Mahler measure and its generalizations

La mesure de Mahler (logarithmique) de P, une fonction rationnelle non nulle à n variables, est définie comme la moyenne arithmétique de log |P| restreinte au tore n-dimensionnel standard (T^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = 1, for all 1 ≤ i ≤ n}) par rapport à la mesure de Haar unique (mesure d'arc normalisée) sur T^n. Elle a des liens avec les hauteurs, les volumes hyperboliques, la dynamique arithmétique et les valeurs spéciales des fonctions L. Il existe plusieurs généralisations de cette définition dans la littérature. Cette thèse se consacre à l'exploration de deux de ces généralisations : premièrement, lorsque le tore unité est remplacé par un tore à rayons arbitraires (T_{a_1, ..., a_n})^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = a_i, for all 1 ≤ i ≤ n} (appelée \textit{mesure de Mahler généralisée}), et deuxièmement, lorsque la mesure d'arc normalisée sur le tore unité est remplacée par la mesure d'aire normalisée sur le disque unité (appelée \textit{mesure de Mahler aréale}). Notre objectif principal est de quantifier le comportement de la mesure de Mahler de $P$ sous de telles modifications. Cette thèse est structurée en cinq projets. 1. Dans le chapitre 1, nous étudions la définition de la mesure de Mahler généralisée pour tous les polynômes de Laurent à n variables lorsqu'ils ne s'annulent pas sur le tore d'intégration. Ce travail a été publié dans [106]. 2. Le chapitre 2 présente des évaluations non triviales de la mesure de Mahler aréale des polynômes à plusieurs variables, définie par Pritsker. Ce travail est réalisé en collaboration avec Lalin, et publié dans [84]. 3. Dans le chapitre 3, nous étudions comment la mesure de Mahler aréale change lorsque l'on effectue un changement de variables par puissance sur les polynômes. Ceci est un travail conjoint avec Lalin, et publié dans [83]. 4. Dans le chapitre 4, nous étudions la mesure de Mahler d'une famille particulière de fonctions rationnelles à un nombre arbitraire de variables et à un degré arbitraire dans l'une des variables. Ce travail est réalisé en collaboration avec Lalín et Nair, et sera publié dans [81]. 5. Le chapitre 5 est consacré à l'évaluation de la mesure de Mahler aréale d'une famille de polynômes en utilisant l'analogue aréal de la mesure de Mahler zêta. Il s'agit d'un travail collaboratif en cours avec Lalin, Nair et Ringeling.The (logarithmic) Mahler measure of a non-zero rational function P in n variables is defined as the arithmetic mean of log |P| restricted to the standard n-torus (T^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = 1, for all 1 ≤ i ≤ n}) with respect to the unique Haar measure (normalized arc measure) on T^n. It has connections to heights, hyperbolic volumes, arithmetic dynamics, and special values of L-functions. Various generalizations of this definition exist in the literature. This thesis is dedicated to exploring two such generalizations: firstly, when the unit torus is substituted by a torus with arbitrary radii (T_{a_1, ..., a_n})^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = a_i, for all 1 ≤ i ≤ n} (referred to as the \textit{generalized Mahler measure}), and secondly, when the normalized arc measure on the unit torus is replaced by the normalized area measure on the unit disk (referred to as the \textit{areal Mahler measure}). Our primary objective is to quantify the behavior of the Mahler measure of P under such alterations. This thesis is structured into five projects. 1. In Chapter 1, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in n-variables when they do not vanish on the integration torus. This work has been published in [106]. 2. In Chapter 2, we exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker. This is a joint work with Lalin, and has been published in [84]. 3. In Chapter 3, we investigate how the areal Mahler measure changes with the power change of variables. This a joint work with Lalin, and has been published in [83]. 4. In Chapter 4, we investigate the Mahler measure of a particular family of rational functions with an arbitrary number of variables and an arbitrary degree in one of the variables. This is a joint work with Lalin and Nair, and will appear in [81]. 5. In Chapter 5, we evaluate the areal Mahler measure of a family of polynomials using the areal analogue of the Zeta Mahler measure. This is an ongoing joint work with Lalin, Nair, and Ringeling

