On the representation of integers by quadratic forms

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.Comment: 33 page

Prescribing the binary digits of squarefree numbers and quadratic residues

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and S\'ark\"ozy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics

Sums of two squares and a power

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a congruence condition is imposed on $z$. These examples are of interest as there is no congruence obstruction itself for the representation of these $n$. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.Comment: 6 pages, to appear in the memorial volume "From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz

Rational points on intersections of cubic and quadric hypersurfaces

We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational number

A Unique Gene Regulatory Network Resets the Human Germline Epigenome for Development.

Resetting of the epigenome in human primordial germ cells (hPGCs) is critical for development. We show that the transcriptional program of hPGCs is distinct from that in mice, with co-expression of somatic specifiers and naive pluripotency genes TFCP2L1 and KLF4. This unique gene regulatory network, established by SOX17 and BLIMP1, drives comprehensive germline DNA demethylation by repressing DNA methylation pathways and activating TET-mediated hydroxymethylation. Base-resolution methylome analysis reveals progressive DNA demethylation to basal levels in week 5-7 in vivo hPGCs. Concurrently, hPGCs undergo chromatin reorganization, X reactivation, and imprint erasure. Despite global hypomethylation, evolutionarily young and potentially hazardous retroelements, like SVA, remain methylated. Remarkably, some loci associated with metabolic and neurological disorders are also resistant to DNA demethylation, revealing potential for transgenerational epigenetic inheritance that may have phenotypic consequences. We provide comprehensive insight on early human germline transcriptional network and epigenetic reprogramming that subsequently impacts human development and disease.W.C.C.T is supported by Croucher Foundation and Cambridge Trust. P.F.C.is a Wellcome Trust Senior Fellow in Clinical Science (101876/Z/13/Z), and a UK NIHR Senior Investigator with additional support from the Wellcome Trust Centre for Mitochondrial Research (096919Z/11/Z). M.A.S. is supported by HFSP and Wellcome Trust Investigator Award.This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.cell.2015.04.05

Nodding syndrome in Tanzania may not be associated with circulating anti-NMDA- and anti-VGKC receptor antibodies or decreased pyridoxal phosphate serum levels-a pilot study

Background: Nodding syndrome (NS) is a seemingly progressive epilepsy disorder of unknown underlying cause. We investigated association of pyridoxal-phosphate serum levels and occurrence of anti-neuronal antibodies against N-methyl-D-aspartate (NMDA) receptor and voltage gated potassium channel (VGKC) complex in NS patients.Methods: Sera of a Tanzanian cohort of epilepsy and NS patients and community controls were tested for the presence of anti-NMDA-receptor and anti-VGKC complex antibodies by indirect immunofluorescence assay. Furthermore pyridoxal-phosphate levels were measured.Results: Auto-antibodies against NMDA receptor or VGKC (LG1 or Caspr2) complex were not detected in sera of patients suffering from NS (n=6), NS plus other seizure types (n=16), primary generalized epilepsy (n=1) and community controls without epilepsy (n=7). Median Pyridoxal-phosphate levels in patients with NS compared to patients with primary generalized seizures and community controls were not significantly different. However, these median pyridoxal-phosphate levels are significantly lower compared to the range considered normal in Europeans.Conclusions: In this pilot study NS was not associated with serum anti-NMDA receptor or anti-VGKC complex antibodies and no association to pyridoxal-phosphate serum levels was found.Key words: nodding syndrome, epilepsy, anti-neuronal antibodies, pyridoxal-phosphat

Murine malaria is associated with significant hearing impairment

<p>Abstract</p> <p>Background</p> <p><it>Plasmodium falciparum </it>malaria has been suspected to cause hearing loss. Developmental, cognitive and language disorders have been observed in children, surviving cerebral malaria. This prospective study aims to evaluate whether malaria influences hearing in mice.</p> <p>Methods</p> <p>Twenty mice were included in a standardized murine cerebral malaria model. Auditory evoked brainstem responses were assessed before infection and at the peak of the illness.</p> <p>Results</p> <p>A significant hearing impairment could be demonstrated in mice with malaria, especially the cerebral form. The control group did not show any alterations. No therapy was used.</p> <p>Conclusion</p> <p>This suggests that malaria itself leads to a hearing impairment in mice.</p