Exact polynomial optimization strengthened with Fritz John conditions

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let $C$ be the set of critical points defined by \begin{equation} C=\{x\in\mathbb R^n\,:\,\text{rank}(\varphi(x))< m\}\quad\text{with}\quad\varphi:=\begin{bmatrix} \nabla g\\ \text{diag}(g) \end{bmatrix}\,. \end{equation} Assume that the image of $C$ under $f$, denoted by $f(C)$, is empty or finite. (Our assumption holds generically since $C$ is empty in a Zariski open set in the space of the coefficients of $g_1,\dots,g_m$ with given degrees.) We provide a sequence of values, returned by semidefinite programs, finitely converges to the minimal value attained by $f$ over the basic semi-algebraic set $S$ defined by \begin{equation} S:=\{x\in\mathbb R^n\,:\,g_j(x)\ge 0\,,\,j=1,\dots,m\}\,. \end{equation} Consequently, we can compute exactly the minimal value of any polynomial with real coefficients in $x$ over one of the following sets: the unit ball, the unit hypercube and the unit simplex. Under a slightly more general assumption, we extend this result to the minimization of any polynomial over a basic convex semi-algebraic set that has non-empty interior and is defined by the inequalities of concave polynomials.Comment: 32 pages and 2 tables, merged with arXiv:2205.0845

Ray Casting for Iso-surface in Volumetric Data

Volume data visualization is an active field of research and development. It can be applied in many areas such as medical, oil and gas exploration, etc... Although volume visualization is highly computational cost, there is a vision of real time volumetric visualization systems based on interactive ray tracing. Over the years, many rendering algorithms have been created and enhanced. The focus of this project is to develop a simple ray casting program for volumetric data. The program will be able to render specific volume data using a single processor in a reasonable amount of time. It is opento improve for implementation on multiprocessors. The thesis will compare some existing algorithms for ray casting in terms of image quality, computing time, complexity and so forth. The thesis includes a proposal of new multisampling algorithm, which significantly reduces rendering time while producing similar quality of image with existing algorithms

A Nichtnegativstellensatz on singular varieties under the denseness of regular loci

Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and Demmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-based representation that characterizes the non-negativity of $f$ on $V$. This representation allows us to build up exact semidefinite relaxations for polynomial optimization problems whose optimal solutions are possibly singularities of the constraint sets.Comment: 20 pages, 1 table, fixed typo

Practical polynomial optimization through positivity certificates with and without denominators

Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé $S$ peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant $S$, sous une certaine condition sur $S$ légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur $S$, et donc cette description alternative fournit un certificat de positivité sur $S$. De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set $S$ can be written as a linear weighted combination of the polynomials describing $S$, under a certain condition on $S$ slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on $S$, and therefore this alternative description provides a certificate of positivity on $S$. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators

La foi catholique et les dispositifs numériques : le cas de la diaspora catholique vietnamienne

Cette étude entend offrir un éclairage sur le rapport entre la foi catholique et les dispositifs numériques dans la situation particulière des catholiques vietnamiens diasporiques. Sur le plan épistémologique, l’approche communicationnelle mettra en évidence l’articulation entre ce qu’offrent les dispositifs numériques et ce qu’en font des usagers. À travers une recherche documentaire sur la constitution de la population catholique vietnamienne d’outre-mer, l’analyse sémiotique de deux sites web et une enquête sociologique par questionnaires et entretiens, ce travail mettra en lumière l’appropriation des médias numériques par cette population et sa façon propre d’imaginer de nouvelles manières de pratiquer sa foi en situation de dispersion. L’enquête montrera une catholicité vietnamienne d’outre-mer en dynamique imaginative qui soulève le défi de repenser la compréhension de l’Église catholique en général sous l’angle du rapport entre le « en ligne » et le « hors ligne », entre l’ancrage solide des traditions et les connections « liquides » contemporaines.Catholic Faith and Digital Media. The case of the Vietnamese Diaspora – This study aims to provide initial insights into the relationship between the Catholic faith and digital media through the particular case of the Vietnamese Catholic diaspora. Drawing on an "epistemology of complexity", this investigation attempts to articulate new possibilities offered by sociotechnical objects with actual usages and practices by Internet users. Through a semiotic analysis of two websites and a sociological survey by questionnaire and interviews, the study will show the way this diasporic population uses digital devices to imagine new practices of their Catholic faith in a situation of geographic dispersion. It highlights a particular Catholicism of overseas Vietnamese characterized by a transnational dimension and an integration of online and offline faith practices. This novel reality poses challenges as well as opening up new perspectives for the Roman Catholic Church in the digital age, now confronted with tensions between long-standing “solid” traditions and contemporary “liquid” connections.La Fe Católica y los dispositivos digitales. El caso de la diáspora católica vietnamita – Este estudio quiere mostrar la relación entre la fe católica y los dispositivos digitales en una situación particular, como es la de los católicos vietnamitas de la diáspora. Sobre el plano epistemológico, el estudio comunicacional que cosntruye una “epistemología de la complejidad” pone en evidencia la articulación entre lo que ofrecen los dispositivos digitales y lo que hacen los usuarios. A través del análisis semiótico de dos sitios web y de una encuesta sociológica mediante un cuestionario y de entrevistas-, este trabajo muestra la apropiación de los medios de comunicación digitales que realiza esta población; a la vez muestra también su forma propia de imaginar nuevas maneras de practicar su fe en situación de dispersión. De la encuesta se deduce una catolicidad vietnamita de ultramar en clave imaginativa que suscita el desafío de repensar la comprensión de la Iglesia católica en general desde el punto de vista de la relación entre el online y el offline, entre la solidez de las tradiciones y las conexiones contemporáneas “líquidas”

