Generalized Orlicz-Lorentz sequence spaces λϕ generated by Musielak-Orlicz functions ϕ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ λ 2 for ϕ is defined in such a way that it guarantees many positive topological and geometric properties of λϕ. The problems of the Fatou property, the order continuity and the Kadec-Klee property with respect to the uniform convergence of the space λϕ are considered. Moreover, some embeddings between λϕ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λϕ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. PreliminariesThroughout this paper R, R + and N denote the sets of reals, nonnegative reals and natural numbers, respectively. The triple (N, 2 N , m) stands for the counting measure space, while l 0 = l 0 (m) denotes the space of all sequences x : N → (−∞, ∞). For every x = (x n ) ∞ n=1 ∈ l 0 (if it is more convenient the notation x(n) is used in place of x n ) we define supp x = {n ∈ N : x n = 0} and |x|(n) = |x(n)| for all n ∈ N , that is, |x| denotes the absolute value of x.The letter e stands for a Banach sequence lattice over the measure space (N, 2 N , m), that is, e is a nontrivial Banach subspace of l 0 (that is e = {0}) satisfying the following conditions (see [23] and [27]):(i) if x ∈ e, y ∈ l 0 and |y| ≤ |x| (i.e., |y n | ≤ |x n | for all n ∈ N ), then y ∈ e and y ≤ x .(ii) there exists a sequence x in e that is strictly positive on the whole N . The positive cone in e is denoted by e + . A Banach lattice e is said to have the Fatou property (e ∈ (FP) for short) if for any x ∈ l 0 and (x m ) ∞ m=1 in e + such that x m x coordinatewise and sup m x m < ∞, we have x ∈ e and x = lim m x m (see [23] and [27]).An element x ∈ e is said to be order continuous if for any sequence (x m ) in e + such that x m ≤ |x| and x m → 0 coordinatewise there holds x m → 0. The subspace e a of all order continuous elements in e is an order ideal of e. A Banach sequence lattice e is said to be order continuous (e ∈ (OC) for short) if e a = e (see [23] and [27]). Notice that e a satisfies condition (ii) from the definition of a Banach sequence lattice, that is, supp e a = N .We say that e has the coordinatewise Kadec-Klee property (e ∈ (H c ) for short) if for any x ∈ e and any sequence (x m ) in e the conditions x m → x and x m (n) → x(n) for all n ∈ N , yield x − x m → 0. We *

We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve E over Q and positive integer k, an asymptotic formula for the number of quadratic twists E d , d positive square-free integers less than X, with finite group E d (Q) and |X(E d (Q))| = k 2 . This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve X 0 (49). In section 8 we exhibit 88 examples of rank zero elliptic curves with |X(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |X(E)| = 1029212 2 .Birch and Swinnerton-Dyer conjecture relates the arithmetic data of E to the behaviour of L(E, s) at s = 1.Conjecture 1 (Birch and Swinnerton-Dyer) (i) L-function L(E, s) has a zero of order r = rank E(Q) at s = 1,If X(E) is finite, the work of Cassels and Tate shows that its order must be a square.The first general result in the direction of this conjecture was proven for elliptic curves E with complex multiplication by Coates and Wiles in 1976 [6], who showed that if L(E, 1) = 0, then the group E(Q) is finite. Gross and Zagier [18] showed that if L(E, s) has a first-order zero at s = 1, then E has a rational point of infinite order. Rubin [26] proves that if E has complex multiplication and L(E, 1) = 0, then X(E) is finite. Let g E be the rank of E(Q) and let r E the order of the zero of L(E, s) at s = 1. Then Kolyvagin [20] proved that, if r E ≤ 1, then r E = g E and X(E) is finite. Very recently, Bhargava, Skinner and Zhang [1] proved that at least 66.48% of all elliptic curves over Q, when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.When E has complex multiplication by the ring of integers of an imaginary quadratic field K and L(E, 1) is non-zero, the p-part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [27] for all primes p which do not divide the order of the group of roots of unity of K. Coates et al. [5] [4], and Gonzalez-Avilés [17] showed that there is a large class of explicit quadratic twists of X 0 (49) whose complex L-series does not vanish at s = 1, and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case p = 2 when K = Q( √ −7)). The deep results by Skinner-Urban ([29], Theorem 2) (see also Theorem 7 in section 8.4 below) allow, in specific cases (still assuming L(E, 1) is non-zero), to establish p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes p (see examples in section 8.4 below, and section 3 in [10]). The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [7], Delaunay [12][13], Watkins [31], Radziwi l l-Soundararajan [25] (see also the papers [11][10] [9], and references therein) substantially extend the systematic tables given by...

Abstract. In this paper we show that at most 2 gcd(m, n) points can be placed with no three in a line on an m × n discrete torus. In the situation when gcd(m, n) is a prime, we completely solve the problem.