Croissance et ensemble nodal de fonctions propres du laplacien sur des surfaces

Dans cette thèse, nous étudions les fonctions propres de l'opérateur de Laplace-Beltrami - ou simplement laplacien - sur une surface fermée, c'est-à-dire une variété riemannienne lisse, compacte et sans bord de dimension 2. Ces fonctions propres satisfont l'équation $\Delta_g \phi_\lambda + \lambda \phi_\lambda = 0$ et les valeurs propres forment une suite infinie. L'ensemble nodal d'une fonction propre du laplacien est celui de ses zéros et est d'intérêt depuis les expériences de plaques vibrantes de Chladni qui remontent au début du 19ème siècle et, plus récemment, dans le contexte de la mécanique quantique. La taille de cet ensemble nodal a été largement étudiée ces dernières années, notamment par Donnelly et Fefferman, Colding et Minicozzi, Hezari et Sogge, Mangoubi ainsi que Sogge et Zelditch. L'étude de la croissance de fonctions propres n'est pas en reste, avec entre autres les récents travaux de Donnelly et Fefferman, Sogge, Toth et Zelditch, pour ne nommer que ceux-là. Notre thèse s'inscrit dans la foulée du travail de Nazarov, Polterovich et Sodin et relie les propriétés de croissance des fonctions propres avec la taille de leur ensemble nodal dans l'asymptotique $\lambda \nearrow \infty$. Pour ce faire, nous considérons d'abord les exposants de croissance, qui mesurent la croissance locale de fonctions propres et qui sont obtenus à partir de la norme uniforme de celles-ci. Nous construisons ensuite la croissance locale moyenne d'une fonction propre en calculant la moyenne sur toute la surface de ces exposants de croissance, définis sur de petits disques de rayon comparable à la longueur d'onde. Nous montrons alors que la taille de l'ensemble nodal est contrôlée par le produit de cette croissance locale moyenne et de la fréquence $\sqrt{\lambda}$. Ce résultat permet une reformulation centrée sur les fonctions propres de la célèbre conjecture de Yau, qui prévoit que la mesure de l'ensemble nodal croît au rythme de la fréquence. Notre travail renforce également l'intuition répandue selon laquelle une fonction propre se comporte comme un polynôme de degré $\sqrt{\lambda}$. Nous généralisons ensuite nos résultats pour des exposants de croissance construits à partir de normes $L^q$. Nous sommes également amenés à étudier les fonctions appartenant au noyau d'opérateurs de Schrödinger avec petit potentiel dans le plan. Pour de telles fonctions, nous obtenons deux résultats qui relient croissance et taille de l'ensemble nodal.

Increased Levels of NOTCH1, NF-kappa B, and Other Interconnected Transcription Factors Characterize Primitive Sets of Hematopoietic Stem Cells

As previously shown, higher levels of NOTCH1 and increased NF-kappa B signaling is a distinctive feature of the more primitive umbilical cord blood (UCB) CD34+ hematopoietic stem cells (HSCs), as compared to bone marrow ( BM). Differences between BM and UCB cell composition also account for this finding. The CD133 marker defines a more primitive cell subset among CD34+ HSC with a proposed hemangioblast potential. To further evaluate the molecular basis related to the more primitive characteristics of UCB and CD133+ HSC, immunomagnetically purified human CD34+ and CD133+ cells from BM and UCB were used on gene expression microarrays studies. UCB CD34+ cells contained a significantly higher proportion of CD133+ cells than BM (70% and 40%, respectively). Cluster analysis showed that BM CD133+ cells grouped with the UCB cells ( CD133+ and CD34+) rather than to BM CD34+ cells. Compared with CD34+ cells, CD133+ had a higher expression of many transcription factors (TFs). Promoter analysis on all these TF genes revealed a significantly higher frequency ( than expected by chance) of NF-kappa B-binding sites (BS), including potentially novel NF-kappa B targets such as RUNX1, GATA3, and USF1. Selected transcripts of TF related to primitive hematopoiesis and self-renewal, such as RUNX1, GATA3, USF1, TAL1, HOXA9, HOXB4, NOTCH1, RELB, and NFKB2 were evaluated by real-time PCR and were all significantly positively correlated. Taken together, our data indicate the existence of an interconnected transcriptional network characterized by higher levels of NOTCH1, NF-kappa B, and other important TFs on more primitive HSC sets.

Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c (q) as the minimum cardinality of a subset H of F 3 q which satisfies the following property: every vector in this space di ff ers in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c (q)

Doubling and Tripling Constructions for Defining Sets in Steiner Triple Systems

A minimal defining set of a Steiner triple system on a points (STS(v)) is a partial Steiner triple system contained in only this STS(v), and such that any of its proper subsets is contained in at least two distinct STS(v)s. We consider the standard doubling and tripling constructions for STS(2v + 1) and STS(3v) from STS(v) and show how minimal defining sets of an STS(v) gives rise to minimal defining sets in the larger systems. We use this to construct some new families of defining sets. For example, for Steiner triple systems on, 3" points; we construct minimal defining sets of volumes varying by as much as 7(n-/-).

Nodal solutions for nonlinear eigenvalue problems

We are concerned with determining values of, for which there exist nodal solutions of the boundary value problems u" + ra(t) f(u) = 0, 0 < t < 1, u(O) = u(1) = 0. The proof of our main result is based upon bifurcation techniques.

Effect of oral creatine supplementation on near-maximal strength and repeated sets of high-intensity bench press exercise

This study examined the effects of 26 days of oral creatine monohydrate (Cr) supplementation on near-maximal muscular strength, high-intensity bench press performance, and body composition. Eighteen male powerlifters with at least 2 years resistance training experience took part in this 28-day experiment. Pre and postmeasurements (Days 1 and 28) were taken of near-maximal muscular strength, body mass, and % body fat. There were two periods of supplementation Days 2 to 6 and Days 7 to 27. ANOVA and t-tests revealed that Cr supplementation significantly increased body mass and lean body mass with no changes in % body fat. Significant increases in 3-RM strength occurred in both groups, both absolute and relative to body mass; the increases were greater in the Cr group. The change in total repetitions also increased significantly with Cr supplementation both in absolute terms and relative to body mass, while no significant change was seen in the placebo (P) group. Creatine supplementation caused significant changes in the number of BP reps in Sets 1, 4, and 5. No changes occurred in the P group. It appears that 26 days of Cr supplementation significantly improves muscular strength and repeated near-maximal BP performance, and induces changes in body composition.

Critical sets for families of Latin squares

In this paper I give details of new constructions for critical sets in latin squares. These latin squares, of order n, are such that they can be partitioned into four subsquares each of which is based on the addition table of the integers module n/2, an isotopism of this or a conjugate.

Maximal sets of Hamilton cycles in Kn,n

In this article, we prove that there exists a maximal set of m Hamilton cycles in K-n,K-n if and only if n/4 < m less than or equal to n/2. (C) 2000 John Wiley & Sons, Inc.

Slant or occlusion: global factors resolve stereoscopic ambiguity in sets of horizontal lines

Perceived slant was measured for horizontal lines aligned on one side and of varying lengths whose length disparity was either a constant linear amount for all lines (consistent with uniocular occlusion) or proportional to line length (consistent with global slant). Although the disparity of any line was ambiguous with respect to these two possibilities, slant of individual lines did not occur in the former case, but a subjective contour in depth was reported along the alignment. For proportional disparity of the set, global slant was seen. Adding a constant length to each line on the invalid eye for occlusion resulted in multiple slants. Smooth uniocular variations in alignment shape elicited subjective contours slanting or curving in depth. Global context can disambiguate the depth status of individual disparate lines. (C) 2004 Elsevier Ltd. All rights reserved.

Critical sets in direct products of back circulant Latin squares

To date very Few families of critical sets for latin squares are known. The only previously known method for constructing critical sets involves taking a critical set which is known to satisfy certain strong initial conditions and using a doubling construction. This construction can be applied to the known critical sets in back circulant latin squares of even order. However, the doubling construction cannot be applied to critical sets in back circulant latin squares of odd order. In this paper a family of critical sets is identified for latin squares which are the product of the latin square of order 2 with a back circulant latin square of odd order. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.

