Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

scholarly journals New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

2021 ◽  
Author(s):  
A. Hantoute ◽  
M. A. López-Cerdá
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Normal Cone ◽  
Countable Family ◽  
Weighted Sums ◽  
Reflexive Banach Spaces ◽  
Finite Dimensional ◽  
Finite Sums ◽  
Effective Domain ◽  
The Relationship

AbstractThis paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family.

scholarly journals Elements of Cartesian Analysis in Saidou Spaces

2021 ◽  
Vol 10 (1) ◽  
pp. 1-10
Author(s):  
Nourrdin SAIDOU
Keyword(s):  
Mathematical Analysis ◽  
Convex Functions ◽  
Analytical Properties ◽  
Functional Part ◽  
Cartesian Geometry ◽  
The One ◽  
Definition Of ◽  
Cartesian Topology ◽  
The Relationship

The objective of this research paper is to follow up on the work already started in order to install a new mathematical analysis, the one we called Cartesian analysis see our previous works[1],[2]. During these last two papers, we started with the definition of cartesian geometry and we defined and introduced to a new Cartesian topology which gave birth to new spaces called Saidou spaces. Thus, and to follow up on this way, we propose to studie the analytical and functional part of this analysis. We will define the notion of a Cartesian function and then its epigraph before to characterize the analytical properties of these functions, for example the continuity and the differentiablity. In an other hand, we will see the relationship between cartesian functions and the convex functions. According to the latest papers and this work we do the asset of the first foundations of the cartesian analysis.

Misreading Egypt

2018 ◽  
Author(s):  
Jesse Schotter
Keyword(s):  
New Media ◽  
Narrative Form ◽  
Experimental Film ◽  
Tower Of Babel ◽  
Avant Garde ◽  
Sergei Eisenstein ◽  
History Of ◽  
The One ◽  
The Relationship ◽  
Modern Technologies

The first chapter of Hieroglyphic Modernisms exposes the complex history of Western misconceptions of Egyptian writing from antiquity to the present. Hieroglyphs bridge the gap between modern technologies and the ancient past, looking forward to the rise of new media and backward to the dispersal of languages in the mythical moment of the Tower of Babel. The contradictory ways in which hieroglyphs were interpreted in the West come to shape the differing ways that modernist writers and filmmakers understood the relationship between writing, film, and other new media. On the one hand, poets like Ezra Pound and film theorists like Vachel Lindsay and Sergei Eisenstein use the visual languages of China and of Egypt as a more primal or direct alternative to written words. But Freud, Proust, and the later Eisenstein conversely emphasize the phonetic qualities of Egyptian writing, its similarity to alphabetical scripts. The chapter concludes by arguing that even avant-garde invocations of hieroglyphics depend on narrative form through an examination of Hollis Frampton’s experimental film Zorns Lemma.

Researching (with) Major Food Retailers

2015 ◽  
Vol 15 (3) ◽  
pp. 33-39 ◽  
Author(s):  
David Evans
Keyword(s):  
Food Systems ◽  
Academic Research ◽  
Social Scientists ◽  
Major Food ◽  
Food Retailers ◽  
Food Retailer ◽  
The One ◽  
The Uk ◽  
The Relationship ◽  
Opening Up

This paper considers the relationship between social science and the food industry, and it suggests that collaboration can be intellectually productive and morally rewarding. It explores the middle ground that exists between paid consultancy models of collaboration on the one hand and a principled stance of nonengagement on the other. Drawing on recent experiences of researching with a major food retailer in the UK, I discuss the ways in which collaborating with retailers can open up opportunities for accessing data that might not otherwise be available to social scientists. Additionally, I put forward the argument that researchers with an interest in the sustainability—ecological or otherwise—of food systems, especially those of a critical persuasion, ought to be empirically engaging with food businesses. I suggest that this is important in terms of generating better understandings of the objectionable arrangements that they seek to critique, and in terms of opening up conduits through which to affect positive changes. Cutting across these points is the claim that while resistance to commercial engagement might be misguided, it is nevertheless important to acknowledge the power-geometries of collaboration and to find ways of leveling and/or leveraging them. To conclude, I suggest that universities have an important institutional role to play in defining the terms of engagement as well as maintaining the boundaries between scholarship and consultancy—a line that can otherwise become quite fuzzy when the worlds of commerce and academic research collide.

Choice Of Technique And The Volume Of Saving (A Re-Examination Of The Cobb-Douglas Model)

1968 ◽  
Vol 8 (4) ◽  
pp. 606-617
Author(s):  
Mohammad Anisur Rahman
Keyword(s):  
Economic Development ◽  
Production Function ◽  
Development Policy ◽  
Social Goals ◽  
The Other ◽  
Economic Development Policy ◽  
Labour Intensity ◽  
The One ◽  
Choice Of Technique ◽  
The Relationship

The purpose of this paper is to re-examine the relationship between the degree of aggregate labour-intensity and the aggregate volume of saving in an economy where a Cobb-6ouglas production function in its traditional form can be assumed to give a good approximation to reality. The relationship in ques¬tion has an obviously important bearing on economic development policy in the area of choice of labour intensity. To the extent that and in the range where an increase in labour intensity would adversely affect the volume of savings, a con¬flict arises between two important social objectives, i.e., higher rate of capital formation on the one hand and greater employment and distributive equity on the other. If relative resource endowments in the economy are such that such a "competitive" range of labour-intensity falls within the nation's attainable range of choice, development planners will have to arrive at a compromise between these two social goals.

