Banach triples with generalized inverses

Penrose showed in [6] that for every complex rectangular matrix aM_nxm() there exists a unique bM_nxm(), the generalized inverse of a, satisfying the following four conditions: i) a a^*b=a, ii) b a^*a=a, iii) b b^*a=b, iv) a b^*b=b.These condition can be expressed in terms of the triple product     

Lucas sequences and quadratic orders

We consider the Lucas sequences (U _n ) _{n ≥ 0} defined by U ₀ = 0, U ₁ = 1, and U _n =  PU _n–1 – QU _n–2 for non-zero integral parameters P, Q such that Δ = P ² – 4Q is not a square. We use the arithmetic of the quadratic order with discriminant Δ to investigate the zeros and the period length of the sequence (U _n ) _{n ≥ 0} modulo a positive integer d coprime to Q. For a prime p not dividing Q, we give precise formulas for p-powers, we determine the p-adic value of U _n , and we connect the results with class number relations for quadratic orders.     

Note on generalized Archimedes–Borchardt algorithm

Aequationes mathematicae - We investigate convergence and invariance properties of the generalized Archimedes–Borchardt algorithm. The main tool is reducing the problem to an appropriate...     

On generalized Fibonacci and Lucas polynomials

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q_h(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.     

New identities involving generalized Fibonacci and generalized Lucas numbers

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.     

Practical numbers in Lucas sequences

Abstract

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (u_n)_n≥0 be the Lucas sequence satisfying u₀ = 0, u₁ = 1, and u_n+2 = au_n+1 + bu_n for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a² + 4b > 0. Also, let be the set of all positive integers n such that |u_n| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ? x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .     

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U₀=0, U₁=1, Un=PUn−1−QUn−2 (n?2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,…,7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,…,12, the only non-degenerate sequences where gcd(P,Q)=1 and Un(P,Q)=□, are given by U₈(1,−4)=²¹², U₈(4,−17)=⁶²⁰², and U₁₂(1,−1)=¹²².     

On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system

Considered herein is the persistence property of the solutions to the generalized two-component integrable Dullin–Gottwald–Holm system, which was derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. Firstly, the persistence properties of the system are investigated in weighted $$L^p$$-spaces for a large class of moderate weights. Then, we establish the new local-in-space blow-up results simplifying and extending earlier blow-up criterion for this system.     

A generalized Lucas sequence and permutation binomials

Let be an odd prime and . Let be an odd positive integer. Let or and . By employing the integer sequence , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials of the finite field .

     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

A class of twisted generalized Reed–Solomon codes

Designs, Codes and Cryptography - Let $${\mathbb F}_{q}$$ be a finite field of size q and $${\mathbb F}_{q}^*$$ the set of non-zero elements of $${\mathbb F}_{q}$$ . In this paper, we study a class...     

Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.     

$$mathcal {F}_pi $$Fπ-residuality of generalized Baumslag–Solitar groups

Archiv der Mathematik - A generalized Baumslag–Solitar group is a finitely generated group that acts on a tree with infinite cyclic edge and vertex stabilizers. A group G is residually a...