MULTIPLICATIVE RENORMALIZABILITY AND THE LAPLACE-BELTRAMI OPERATOR

1991 ◽  
Vol 06 (06) ◽  
pp. 955-976
Author(s):  
D. OLIVIER ◽  
G. VALENT
Keyword(s):  
Irreducible Representation ◽  
Isometry Group ◽  
Zero Mode ◽  
Beltrami Operator ◽  
Sufficient Condition ◽  
Representation Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Multiplicative Renormalizability

For some rank 1 non-linear σ models we prove that a necessary and sufficient condition of multiplicative renormalizability for composite fields is that they should be eigenfunctions of the coset Laplace-Beltrami operator. These eigenfunctions span the irreducible representation space of the isometry group and may be finite- or infinite-dimensional. The zero mode of the Laplace-Beltrami operator plays a particular role since it is not renormalized at all.

scholarly journals Characterizations of majorization on summable sequences

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2193-2202
Author(s):  
Kosuru Raju ◽  
Subhajit Saha
Keyword(s):  
Sequence Space ◽  
Dimensional Space ◽  
Sufficient Condition ◽  
Infinite Dimensional Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Summable Sequence ◽  
Necessary And Sufficient

In this paper, we prove a necessary and sufficient condition for majorization on the summable sequence space. For this we redefine the notion of majorization on an infinite dimensional space and therein investigate properties of the majorization. We also prove the infinite dimensional Schur-Horn type and Hardy-Littlewood-P?lya type theorems.

scholarly journals The point spectrum and residual spectrum of upper triangular operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1759-1771
Author(s):  
Xiufeng Wu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Hilbert Spaces ◽  
Point Spectrum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let H,K be infinite dimensional complex separable Hilbert spaces and write MX = (AX0B) ? B(H?K). For given operators A ? B(H) and B ? B(K), the sets ? X?B(K,H) ?+,i(MX)(+ = p,r;i = 1,2), are characterized. Moreover, we obtain some necessary and sufficient condition such that ?*,i(MX) = ?*,i(A) ?*,i(B) (* = p,r;i = 1,2) for every X ? B(K,H).

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

Isometry Groups with Surface-Orthogonal Trajectories

1967 ◽  
Vol 22 (9) ◽  
pp. 1351-1355 ◽  
Author(s):  
Bernd Schmidt
Keyword(s):  
Abelian Group ◽  
Tangent Space ◽  
Isometry Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Isometry Groups ◽  
Linearly Independent ◽  
Necessary And Sufficient

It is shown that the trajectories of an isometry group admit orthogonal surfaces if the sub-group of stability leaves no vector in the tangent space of the trajectories fixed. A necessary and sufficient condition is given that the trajectories of an Abelian group admit orthogonal surfaces.In spacetimes which admit an Abelian G2 of isometries, the trajectories admit orthogonal 2-surfaces if a timelike congruence exists with the following properties: the curves lie in the trajectories and are invariant under G2; ωα and üα are linearly independent and orthogonal to the trajectories.

scholarly journals The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Shifang Zhang ◽  
Huaijie Zhong ◽  
Long Long
Keyword(s):  
Separable Hilbert Space ◽  
Invertible Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Infinite Dimensional ◽  
Lower Semi ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

WhenA∈B(H)andB∈B(K)are given, we denote byMCthe operator acting on the infinite-dimensional separable Hilbert spaceH⊕Kof the formMC=(AC0B). In this paper, it is proved that there exists some operatorC∈B(K,H)such thatMCis upper semi-Browder if and only if there exists some left invertible operatorC∈B(K,H)such thatMCis upper semi-Browder. Moreover, a necessary and sufficient condition forMCto be upper semi-Browder for someC∈G(K,H)is given, whereG(K,H)denotes the subset of all of the invertible operators ofB(K,H).

Wandering phenomena in infinite-allelic diffusion models

1982 ◽  
Vol 14 (3) ◽  
pp. 457-483 ◽  
Author(s):  
Tokuzo Shiga
Keyword(s):  
Population Genetics ◽  
Ergodic Theorem ◽  
Stationary Distribution ◽  
Diffusion Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Individual Ergodic Theorem ◽  
Necessary And Sufficient

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Wandering phenomena in infinite-allelic diffusion models

1982 ◽  
Vol 14 (03) ◽  
pp. 457-483 ◽  
Author(s):  
Tokuzo Shiga
Keyword(s):  
Population Genetics ◽  
Ergodic Theorem ◽  
Stationary Distribution ◽  
Diffusion Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Individual Ergodic Theorem ◽  
Necessary And Sufficient

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Ck centre unstable manifolds

1988 ◽  
Vol 108 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
Shui-Nee Chow ◽  
Ke ning Lu
Keyword(s):  
Unstable Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Lipschitz Map ◽  
Unstable Manifolds ◽  
Necessary And Sufficient

SynopsisWe consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps. We show that every global centre unstable manifold can be expressed as a graph of a Ck map, provided that the nonlinearities are Ck smooth. The proofs are based on a lemma by D. Henry on a necessary and sufficient condition for a Lipschitz map to be continuously differentiable.

scholarly journals Variational necessary and sufficient stability conditions for inviscid shear flow

2014 ◽  
Vol 470 (2172) ◽  
pp. 20140322 ◽  
Author(s):  
M. Hirota ◽  
P. J. Morrison ◽  
Y. Hattori
Keyword(s):  
Shear Flow ◽  
Adjoint Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Sufficient Stability ◽  
Parallel Shear Flow ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Sufficient Stability Conditions

A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-self-adjoint eigenvalue problem) can be associated with positive eigenvalues of a certain self-adjoint operator. The stability is therefore determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving Rayleigh's equation. This variational stability criterion is based on the understanding of Kreĭn signature for continuous spectra and is applicable to other stability problems of infinite-dimensional Hamiltonian systems.

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