Bastard Feudalism and the Framing of Thirteenth-Century England

2018 ◽  
Author(s):  
Peter Coss
Keyword(s):  
Middle Ages ◽  
Thirteenth Century ◽  
Twelfth Century ◽  
The State ◽  
The Other ◽  
Great Work ◽  
Early Middle Ages ◽  
The One ◽  
The Relationship

In the introduction to his great work of 2005, Framing the Early Middle Ages, Chris Wickham urged not only the necessity of carefully framing our studies at the outset but also the importance of closely defining the words and concepts that we employ, the avoidance ‘cultural sollipsism’ wherever possible and the need to pay particular attention to continuities and discontinuities. Chris has, of course, followed these precepts on a vast scale. My aim in this chapter is a modest one. I aim to review the framing of thirteenth-century England in terms of two only of Chris’s themes: the aristocracy and the state—and even then primarily in terms of the relationship between the two. By the thirteenth century I mean a long thirteenth century stretching from the period of the Angevin reforms of the later twelfth century on the one hand to the early to mid-fourteenth on the other; the reasons for taking this span will, I hope, become clearer during the course of the chapter, but few would doubt that it has a validity.

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

scholarly journals A New Paradigm in the Relationship between Melatonin and Breast Cancer: Gut Microbiota Identified as a Potential Regulatory Agent

Cancers ◽  
2021 ◽  
Vol 13 (13) ◽  
pp. 3141
Author(s):  
Aurora Laborda-Illanes ◽  
Lidia Sánchez-Alcoholado ◽  
Soukaina Boutriq ◽  
Isaac Plaza-Andrades ◽  
Jesús Peralta-Linero ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Gut Microbiota ◽  
Bacterial Population ◽  
Kynurenine Pathway ◽  
New Paradigm ◽  
Bacterial Composition ◽  
Glucuronidase Activity ◽  
The One ◽  
The Relationship ◽  
Melatonin Production

In this review we summarize a possible connection between gut microbiota, melatonin production, and breast cancer. An imbalance in gut bacterial population composition (dysbiosis), or changes in the production of melatonin (circadian disruption) alters estrogen levels. On the one hand, this may be due to the bacterial composition of estrobolome, since bacteria with β-glucuronidase activity favour estrogens in a deconjugated state, which may ultimately lead to pathologies, including breast cancer. On the other hand, it has been shown that these changes in intestinal microbiota stimulate the kynurenine pathway, moving tryptophan away from the melatonergic pathway, thereby reducing circulating melatonin levels. Due to the fact that melatonin has antiestrogenic properties, it affects active and inactive estrogen levels. These changes increase the risk of developing breast cancer. Additionally, melatonin stimulates the differentiation of preadipocytes into adipocytes, which have low estrogen levels due to the fact that adipocytes do not express aromatase. Consequently, melatonin also reduces the risk of breast cancer. However, more studies are needed to determine the relationship between microbiota, melatonin, and breast cancer, in addition to clinical trials to confirm the sensitizing effects of melatonin to chemotherapy and radiotherapy, and its ability to ameliorate or prevent the side effects of these therapies.

Pensare Dio. Spunti di riflessione in dialogo con Anca Vasiliu

Elenchos ◽  
2020 ◽  
Vol 41 (1) ◽  
pp. 181-194
Author(s):  
Angela Longo
Keyword(s):  
Late Antiquity ◽  
Human Being ◽  
Common Thread ◽  
Critical Issues ◽  
Relevant Topic ◽  
The Common ◽  
The One ◽  
Relationship Of ◽  
The Relationship ◽  
The Trinity

AbstractThe following work features elements to ponder and an in-depth explanation taken on the Anca Vasiliu’s study about the possibilities and ways of thinking of God by a rational entity, such as the human being. This is an ever relevant topic that, however, takes place in relation to Platonic authors and texts, especially in Late Antiquity. The common thread is that the human being is a God’s creature who resembles him and who is image of. Nevertheless, this also applies within the Christian Trinity according to which, not without problems, the Son is the image of the Father. Lastly, also the relationship of the Spirit with the Father and the Son, always within the Trinity, can be considered as a relationship of similarity, but again not without critical issues between the similarity of attributes, on the one hand, and the identity of nature, on the other.

scholarly journals A note on generalized convex functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Syed Zaheer Ullah ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Generalized Convex Functions

Abstract In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$(η1,η2)-convex function and establish its Hermite–Hadamard type inequality.

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