Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space

Funding for open access publishing: Universidad de Granada/CBUA Funding for open access charge: Universidad de Granada/CBUA.Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, BF(k) and BT(k) . We will classify real hypersurfaces in complex projective space for which BF(k) and BT(k) either take values in the maximal holomorphic distribution D or are parallel to the structure vector field ξ , in the particular case of B= A , where A denotes the shape operator of M. We also introduce the concept of AF(k) and AT(k) being D -recurrent and classify real hypersurfaces such that either AF(k) or AT(k) are D -recurrent.MICINN: PID 2020-116126GB-I00Universidad de Granada/CBU

Wages: A working conditions and industrial relations perspective

[Excerpt] This paper looks at wages from two different angles: from the perspective of individual employees, discussed in conjunction with their working conditions, and from the perspective of the industrial relations system. After a brief overview of EU-level policy developments with a potential impact on national level pay determination, this report gives a comparative overview of the levels of collective wage setting and how they are set throughout Europe and goes on to report on reforms, changes or debates linked to these processes between the different actors at both the Member State and the European level in 2011 and 2012. This includes, for instance, debates on potential changes of indexation mechanisms in Belgium, Luxembourg and Cyprus, as a result of the Commission’s recommendations within the Euro Plus Pact. While in some countries (Estonia, Bulgaria) social partners resumed collective bargaining (either on wages or on minimum wages) and came to agreements, in other countries (Lithuania, Romania), no agreements could be reached. Some changes in the way collective bargaining is organised were recently introduced in Spain, Romania, Greece, Ireland and Portugal. Information on these issues stems from the European Industrial Relations Observatory (EIRO). Furthermore, this paper also summarises the current changes and debates among the social partners and governments in relation to the setting and the level of minimum wages across Europe. It then looks into the area of ‘working poor’, that is people who are in employment but still at risk of poverty. To what extent do governments and social partners put forth policy responses to address the issue of in-work-poverty? And to what extent did the crisis have an impact? In order to answer these questions, the paper draws on Eurofound research on the situation of the ‘Working poor’. Finally the paper presents a collection of recent ‘information updates’ from the European Industrial Relations Observatory (EIRO) on various issues in relation to pay from Member States

Some Bounds for the Number of Blocks III

Let $\mathcal D=(\Omega, \mathcal B)$ be a pair of $v$ point set $\Omega$ and a set $\mathcal B$ consists of $k$ point subsets of $\Omega$ which are called blocks. Let $d$ be the maximal cardinality of the intersections between the distinct two blocks in $\mathcal B$. The triple $(v,k,d)$ is called the parameter of $\mathcal B$. Let $b$ be the number of the blocks in $\mathcal B$. It is shown that inequality ${v\choose d+2i-1}\geq b\{{k\choose d+2i-1} +{k\choose d+2i-2}{v-k\choose 1}+....$ $.+{k\choose d+i}{v-k\choose i-1} \}$ holds for each $i$ satisfying $1\leq i\leq k-d$, in the paper: Some Bounds for the Number of Blocks, Europ. J. Combinatorics 22 (2001), 91--94, by R. Noda. If $b$ achieves the upper bound, $\mathcal D$ is called a $\beta(i)$ design. In the paper, an upper bound and a lower bound, $\frac{(d+2i)(k-d)}{i}\leq v \leq \frac{(d+2(i-1))(k-d)}{i-1}$, for $v$ of a $\beta(i)$ design $\mathcal D$ are given. In the present paper we consider the cases when $v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for $v$ respectively. We apply this bound to the problem of the perfect $e$-codes in the Johnson scheme, and improve the bound given by Roos in 1983.Comment: Stylistic corrections are made. References are adde

On the divisibility of the rank of appearance of a Lucas sequence

Let U = (U-n)(n >= 0) be a Lucas sequence and, for every prime number p, let rho(U)(p) be the rank of appearance of p in U, that is, the smallest positive integer k such that p divides U-k, whenever it exists. Furthermore, let d be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes p <= x such that d divides p(U)(p), as x -> +infinity

Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the rr-Metastable Range

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps from K to R^d without r-intersection points among any set of r pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions (r d > (r+1) dim K +2). This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 3).Comment: 35 pages, 10 figures (v2: reference for the algorithmic aspects updated & appendix on Block Bundles added