Lineability on nets and uncountable sequences of functions in measure theory

In general, some of the well known results of measure theory dealing with the convergence of sequences of functions such as the Dominated Convergence Theorem or the Monotone Convergence Theorem are not true when we consider arbitrary nets of functions instead of sequences.In this paper, we study the algebraic genericity of families of nets of functions that do not satisfy important results of measure theory, and we also analyze the particular case of uncountable sequences

Linearly continuous maps discontinuous on the graphs of twice differentiable functions

A function g : R n → R is linearly continuous provided its restriction g ` to every straight line ` ⊂ R n is continuous. It is known that the set D(g) of points of discontinuity of any linearly continuous g : R n → R is a countable union of isometric copies of (the graphs of) f P, where f : R n−1 → R is Lipschitz and P ⊂ R n−1 is compact nowhere dense. On the other hand, for every twice continuously differentiable function f : R → R and every nowhere dense perfect P ⊂ R there is a linearly continuous g : R 2 → R with D(g) = f P. The goal of this paper is to show that this last statement fails, if we do not assume that f 00 is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f : R → R with nowhere monotone derivative, which includes twice differentiable functions f with such property. This generalizes a recent result of professor Ludek Zajicek and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e Innovación (MICINN)/FEDERunpu

A note concerning polyhyperbolic and related splines

This note concerns the finite interpolation problem with two parametrized families of splines related to polynomial spline interpolation. We address the questions of uniqueness and establish basic convergence rates for splines of the form $s_\alpha = p\cosh(\alpha\cdot)+q\sinh(\alpha \cdot)$ and $t_\alpha = p+q\tanh(\alpha \cdot)$ between the nodes where $p,q\in\Pi_{k-1}$.Comment: 13 pages, updated to include fundin

Sequence and analysis of the genome of the pathogenic yeast Candida orthopsilosis

Candida orthopsilosis is closely related to the fungal pathogen Candida parapsilosis. However, whereas C. parapsilosis is a major cause of disease in immunosuppressed individuals and in premature neonates, C. orthopsilosis is more rarely associated with infection. We sequenced the C. orthopsilosis genome to facilitate the identification of genes associated with virulence. Here, we report the de novo assembly and annotation of the genome of a Type 2 isolate of C. orthopsilosis. The sequence was obtained by combining data from next generation sequencing (454 Life Sciences and Illumina) with paired-end Sanger reads from a fosmid library. The final assembly contains 12.6 Mb on 8 chromosomes. The genome was annotated using an automated pipeline based on comparative analysis of genomes of Candida species, together with manual identification of introns. We identified 5700 protein-coding genes in C. orthopsilosis, of which 5570 have an ortholog in C. parapsilosis. The time of divergence between C. orthopsilosis and C. parapsilosis is estimated to be twice as great as that between Candida albicans and Candida dubliniensis. There has been an expansion of the Hyr/Iff family of cell wall genes and the JEN family of monocarboxylic transporters in C. parapsilosis relative to C. orthopsilosis. We identified one gene from a Maltose/Galactoside O-acetyltransferase family that originated by horizontal gene transfer from a bacterium to the common ancestor of C. orthopsilosis and C. parapsilosis. We report that TFB3, a component of the general transcription factor TFIIH, undergoes alternative splicing by intron retention in multiple Candida species. We also show that an intein in the vacuolar ATPase gene VMA1 is present in C. orthopsilosis but not C. parapsilosis, and has a patchy distribution in Candida species. Our results suggest that the difference in virulence between C. parapsilosis and C. orthopsilosis may be associated with expansion of gene families

Linear structures in the set of non-norm-attaining operators on Banach spaces

We push further the study of lineability properties related to sets from norm-attaining theory. As a matter of fact, we provide several results in the context of lineability, spaceability, maximal-spaceability, and $(\alpha, \beta)$-spaceability for sets of non-norm-attaining bounded linear operators whenever such sets are non-empty. Our results concerning the cardinality of the linear subspaces are presented in a more general framework, where classical spaces of bounded linear operators between Banach spaces and their subsets are considered as special cases. This leads to new results and generalizes several ones from the literature.Comment: 34 page

Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions

A function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC, if each open set G ⊂ R2 containing the graph of f contains also the graph of a continuous function g : R → R; Sierpiński-Zygmund, f ∈ SZ (or, more generally, f ∈ SZ(Bor)), provided its restriction f M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński-Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov(M) = c, that is, that R cannot be covered by less than c-many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ∩AC is also consistent with ZFC plus the negation of cov(M) = c. More precisely, we show that it is consistent with ZFC+cov(M) < c (follows from the assumption that non(N ) < cov(N ) = c) that there is an f ∈ SZ(Bor)∩AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov(M) = c or non(N ) < cov(N ) = c, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński-Zygmund functions. Several open problems are also stated

CDC5 Inhibits the Hyperphosphorylation of the Checkpoint Kinase Rad53, Leading to Checkpoint Adaptation

The mechanistic role of the yeast kinase CDC5, in allowing cells to adapt to the presence of irreparable DNA damage and continue to divide, is revealed

Lineability, differentiable functions and special derivatives

The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (3) the family of functions from (0, 1) to R that are not derivatives, or (4) the family of mappings that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results

Additivity coefficients for all classes in the algebra of Darboux-Like maps on R

The class D of generalized continuous functions on R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra A(D) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient A(F) for all atoms F in the algebra A(D). We also determine the values A(F) for a lot of other families F∈A(D). Open questions and new directions of research shall also be provided