The Hilbert functions ofk-configurations in\mathbb{P}^2 and\mathbb{P}^3

In this paper, we proved the set of points which are the vertices of then-gon in $\mathbb{P}^2 $ (n ≥ 3) has the Uniform Position Property and what the graded free resolutions of the ideals ofk-configurations in $\mathbb{P}^3 $ are.     

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.     

Superficie cubiche di\mathbb{A}_\mathbb{C}^3 a curve sottoinsieme intersezione completa

Summary In this paper we classify the algebraic cubic surfaces of the affine space is the complex field, whose algebraic curves are set-theoretic complete intersections of ; in other words surfaces such that every prime ideal of height 1in the coordinate ring [] of is the radical of a principal ideal; if is non singular in codimension 1this means that [] is semifactorial. We give the equations of such surfaces within linear isomorphisms of providing also methods by which one can construct the equations of the surfaces cutting on its curves as set-theoretic complete intersections. Moreover for each of these surfaces we determine the minimum positive number such that every algebraic curve of with multiplicity of intersection , is complete intersection of itself with another surface § 8where the results are summarized). We tackle also the problem of such a classification over algebraically closed fields k different from .

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R.     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

Linear codes over {\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}

In this work, we investigate linear codes over the ring ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes. The duals and the complete weight enumerators for ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ are obtained.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are classified, up to automorphisms of balls.     

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ to $$\mathbb {Z}_{m}$$Zm

In this paper, several nonexistence results on generalized bent functions \(f:\mathbb {Z}_{2}^{n} \rightarrow \mathbb {Z}_{m}\) are presented by using the knowledge on cyclotomic number fields and their imaginary quadratic subfields.     

\mathbb{Z }_2\mathbb{Z }_4-Additive formally self-dual codes

We study odd and even \(\mathbb{Z }_2\mathbb{Z }_4\) formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

On Hamiltonian Minimality of Isotropic Nonhomogeneous Tori in $$\mathbb{H}^n$$ and $$\mathbb{C} \mathrm{P}^{2n+1}$$

Mathematical Notes - We construct a family of flat isotropic nonhomogeneous tori in $$\mathbb{H}^n$$ and $$\mathbb{C}\mathrm{P}^{2n+1}$$ and find necessary and sufficient conditions for their...     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

On two-weight $$\mathbb {Z}_{2^k}$$-codes

We determine the possible homogeneous weights of regular projective two-weight codes over \(\mathbb {Z}_{2^k}\) of length \(n>3\), with dual Krotov distance \(d^{\lozenge }\) at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When \(k=2\), we characterize the parameters of such codes as those of the inverse Gray images of \(\mathbb {Z}_4\)-linear Hadamard codes, which have been characterized by their types by several authors.     

Classifying ACM sets of points in $${\mathbb{P}^{1} \times \mathbb{P}^{1}}$$ via separators

The purpose of this note is to give a new, short proof of a classification of ACM sets of points in in terms of separators.     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.