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.     

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.     

A space bound for one-tape multidimensional Turing machines

Let L be a language recognized by a nondeterministic d-dimensional Turing machine with one worktape head of time complexity T(n). Then L can be recognized by a deterministic Turing machine of space complexity $\text{(T(n)}\text{log}\text{T(n))}{}^{\mathrm{<mtext>d</mtext><mtext>(d+1)</mtext>}}$. The proof employs a generalization of crossing sequences.     

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.     

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.     

The complexity of a multiprocessor task assignment problem without deadlines

We construct a sequence of monotone Boolean functions h_n :{0, 1}ⁿ→{0, 1}ⁿ, such that the monotone complexity of h_n is of order $\text{n}{}^2\text{log n}$. This result includes the largest known lower bound of this kind. Previously there were an $\text{Ω(n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{)}$ bound for the Boolean matrix product, an $\text{Ω(n}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}\text{)}$ bound for Boolean sums and an $\text{Ω(}\text{n}{}^2\text{log}{}^2\text{n}\text{)}$ bound by the author for the same functions h_n. This new lower bound is proved by new methods which probably will turn out to be useful also for other problems.     

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.     

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.     

On time versus space II

Every t(n)-time bounded RAM (assuming the logarithmic cost measure) can be simulated by a t(n)/log t(n)-space bounded Turing machine and every t(n)-time bounded Turing machine with d-dimensional tapes by a bounded machine, where n is the length of the input. A class $\text{E}$ of storage structures which generalizes multidimensional tapes is defined. Every t(n)-time bounded Turing machine whose storage structures are in $\text{E}$ can be simulated by a t(n) loglog t(n)/log t(n)-space bounded Turing machine.     

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.     

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.     

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.     

Sorting using networks of deques

The number of required deques for sorting all sequences of n items in a parallel or series network of deques is considered. It is shown that the optimal number of required deques is $\text{O(n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ for a parallel network, while it is O(log n) for a series network. These orders, $\text{O(n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and O(log n), also remain valid for the networks of restricted deques.