Approaches towards fighting the COVID19 pandemic in Vietnam

My placement during the Master of Applied Epidemiology (MAE) program was in the Department of Epidemiology of the National Institute of Hygiene and Epidemiology, Hanoi, Vietnam from February 2020 to October 2021. During this time, I conducted four main projects to meet the MAE requirements. In addition, I contributed to seven peer-reviewed publications, another three advanced manuscripts, which have been submitted for publication, and five presentations at international conferences. This thesis contains seven chapters dedicated to my work during the COVID-19 epidemic in Vietnam during my MAE fellowship. Chapter 1 presents an overview of the thesis, providing an overview of all projects and how they relate to MAE requirements. Chapter 2 discusses the MAE requirement for outbreak investigation. I provide the investigation of the first COVID-19 outbreak in a healthcare setting in Vietnam. I also discuss the application and usefulness of Social Network Analysis in curbing the COVID-19 transmission in the hospital setting. Chapter 3 discusses the analysis of the public health dataset using the contact tracing data of more than 2,000 passengers on flights with at least one COVID-19 confirmed case during the beginning of the COVID-19 pandemic in Vietnam. Chapter 4 discuss the evaluation of the COVID-19 surveillance system at Points of Entry. This project used data from a country-led workshop that was designed based on WHO's guidance for Intra-Action Review during the COVID-19 pandemic. Chapter 5 includes an epidemiological study about the mental health problem among community health workers in Vietnam during COVID-19 pandemic. Chapter 6 includes a systematic review to assess the robustness of COVID-19 efficacy vaccines trial results. Chapter 7 includes materials for a lay audience report, lessons from the field, teaching from the field, and a side project

A Comparison between Male and Female Lab Rats in Operant Conditioning

Undergraduate Basi

Vulnerability of Rice Production in Mekong River Delta under Impacts from Floods, Salinity and Climate Change

This study sought to estimate the floods and salinity impact index and climate change vulnerability index for the rice farming provinces in Mekong River Delta. In both indexes, Tra Vinh province and the communes within it have the highest index value, followed by other coastal provinces. The estimation showed that the rice production in these areas are being threatened and will be worsen if there is no appropriate plan to cope with the changes in climate condition and extreme events. The results for the simulation model of paddy yield under different scenarios showed decreases in the paddy yield in Mekong River Delta. Specifically, the yield of Spring paddy decreases 6%, Autumn paddy decreases 2%, Winter paddy decreases 4% and Autumn-winter paddy decreases 4% in 2050. From these results, the climate change adaptation and mitigation policies in this delta is suggested to be focused reducing the exposure to sea level rise; upgrading the irrigation system for paddy planting since the coastal provinces have high rate of rain-fed paddy, vulnerability can also be reduced by enhancing the adaptive capacity of provinces through subsidizing and providing farmers with new paddy varieties which are more tolerant to salinity

COSMETIC CONSUMPTION TRENDS OF VIETNAMESE YOUNG PEOPLE – A CASE STUDY OF HIGH SCHOOL AND UNIVERSITY STUDENTS IN HANOI

Our beauty demands are significantly increasing with the rapid development of the Global Economy. The use of cosmetic products has become prevalent among Vietnamese youth. In this article, the research team concentrates on collecting and analyzing data on cosmetics consumer trends of Vietnamese teenagers - research of high school and university students in Hanoi. The research yields some important results: Young people are becoming more interested in natural and time-saving trends toward beauty. The top concerns in using cosmetic products are skincare, haircare, and body odor. Make-up is also catching teenagers’ attention, but the number of young people frequently doing makeup is inconsiderable, mainly due to the lack of knowledge, interest, and time for this field. In addition, the use of cosmetic products is mostly influenced by personal preferences, opinions of friends/relatives, and cultural tastes. The purchase of cosmetic products is driven by prices, brand names, and ingredients. Adolescents also tend to use some healthy products, such as cosmeceutical or vegan ones, to overcome the problems of unclear origins or irritating ingredients. Besides, high school and university students have low financial capacities; therefore, their money spent on cosmetic products is not too high, and the number of cosmetic products owned is not too many. Finally, the improvement of technology has led to two trends among teenagers: the convenience of cosmetic-related information searching and the widespread presence of shopping through e-commerce platforms