Fuzzy Information On Discrete And Continuous Domains: Approximation Results

Measures of information based on fuzzy sets have been defined both for finite and for continuous universes. In the continuous case, the measure of information I(f) depends on the concept of non-increasing rearrangement of the function f. It has been observed that I(f) can be obtained as a limit of discrete distributions π^(N) approximating f. We consider the approximation problem in more detail, and study the convergence of I(π^(N)) to I(f) in terms of the smoothness properties of f itself (modulus of continuity and Lipschitz exponent).     

Diagonal polynomials for small dimensions

HereR andN denote respectively the real numbers and the nonnegative integers. Also 0 <n εN, ands(x) =x ₁+...+x _n when x = (x ₁,...,x _n) εR ⁿ. Adiagonal function of dimensionn is a mapf onN ⁿ (or any larger set) that takesN ⁿ bijectively ontoN and, for all x, y inN ⁿ, hasf(x) <f(y) whenevers(x) <s(y). We show that diagonalpolynomials f of dimensionn all have total degreen and have the same terms of that degree, so that the lower-degree terms characterize any suchf. We call two polynomialsequivalent if relabeling variables makes them identical. Then, up to equivalence, dimension two admits just one diagonal polynomial, and dimension three admits just two.     

A new quantum claw-finding algorithm for three functions

Fork functionsf ₁, ...f _k, ak-tuple (x ₁, ...x _k) such thatf ₁(x ₁)=...=f _k(x _k) is called a claw off ₁, ...,f _k. In this paper, we construct a new quantum claw-finding algorithm for three functions that is efficient when the numberM of intermediate solutions is small. The known quantum claw-finding algorithm for three functions requiresO(N ^7/8 logN) queries to find a claw, but our algorithm requiresO(N ^3/4 logN) queries ifM ≤ √N andO(N ^7/12 M ^1/3 logN) queries otherwise. Thus, our algorithm is more efficient ifM≤N ^7/8. Kazuo Iwama, Ph.D.: Professor of Informatics, Kyoto University, Kyoto 606-8501, Japan. Received BE, ME, and Ph.D. degrees in Electrical Engineering from Kyoto University in 1978, 1980 and 1985, respectively. His research interests include algorithms, complexity theory and quantum computation. Editorial board of Information Processing Letters and Parallel Computing. Council Member of European Association for Theoretical Computer Science (EATCS). Akinori Kawachi: Received B.Eng. and M.Info. from Kyoto University in 2000 and 2002, respectively. His research interests are quantum computation and distributed computation.     

Crown graphs and subdivision of ladders are odd graceful

A graph G of size q is odd graceful, if there is an injection φ from V(G) to {0, 1, 2, …, 2q?1} such that, when each edge xy is assigned the label or weight |f(x)?f(y)|, the resulting edge labels are {1, 3, 5, …, 2q?1}. This definition was introduced in 1991 by Gnanajothi [3], who proved that the graphs obtained by joining a single pendant edge to each vertex of C _n are odd graceful, if n is even. In this paper, we generalize Gnanajothi's result on cycles by showing that the graphs obtained by joining m pendant edges to each vertex of C _n are odd graceful if n is even. We also prove that the subdivision of ladders S(L _n ) (the graphs obtained by subdividing every edge of L _n exactly once) is odd graceful.     

An enlarged family of packing polynomials on multidimensional lattices

HereN = {0, 1, 2, ...}, while a functionf onN ^m or a larger domain is apacking function if its restrictionf|N ^m is a bijection ontoN. (Packing functions generalize Cantor's [1]pairing polynomials, and yield multidimensional-array storage schemes.) We call two functionsequivalent if permuting arguments makes them equal. Alsos(x) =x ₁ + ... +x _m when x = (x ₁,...,x _m); and such anf is adiagonal mapping iff(x) <f(y) whenever x, y εN ^m ands(x) <s(y). Lew [7] composed Skolem's [14], [15] diagonal packing polynomials (essentially one for eachm) to constructc(m) inequivalent nondiagonal packing polynomials on eachN ^m. For eachm > 1 we now construct 2^m−2 inequivalent diagonal packing polynomials. Then, extending the tree arguments of the prior work, we obtaind(m) inequivalent nondiagonal packing polynomials, whered(m)/c(m) → ∞ asm → ∞. Among these we count the polynomials of extremal degree.     

On the asymptotic complexity of rectangular matrix multiplication

The number of essential multiplications required to multiply matrices of size N×N and N×N^x is studied as a function f(x). Bounds to f(x) sharper than trivial ones are presented and the asymptotic behaviour of f(x) is studied. An analogous investigation is performed for the problem of multiplying matrices of size N×N^x and N^x×N^y.     

Exponentially Accurate Approximations to Periodic Lipschitz Functions Based on Fourier Series Partial Sums

The problem of accurately reconstructing a piecewise smooth. 2-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. The locations of the singularities are approximated to within O(N ^–M–2), and the associated jump of the kth derivative of f is approximated to within O(N ^–M–1+k ), as N , and the method is robust. These estimates are then used with a class of singular basis functions, which have certain built-in singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M . The technique is illustrated with several examples.     

Efficient parallel and sequential algorithms for 4-coloring perfect planar graphs

We present an efficient algorithm for 4-coloring perfect planar graphs. The best previously known algorithm for this problem takesO(n ^3/2) sequential time, orO(log⁴ n) parallel time withO(n³) processors. The sequential implementation of our algorithm takesO(n logn) time. The parallel implementation of our algorithm takesO(log³ n) time withO(n) processors on a PRAM.     

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

It is known that if a Boolean function f in n variables has a DNF and a CNF of size then f also has a (deterministic) decision tree of size exp(O(log n log² N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp where N is the total number of monomials in minimal DNFs for f and ?f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC⁰. Received: June 5 1997.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.