A Meinardus theorem with multiple singularities

Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size $n$, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.Comment: 26 pages. This version incorporates the following two changes implied by referee's remarks: (i) We made changes in the proof of Proposition 1; (ii) We provided an explanation to the argument for the local limit theorem. The paper is tentatively accepted by "Communications in Mathematical Physics" journa

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc

Mathematics education and technology

Recent international surveys such as the Organisation for Economic Co-operation and Development (OECD) report Students, Computers and Learning ( 2015) highlight the wide gap in students’ access to, and use of, technology in secondary mathematics in participating countries. The OECD “snapshot” methodology in which 15-year-old students were asked if they (or their teachers) had performed a range of mathematical tasks using computers in the month preceding their completion of the Programme for International Student Assessment (PISA) survey revealed low levels of technology use (see Fig. 1)

On orientational relief of inter-molecular potential and the structure of domain walls in fullerite C60

A simple planar model for an orientational ordering of threefold molecules on a triangular lattice modelling a close-packed (111) plane of fullerite is considered. The system has 3-sublattice ordered ground state which includes 3 different molecular orientations. There exist 6 kinds of orientational domains, which are related with a permutation or a mirror symmetry. Interdomain walls are found to be rather narrow. The model molecules have two-well orientational potential profiles, which are slightly effected by a presence of a straight domain wall. The reason is a stronger correlation between neighbour molecules in triangular lattice versus previously considered square lattice A considerable reduction (up to one order) of orientational interwell potential barrier is found in the core regions of essentially two-dimentional potential defects, such as a three-domain boundary or a kink in the domain wall. For ultimately uncorrelated nearest neighbours the height of the interwell barrier can be reduced even by a factor of 100.Comment: 11 pages, 13 figures, LaTeX, to appear in Low Temperature Physic

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

Linear forms and quadratic uniformity for functions on ZN\mathbb{Z}_N

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].Comment: 66 page

Bibliometric and authorship trends over a 30 year publication history in two representative US sports medicine journals

Bibliometric studies are important to understand changes and improvement opportunities in academia. This study compared bibliometric trends for two major sports medicine/arthroscopy journals, the American Journal of Sports Medicine® (AJSM®) and Arthroscopy® over the past 30 years. Trends over time and comparisons between both journals were noted for common bibliometric variables (number of authors, references, pages, citations, and corresponding author position) as well as author gender and continental origin. Appropriate statistical analyses were performed. A p < 0.001 was considered statistically significant. One representative year per decade was used. There were 814 manuscripts from AJSM® and 650 from Arthroscopy®. For AJSM® the number of manuscripts steadily increased from 86 in 1986 to 350 in 2016; for Arthroscopy® the number of manuscripts increased from 73 in 1985/1986, to 267 in 2006, but then dropped to 229 in 2016. There were significant increases in all bibliometric variables, except for the number of citations which decreased in Arthroscopy®. There were significant differences in manuscript region of origin by journal (p = 0.000002). Arthroscopy® had a greater percentage of manuscripts from Asia than AJSM® (19.3% vs 11.5%) while AJSM® had a greater percentage from North America (70.3% vs 59.2%); both journals had similar percentages from Europe (18.2% for AJSM® and 21.6% for Arthroscopy®). For AJSM® the average percentage of female first authors was 13.3%, increasing from 4.7% in 1986 to 19.3% in 2016; the average percentage of female corresponding authors was 7.3%. For Arthroscopy®, the average percentage of female first authors was 8.1%, increasing from 2.8% in 1985/1986 to 15.7% in 2016 (p = 0.00007). In conclusion, AJSM® and Arthroscopy® showed an increase in most variables analyzed. Although Arthroscopy® is climbing at a higher rate than AJSM® for female authors, AJSM® has an overall greater percentage of female authors

Inhibition of DUSP6 sensitizes ovarian cancer cells to chemotherapeutic agents via regulation of ERK signaling response genes

Dual specificity phosphatase 6 (DUSP6) is a protein phosphatase that deactivates extracellular-signal-regulated kinase (ERK). Since the ovarian cancer biomarker human epididymis protein 4 (HE4) interacts with the ERK pathway, we sought to determine the relationship between DUSP6 and HE4 and elucidate DUSP6’s role in epithelial ovarian cancer (EOC). Viability assays revealed a significant decrease in cell viability with pharmacological inhibition of DUSP6 using (E/Z)-BCI hydrochloride in ovarian cancer cells treated with carboplatin or paclitaxel, compared to treatment with either agent alone. Quantitative PCR was used to evaluate levels of ERK pathway response genes to BCI in combination with recombinant HE4 (rHE4), carboplatin, and paclitaxel. Expression of EGR1, a promoter of apoptosis, was higher in cells co-treated with BCI and paclitaxel or carboplatin than in cells treated with chemotherapeutic agents alone, while expression of the proto-oncogene c-JUN was decreased with co-treatment. The effect of BCI on the expression of these two genes opposed that of rHE4. Pathway focused quantitative PCR also revealed suppression of ERBB3 in cells co-treated with BCI plus carboplatin or paclitaxel. Finally, expression levels of DUSP6 in EOC tissue were evaluated by immunohistochemistry, revealing significantly increased levels of DUSP6 in serous EOC tissue compared to adjacent normal tissue. A positive correlation between HE4 and DUSP6 levels was determined by Spearman Rank correlation. In conclusion, DUSP6 inhibition sensitizes ovarian cancer cells to chemotherapeutic agents and alters gene expression of ERK response genes, suggesting that DUSP6 could plausibly function as a novel therapeutic target to reduce chemoresistance in EOC

MicroRNA-335-5p suppresses voltage-gated sodium channel expression and may be a target for seizure control

There remains an urgent need for new therapies for treatment-resistant epilepsy. Sodium channel blockers are effective for seizure control in common forms of epilepsy, but loss of sodium channel function underlies some genetic forms of epilepsy. Approaches that provide bidirectional control of sodium channel expression are needed. MicroRNAs (miRNA) are small noncoding RNAs which negatively regulate gene expression. Here we show that genome-wide miRNA screening of hippocampal tissue from a rat epilepsy model, mice treated with the antiseizure medicine cannabidiol, and plasma from patients with treatment-resistant epilepsy, converge on a single target-miR-335-5p. Pathway analysis on predicted and validated miR-335-5p targets identified multiple voltage-gated sodium channels (VGSCs). Intracerebroventricular injection of antisense oligonucleotides against miR-335-5p resulted in upregulation of Scn1a, Scn2a, and Scn3a in the mouse brain and an increased action potential rising phase and greater excitability of hippocampal pyramidal neurons in brain slice recordings, consistent with VGSCs as functional targets of miR-335-5p. Blocking miR-335-5p also increased voltage-gated sodium currents and SCN1A, SCN2A, and SCN3A expression in human induced pluripotent stem cell-derived neurons. Inhibition of miR-335-5p increased susceptibility to tonic-clonic seizures in the pentylenetetrazol seizure model, whereas adeno-associated virus 9-mediated overexpression of miR-335-5p reduced seizure severity and improved survival. These studies suggest modulation of miR-335-5p may be a means to regulate VGSCs and affect neuronal excitability and seizures. Changes to miR-335-5p may reflect compensatory mechanisms to control excitability and could provide biomarker or therapeutic strategies for different types of treatment-resistant epilepsy