Optimality of generalized Bernstein operators

AbstractWe show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems

Fischer decompositions for entire functions and the Dirichlet problem for parabolas

Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left( m+D\right) ^{\alpha }}\left\langle f_{m},f_{m}\right\rangle_{\mathbb{S}^{d-1}} \end{equation*} for all homogeneous polynomials $f_{m}$ of degree $m.$ Assume that $P_{j}$ for $j=0, \dots ,\beta <2k$ are homogeneous polynomials of degree $j$. The main result of the paper states that for any entire function $f$ of order there exist entire functions $q$ and $h$ of order bounded by $\rho$ such that \begin{equation*} f=\left( P_{2k}-P_{\beta }- \dots -P_{0}\right) q+h\text{ and }\Delta ^{h}r=0. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than $\frac{1}{2}$.Comment: 27 page

Bernstein operators for exponential polynomials

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b]$ is smaller than $\pi /M_{n}$, where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\} , then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $$ is positive on the open interval $(a,b) .$ Under the additional assumption that $\lambda_{0}$ and $\lambda_{1}$ are real and distinct, our first main result states that there exist points $a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $\alpha_{0},..,\alpha_{n}$%, such that the operator \begin{equation*} B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies $B_{n}e^{\lambda_{j}x}=e^{\lambda_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\in C[ a,b]$.Comment: A very similar version is to appear in Constructive Approximatio

Control Loop for a Pulse Generator of a Fast Septum Magnet using DSP and Fuzzy Logic

A prototype of a fast pulsed eddy current septum magnet for one of thebeam extraction's from the SPS towards LHC is under development. The precision of the magnetic field must be better than ±1.0 10-4 during a flat top of 30 µs. The current pulse is generated by discharging the capacitors of a LC circuit that resonates on the 1st and on the 3rd harmonic of a sine wave with a repetition rate of 15 s. The parameters of the circuit and the voltage on the capacitors must be carefully adjusted to meet the specifications. Drifts during operation must be corrected between two pulses by mechanically adjusting the inductance of the coil in the generator as well as the primary capacitor voltage. This adjustment process is automated by acquiring the current pulse waveform with sufficient time and amplitude resolution, calculating the corrections needed and applying these corrections to the hardware for the next pulse. A very cost-effective and practical solution for this adjustment process is the integration of off-the-shelf commercially available boards into an active digital control loop. A 16-bit fixed point, 33 MIPS, DSP together with a 12-bit, 500 kSPS, ADC (total cost of under 250 $) has been used for this control process. The correction algorithm developed for the DSP uses Fuzzy Logic reasoning

Some inequalities on generalized entropies

We give several inequalities on generalized entropies involving Tsallis entropies, using some inequalities obtained by improvements of Young's inequality. We also give a generalized Han's inequality.Comment: 15 page

Functional significance may underlie the taxonomic utility of single amino acid substitutions in conserved proteins

We hypothesized that some amino acid substitutions in conserved proteins that are strongly fixed by critical functional roles would show lineage-specific distributions. As an example of an archetypal conserved eukaryotic protein we considered the active site of ß-tubulin. Our analysis identified one amino acid substitution—ß-tubulin F224—which was highly lineage specific. Investigation of ß-tubulin for other phylogenetically restricted amino acids identified several with apparent specificity for well-defined phylogenetic groups. Intriguingly, none showed specificity for “supergroups” other than the unikonts. To understand why, we analysed the ß-tubulin Neighbor-Net and demonstrated a fundamental division between core ß-tubulins (plant-like) and divergent ß-tubulins (animal and fungal). F224 was almost completely restricted to the core ß-tubulins, while divergent ß-tubulins possessed Y224. Thus, our specific example offers insight into the restrictions associated with the co-evolution of ß-tubulin during the radiation of eukaryotes, underlining a fundamental dichotomy between F-type, core ß-tubulins and Y-type, divergent ß-tubulins. More broadly our study provides proof of principle for the taxonomic utility of critical amino acids in the active sites of conserved proteins

SOX9 Triggers Different Epithelial to Mesenchymal Transition States to Promote Pancreatic Cancer Progression.

BACKGROUND Pancreatic ductal adenocarcinoma (PDAC) is one of the most lethal cancers mainly due to spatial obstacles to complete resection, early metastasis and therapy resistance. The molecular events accompanying PDAC progression remain poorly understood. SOX9 is required for maintaining the pancreatic ductal identity and it is involved in the initiation of pancreatic cancer. In addition, SOX9 is a transcription factor linked to stem cell activity and is commonly overexpressed in solid cancers. It cooperates with Snail/Slug to induce epithelial-mesenchymal transition (EMT) during neural development and in diseases such as organ fibrosis or different types of cancer. METHODS We investigated the roles of SOX9 in pancreatic tumor cell plasticity, metastatic dissemination and chemoresistance using pancreatic cancer cell lines as well as mouse embryo fibroblasts. In addition, we characterized the clinical relevance of SOX9 in pancreatic cancer using human biopsies. RESULTS Gain- and loss-of-function of SOX9 in PDAC cells revealed that high levels of SOX9 increased migration and invasion, and promoted EMT and metastatic dissemination, whilst SOX9 silencing resulted in metastasis inhibition, along with a phenotypic reversion to epithelial features and loss of stemness potential. In both contexts, EMT factors were not altered. Moreover, high levels of SOX9 promoted resistance to gemcitabine. In contrast, overexpression of SOX9 was sufficient to promote metastatic potential in K-Ras transformed MEFs, triggering EMT associated with Snail/Slug activity. In clinical samples, SOX9 expression was analyzed in 198 PDAC cases by immunohistochemistry and in 53 patient derived xenografts (PDXs). SOX9 was overexpressed in primary adenocarcinomas and particularly in metastases. Notably, SOX9 expression correlated with high vimentin and low E-cadherin expression. CONCLUSIONS Our results indicate that SOX9 facilitates PDAC progression and metastasis by triggering stemness and EMT