Pseudo-linear algebra is the study of common properties of linear differential and difference operators. We introduce in this paper its basic objects (pseudo-derivations, skew polynomials, and pseudo-linear operators) and describe several recent algorithms on them, which, when applied in the differential and difference cases, yield algorithms for uncoupling and solving systems of linear differe...

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...

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation r t xΔ t γ Δ p t x g t 0 on an arbitrary time scale T with sup T ∞, where g t ≥ t and ∫∞ to Δs/ r1/γ s < ∞. Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify th...

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...

We then define the Pfaffian transformation of (a0, a1, a2, ...) to be the sequence of Pfaffians (Pf(A0), P f(A1), P f(A2), ...), where the Pfaffian of a skew-symmetric matrix is the positive or negative square root of its determinant. (For a precise definition of the Pfaffian, see section 2.) The Pfaffian transformation is thus a function from sequences to sequences. We begin by observing the e...

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...

Let E be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let P = (x1/z 2 1 , y1/z 3 1) be a rational point of infinite order on E, where x1, y1, z1 are coprime integers. We show that the integer sequence (zn)n>1 defined by nP = (xn/z 2 n, yn/z 3 n) for all n > 1 does not eventually coincide with (un2)n>1 for any choice of linear recurrence sequence (un)n>1 with...

We prove that for any base b ≥ 2 and for any linear homogeneous recurrence sequence {an}n≥1 satisfying certain conditions, there exits a positive constant c > 0 such that #{n ≤ x : an is palindromic in base b} x1−c.

An M -partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M -partition...