Analysis of Convex Functions

Convexity is an old subject in mathematics. The �rst speci�c de�nition of convexity was given by Herman Minkowski in 1896. Convex functions were introduced by Jensen in 1905. The concept appeared intermittently through the centuries, but the subject was not really formalized until the seminal 1934 tract Theorie der konverxen Korper of Bonneson and Fenchel.Today convex geometry is a mathematical subject in its own right. Classically oriented treatments, like the work done by Frederick Valentine form the elementary de�nition, which is that a domain K in the plane or in RN is convex if for all P;Q 2 K, then the segment PQ connecting P to Q also lies in K. In fact this very simple idea gives forth a very rich theory. But it is not a theory that interacts naturally with mathematical analysis. For analysis, one would like a way to think about convexity that is expressed in the language of functions and perhaps its derivatives. Our goal in this thesis is to present and to study convexity in a more analytic way. Through Chapter 1, Chapter 2 and Chapter 3, I have tried to point out the important role of convex sets and its associated convex functions in Mathematical Analysis. Chapter 1 is devoted to Convex sets and some geometric properties achieved by these objects in �nite Euclidean spaces. The emphasis is given on establishing a criteria for convexity. Various useful examples are given, and it is shown how further examples can be generated from these by means of operations such as addition or taking convex hulls. The fundamental idea to be understood is that the convex functions on Rn can be identi�ed with certain convex subsets of Rn+1 (their epigraphs), while the convex sets in Rn can be identi�ed with certain convex functions on Rn (their indicators). These identi�cations make it easy to pass back and forth between a geometric approach and an analytic approach. Chapter 2 begins with idea of convexity of functions in a �nite dimensional space. Convex functions are an important device for the study of extremal problems. They are also important analytic tools. The fact that a convex function can have at most one minimum and no maxima is a notable piece of information that proves to be quite useful. A convex function is also characterized by the non negativity of its second derivative. This useful information interacts nicely with the ideas of calculus. We relate convex functions to an elegant characterisation of Gamma functions by Bohr Mollerup Theorem. Chapter 3 provides an introduction to convex analysis, the properties of sets and functions in in�nite dimensional space. We start by taking the convexity of the epigraph to be the definition of a convex function, and allow convex functions to be extended -real valued. One of the main themes of this chapter is the maximization of linear functions over non empty convex sets. Here we relate the subdi�erential to the directional derivative of a function. There are several modern works on convexity that arise from studies of functional analysis. One of the nice features of the analytic way of looking at convexity is the Bishop-Phelps Theorem, it says that in a Banach Space, a convex function has a subgradient on a dense subset of its e�ective

Role of Hoxa2 and its Putative Downstream Target Htra1 in Osteogenic Differentiation of the Palatal Mesenchyme

Cleft palate is one of the most common structural birth defects in humans. Hoxa2 is the most anteriorly expressed gene of the homeobox family that define the anterior-posterior axis during embryonic development. Hoxa2 is expressed in the palatal shelves and plays an intrinsic role regulating cell proliferation. Hoxa2-/- mice develop cleft palate, albeit the cellular and molecular mechanisms remain to be explored. In this thesis, I have tested the hypothesis that Hoxa2 inhibits osteogenic differentiation of the palatal mesenchyme. I present here evidence that loss of Hoxa2 results in increased canonical BMP signaling dependent osteogenic program spatially and temporally in the developing Hoxa2-/- palatal shelves in vivo and in Hoxa2-/- MEPM cells in vitro. In the second part of this thesis, I investigated the role of a serine protease Htra1, a putative downstream target of Hoxa2 in osteogenic differentiation of the palatal mesenchyme. The results indicate that Htra1 is a positive regulator of osteogenic differentiation and is upregulated in the Hoxa2-/- palatal shelves. In addition, I identified that Runx2, a master regulator of osteogenic differentiation, binds to the proximal promoter region of Htra1 to induce its expression. Collectively, the data reveals that aberrant osteogenic signaling in the Hoxa2-/- palatal shelves due to increased osteoprogenitor commitment and proliferation may lead to cleft palate in these mice. In addition, I show for the first time that Htra1 is a novel direct downstream target of Runx2 during osteogenic differentiation. In summary, this thesis contributes to a better understanding of the cellular and molecular mechanisms governing osteogenic differentiation in the palatal mesenchyme during normal palate development and in cleft palate pathogenesis

