Variational Relations in Abstract Convex Spaces

Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.

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ON THE PRINCIPLE OF UNIFORM BOUNDEDNESS FOR LSC CONVEX PROCESSES IN A STRICTLY N LOCALLY CONVEX SPACES

Demonstratio Mathematica ◽

10.1515/dema-2008-0117 ◽

2008 ◽

Vol 41 (1) ◽

Author(s):

S. Lahrech ◽

A. Jaddar ◽

J. Hlal ◽

A. Ouahab ◽

A. Mbarki

Keyword(s):

Uniform Boundedness ◽

Locally Convex Spaces ◽

Convex Processes ◽

Locally Convex ◽

Convex Spaces

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(L, M)- fuzzy (restricted) convex derived hull spaces

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-211115 ◽

2021 ◽

pp. 1-13

Author(s):

Xiu-Yun Wu ◽

Chun-Yan Liao ◽

Yan-Hui Zhao

Keyword(s):

Convex Hull ◽

Convex Spaces

In this paper, the notion of (L, M)- fuzzy convex derived hull spaces is introduced. It is proved that the category of (L, M)- fuzzy convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy convex spaces and the category of (L, M)- fuzzy convex enclosed relation spaces. Based on this, the notion of (L, M)- fuzzy restricted convex derived hull spaces is introduced. It is further proved that the category of (L, M)- fuzzy restricted convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy restricted convex hull spaces.

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M-Hazy Vector Spaces over M-Hazy Field

Mathematics ◽

10.3390/math9101118 ◽

2021 ◽

Vol 9 (10) ◽

pp. 1118

Author(s):

Faisal Mehmood ◽

Fu-Gui Shi

Keyword(s):

Vector Space ◽

Linear Transformation ◽

Binary Operation ◽

Vector Spaces ◽

Fundamental Properties ◽

Classical Algebra ◽

Important Development ◽

Fuzzy Algebra ◽

Convex Spaces

The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called M-hazy vector spaces over M-hazy field. Some fundamental properties of M-hazy field, M-hazy vector spaces, and M-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of M-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that M-fuzzifying convex spaces are induced by an M-hazy subspace of M-hazy vector space.

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