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ABSTRACT We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also.
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(joint work with Simone Costa; Fiorenza Morini; Marco A. Pellegrini) It is well known that difference methods have a primary role in the construction of combinatorial designs of various kinds. The continuous search for more efficient... more
(joint work with Simone Costa; Fiorenza Morini; Marco A. Pellegrini) It is well known that difference methods have a primary role in the construction of combinatorial designs of various kinds. The continuous search for more efficient ways to use these methods often leads to intriguing problems which are very difficult despite their easy statements. Some examples are the conjectures proposed by Archdeacon et al. [1], by Buratti et al. [3] and by Meszka et al. [4]. In this talk I will present some results about the following conjecture (see [2]): Let $(G,+)$ be an abelian group. Let $A\neq\emptyset$ be a finite subset of $G\setminus\{0\}$ such that no $2$-subset $\{x,-x\}$ is contained in $A$ and with the property $\sum _ {a\in A} a=0$. Then there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Bibliography [1] {\sc D.S. Archdeacon, J.H. Dinitz, A. Mattern, D.R. Stinson}, On partial sums in cyclic groups, {\em J. Combin. Math. Combin. Comput.}, 98 (2016), 327--342. [2] {\sc S. Costa, F. Morini, A.Pasotti, M.A. Pellegrini}, A problem on partial sums in abelian groups, in preparation. [3] {\sc P. Horak, A. Rosa}, On a problem of Marco Buratti, {\em Electron. J. Comb.}, 16 (2009), $\sharp$R20. [4] {\sc A. Pasotti, M.A. Pellegrini}, A generalization of the problem of Mariusz Meszka, {\em Graphs and Combin.}, 32 (2016), 333--350.
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Two $k$-cycle decompositions $\mathcal{C}$ and $\mathcal{C}'$ of a graph $\Gamma$ are said to be orthogonal if any cycle $C\in \mathcal{C}$ intersects any cycle $C'\in \mathcal{C}'$ in at most one edge. In this paper we... more
Two $k$-cycle decompositions $\mathcal{C}$ and $\mathcal{C}'$ of a graph $\Gamma$ are said to be orthogonal if any cycle $C\in \mathcal{C}$ intersects any cycle $C'\in \mathcal{C}'$ in at most one edge. In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In particular we provide explicit constructions of such decompositions for cycles of length $k\leq 10$.
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The concept of Heffter array has been introduced by A. Archdeacon in [1] where he showed various of its applications. This leads several authors to investigate the existence problem (see, for example, [2] and [4]). Here we are interested... more
The concept of Heffter array has been introduced by A. Archdeacon in [1] where he showed various of its applications. This leads several authors to investigate the existence problem (see, for example, [2] and [4]). Here we are interested in the relationship between Heffter arrays and orthogonal cyclic cycle systems. In this regard we introduce, in [3], the class of globally simple Heffter arrays whose existence assures the one of orthogonal cyclic k-cycle decompositions of complete graphs and of cocktail party graphs. Bibliography [1] Archdeacon D.S., Heffter arrays and biembedding graphs on surfaces, Electron. J.Combin., 22 (2015) #P1.74. [2] Archdeacon D.S., Dinitz J.H., Donovan D.M. and Yazici E.S., Square integer Heffter arrays with empty cells, Des. Codes Cryptogr., 77 (2015) 409-426. [3] Costa S., Morini F., Pasotti A., Pellegrini M.A., Simple Heffter arrays and orthogonal cyclic cycle systems, in preparation. [4] Dinitz J.H., Wanless I.M., The existence of square integer Heff...
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In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let be a positive integer, where divides , and let be the... more
In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let be a positive integer, where divides , and let be the subgroup of of order . A Heffter array over relative to is an partially filled array with elements in such that (a) each row contains filled cells and each column contains filled cells; (b) for every , either or appears in the array; and (c) the elements in every row and column sum to . Here we study the existence of square integer (i.e., with entries chosen in and where the sums are zero in ) relative Heffter arrays for , denoted by . In particular, we prove that for , with , there exists an integer if and only if one of the following holds: (a) is odd and ; (b) and is even; (c) . Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.
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Research Interests: Mathematics and Algebra
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A nearring (N; +; ) is called weakly divisible i for all elements a;b2 N there exists an elementx2 N such thatx a = b orx b = a. All such nite zerosymmetric nearrings are determined.
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ABSTRACT Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs)... more
ABSTRACT Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. In [12] the authors define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD, called disk-design. In this paper we present a method for the construction of an association scheme which makes the disk-design, in some interesting cases, an union of partially incomplete block designs (PBIBDs). Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters and to prove that in some cases they are also cyclic.
