Space Notation Research Articles

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

Limit cycles in relay systems with fractional order plants

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.

Gaupta-Bleuler triplet for massless spin- field in de Sitter space

Physicists have been interested in quantization of spinor and vector free fields in 4-dimensional de Sitter space-time, in ambient space notation. The Gupta-Bleuler formalism has been extensively applied to the quantization of gauge invariant theories. The field equation of the massless spin- fields is gauge invariant in de Sitter space. In this paper, we study the quantization of massless spin- gauge fields in de Sitter space-time by the Gupta-Bleuler formalism. This triplet carries an indecomposable representation of the de Sitter group.

Group theoretical interpretation of the modified gravity in de Sitter space

A frame work has been presented for theoretical interpretation of various modified gravitational models which is based on the group theoretical approach and unitary irreducible representations (UIR's) of de Sitter (dS) group. In order to illustrate the application of the proposed method, a model of modified gravity has been investigated. The background field method has been utilized and the linearized modified gravitational field equation has been obtained in the 4-dimensional dS space-time as the background. The field equation has been written as the eigne-value equation of the Casimir operators of dS space using the flat 5-dimensional ambient space notations. The Minkowskian correspondence of the theory has been obtained by taking the zero curvature limit. It has been shown that under some simple conditions, the linearized modified field equation transforms according to two of the UIR's of dS group labeled by $\Pi^\pm_{2,1}$ and $\Pi^\pm_{2,2}$ in the discrete series. It means that the proposed modified gravitational theory can be a suitable one to describe the quantum gravitational effects in its linear approximation on dS space. The field equation has been solved and the solution has been written as the multiplication of a symmetric rank-2 polarization tensor and a massless scalar field using the ambient space notations. Also the two-point function has been calculated in the ambient space formalism. It is dS invariant and free of any theoretical problem.

Covariant and infrared-free graviton two-point function in de Sitter spacetime

In this paper, the two-point function of linearized gravitons on de Sitter (dS) space is presented. Technically, respecting the dS ambient space notation, the field equation is given by the coordinate-independent Casimir operators of the de Sitter group. Analogous to the quantization of the electromagnetic field in Minkowski space, the field equation admits gauge solutions. The notation allows us to exhibit the formalism of Gupta-Bleuler triplets for the present field in exactly the same manner as it occurs for the electromagnetic field. In this regard, centering on the spin-two part (the traceless part, ${\cal{K}}^t$), the field solution is written as a product of a generalized polarization tensor and a minimally coupled massless scalar field. Then, admitting a de Sitter-invariant vacuum through the so-called "Krein space quantization", the de Sitter fully covariant two-point function is calculated. This function is interestingly free of pathological large distance behavior (infrared divergence). Moreover, the spin-zero part (the pure-trace part; ${\cal{K}}^{pt}$) of the field is discussed in this paper. It is shown that the implications of the dS group unitary irreducible representations restrict the gauge-fixing parameter to the optimal value, which remarkably results in the pure-trace part be written in terms of a conformally coupled massless scalar field.

Conformally invariant spin- $$\frac{3}{2}$$ 3 2 field equation in de Sitter space–time

AbstractIn the previous paper (Behroozi et al., Phys Rev D 74:124014, 2006; Dehghani et al., Phys Rev D 77:064028, 2008), conformal invariance for massless tensor fields (scalar, vector and spin-2 fields) was studied and the solutions of their wave equations and two-point functions were obtained. In the present paper, conformally invariant wave equation for massless spinor field in de Sitter space–time has been obtained. For this propose, we use Dirac’s six-cone formalism. The solutions of massless spin-12documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{2}$$end{document} and -32documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{3}{2}$$end{document} equations, in the ambient space notation, have been calculated.

A group theoretical approach to graviton two-point function

Respecting the group theoretical approach, it is debated that the theory of linear conformal gravity should be formulated through a tensor field of rank-3 and mixed symmetry \cite{binegar}. Pursuing this path, such a field equation was obtained in de Sitter space \cite{takook}. In present work, considering the de Sitter ambient space notation, a proper solution to the physical part of this field equation is obtained. We have also calculated the related two-point function, which is interestingly de Sitter invariant and free of infrared divergence.

Conformal graviton two-point function in de Sitter space

The conformally invariant linearized de Sitter gravitational wave equation has recently received a lot of attention. In this article, using the ambient space notations, we solve this field equation in five various cases. In each case, it has been shown that the solution can be written as the product of a generalized symmetric polarization tensor of rank 2 and a massless minimally or conformally coupled scalar field in de Sitter space-time. The conformally covariant graviton two-point functions have been calculated in terms of the massless minimally or conformally coupled scalar two-point functions, using ambient space notations. In the case of massless minimally coupled scalar field, the Krien space quantization has been used to avoid the violation of de Sitter invariance. The two-point functions are expressed in terms of de Sitter intrinsic coordinates from their ambient space counterparts, which are free of any theoretical problem.