Scope of Social Work Profession in Medical Setting

Social Work in today’s world has established itself as a significant full-fledged professional at par with any other profession.  As the world is become a materialistic; the devoid of the human sentiments and with emotions, diseas and illness with people are largly growing more and more self-centred.  It is understood a sick and happy child cannot learn, and cannot produce.  Good health is very important both to the nation and an individual.  Health is  state of complete physical, mental and social well being and not merely an absence of disease or infirmity.  It is the yardstick of measuring an individual progress and devlopment.  Among other factors, social factors create a predisposition to disease, directly cause disease, transmit the cause of disease and influence the course of disease.  Therefore the medical social workers helps the patients from the moment they enters he hospital, up to the adjustment with the past normal life.  The main aim of medical social worker is to helps us prevention of disease and rehabilitation treatment plan.  The scope of social workers is widely bigger in nature but simentanously lots of difficulties and challenges they have been facing  in the field of health.  The cirriculum of the social work was carefully developed to attain the needs of the society. This Paper examines the methodology of appointment of Social Worker, challenges, roles in the hospitals (scope of Practice in hospital social work), department where the social workers enhance their services

A Future Internet Architecture Based on De-Conflated Identities

We present a new Internet architecture based on de-conflated identities (ADI) that explicitly establishes the separation of ownership of hosts from the underlying infrastructure connectivity. A direct impact of this de-conflated Internet architecture is the ability to express organizational policies separately and thus more naturally, from the underlying infrastructure routing policies. Host or organizational accountability is separated from the infrastructure accountability, laying the foundations of a cleaner security and policy enforcement framework. Also, it addresses the present Internet routing problems of mobility, multihoming, and traffic engineering more naturally by making a clear distinction of host and infrastructure responsibilities and thus defining these functions as a set of primitives governed by individual policies. In this paper, we instantiate the primitive mechanisms related to the issues of end-to-end policy enforcements, mobility, multihoming, traffic engineering, etc., within the context of our architecture to emphasize the relevance of a de-conflated Internet architecture on these functions

Integrated Correlators in N=4\mathcal{N}=4 SYM via SL(2,Z)SL(2,\mathbb{Z}) Spectral Theory

We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$. These observables, defined by a certain spacetime integral of $\langle\mathcal{O}_2\mathcal{O}_2\mathcal{O}_p\mathcal{O}_p\rangle$ where $\mathcal{O}_p$ is a superconformal primary of charge $p$, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling $\tau$. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of $N$ and $\tau$. Their physical and formal properties are greatly illuminated upon employing the $SL(2,\mathbb{Z})$ spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different $N$ and $p$, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the $SU(2)$ theory. Our computations include the full matrix of integrated correlators at low values of $p$, and a certain infinite class involving operators of arbitrary $p$. The latter satisfy an open lattice chain equation for all $N$, reminiscent of the Toda equation obeyed by extremal correlators in $\mathcal{N}=2$ superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large $N$, confirming and predicting features of semiclassical AdS$_5\, \times$ S$^5$ supergravity amplitudes.Comment: 42+24 page

Architectures for the Future Networks and the Next Generation Internet: A Survey

Networking research funding agencies in the USA, Europe, Japan, and other countries are encouraging research on revolutionary networking architectures that may or may not be bound by the restrictions of the current TCP/IP based Internet. We present a comprehensive survey of such research projects and activities. The topics covered include various testbeds for experimentations for new architectures, new security mechanisms, content delivery mechanisms, management and control frameworks, service architectures, and routing mechanisms. Delay/Disruption tolerant networks, which allow communications even when complete end-to-end path is not available, are also discussed