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ABSTRACT Let ${(N, \Phi)}$ be a finite circular Ferrero pair. We define the disk with center b and radius ${a, \mathcal{D}(a;b)}$ , as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap (... more
ABSTRACT Let ${(N, \Phi)}$ be a finite circular Ferrero pair. We define the disk with center b and radius ${a, \mathcal{D}(a;b)}$ , as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap ( \Phi(a)+b)|=1\}.$$ Using this definition we introduce the concept of interior part of a circle, ${\Phi(a)+b}$ , as the set ${\mathcal{I}(\Phi (a)+b)=\mathcal{D} (a; b) \setminus (\Phi (a)+b)}$ . Moreover, if ${\mathcal{B}^{\mathcal{D}}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure ${(N, \mathcal{B}^{\mathcal{D}}, \in)}$ is actually a balanced incomplete block design and we are able to calculate its parameters depending on |N| and ${|\Phi|}$ .
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ABSTRACT We consider zerosymmetric right nearrings whose lattice of N-subgroups is linearly ordered by inclusion and in which for every n ∈ N, there is x ∈ N such that x * n = n. All such nearrings with finitely many N-subgroups are... more
ABSTRACT We consider zerosymmetric right nearrings whose lattice of N-subgroups is linearly ordered by inclusion and in which for every n ∈ N, there is x ∈ N such that x * n = n. All such nearrings with finitely many N-subgroups are constructed.
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We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number... more
We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.
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Let \({(N, \Phi)}\) be a finite circular Ferrero pair. We define the disk with center b and radius \({a, \mathcal{D}(a;b)}\), as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap (... more
Let \({(N, \Phi)}\) be a finite circular Ferrero pair. We define the disk with center b and radius \({a, \mathcal{D}(a;b)}\), as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap ( \Phi(a)+b)|=1\}.$$ Using this definition we introduce the concept of interior part of a circle, \({\Phi(a)+b}\), as the set \({\mathcal{I}(\Phi (a)+b)=\mathcal{D} (a; b) \setminus (\Phi (a)+b)}\). Moreover, if \({\mathcal{B}^{\mathcal{D}}}\) is the set of all disks, then, in some interesting cases, we show that the incidence structure \({(N, \mathcal{B}^{\mathcal{D}}, \in)}\) is actually a balanced incomplete block design and we are able to calculate its parameters depending on |N| and \({|\Phi|}\).
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It is well known that a permutation group of degree $ n \neq 3 $ can be generated by $ [\frac{n}{2}] $ elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree n with $... more
It is well known that a permutation group of degree $ n \neq 3 $ can be generated by $ [\frac{n}{2}] $ elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree n with $ [\frac{n}{2}] $ elements. In particular we prove that if n is large enough and $ [\frac{n}{2}]
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ABSTRACT We study the asymptotic behavior of the probability of generating a finite completely reducible linear group G of degree n with [ n] elements. In particular we prove that if 3/2 and n is large enough then [ n] randomly chosen... more
ABSTRACT We study the asymptotic behavior of the probability of generating a finite completely reducible linear group G of degree n with [ n] elements. In particular we prove that if 3/2 and n is large enough then [ n] randomly chosen elements that generate G modulo O2(G) almost certainly generate G itself.
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Let $m,n,s,k$ be integers such that $4\leq s\leq n$, $4\leq k \leq m$ and $ms=nk$. Let $\lambda$ be a divisor of $2ms$ and let $t$ be a divisor of $\frac{2ms}{\lambda}$. In this paper we construct magic rectangles $MR(m,n;s,k)$, signed... more
Let $m,n,s,k$ be integers such that $4\leq s\leq n$, $4\leq k \leq m$ and $ms=nk$. Let $\lambda$ be a divisor of $2ms$ and let $t$ be a divisor of $\frac{2ms}{\lambda}$. In this paper we construct magic rectangles $MR(m,n;s,k)$, signed magic arrays $SMA(m,n;s,k)$ and integer $\lambda$-fold relative Heffter arrays ${}^\lambda H_t(m,n;s,k)$ where $s,k$ are even integers. In particular, we prove that there exists an $SMA(m,n;s,k)$ for all $m,n,s,k$ satisfying the previous hypotheses. Furthermore, we prove that there exist an $MR(m,n;s,k)$ and an integer ${}^\lambda H_t(m,n;s,k)$ in each of the following cases: $(i)$ $s,k \equiv 0 \pmod 4$; $(ii)$ $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$; $(iii)$ $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$; $(iv)$ $s,k\equiv 2 \pmod 4$ and $m,n$ both even.