Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.Comment: 7 pages, 0 figure

The set of badly approximable vectors is strongly C1C^1 incompressible

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's $(\alpha, \beta)$-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.Comment: 18 page

Itinerant-electron Ferromagnetism in W(Nb)O3-d

The crystal structure and the magnetic properties of the W1-xNbxO3-d, (x<0.03) system have been investigated. In contrast to the orthorhombic diamagnetic WO3, the material with x=0.01 is paramagnetic down to 5 K. Introducing of 2.5 at. % of Nb into WO3 leads to a tetragonal structure and to a weak itinerant ferromagnetic ordering below TC= 225 K. The saturation magnetic moment at 5 K is 1.07*10-3 mB, whereas the paramagnetic effective moment is 0.06 mB per mole. This high ratio indicates itinerant ferromagnetism in W0.975Nb0.025O3-d.Comment: accepted to Physica

Viral infection reveals hidden sharing of TCR CDR3 sequences between individuals

The T cell receptor is generated by a process of random and imprecise somatic recombination. The number of possible T cell receptors which this process can produce is enormous, greatly exceeding the number of T cells in an individual. Thus, the likelihood of identical TCRs being observed in multiple individuals (public TCRs) might be expected to be very low. Nevertheless such public TCRs have often been reported. In this study we explore the extent of TCR publicity in the context of acute resolving Lymphocytic choriomeningitis virus (LCMV) infection in mice. We show that the repertoire of effector T cells following LCMV infection contains a population of highly shared TCR sequences. This subset of TCRs has a distribution of naive precursor frequencies, generation probabilities, and physico-chemical CDR3 properties which lie between those of classic public TCRs, which are observed in uninfected repertoires, and the dominant private TCR repertoire. We have named this set of sequences "hidden public" TCRs, since they are only revealed following infection. A similar repertoire of hidden public TCRs can be observed in humans after a first exposure to SARS-COV-2. The presence of hidden public TCRs which rapidly expand following viral infection may therefore be a general feature of adaptive immunity, identifying an additional level of inter-individual sharing in the TCR repertoire which may form an important component of the effector and memory response

Intrinsic approximation on Cantor-like sets, a problem of

Abstract In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets. The resulting approximation function is analogous to that for R d , but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead

Viral infection reveals hidden sharing of TCR CDR3 sequences between individuals

The T cell receptor is generated by a process of random and imprecise somatic recombination. The number of possible T cell receptors which this process can produce is enormous, greatly exceeding the number of T cells in an individual. Thus, the likelihood of identical TCRs being observed in multiple individuals (public TCRs) might be expected to be very low. Nevertheless such public TCRs have often been reported. In this study we explore the extent of TCR publicity in the context of acute resolving Lymphocytic choriomeningitis virus (LCMV) infection in mice. We show that the repertoire of effector T cells following LCMV infection contains a population of highly shared TCR sequences. This subset of TCRs has a distribution of naive precursor frequencies, generation probabilities, and physico-chemical CDR3 properties which lie between those of classic public TCRs, which are observed in uninfected repertoires, and the dominant private TCR repertoire. We have named this set of sequences “hidden public” TCRs, since they are only revealed following infection. A similar repertoire of hidden public TCRs can be observed in humans after a first exposure to SARS-COV-2. The presence of hidden public TCRs which rapidly expand following viral infection may therefore be a general feature of adaptive immunity, identifying an additional level of inter-individual sharing in the TCR repertoire which may form an important component of the effector and memory response

A hierarchy of selection pressures determines the organization of the T cell receptor repertoire

We systematically examine the receptor repertoire in T cell subsets in young, adult, and LCMV-infected mice. Somatic recombination generates diversity, resulting in the limited overlap between nucleotide sequences of different repertoires even within the same individual. However, statistical features of the repertoire, quantified by the V gene and CDR3 k-mer frequency distributions, are highly conserved. A hierarchy of immunological processes drives the evolution of this structure. Intra-thymic divergence of CD4+ and CD8+ lineages imposes subtle but dominant differences observed across repertoires of all subpopulations in both young and adult mice. Differentiation from naive through memory to effector phenotype imposes an additional gradient of repertoire diversification, which is further influenced by age in a complex and lineage-dependent manner. The distinct repertoire of CD4+ regulatory T cells is more similar to naive cells in young mice and to effectors in adults. Finally, we describe divergent (naive and memory) and convergent (CD8+ effector) evolution of the repertoire following acute infection with LCMV. This study presents a quantitative framework that captures the structure of the repertoire in terms of its fundamental statistical properties and describes how this structure evolves as individual T cells differentiate, migrate and mature in response to antigen exposure

Mnk2 Alternative Splicing Modulates the p38-MAPK Pathway and Impacts Ras-Induced Transformation

Summary: The kinase Mnk2 is a substrate of the MAPK pathway and phosphorylates the translation initiation factor eIF4E. In humans, MKNK2, the gene encoding for Mnk2, is alternatively spliced yielding two splicing isoforms with differing last exons: Mnk2a, which contains a MAPK-binding domain, and Mnk2b, which lacks it. We found that the Mnk2a isoform is downregulated in breast, lung, and colon tumors and is tumor suppressive. Mnk2a directly interacts with, phosphorylates, activates, and translocates p38α-MAPK into the nucleus, leading to activation of its target genes, increasing cell death and suppression of Ras-induced transformation. Alternatively, Mnk2b is pro-oncogenic and does not activate p38-MAPK, while still enhancing eIF4E phosphorylation. We further show that Mnk2a colocalization with p38α-MAPK in the nucleus is both required and sufficient for its tumor-suppressive activity. Thus, Mnk2a downregulation by alternative splicing is a tumor suppressor mechanism that is lost in some breast, lung, and colon tumors. : The Mnk2 kinase is a MAPK pathway substrate and phosphorylates the translation initiation factor eIF4E. Mnk2 is alternatively spliced yielding two isoforms: Mnk2a and Mnk2b. Maimon et al. now report that Mnk2a is downregulated in many tumors and behaves like a tumor suppressor. They show that whereas Mnk2b is pro-oncogenic and does not activate p38α-MAPK, Mnk2a phosphorylates and activates p38α-MAPK leading to induction of its target genes, cell death, and suppression of Ras-induced transformation