Skyscrapers: Origin, History, Evolution and Future

From humble cottages to super tall structures, human beings have progressively developed the living and working space with time. This paper describes the origin and growth of modern skyscrapers, the subsequent challenges faced, and the way it was outdone. Research papers and case studies have been thoroughly studied and important excerpts from them have been explained to show how the modern structures have been evolved. The sources and causes of evolution is debatable among researchers, this paper has taken into account 7 most vital milestones in the growth of current generation skyscrapers and their contribution to the construction industry and concludes with the ideas and scopes where growth is still possible and challenges need to be solved

Trasformazione dell’area in prossimità di Berhampore: da zona concentrica a zone a nuclei multipli

Urbanization is the process of population growth in urban areas. Urban out growth is a contiguous part of urban area such as statutory towns or census towns. Those towns are express as a local service center. Berhampore Municipality is an oldest municipality in Murshidabad district. Within a decade numbers of census towns are increased from 3 to 9 around Berhampore municipality. Through this paper we are trying to show the delimitation of urban growth with three indicators and identify the transformation of urban city center from concentric zone to multiple nuclei zone. After analysis of whole things with the help of quantitative statistic it is identified that – star like linear expansion of town along with N.H.34 and S.H.11 road due to their functionalityL’urbanizzazione è il processo di crescita della popolazione nelle aree urbane. La crescita urbana avviene nelle zone contigue alle città di fondazione o alle città di recente impianto caratterizzate da un centro di servizi locale. Il Comune di Berhampore è il comune più antico del distretto di Murshidabad. In un decennio il numero di città censite al suo intorno è aumentato da 3 a 9. Attraverso questo articolo si cerca di mostrare la delimitazione della crescita urbana con tre indicatori ed identificare la trasformazione del centro urbano da zona concentrica a zona a più nuclei. Dopo un’analisi statistica di tipo quantitativa, si è identificato che l’espansione lineare della città va di pari passo allo sviluppo delle strade N.H.34 e S.H.11 particolamente funzionali

An Experimental Study on Ground Improvement by Application of Fly Ash and Lime on Clayey and Sandy Soil

Construction on locally available clayey soil is often problematic due to its swelling and shrinkage nature. Pavements are most affected as the upthrust due to regional swelling of clayey soil during monsoon season and shrinking during dry season causes unwanted cracks in the pavement. As a consequence, the pavement gets damaged. In places having extensive deposit of clayey soil, soil replacement becomes time-consuming and uneconomical. Hence the clayey soil to be considered as subgrade needs to be pre-treated. Fly ash, an industrial waste can be used for such treatment. To improve the engineering properties of on-site available clayey soil and sandy soil with lime and fly ash was studied. Based on the results obtained from experiments the suitability of fly ash and lime to be considered as additives to improve local clayey and sandy soil properties has been analyzed. It was observed that the on the addition of fly ash within 40-60% range can be satisfactorily used to replace the local clayey soil and fly ash percentage within 20-40% can be used to replace the local sand. Lime content in the range of 4-8% can be satisfactorily used in both in situ available soil and local sand with fly ash mixtures for the improvement of strength in terms of shear strength as well as CBR value

Targeting Nox4 and Interleukin-6 crosstalk can be a potential strategy for Gastric Cancer prevention

Over 950000 million people are being affected in Gastric Cancer (GC) in each year. And so many patients are dying for its lethality. That means prognosis and treatment strategies are not well enough for its treatment. So, in the present scenario research work on this disease is truly significant and necessary. Nicotinamide adenine dinucleotide phosphate oxidase (NADPH oxidase or NOX) is a family of enzyme consists of seven NOX isoforms members, such as- NOX1, NOX2, NOX3, NOX4, NOX5 and dual oxidase (DUOX) 1 and 2, that possess similarities in terms of enzyme function and structure. All these seven isoforms are transmembrane proteins (having six transmembrane domains) with a NADPH binding site, a FAD-binding site and four heme-binding histidines. Among these enzymes (NOX family), NOX4 is highly expressed in Gastric Cancer (GC) cells. As well as it was observed that high expression of NOX4 is correlated with worse overall survival (OS) in all GC patients. Interestingly, high mRNA expression of NOX4 indicates a worse OS in stage III/IV GC patients, but not in stage I/II GC patients. That is suggesting, NOX4 may contribute to the GC progression and play a crucial role in its (GC) late-stage. And it is observed that NOX4 is related with the cell invasion by regulating the JAK2/STAT3 signaling pathway. Functionally this enzyme family is leading with production of Reactive Oxygen Species (ROS) by utilizing NADPH. Few common examples of ROS are hydrogen peroxide (H2O2), superoxide anion (O2-) and hydroxyl radicles (OH) etc. These NOX producing ROS are connected to carcinogenesis as it is involved with so many cellular processes like cell proliferation, DNA damage and angiogenesis etc. On the other hand, recent studies show that Interleukin-6 (IL-6) is significantly related with the cell invasion and JAK2/STAT3 activation. That means both compounds (NOX4 and IL-6) are related with JAK2/STAT3 signaling pathway. And this JAK2/STAT3 signaling actually activates those genes which are associated with cell proliferation and anti-apoptosis. So, it is suggested that, targeting both Nox4 and IL-6 may be a fruitful strategy in GC treatmen

