scholarly journals Dichotomy theorems for countably infinite dimensional analytic hypergraphs

2011 ◽  
Vol 162 (7) ◽  
pp. 561-565 ◽  
Author(s):  
B.D. Miller
Keyword(s):  
Infinite Dimensional ◽  
Dichotomy Theorems ◽  
Countably Infinite

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

Isomorphism classes of commutative algebras generated by idempotents

2019 ◽  
pp. 2150008
Author(s):  
Kazuyo Inoue ◽  
Hideyasu Kawai ◽  
Nobuharu Onoda
Keyword(s):  
Nonnegative Integer ◽  
Integral Domain ◽  
Algebra Isomorphism ◽  
Infinite Dimensional ◽  
Isomorphism Classes ◽  
Commutative Algebras ◽  
Primitive Idempotents ◽  
Countably Infinite ◽  
Torsion Free

We study commutative algebras generated by idempotents with particular emphasis on the number of primitive idempotents. Let [Formula: see text] be an integral domain with the field of fractions [Formula: see text] and let [Formula: see text] be an [Formula: see text]-algebra which is torsion-free as an [Formula: see text]-module. We show that if [Formula: see text] satisfies the three conditions: [Formula: see text] is generated by idempotents over [Formula: see text]; [Formula: see text] is countably infinite dimensional over [Formula: see text]; [Formula: see text] has [Formula: see text] primitive idempotents for a nonnegative integer [Formula: see text], then [Formula: see text] is uniquely determined up to [Formula: see text]-algebra isomorphism. We also consider the case where [Formula: see text] has countably many primitive idempotents.

scholarly journals Latent Complete-Lattice Structure of Hilbert-Space Projectors

Quanta ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Fedor Herbut
Keyword(s):  
Hilbert Space ◽  
Complete Lattice ◽  
Lattice Structure ◽  
Infinite Dimensional ◽  
Mechanical Formalism ◽  
Quantum Mechanical Formalism ◽  
The One ◽  
Infinite Sums ◽  
Adjoint Operators ◽  
Countably Infinite

To uncover the hidden complete-lattice structure of Hilbert-space projectors, which is not seen by the operator operations and relations (algebraically), resort is taken to the ranges of projectors (to subspaces—to geometry). Taking the range of a projector is completed into a bijection of all projectors onto all subspaces of any finite or countably infinite dimensional Hilbert space. As a second step, this basic bijection is upgraded into an isomorphism of partially ordered sets utilizing the sub-projector relation on the one hand, and the subspace relation on the other. As a third and final step, the basic bijection is further upgraded to isomorphism of complete lattices. The complete-lattice structure is derived for subspaces, then, using the basic bijection, it is transferred to the set of all projectors. Some consequences in the quantum-mechanical  formalism are examined with particular attention to the infinite sums appearing in spectral decompositions of discrete self-adjoint operators with infinite spectra.Quanta 2019; 8: 1–10.

Infinite-dimensional stochastic difference equations for particle systems and network flows

1989 ◽  
Vol 26 (02) ◽  
pp. 325-344
Author(s):  
R. W. R. Darling
Keyword(s):  
Neural Network ◽  
Flow Model ◽  
Network Flows ◽  
Particle Systems ◽  
Stochastic Flow ◽  
Random Vectors ◽  
Infinite Dimensional ◽  
Countably Infinite ◽  
Infinite Set ◽  
Number Of Births

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn = AnXn – 1 + ζ n , for i.i.d. random matrices {An } and i.i.d. random vectors {ζ n }. Interpretation: site x in V is occupied by Xn (x) particles at time n; An describes random transport of existing particles, and ζ n (x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn } to equilibrium, and (2) a central limit theorem for n–1/2(X 1 + · ·· + Xn ), respectively. When the matrices {An } consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.

Order arguments on the dimension of vector lattices

1981 ◽  
Vol 89 (1-2) ◽  
pp. 51-53
Author(s):  
John T. Annulis
Keyword(s):  
Complete Space ◽  
Infinite Dimensional ◽  
Vector Lattices ◽  
Countably Infinite ◽  
Infinite Set

SynopsisThe main result asserts that the base of an infinite dimensional Dedekind complete space with unit contains an infinite set of disjoint elements. From this result it can be shown that the dimension of Dedekind σ -complete spaces with unit is not countably infinite.

scholarly journals Recursive equivalence types of vector spaces

1974 ◽  
Vol 18 (3) ◽  
pp. 376-384 ◽  
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

Infinite-dimensional stochastic difference equations for particle systems and network flows

1989 ◽  
Vol 26 (2) ◽  
pp. 325-344 ◽  
Author(s):  
R. W. R. Darling
Keyword(s):  
Neural Network ◽  
Flow Model ◽  
Network Flows ◽  
Particle Systems ◽  
Stochastic Flow ◽  
Random Vectors ◽  
Infinite Dimensional ◽  
Countably Infinite ◽  
Infinite Set ◽  
Number Of Births

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn = AnXn– 1 + ζn, for i.i.d. random matrices {An} and i.i.d. random vectors {ζ n}. Interpretation: site x in V is occupied by Xn(x) particles at time n; An describes random transport of existing particles, and ζ n(x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn} to equilibrium, and (2) a central limit theorem for n–1/2(X1 + · ·· + Xn), respectively. When the matrices {An} consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.

Bases and α-dimensions of countable vector spaces with recursive operations

1970 ◽  
Vol 35 (1) ◽  
pp. 85-96
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

Sign in / Sign up

Export Citation Format

Share Document