Abstract:
We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\frac{d}{dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of $\E{Z_{p_N}\wedge C_{p_N}},$ the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{p_N}$ and $C_{p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like $e^{-N\abs{z-w}^2}.$ With $\abs{z-w}$ on the order of $N^{-1/2},$ however, $\E{Z_{p_N}\wedge C_{p_N}}(z,w)$ is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.

Abstract:
We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(\xi)=0$ for a fixed $\xi \in \C\backslash\set{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-\ep}<\abs{z-\xi}< N^{-1+\ep}}$ and no critical points closer to $\xi$ with probability at least $1-O(N^{-3/2+3\ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

Abstract:
We obtain new off-diagonal remainder estimates for the kernel of the spectral projector of the Laplacian onto frequencies up to \lambda. A corollary is that the kernel of the spectral projector onto frequencies (\lambda, \lambda+1] has a universal scaling limit as \lambda go to infinity at any non self-focal point. Our results also imply that immersions of manifolds without conjugate points into Euclidean space by arrays of eigenfunctions with frequencies in (\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large. Finally, we find precise asymptotics for sup norms of gradients of linear combinations of eigenfunctions with frequencies in ({\lambda}, {\lambda} + 1].

Abstract:
A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.

Abstract:
This article concerns upper bounds for $L^\infty$-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold $(M,g).$ We study $f_{\lambda}$ chosen uniformly at random from the space of $L^2$-normalized linear combinations of Laplace eigenfunctions with eigenvalues in the interval $(\lambda^2, \lr{\lambda+1}^2].$ Our main result is that the expected value of $\norm{f_\lambda}_\infty$ grows at most like $C \sqrt{\log \lambda}$ as $\lambda \to \infty$, where $C$ is an explicit constant depending only on the dimension and volume of $(M,g).$ In addition, we obtain concentration of the $L^\infty$-norm around its mean and median and study the analogous problems for Gaussian random waves on $(M,g).$

Abstract:
We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.

Abstract:
Nowadays it is widely accepted that mathematics and, especially, problemsolving tasks, are particularly concerned by the issue of emotions. Yet, educational interventions designed to improve students’ problem-solving competence and performance still mainly focus on cognitive and metacognitive knowledge and skills. The main purpose of this study was to design and assess the benefit of a training program that promotes the development of not only cognitive but also emotional knowledge and skills. This benefit was assessed using four variables, namely, problem-solving performance, problem-solving competence, academic emotions and emotion regulation strategies. 428 fifth and sixth graders took part in the study, split into four conditions: 1) a “cognition” condition which received an intervention on an eight-step problemsolving process; 2) an “emotion” condition in which emotional knowledge and skills were developed through various activities; 3) an “emotion and cognition” condition overlapping the two previous ones, and 4) a “control” condition. The findings showed that the “emotion and cognition” condition and the “cognition” condition had equivalent cognitive efficiency. However, only the former reduced negative emotions, aroused the emergence of positive ones, promoted the use of adaptive emotion regulation strategies and discouraged the use of maladaptive ones. The practical implications for educational practices and possible avenues for further research are discussed.

Abstract:
We discuss philosophical, methodological, and biomedical grounds for the traditional paradigm of cancer and some of its critical flaws. We also review some potentially fruitful approaches to understanding cancer and its treatment. This includes the new paradigm of cancer that was developed over the last 15 years by Michael Retsky, Michael Baum, Romano Demicheli, Isaac Gukas, William Hrushesky and their colleagues on the basis of earlier pioneering work of Bernard Fisher and Judah Folkman. Next, we highlight the unique and pivotal role of mathematical modeling in testing biomedical hypotheses about the natural history of cancer and the effects of its treatment, elaborate on model selection criteria, and mention some methodological pitfalls. Finally, we describe a specific mathematical model of cancer progression that supports all the main postulates of the new paradigm of cancer when applied to the natural history of a particular breast cancer patient and fit to the observables.

Abstract:
Background: Increasing clarithromycin resistance has undermined the effectiveness of traditional clarithromycin-containing triple eradication therapy of Helicobacter pylori infections. Sequential and concomitant therapies show improved outcome with clarithromycin resistance. Aim: To evaluate the effectiveness of sequential and concomitant 4-drug non-bismuth therapies for eradication of Helicobacter pylori in a prospective, randomized, clinical trial conducted in Palestine. Patients and Methods: Patients who underwent upper endoscopy for a clinical indication and tested positive for rapid urease test were included. Subjects randomly allocated into two groups: One received a modified sequential therapy: esomeprazole 40 mg OD and amoxicillin 1 g BID for 5 days then esomeprazole 40 mg OD, clarithromycin 500 mg BID and tinidazole 500 mg BID for another 5 days. The other group received concomitant therapy in which the same 4 drugs and doses were all given daily for 10 days. Stool antigen was tested 4 weeks after completion of treatment. Results: Five hundred thirty three (533) patients were tested for H. pylori and 180 (34%) were positive; 141 patients were included in the study and 112 patients completed. The overall per protocol eradication rate was (74%; 95% CI = 65.9% - 82.1%). The eradication rates for sequential therapy was, (70.9%; 95% CI = 58.9% - 82.9%) and for concomitant therapy (77.2%; 95% CI = 66.3% - 88.1%). The results intention-to-treat were: sequential 61%, concomitant 57%. Conclusion: Neither sequential nor concomitant therapy achieved an acceptable H. pylori eradiation rate in Palestine.

Abstract:
A general framework for solving identification problem for a broad class of deterministic and stochastic models is discussed. This methodology allows for a unified approach to studying identifiability of various stochastic models arising in biology and medicine including models of spontaneous and induced Carcinogenesis, tumor progression and detection, and randomized hit and target models of irradiated cell survival. A variety of known results on parameter identification for stochastic models is reviewed and several new results are presented with an emphasis on rigorous mathematical development.