Influence of the interaction between nodal fibroblast and breast cancer cells on gene expression

Our aim was to evaluate the interaction between breast cancer cells and nodal fibroblasts, by means of their gene expression profile. Fibroblast primary cultures were established from negative and positive lymph nodes from breast cancer patients and a similar gene expression pattern was identified, following cell culture. Fibroblasts and breast cancer cells (MDA-MB231, MDA-MB435, and MCF7) were cultured alone or co-cultured separated by a porous membrane (which allows passage of soluble factors) for comparison. Each breast cancer lineage exerted a particular effect on fibroblasts viability and transcriptional profile. However, fibroblasts from positive and negative nodes had a parallel transcriptional behavior when co-cultured with a specific breast cancer cell line. The effects of nodal fibroblasts on breast cancer cells were also investigated. MDA MB-231 cells viability and migration were enhanced by the presence of fibroblasts and accordingly, MDA-MB435 and MCF7 cells viability followed a similar pattern. MDA-MB231 gene expression profile, as evaluated by cDNA microarray, was influenced by the fibroblasts presence, and HNMT, COMT, FN3K, and SOD2 were confirmed downregulated in MDA-MB231 co-cultured cells with fibroblasts from both negative and positive nodes, in a new series of RT-PCR assays. In summary, transcriptional changes induced in breast cancer cells by fibroblasts from positive as well as negative nodes are very much alike in a specific lineage. However, fibroblasts effects are distinct in each one of the breast cancer lineages, suggesting that the inter-relationships between stromal and malignant cells are dependent on the intrinsic subtype of the tumor.

Lymph Node Size in Rectal Cancer Following Neoadjuvant Chemoradiation-Can We Rely on Radiologic Nodal Staging After Chemoradiation?

OBJECTIVES: Local excision is currently being considered as an alternative strategy for ypT0-2 rectal cancer. However, patient selection is crucial to rule out nodal disease and is performed by radiologic studies that consider size as a surrogate marker for positive nodes. The purpose of this study was to determine the difference in size between metastatic and nonmetastatic nodes and the critical lymph node size after neoadjuvant chemoradiation therapy. METHODS: The 201 lymph nodes available from 31 patients with ypT0-2 rectal cancer were reviewed and measured. Lymph nodes were compared according to the presence of metastases and size. RESULTS: There was a mean of 6.5 lymph nodes per patient and 12 positive nodes of the 201 recovered (6%). Ninety-five percent of all lymph nodes were <5 mm, whereas 50% of positive lymph nodes were <3 mm. Metastatic lymph nodes were significantly greater in size (5.0 vs. 2.5mm; P = 0.02). Lymph nodes >4.5 mm had a greater risk of harboring metastases (P = 0.009). CONCLUSIONS: Patients with ypT0-2 rectal cancer following neoadjuvant chemoradiation have very small perirectal nodes. Individual metastatic lymph nodes are significantly larger. However, a significant number of lymph nodes after neoadjuvant chemoradiation (negative and positive) are <3 mm. Individual lymph node size is not a good predictor of nodal metastases and may lead to inaccurate radiologic staging.

Individual risk prediction of nodal and distant metastasis for patients with typical bronchial carcinoid tumors

Objective: Bronchial typical carcinoid tumors are tow-grade malignancies. However, metastases are diagnosed in some patients. Predicting the individual risk of these metastases to determine patients eligible for a radical lymphadenectomy and patients to be followed-up because of distant metastasis risk is relevant. Our objective was to screen for predictive criteria of bronchial typical carcinoid tumor aggressiveness based on a logistic regression model using clinical, pathological and biomolecular data. Methods: A multicenter retrospective cohort study, including 330 consecutive patients operated on for bronchial typical carcinoid tumors and followed-up during a period more than 10 years in two university hospitals was performed. Selected data to predict the individual risk for both nodal and distant metastasis were: age, gender, TNM staging, tumor diameter and location (central/peripheral), tumor immunostaining index of p53 and Ki67, Bcl2 and the extracellular density of neoformed microvessels and of collagen/elastic extracellular fibers. Results: Nodal and distant metastasis incidence was 11% and 5%, respectively. Univariate analysis identified all the studied biomarkers as related to nodal metastasis. Multivariate analysis identified a predictive variable for nodal metastasis: neo angiogenesis, quantified by the neoformed pathological microvessels density. Distant metastasis was related to mate gender. Discussion: Predictive models based on clinical and biomolecular data could be used to predict individual risk for metastasis. Patients under a high individual risk for lymph node metastasis should be considered as candidates to mediastinal lymphadenectomy. Those under a high risk of distant metastasis should be followed-up as having an aggressive disease. Conclusion: Individual risk prediction of bronchial typical carcinoid tumor metastasis for patients operated on can be calculated in function of biomolecular data. Prediction models can detect high-risk patients and help surgeons to identify patients requiring radical lymphadenectomy and help oncologists to identify those as having an aggressive disease requiring prolonged follow-up. (C) 2008 European Association for Cardio-Thoracic Surgery. Published by Elsevier B.V. All rights reserved.