Concrete stream calculus: An extended study

2010 ◽  
Vol 20 (5-6) ◽  
pp. 463-535 ◽  
Author(s):  
RALF HINZE
Keyword(s):  
Generating Functions ◽  
Proof Technique ◽  
Extended Study ◽  
Equational Reasoning ◽  
The Rich ◽  
Infinite Sequences

AbstractThis paper shows how to reason about streams concisely and precisely. Streams, infinite sequences of elements, live in a coworld: they are given by a coinductive datatype, operations on streams are implemented by corecursive programs, and proofs are typically concocted using coinduction. This paper offers an alternative to coinduction. Suitably restricted, stream equations possessunique solutions. This property gives rise to a simple and attractive proof technique, essentially bringing equational reasoning to the coworld. We redevelop the theory of recurrences, finite calculus and generating functions using streams and stream operators, building on the cornerstone of unique solutions. The paper contains a smörgåsbord of examples: we study recursion elimination, investigate the binary carry sequence, explore Sprague-Grundy numbers and present two proofs of Moessner's Theorem. The calculations benefit from the rich structure of streams. As the type of streams is an applicative functor we can effortlessly lift operations and their properties to streams. In combination with Haskell's facilities for overloading, this greatly contributes to conciseness of notation. The development is indeed constructive: streams and stream operators are implemented in Haskell, usually by one-liners. The resulting calculus or library, if you wish, is elegant and fun to use.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals Complete k-ary trees and generalized meta-Fibonacci sequences

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Chris Deugau ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
International Audience ◽  
Number Of Leaves ◽  
Fibonacci Sequences ◽  
Infinite Sequences

International audience We show that a family of generalized meta-Fibonacci sequences arise when counting the number of leaves at the largest level in certain infinite sequences of k-ary trees and restricted compositions of an integer. For this family of generalized meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations.

scholarly journals Lambda calculus with algebraic simplification for reduction parallelisation: Extended study

2021 ◽  
Vol 31 ◽  
Author(s):  
AKIMASA MORIHATA
Keyword(s):  
Parallel Programming ◽  
Type System ◽  
Lambda Calculus ◽  
Extended Study ◽  
Equational Reasoning ◽  
Typed Lambda Calculus ◽  
Parallel Reduction ◽  
Algebraic Properties ◽  
Algebraic Simplification

Abstract Parallel reduction is a major component of parallel programming and widely used for summarisation and aggregation. It is not well understood, however, what sorts of non-trivial summarisations can be implemented as parallel reductions. This paper develops a calculus named λAS, a simply typed lambda calculus with algebraic simplification. This calculus provides a foundation for studying a parallelisation of complex reductions by equational reasoning. Its key feature is δ abstraction. A δ abstraction is observationally equivalent to the standard λ abstraction, but its body is simplified before the arrival of its arguments using algebraic properties such as associativity and commutativity. In addition, the type system of λAS guarantees that simplifications due to δ abstractions do not lead to serious overheads. The usefulness of λAS is demonstrated on examples of developing complex parallel reductions, including those containing more than one reduction operator, loops with conditional jumps, prefix sum patterns and even tree manipulations.

scholarly journals GENERALIZED COHERENT STATES FOR THE SPHERICAL HARMONICS $Y_{m}^{m}(\theta,\phi)$

2010 ◽  
Vol 25 (10) ◽  
pp. 2165-2170 ◽  
Author(s):  
H. FAKHRI ◽  
B. MOJAVERI
Keyword(s):  
Spherical Harmonics ◽  
Generating Functions ◽  
Coherent States ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Rodrigues Formula ◽  
Generalized Coherent States ◽  
Infinite Sequences

The associated Legendre functions [Formula: see text] for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics [Formula: see text].

Infinite sets of identities for classical polynomials and Bessel functions

1968 ◽  
Vol 64 (1) ◽  
pp. 105-112 ◽  
Author(s):  
Mostafa A. Abdelkader
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Bessel Functions ◽  
Integral Transforms ◽  
Delta Function ◽  
Trigonometric Functions ◽  
Power Functions ◽  
Classical Polynomials ◽  
Infinite Sets ◽  
Infinite Sequences

AbstractAn identity for a rational function is used, in conjunction with generating functions for a certain class of polynomials, to derive infinite sets of identities for these polynomials. Identities are given for the polynomials of Legendre, Gegenbauer, Hermite, Tchebycheff and L. Carlitz. A novel type of classification for these polynomials is indicated. Next, infinite sequences of identities for various Bessel functions and for power functions are derived. New identities for trigonometric functions are also given, as well as brief miscellanea on: functions resembling Dirac's delta function, integral transforms and a functional equation.

scholarly journals Political Jurisprudence

2018 ◽  
Author(s):  
Martin Loughlin
Keyword(s):  
Generating Functions ◽  
State Building ◽  
Public Law ◽  
Legal Order ◽  
The Political ◽  
The World ◽  
Reason Of State ◽  
Constitutional Legality ◽  
The Rich ◽  
Political Right

Political jurisprudence is the branch of jurisprudence that treats law as an aspect of human experience called ‘the political’. This is an approach that many contemporary jurists, those whose work presupposes the autonomy of legal order, tend to suppress. In this book, Martin Loughlin assesses the contribution made by political jurists and explains its contemporary significance. Political jurists maintain that the essential characteristics of modern legal order can only be revealed by considering how political authority is constituted. The political is orientated to the fact that people are organized into territorially-bounded units within which authoritative governing arrangements have been established, but the authority of this way of viewing the world is strengthened only through institutional-building. Law may be an aspect of the political, but to perform its authority-generating functions effectively it must operate relatively autonomously. The political and the legal operate relationally, without one being reduced to the other. Loughlin introduces the rich literature of political jurisprudence through essays on innovative political jurists such as Hobbes, Burke, Constant, Romano, and Schmitt, and on such central themes as political right, institutionalism, constitutional legality, and reason of state. Building on his earlier books, The Idea of Public Law (OUP 2003) and Foundations of Public Law (OUP 2010), this collection extends his account of this influential strand of European legal thought.

Poor and Hidden Among the Rich and Not So Rich

PsycCRITIQUES ◽  
2006 ◽  
Vol 51 (49) ◽  
Author(s):  
Patricia M. Berliner
Keyword(s):  
The Rich

Institutional Theory of Endless Redistribution

2005 ◽  
pp. 4-18 ◽  
Author(s):  
K. Sonin
Keyword(s):  
Property Rights ◽  
Institutional Theory ◽  
Growth Rates ◽  
Rent Seeking ◽  
Economic Institutions ◽  
Wide Spread ◽  
Grass Roots ◽  
Protection Systems ◽  
Private Protection ◽  
The Rich

In unequal societies, the rich may benefit from shaping economic institutions in their favor. This paper analyzes the dynamics of institutional subversion by focusing on public protection of property rights. If this institution functions imperfectly, agents have incentives to invest in private protection of property rights. The ability to maintain private protection systems makes the rich natural opponents of public protection of property rights and precludes grass-roots demand to drive the development of the market-friendly institution. The economy becomes stuck in a bad equilibrium with low growth rates, high inequality of income, and wide-spread rent-seeking. The Russian oligarchs of the 1990s, who controlled large stakes of newly privatized property, provide motivation for this paper.

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