))), l(x, y) ? y. (ii) The class of all languages which are acceptable in exponential time is equal to the class of all languages generated by context-sensitive grammars with context-free control sets.     

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).