Differentials and Convex Analysis

2020 ◽  
pp. 7-64
Author(s):  
Robert G. Chambers
Keyword(s):  
Convex Analysis ◽  
Duality Theorem ◽  
Convex Sets ◽  
Directional Derivatives ◽  
Conjugate Duality ◽  
Support Functions ◽  
Essential Properties ◽  
Global Optima ◽  
Maximum Theorem ◽  
The Relationship

Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.

The Trade Mark, Other Product Designations, and Goodwill

2017 ◽  
Author(s):  
Justine Pila
Keyword(s):  
Subject Matter ◽  
Common Law ◽  
Trade Mark ◽  
Geographical Indication ◽  
Passing Off ◽  
Rights Protection ◽  
Essential Properties ◽  
The Subject ◽  
The Common ◽  
The Relationship

This chapter considers the meaning of the terms that appropriately denote the subject matter protectable by registered trade mark and allied rights, including the common law action of passing off. Drawing on the earlier analyses of the objects protectable by patent and copyright, it defines the trade mark, designation of origin, and geographical indication in their current European and UK conception as hybrid inventions/works in the form of purpose-limited expressive objects. It also considers the relationship between the different requirements for trade mark and allied rights protection, and related principles of entitlement. In its conclusion, the legal understandings of trade mark and allied rights subject matter are presented as answers to the questions identified in Chapter 3 concerning the categories and essential properties of the subject matter in question, their method of individuation, and the relationship between and method of establishing their and their tokens’ existence.

The Subject Matter of Intellectual Property

2017 ◽  
Author(s):  
Justine Pila
Keyword(s):  
Intellectual Property ◽  
Subject Matter ◽  
Final Stage ◽  
Theoretical Framework ◽  
Trade Mark ◽  
Second Stage ◽  
Essential Properties ◽  
The Subject ◽  
Three Stages ◽  
The Relationship

This book offers a study of the subject matter protected by each of the main intellectual property (IP) regimes. With a focus on European and UK law particularly, it considers the meaning of the terms used to denote the objects to which IP rights attach, such as ‘invention’, ‘authorial work’, ‘trade mark’, and ‘design’, with reference to the practice of legal officials and the nature of those objects specifically. To that end it proceeds in three stages. At the first stage, in Chapter 2, the nature, aims, and values of IP rights and systems are considered. As historically and currently conceived, IP rights are limited (and generally transferable) exclusionary rights that attach to certain intellectual creations, broadly conceived, and that serve a range of instrumentalist and deontological ends. At the second stage, in Chapter 3, a theoretical framework for thinking about IP subject matter is proposed with the assistance of certain devices from philosophy. That framework supports a paradigmatic conception of the objects protected by IP rights as artifact types distinguished by their properties and categorized accordingly. From this framework, four questions are derived concerning: the nature of the (categories of) subject matter denoted by the terms ‘invention’, ‘authorial work’, ‘trade mark’, ‘design’ etc, including their essential properties; the means by which each subject matter is individuated within the relevant IP regime; the relationship between each subject matter and its concrete instances; and the manner in which the existence of a subject matter and its concrete instances is known. That leaves the book’s final stage, in Chapters 3 to 7. Here legal officials’ use of the terms above, and understanding of the objects that they denote, are studied, and the results presented as answers to the four questions identified previously.

On the Semigroup of Probability Measures of a Locally Compact Semigroup II

1987 ◽  
Vol 30 (3) ◽  
pp. 273-281 ◽  
Author(s):  
James C. S. Wong
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Compact Semigroups ◽  
Fixed Point Properties ◽  
The Relationship ◽  
Locally Compact Semigroups

AbstractThis is a sequel to the author's paper "On the semigroup of probability measures of a locally compact semigroup." We continue to investigate the relationship between amenability of spaces of functions and functionals associated with a locally compact semigroups S and its convolution semigroup MO(S) of probability measures and fixed point properties of actions of S and MO(S) on compact convex sets.

