Lucjan Szymaszkiewicz

• PhD
• University of Szczecin

Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such $q$'s up to some reasonable ranges for all primes $q$ congruent to $7$ modulo $8$. All these values are non-zero,...

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that,...

Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (...

This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calc...

This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calc...

An arc in (Formula presented.) is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small n.

We present the results of our search for the orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves.

We present the results of our search for the orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves.

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 $\Bbb Q$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$,...

We present the results of our search for the orders of Tate–Shafarevich groups for the quadratic twists of \(E=X_0(49)\).

By an arc in $\Z^2_n$ we mean the set where no three points are on a line. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.

In this paper we show that at most $2 \gcd(m,n)$ points can be placed with no three in a line on an $m\times n$ discrete torus. This limit is attained for infinitely many cases.

In this paper we prove that gamma(r2) (C-n square C-5) >= 2n. This, together with the result of Stcpien and Zwierzchowski (2014), gives gamma(r2)(C-n square C-5) = 2n. Since for n = 5k we have gamma(C-n square C-5) = n (see Klavzar and Seifter (1995)), it follows that a product C-n square C-5 is an example of a graph class for which gamma(r2) = 2 g...

Criteria in order that a Musielak-Orlicz function space L Φ as well as Musielak-Orlicz sequence space l Φ contains an asymptotically isometric copy of c0 are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of c0 in the subspaces of order continuous elements in Orlicz spaces, Journa...

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G□K2 (see Brešar et al., Taiwan J Math 12:213–225, 2008). Hence γr2(Cm□Cn)≥mn3. In this paper we give full characterization of graphs Cm□Cn with γr2(Cm□Cn)=mn3.

Orlicz spaces lφ(Γ) over an arbitrary set Γ, being a natural generalizations of Orlicz sequence spaces are studied. The following problems in these spaces are considered: relationships between the Luxemburg norm and the modular, Fatou property, relationships between the Luxemburg norm and the Orlicz norm, equality of the Orlicz norm and the Amemiya...

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements,...

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 δλ2 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 o...

Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity, and uniform monotonicity of Musielak–Orlicz spaces over any σ-finite and complete measure space, endowed with the Amemiya norm are given. The fact that the spaces are considered over arbitrary σ-finite measure space is essential because, as it is s...