Given Number Of Variables Research Articles

Mean and variance of the number of subfunctions of random Boolean function which are close to the affine functions set

Abstract For random equiprobable Boolean functions we investigate the distribution of the number of subfunctions which have a given number of variables and are close to the set of affine Boolean functions. It is shown, for example, that for Boolean functions of n variables the mean number of subfunctions having s ⩾ 3 + log2 n variables and the Hamming distance to the set of affine functions smaller than 2 s−2 tends to 0 as n → ∞.

Connection between Petri nets and Polish notation

We propose Petri nets that produce languages of Polish notation and reverse Polish notation for propositional formulas and mathematical expressions. Propositional formulas can contain a given number of variables and mathematical expressions. Arithmetic expressions can contain a given number of variables and constants. We also propose inhibitor nets that produce the fixed-point binary numbers in mathematical expressions for above-mentioned languages. The technique of the nets construction allows to use arbitrary functions with a given arity. We also propose a coloured Petri net for calculating values of propositional formulas in reverse Polish notation. The technique of the net construction allows to use arbitrary functions with a given arity using a truth table of a corresponding function.

Generic primal-dual solvability in continuous linear semi-infinite programming†

In this article, we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e. finite valued), but either unsolvable or having an unbounded optimal set, and unbounded (i.e. with infinite optimal value), giving rise to the so-called refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haar's dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable. †In celebration of Prof. Dr Hubertus Th. Jongen's 60th Birthday.

The discarding of variables in multivariate analysis

In many multivariate situations we are presented with more variables than we would like, and the question arises whether they are all necessary and if not which can be discarded. In this paper we consider two such situations. (a)Regression analysis. The problem here is whether any variables can be discarded as adding little or nothing to the accuracy with which the regression equation correlates with the dependent variable. (b)Interdependence analysis. The problem is whether a constellation in p dimonsions collapses, exactly or approximately, into fewer dimensions, and if so whether any of the original variables can be discarded. We may define the best solution to (a) using any given number of variables as the one that maximizes the multiple correlation between the selected variables and the dependent variable, and similarly for (b) as the one that maximizes the smallest multiple correlation with any of the rejected variables. In practice it is usual to accept an approximate solution to (a) based on ‘step-wise’ multiple regression: we know of no standard program for (b). We have developed cut-off rules that enable us to find the best solution to both problems by partial enumeration. The paper discusses the details of this approach, and computational experience.