Evolving perspectives in reverse cardio-oncology: A review of current status, pathophysiological insights, and future directives

Cardiovascular disease (CVD) and cancer are leading causes of mortality worldwide, traditionally linked through adverse effects of cancer therapies on cardiovascular health. However, reverse cardio-oncology, a burgeoning field, shifts this perspective to examine how cardiovascular diseases influence the onset and progression of cancer. This novel approach has revealed a higher likelihood of cancer development in patients with pre-existing cardiovascular conditions, attributed to shared risk factors such as obesity, a sedentary lifestyle, and smoking. Underlying mechanisms like chronic inflammation and clonal hematopoiesis further illuminate the connections between cardiovascular ailments and cancer. This comprehensive narrative review, spanning a broad spectrum of studies, outlines the syndromic classification of cardio-oncology, the intersection of cardiovascular risk factors and oncogenesis, and the bidirectional dynamics between CVD and cancer. Additionally, the review also discusses the pathophysiological mechanisms underpinning this interconnection, examining the roles of cardiokines, genetic factors, and the effects of cardiovascular therapies and biomarkers in cancer diagnostics. Lastly, it aims to underline future directives, emphasising the need for integrated healthcare strategies, interdisciplinary research, and comprehensive treatment protocols

Generalized Mahler measure of a family of polynomials

Le présent mémoire traite une variation de la mesure de Mahler où l'intégrale de définition est réalisée sur un tore plus général. Notre travail est basé sur une famille de polynômes tempérée originellement étudiée par Boyd, P_k (x, y) = x + 1/x + y + 1/y + k avec k ∈ R_{>4}. Pour le k = 4 cas, nous utilisons des valeurs spéciales du dilogarithme de Bloch-Wigner pour obtenir la mesure de Mahler de P_4 sur un tore arbitraire (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } avec a, b ∈ R_{> 0}. Ensuite, nous établissons une relation entre la mesure de Mahler de P_8 sur un tore (T_ {a, √a} )^2 et sa mesure de Mahler standard. La combinaison de cette relation avec des résultats de Lalin, Rogers et Zudilin conduit à une formule impliquant les mesures de Mahler généralisées de ce polynôme données en termes de L' (E, 0). Au final, nous proposons une stratégie pour prouver des résultats similaires dans le cas général k> 4 sur (T_ {a, b})^2 avec certaines restrictions sur a, b.In this thesis we consider a variation of the Mahler measure where the defining integral is performed over a more general torus. Our work is based on a tempered family of polynomials originally studied by Boyd, Boyd P_k (x, y) = x + 1/x + y + 1/y + k with k ∈ R_{>4}. For the k = 4 case we use special values of the Bloch-Wigner dilogarithm to obtain the Mahler measure of P_4 over an arbitrary torus (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } with a, b ∈ R_{> 0}. Next we establish a relation between the Mahler measure of P_8 over a torus(T_ {a, √a} )^2 and its standard Mahler measure. The combination of this relation with results due to Lalin, Rogers, and Zudilin leads to a formula involving the generalized Mahler measure of this polynomial given in terms of L'(E, 0). In the end, we propose a strategy to prove some similar results for the general case k > 4 over (T_ {a, b})^2 with some restrictions on a, b