scholarly journals Conjugate duality in stochastic controls with delay

2017 ◽  
Vol 49 (4) ◽  
pp. 1011-1036
Author(s):  
Zimeng Wang ◽  
David J. Hodge ◽  
Huiling Le
Keyword(s):  
Optimal Control ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Maximum Principles ◽  
Conjugate Duality ◽  
Control Problems ◽  
Convex Problem ◽  
Stochastic Controls ◽  
The Relationship ◽  
Equations With Delay

AbstractIn this paper we use the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem with delay and derive the expression for the corresponding dual problem. This enables us to obtain the relationship between the optimalities for the two problems. Then, by linking stochastic optimal control problems with delay with a particular type of stochastic convex problem, the result for the latter leads to sufficient maximum principles for the former.

The Invention and Plant Variety

2017 ◽  
Author(s):  
Justine Pila
Keyword(s):  
Subject Matter ◽  
Individual Subject ◽  
Plant Variety ◽  
Essential Properties ◽  
The Subject ◽  
The Law ◽  
The Relationship

This chapter seeks to define the terms ‘invention’ and ‘plant variety’ as used by European and UK legal officials particularly. To that end, the nature of the objects identified by the legislature and courts as inventions and plant varieties is considered, as well as the nature of the objects excluded from protection, whether with reference to the requirement for an invention or plant variety or on other statutory grounds. The chapter also considers the law governing patent and plant variety right entitlement, and its implications for legal conceptions of the invention and plant variety. In its conclusion, the legal understandings of inventions and plant varieties are presented as answers to the questions identified in Chapter 3 concerning the categories and essential properties of the subject matter in question, their method of individuation, and the relationship between and method of establishing the existence of each individual subject matter and its tokens.

The Structure of C*-Convex Sets

1994 ◽  
Vol 46 (5) ◽  
pp. 1007-1026 ◽  
Author(s):  
Phillip B. Morenz
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Structural Elements ◽  
The Right ◽  
The Relationship ◽  
Carathéodory Theorem ◽  
Convex Subsets

AbstractCompact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

scholarly journals Structure-Activity Relationships of the Human Immunodeficiency Virus Type 1 Maturation Inhibitor PF-46396

2016 ◽  
Vol 90 (18) ◽  
pp. 8181-8197 ◽  
Author(s):  
Christopher Murgatroyd ◽  
Lisa Pirrie ◽  
Fanny Tran ◽  
Terry K. Smith ◽  
Nicholas J. Westwood ◽  
...  
Keyword(s):  
Antiviral Activity ◽  
Betulinic Acid ◽  
Proteolytic Processing ◽  
Structure Activity Relationships ◽  
Modes Of Action ◽  
Structure Activity ◽  
Essential Properties ◽  
Maturation Inhibitor ◽  
The Relationship ◽  
Hiv 1

ABSTRACTHIV-1 maturation inhibitors are a novel class of antiretroviral compounds that consist of two structurally distinct chemical classes: betulinic acid derivatives and the pyridone-based compound PF-46396. It is currently believed that both classes act by similar modes of action to generate aberrant noninfectious particles via inhibition of CA-SP1 cleavage during Gag proteolytic processing. In this study, we utilized a series of novel analogues with decreasing similarity to PF-46396 to determine the chemical groups within PF-46396 that contribute to antiviral activity, Gag binding, and the relationship between these essential properties. A spectrum of antiviral activity (active, intermediate, and inactive) was observed across the analogue series with respect to CA-SP1 cleavage and HIV-1 (NL4-3) replication kinetics in Jurkat T cells. We demonstrate that selected inactive analogues are incorporated into wild-type (WT) immature particles and that one inactive analogue is capable of interfering with PF-46396 inhibition of CA-SP1 cleavage. Mutations that confer PF-46396 resistance can impose a defective phenotype on HIV-1 that can be rescued in a compound-dependent manner. Some inactive analogues retained the capacity to rescue PF-46396-dependent mutants (SP1-A3V, SP1-A3T, and CA-P157S), implying that they can also interact with mutant Gag. The structure-activity relationships observed in this study demonstrate that (i) thetert-butyl group is essential for antiviral activity but is not an absolute requirement for Gag binding, (ii) the trifluoromethyl group is optimal but not essential for antiviral activity, and (iii) the 2-aminoindan group is important for antiviral activity and Gag binding but is not essential, as its replacement is tolerated.IMPORTANCECombinations of antiretroviral drugs successfully treat HIV/AIDS patients; however, drug resistance problems make the development of new mechanistic drug classes an ongoing priority. HIV-1 maturation inhibitors are novel as they target the Gag protein, specifically by inhibiting CA-SP1 proteolytic cleavage. The lack of high-resolution structural information of the CA-SP1 target in Gag has hindered our understanding of the inhibitor-binding pocket and maturation inhibitor mode of action. Therefore, we utilized analogues of the maturation inhibitor PF-46396 as chemical tools to determine the chemical components of PF-46396 that contribute to antiviral activity and Gag binding and the relationship between these essential properties. This is the first study to report structure-activity relationships of the maturation inhibitor PF-46396. PF-46396 is chemically distinct from betulinic acid-derived maturation inhibitors; therefore, our data provide a foundation of knowledge that will aid our understanding of how structurally distinct maturation inhibitors act by similar modes of action.

