scholarly journals Resource convertibility and ordered commutative monoids

2015 ◽  
Vol 27 (6) ◽  
pp. 850-938 ◽  
Author(s):  
TOBIAS FRITZ
Keyword(s):  
Information Theory ◽  
Communication Channel ◽  
Linear Logic ◽  
Commutative Monoids ◽  
Science And Engineering ◽  
Von Neumann ◽  
Torsion Freeness ◽  
Equality Relation ◽  
Image Position ◽  
Reaction Equations

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as\mathrm{2H_2 + O_2} \lra \mathrm{2H_2O,}are concerned both with a convertibility relation ‘→’ and a combination operation ‘+.’ We study ordered commutative monoids from an algebraic and functional-analytic perspective and derive a wealth of results which should have applications to concrete resource theories, such as a formula for rates of conversion. As a running example showing that ordered commutative monoids are also of purely mathematical interest without the resource-theoretic interpretation, we exemplify our results with the ordered commutative monoid of graphs.While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on the foundations of thermodynamics.Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

Information mechanics

2017 ◽  
Author(s):  
Sandip Tiwari
Keyword(s):  
Information Flow ◽  
Irreversible Processes ◽  
State Machines ◽  
Science And Engineering ◽  
Von Neumann ◽  
Principle Of Maximum Entropy ◽  
Information Manipulation ◽  
Classical Quantum ◽  
Physical Laws ◽  
Principle Of Maximum

Information is physical, so its manipulation through devices is subject to its own mechanics: the science and engineering of behavioral description, which is intermingled with classical, quantum and statistical mechanics principles. This chapter is a unification of these principles and physical laws with their implications for nanoscale. Ideas of state machines, Church-Turing thesis and its embodiment in various state machines, probabilities, Bayesian principles and entropy in its various forms (Shannon, Boltzmann, von Neumann, algorithmic) with an eye on the principle of maximum entropy as an information manipulation tool. Notions of conservation and non-conservation are applied to example circuit forms folding in adiabatic, isothermal, reversible and irreversible processes. This brings out implications of fluctuation and transitions, the interplay of errors and stability and the energy cost of determinism. It concludes discussing networks as tools to understand information flow and decision making and with an introduction to entanglement in quantum computing.

scholarly journals Coordinates for analytic operator algebras

1989 ◽  
Vol 31 (1) ◽  
pp. 31-47
Author(s):  
Baruch Solel
Keyword(s):  
Abelian Group ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Dual Group ◽  
Positive Semigroup ◽  
Analytic Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Analytic Algebra ◽  
Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

The Role of the Information Channel in Visual Computing

2020 ◽  
pp. 464-490
Author(s):  
Miquel Feixas ◽  
Mateu Sbert
Keyword(s):  
Information Theory ◽  
Communication Channel ◽  
Probability Distributions ◽  
Global Illumination ◽  
Three Dimensional ◽  
Information Channel ◽  
Visual Computing ◽  
The Social ◽  
Shared Information

Around seventy years ago, Claude Shannon, who was working at Bell Laboratories, introduced information theory with the main purpose of dealing with the communication channel between source and receiver. The communication channel, or information channel as it later became known, establishes the shared information between the source or input and the receiver or output, both of which are represented by random variables, that is, by probability distributions over their possible states. The generality and flexibility of the information channel concept can be robustly applied to numerous, different areas of science and technology, even the social sciences. In this chapter, we will present examples of its application to select the best viewpoints of an object, to segment an image, and to compute the global illumination of a three-dimensional virtual scene. We hope that our examples will illustrate how the practitioners of different disciplines can use it for the purpose of organizing and understanding the interplay of information between the corresponding source and receiver.

Fubini's Theorem for Ultraproducts of Noncommutative Lp-Spaces

2004 ◽  
Vol 56 (5) ◽  
pp. 983-1021 ◽  
Author(s):  
Marius Junge
Keyword(s):  
Von Neumann Algebras ◽  
Type Result ◽  
Von Neumann ◽  
Lp Spaces ◽  
Fubini’S Theorem ◽  
Image Position ◽  
Local Reflexivity ◽  
Fubini's Theorem

AbstractLet (ℳi)i∈I, be families of von Neumann algebras and be ultrafilters in I, J, respectively. Let 1 ≤ p < ∞ and n ∈ ℕ. Let x1,… ,xn in ΠLp(ℓi ) and y1,… ,yn in be bounded families. We show the following equalityFor p = 1 this Fubini type result is related to the local reflexivity of duals of C*-algebras. This fails for p = ∞.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

Ergodic theorems for semifinite von Neumann algebras: II

1980 ◽  
Vol 88 (1) ◽  
pp. 135-147 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Unbounded Operators ◽  
Von Neumann ◽  
The Mean ◽  
Image Position

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averagesconverge in norm.

On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

John von Neumann: The Founding Father of Artificial Life

1998 ◽  
Vol 4 (3) ◽  
pp. 229-235 ◽  
Author(s):  
Pierre Marchal
Keyword(s):  
Artificial Life ◽  
Double Helix ◽  
Genomic Information ◽  
Founding Father ◽  
Science And Engineering ◽  
Computing Systems ◽  
John Von Neumann ◽  
Von Neumann ◽  
Founding Fathers ◽  
Mathematics And Physics

Aside from being known for his contributions to mathematics and physics, John von Neumann is considered one of the founding fathers of computer science and engineering. Not only did he do pioneering work on sequential computing systems, but he also carried out a major investigation of parallel architectures, leading to his work on cellular automata. His exceptional vision and daring, borrowing from biology the concept of genomic information even before the discovery of DNA's double helix, led him to propose the concept of self-reproducing automata.

Recent results in information theory

1966 ◽  
Vol 3 (1) ◽  
pp. 1-93 ◽  
Author(s):  
Samuel Kotz
Keyword(s):  
Information Theory ◽  
Communication Channel ◽  
Quantitative Measure ◽  
Strict Sense ◽  
Bell System Technical Journal ◽  
Noisy Channels ◽  
Basic Theorem ◽  
Reliable Transmission ◽  
Transmission Of Information ◽  
Communications System

Information theory, in the strict sense, is a rapidly developing branch of probability theory originating from a paper by Claude E. Shannon in the Bell System Technical Journal in 1948,in which anew mathematical model ofcommunications systems was proposed and investigated.One of the central innovations of this model was in regarding the prime components of a communications system (the source of messages and the communication channel) as probabilistic entities. Shannon also proposed a quantitative measure of the amount of information based on his notion of entropy and proved the basic theorem of this theory concerning the possi bility of reliable transmission of information over a particular class of noisy channels.

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