So far the only method we have to solve the propositional satisfiability problem is to use truth tables, which takes exponential time in the formula size in the worst case. In this lecture we show that for Horn formulas and 2-CNF formulas satisfiability can be decided in polynomial time, whereas for 3-CNF formulas satisfiability is as hard as the general case. We also show that if we replace di...

A multidimesional function y(~r) defined by a sample of points {~ri, yi} is approximated by a differentiable function ỹ(~r). The problem is solved by using the GaussHermite folding method developed in the nuclear shell correction method by Strutinsky.

This manuscript describes a review and analysis of the correction factor for come-up time (CUT), introduced by Dr. C. Olin Ball in his famous Formula Method for thermal process calculations. This correction factor has commonly been considered applicable only to the Ball Formula method. In the alternative General Method, the effect of CUT is automatically included in the calculated lethality val...

We establish an integral formula for the duality between multilinear forms/homogeneous polynomials and tensor products for dual spaces with the approximation property and for which the injective tensor products of their preduals is separable and does not contain a copy of `1. We deduce some multilinear Bishop-Phelps-type results. Although spaces of multilinear forms and homogeneous polynomials ...

We adapt a formula due to T.V. Narayana to count ray paths from a source toa receiver according to the number of times the paths change direction. In the simplest case we show that Narayana's formula leads to a hook-sum formula for L(rn, n) posets. §l. Introduction and Motivation This paper arose in an attempt to solve a problem of counting certain ray paths in seismology. The key ingredient of...

a Bk({1− t}) k! f (t)dt where a and b are arbitrary real numbers with difference b − a being a positive integer number, Bn and bn are Bernoulli polynomials and numbers, respectively, and k is any positive integer. The condition we impose on the real function f is that it should have continuous k-th derivative. The symbol {x} for a real number x denotes the fractional part of x. Proof of this th...

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra Uq(X (1) n ) and discuss several aspects related to representation theories and combinatorics. They include crystal base theor...

Let H be Monge-Ampère singular integral operator, [Formula: see text], and [Formula: see text]. It is proved that the commutator [Formula: see text] is bounded from [Formula: see text] to [Formula: see text] for [Formula: see text] and from [Formula: see text] to [Formula: see text] for [Formula: see text]. For the extreme case [Formula: see text], a weak estimate is given.