Weak Convergence of Distributions

2021 ◽  
pp. 495-519
Author(s):  
James Davidson
Keyword(s):  
Weak Convergence ◽  
Representation Theorem ◽  
Stable Distribution ◽  
Characteristic Functions ◽  
Random Sums ◽  
Skorokhod Representation Theorem

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved and the chapter then considers the preservation of weak convergence under transformations. Next, the role of moments and characteristic functions is considered. In the leading case of random sums, the criteria for weak convergence and the concept of a stable distribution are studied.

scholarly journals The Role of Neuropeptide Y in the Nucleus Accumbens

2021 ◽  
Vol 22 (14) ◽  
pp. 7287
Author(s):  
Masaki Tanaka ◽  
Shunji Yamada ◽  
Yoshihisa Watanabe
Keyword(s):  
Nervous System ◽  
Nucleus Accumbens ◽  
Neuropeptide Y ◽  
Emotional Behavior ◽  
Characteristic Functions ◽  
Receptor Subtypes ◽  
Neuroendocrine Mechanisms ◽  
Fear And Anxiety ◽  
The Brain

Neuropeptide Y (NPY), an abundant peptide in the central nervous system, is expressed in neurons of various regions throughout the brain. The physiological and behavioral effects of NPY are mainly mediated through Y1, Y2, and Y5 receptor subtypes, which are expressed in regions regulating food intake, fear and anxiety, learning and memory, depression, and posttraumatic stress. In particular, the nucleus accumbens (NAc) has one of the highest NPY concentrations in the brain. In this review, we summarize the role of NPY in the NAc. NPY is expressed principally in medium-sized aspiny neurons, and numerous NPY immunoreactive fibers are observed in the NAc. Alterations in NPY expression under certain conditions through intra-NAc injections of NPY or receptor agonists/antagonists revealed NPY to be involved in the characteristic functions of the NAc, such as alcohol intake and drug addiction. In addition, control of mesolimbic dopaminergic release via NPY receptors may take part in these functions. NPY in the NAc also participates in fat intake and emotional behavior. Accumbal NPY neurons and fibers may exert physiological and pathophysiological actions partly through neuroendocrine mechanisms and the autonomic nervous system.

The Characteristic Function, a Method-Specific Alternative to the Horwitz Function

2012 ◽  
Vol 95 (6) ◽  
pp. 1803-1806 ◽  
Author(s):  
Michael Thompson
Keyword(s):  
Detection Limit ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
Mathematical Form ◽  
Essential Difference ◽  
Food Sector ◽  
Wide Range ◽  
Specific Alternative ◽  
Additional Role

Abstract The Horwitz function is compared with the characteristic function as a descriptor of the precision of individual analytical methods. The Horwitz function describes the trend of reproducibility SDs observed in collaborative trials in the food sector over a wide range of concentrations of the analyte. However, it is imperfectly adaptable for describing the precision of individual methods, which is the role of the characteristic function. An essential difference between the two functions is that the characteristic function can accommodate a detection limit. This makes it a useful alternative when the precision of a method down to a detection limit is of interest. Many characteristic functions have a simple mathematical form, the parameters of which can be estimated with the usual resources. The Horwitz function serves an additional role as a fitness-for-purpose criterion in the form of the Horwitz ratio (HorRat). This use also has some shortcomings. The functional form of the characteristic function (with suitable prescribed parameters) is better adapted to this task.

scholarly journals Estructuras y representaciones

1990 ◽  
Vol 22 (64) ◽  
pp. 3-22
Author(s):  
Adolfo García de la Sienra
Keyword(s):  
Real World ◽  
Representation Theorem ◽  
Mathematical Structures ◽  
The Real ◽  
Conceptual Apparatus ◽  
Philosophical Basis ◽  
Aristotelian Metaphysics

The aim of the present paper is to set a philosophical basis in order to discuss the type of representation that holds between mathematical structures and those aspects of the real world which they represent. It is maintained that an actualized version of Aristotelian metaphysics is suited for this purpose. The connection between the abstract, rigid concepts of mathematics, and the concepts of metaphysics is attempted through the concept of a fundamental measurement. The existence and degree of uniqueness of a fundamental measurement is established as a representation theorem asserting the existence of a homomorphism from what I call an ontological structure into a numerical one. An ontological structure contains as elements real beings, and its relations represent —in a sense made precise thereof— real relations among these beings. The role of metaphysics in the establishment of a representation theorem is to provide the conceptual apparatus required to discuss and formulate the ontological axioms required to derive the theorem. The paper contains a very complete example of a fundamental measurement in the sense described, namely, the measurement of the height of a physical parallelepiped and that of its potential parts.

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