scholarly journals The relationship between income and food insecurity among Oregon residents: does social support matter?

2009 ◽  
Vol 12 (11) ◽  
pp. 2104-2112 ◽  
Author(s):  
Molly De Marco ◽  
Sheryl Thorburn
Keyword(s):  
Social Support ◽  
Social Network ◽  
Food Insecurity ◽  
Low Income ◽  
Negative Impact ◽  
Mail Survey ◽  
Support Functions ◽  
Hierarchical Logistic Regression ◽  
Confounding Variables ◽  
The Relationship

AbstractObjectiveMillions of US households experienced food insecurity in 2005. Research indicates that low wages and little social support contribute to food insecurity. The present study aimed to examine whether social support moderates the relationship between income and food insecurity.DesignUsing a mail survey, we collected data on social support sources (social network, intimate partner and community) and social support functions from a social network (instrumental, informational and emotional). We used hierarchical logistic regression to examine the potential moderation of various measures of social support on the relationship between income and food insecurity, adjusting for potential confounding variables.SettingOregon, USA.SubjectsA stratified random sample of Oregonians aged 18–64 years (n 343).ResultsWe found no evidence of an association between social support and food insecurity, nor any evidence that social support acts as a moderator between income and food insecurity, regardless of the measure of social support used.ConclusionsAlthough previous research suggested that social support could offset the negative impact of low income on food security, our study did not find support for such an effect.

scholarly journals Economic or Organisation Perspective?: Triangulation of Workplace Spirituality, Organisational Learning Capacity (OLC) and Organisational Trust

2018 ◽  
Vol III (II) ◽  
pp. 20-33
Author(s):  
Affaf Zahra ◽  
Muhammad Zia-ur Rehman ◽  
Asia Baig
Keyword(s):  
Organizational Learning ◽  
Structural Equation ◽  
Positive Impact ◽  
Organizational Trust ◽  
Workplace Spirituality ◽  
Maximum Probability ◽  
Support Functions ◽  
Learning Capacity ◽  
Organizational Learning Capacity ◽  
The Relationship

The study examines the relationship between workplace spirituality, organizational learning capacity, and organizational trust. Within the organization, trust plays a significant role in establishing learning capacity. The study is crosssectional in nature with a simple random sampling technique. Information is gathered from Training/learning and OD staff of government and commercial banks operating in Pakistan via selfadministered questionnaires. Data were analyzed through “Structural Equation Modelling (SEM)” and through maximum probability and likelihood. Findings indicate that workplace spirituality had a substantial positive effect on organizational learning capacity and organizational trust acts as moderator. Hence, organizational trust also has a significant positive impact on the relationship between workplace spirituality and organizational learning capacity. Middle and upper-level Managers working in the bank, different industry associations, multiple In-house and Ex-house training providing firms, support functions and research institutions may utilize findings to bring major improvements in learning capacity mechanisms and employee behavior to enhance the overall adequacy of the associations.

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