where A = [uVlnxn with aU = 6i,ji + hi_ ,,j, C = I n-‘J, I is the n x n identity matrix and J is the n x n matrix of all 1’s. This determinant arises in the calculation of cumulants of a statistic analogous to Pearson’s chisquared for a multinomial sample (Sibson [5]), and its value has been conjectured by Good [2]. In this paper we use enumerative methods to prove Good’s conjecture. The partic...

The second order linear difference equation (1) ∆(rk∆xk) + ckxk+1 = 0, where rk 6= 0 and k ∈ , is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Bor̊uvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investiga...

We describe all the solutions of a rational difference equation from Putnam’s mathematical competition, which are eventually equal to its positive equilibrium x\over \tilda=1. As a consequence we give a new, elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one. Moreover, we show that almost all solutions of the equation are not event...

The SR-based critic learns an estimate of the value function, using the SR as its feature representation. Unlike standard actor-critic methods, the critic does not use reward-based temporal difference errors to update its value estimate; instead, it relies on the fact that the value function is given by V (s) = ∑ s′ M(s, s ′)R(s′), where M is the successor representation andR is the expected re...

The 321,hexagon–avoiding (321–hex) permutations were introduced and studied by Billey and Warrington in [4] as a class of elements of Sn whose Kazhdan– Lusztig and Poincaré polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7–term linear recurrence relation leading to an explicit enumeration of the 321–hex pe...

A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients, which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is ...

A nonhomogeneous system of linear recurrence equations can be recognized by an automaton A over a one-letter alphabet A = {z}. Conversely, the automaton A generates precisely this nonhomogeneous system of linear recurrence equations. We present the solutions of these systems and apply the z-transform to these solutions to obtain their series representation. Finally, we show some results that si...

Global asymptotic stability of rational difference equations is an area of research that has been well studied. In contrast to the many current methods for proving global asymptotic stability, we propose an algorithmic approach. The algorithm we summarize here employs the idea of contractions. Given a particular rational difference equation, defined by a function Q : R → R, we attempt to find a...