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In a paper by J. V. Pereira and the author it was proved that any pencil of plane curves of degree $d>1$ with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such fibers had ever been found. Recently Janis Stipins has proved that if a pencil has a base of $d^2$ points, then it cannot have five completely reducible fibers. In this paper we generalize Stipins' result to arbitrary pencils. We also include into consideration more general special fibers that are the unions of lines and non-reduced curves. These fibers are important for characteristic varieties of hyperplane complements. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $ \mathfrak{g}\otimes \mathbb{C}[t,t^{-1},u\vert u^2=(t^2-b^2)(t^2-c^2)]$ [ABSTRACT FROM AUTHOR]

In this paper we discuss various refinements and generalizations of a theorem of Sankar Dutta and Paul Roberts. Their theorem gives a criterion for $ d$-dimensional Noetherian Cohen-Macaulay local ring to be a system of parameters, i.e., to have height $ d$ [ABSTRACT FROM AUTHOR]

In this paper we study an operator $ T$ which is $ M$ of $ E$-hypercyclic and use it to answer negatively two questions asked by Madore and Martínez-Avendaño. We also give a sufficient condition for $ T$-hypercyclic for all finite co-dimensional subspaces $ M$. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that there exist measures $ d\mu(x)=\gamma(x)dx$ $ \gamma(x)=\gamma_{0}(\vert x\vert)$ $ \gamma_{0}$ $ \mathcal{M}_{\mu}$ does not map $ L^{p}_{\mu}(\mathbb{R}^{n})$ $ L^{p}_{\mu}(\mathbb{R}^{n})$. This result answers an open question of P. Sjögren and F. Soria. [ABSTRACT FROM AUTHOR]

In the present paper, by making use of the concatenation of $17^2-1=288$ points on the sphere of radius $4$ in $mathbb {R}^{16}$ and subcodes of algebraic geometry codes over $mathbb {F}_{17^2}$, we improve the best-known constructive bound on kissing numbers by A. Vardy. [ABSTRACT FROM AUTHOR]

Let $N(a,m)$ be the least integer $n$ (if it exists) such that $varphi (n)equiv apmod m$. Friedlander and Shparlinski proved that for any $varepsilon >0$ there exists $A=A(varepsilon )>0$ such that for any positive integer $m$ which has no prime divisors $p<(log m)^A$ and any integer $a$ with $gcd (a,m)=1,$ we have the bound $N(a,m)ll m^{3+varepsilon }.$ In the present paper we improve this bound to $N(a,m)ll m^{2+varepsilon }.$ [ABSTRACT FROM AUTHOR]

In this paper, we present a number of examples of commutative Banach algebras with various Arens irregularity properties. These examples illustrate in particular that strong Arens irregularity and extreme non-Arens regularity, the two natural concepts of ``maximal'' Arens irregularity for general Banach algebras as introduced by Dales-Lau and Granirer, respectively, are indeed distinct. Thereby, an open question raised by several authors is answered. We also link these two properties to another natural Arens irregularity property. [ABSTRACT FROM AUTHOR]

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of ``generalized Souslin lines", \textit{i.e.,} linearly ordered spaces in which every collection of disjoint open intervals is $\sigma$-discrete, but which do not have a $\sigma$-discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace. [ABSTRACT FROM AUTHOR]

Let Fp be a prime field of order p > 2, and let A be a set in Fp with very small size in terms of p. In this note, we show that the number of distinct cubic distances determined by points in A × A satisfies |(A - A)³ + (A - A)³| ≫ |A|8/7, which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that max {|A + A|, |f(A,A)|} ≫ |A|6/5, where f(x, y) is a quadratic polynomial in Fp[x, y] that is not of the form g(αx + βy) for some univariate polynomial g. [ABSTRACT FROM AUTHOR]

We prove an infinite-dimensional KAM theorem with dense normal frequencies. In this theorem, we relax the separation condition on normal frequencies which is required by the KAM theorem. [ABSTRACT FROM AUTHOR]