On the fast computation of the weight enumerator polynomial and the tt value of digital nets over finite abelian groups

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$ dimensional unit cube $[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further

Eigenvalue Estimates for submanifolds of N×RN \times \mathbb{R} with locally bounded mean curvature

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq \kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.Comment: 9 page

The prime at infinity and the rank of the class group in global function fields

AbstractIn this paper we construct, for any integers m and n, and 2⩽g⩽m-1, infinitely many function fields K of degree m over F(T) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to (Z/nZ)m-g. This extends previous results of the author and Lee for the cases g=1 and g=m

Collective discussions for the development of Interpretative Knowledge in Mathematics Teacher Education

We start from the assumption that teachers need a deep and broad mathematical knowledge —called Interpretative Knowledge (IK)—that allows them to support students in building their mathematical knowledge from their own reasoning and productions. In the present study, we aimed to ascertain how collective discussions focusing on the interpretation of students’ productions engage Prospective Teachers (PTs) and impact their IK development. In particular, we observe how this form of collaborative work among PTs allows for the emergence of novel insights into the mathematical aspects of students’ productions that were not considered during previous individual work, and produce changes in PTs’ attitudes towards students’ productions

Effect of obesity and metabolic syndrome on plasma oxysterols and fatty acids in human

BACKGROUND: Obesity and the related entity metabolic syndrome are characterized by altered lipid metabolism and associated with increased morbidity risk for cardiovascular disease and cancer. Oxysterols belong to a large family of cholesterol-derived molecules known to play crucial role in many signaling pathways underlying several diseases. Little is known on the potential effect of obesity and metabolic syndrome on oxysterols in human. OBJECTIVES: In this work, we questioned whether circulating oxysterols might be significantly altered in obese patients and in patients with metabolic syndrome. We also tested the potential correlation between circulating oxysterols and fatty acids. METHODS: 60 obese patients and 75 patients with metabolic syndrome were enrolled in the study along with 210 age- and sex-matched healthy subjects, used as control group. Plasma oxysterols were analyzed by isotope dilution GC/MS, and plasma fatty acids profiling was assessed by gas chromatography coupled with flame ionization detection. RESULTS: We found considerable differences in oxysterols profiling in the two disease groups that were gender-related. Compared to controls, males showed significant differences only in 4α- and 4β-hydroxycholesterol levels in obese and metabolic syndrome patients. In contrast, females showed consistent differences in 7-oxocholesterol, 4α-hydroxycholesterol, 25-hydroxycholesterol and triol. Concerning fatty acids, we found minor differences in the levels of these variables in males of the three groups. Significant changes were observed in plasma fatty acid profile of female patients with obesity or metabolic syndrome. We found significant correlations between various oxysterols and fatty acids. In particular, 4β-hydroxycholesterol, which is reduced in obesity and metabolic syndrome, correlated with a number of saturated and mono-unsaturated fatty acids that are end-products of de novo lipogenesis. CONCLUSIONS: Our data provide the first evidence that obesity and metabolic syndrome are associated with major, gender-specific, changes in circulating oxysterols and fatty acids. These findings suggest a metabolic link between oxysterols and fatty acids, and that oxysterols may contribute to the epidemic diseases associated with obesity and metabolic syndrome in female

On submanifolds with tamed second fundamental form

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq \kappa \leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.Comment: 10 page

Universal mean-field upper bound for the generalization gap of deep neural networks

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalization gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimized given a specific realization of the remaining weights. We show how these results-combined with ideas from statistical learning theory-provide a stringent asymptotic upper bound on the generalization gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N-out (where N-out is the size of the last layer) and N-out << P, the generalization gap approaches zero faster than 2N(out)/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks

Serum cholesterol levels in neutropenic patients with fever

Hypocholesterolemia, which often accompanies infectious diseases has been suggested to serve as a prognostic marker in hospitalized patients. Even though patients with chemotherapyinduced leukopenia are at high risk of infection and mortality, only limited information is available on serum cholesterol levels in these patients. We therefore measured serum cholesterol levels in 17 patients with hematological malignancies during chemotherapyinduced neutropenia and correlated it with clinical outcome. Patients with fever (>38.5 degreesC) showed a significant decrease in serum cholesterol levels within 24 hours. Eight days after onset of the fever nonsurvivors had significantly lower serum cholesterol levels (median 2.09 mmol/l, range 0.492.79, n=6) compared to survivors (median 3.23 mmol/l, range 1.684.86, n=11). Cholesterol levels in survivors returned to baseline levels at the time of discharge from the hospital. At the onset of fever, serum levels of inflammatory cytokines interleukin-6, tumor necrosis factor (TNF) and soluble TNF receptors p55 and p75 were elevated in all patients, but only TNF and TNF receptor p75 levels were significantly different in survivors and nonsurvivors. Our data suggest that a decrease in serum cholesterol levels is a prognostic marker in neutropenic patients with fever. Release of inflammatory cytokines may in part be responsible for hypocholesterolemia in these patients