A NEW PHYSICAL STATE FOR DE SITTER LINEAR GRAVITY

Based on conformal invariance and using Dirac's six-cone formalism, a new conformally invariant physical field equation for de Sitter (dS) linear gravity has been obtained, which corresponds to one of the unitary irreducible representations of the dS group and is denoted by [Formula: see text] in the sense of discrete series. Using ambient space notations, it has been shown that the solution to this new field equation can be written as the multiplication of a generalized symmetric polarization tensor of rank 2 and a massless conformally coupled scalar field in dS space–time. The physical tensor two-point function has been calculated in terms of the conformally coupled scalar two-point function in the ambient space formalism. It has been expressed in terms of dS intrinsic coordinates from its ambient space counterpart, which is dS-invariant and free of any divergences.

Linear Weyl gravity in de Sitter universe

In this paper we obtain the linear Weyl gravity in de Sitter background. Its linear form in five-dimensional ambient space notation has also been obtained. From the group theoretical point of view, we show that this conformal invariant fourth order theory in its linear form does not transform as an unitary irreducible representation of the de Sitter group.

EVALUATION OF THE GRAVITON TWO-POINT FUNCTION IN DE SITTER SPACE

After a brief review of the linearized gravity in de Sitter (dS) four-dimensional space and ambient flat five-dimensional notations, the linearized field equation is written in terms of the Casimir operators of dS group. It is shown that the field equation is gauge invariant under some special gauge transformations. Because of this gauge freedom, a gauge-fixing parameter c is inserted in the field equation. It is shown that the solution to the field equation can be written as the multiplication of a generalized symmetric rank-2 polarization tensor and a massless minimally coupled scalar field in ambient space notations. The graviton two-point function has been thoroughly calculated, which is dS-invariant and free of any divergences. This two-point function has been expressed in terms of the intrinsic dS coordinates, from its ambient space counterpart, which is clearly dS-invariant and free of any divergences again.

COVARIANT TWO-POINT FUNCTION FOR MASSLESS SPIN-2 FIELD IN DE SITTER SPACE

The linearized gravitational field equation in de Sitter (dS) four-dimensional space is gauge invariant under some special gauge transformations. It is also gauge invariant in ambient space notations in which the field equation is written in terms of the Casimir operators of dS group. In this paper the field equation is solved in terms of a gauge-fixed value (i.e. in the minimal case). It is shown that the solution can be written as the multiplication of a generalized symmetric rank-2 polarization tensor and a massless minimally coupled scalar field in ambient space notations. The two-point function is calculated in ambient space notations, which is dS-invariant and free of any divergences. This two-point function has been expressed in terms of dS intrinsic coordinates, from its ambient space counterpart, which is clearly dS-invariant and free of any divergences again.

A Fixed Point Theorems in L-Fuzzy Quasi-Metric Spaces

At first we considered the L-fuzzy metric space notation which is useful in modeling some phenomena where it is necessary to study the relationship between two probability functions as well observed in Gregori et al. (A note on intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2006; 28: 902-905). Then we introduced the concept of fixed point theorem in L-fuzzy metric space and finally, showed that every contractive mapping on an L-fuzzy metric space has a unique fixed point.

The First Quantization of Spin $\frac{3}{2}$ Field in de Sitter Space

The de Sitter solution to the positive cosmological Einstein field equation has been viewed as a one-sheeted hyperboloid embedded in a five dimensional Minkowski space. To find Lagrangian equation of supersymmetry-group in the de Sitter space, the different spinor field’s quantization have been demonstrated. In this work, the first quantization of spin \(\frac{3}{2}\) field in the time-space de Sitter universe with ambient space notation has been done.

N=2 de Sitter Super-symmetry Algebra

It was shown that N=1 super-symmetry algebra can be constructed in de Sitter space, through calculation of charge conjugation in the ambient space notation. Calculation of N=2 super-symmetry algebra constitutes the main frame of this paper. N=2 super-symmetry algebra was presented in. In this paper, we obtain an alternative N=2 super-symmetry algebra.

Auditory sketching as a tool in architectural design education

Architectural spaces are perceived through embodied experiences that are made up of both visual and non-visual stimuli. However, the conventional means that a designer uses to depict non -visual stimuli such as aural phenomena contain a visual bias. We have introduced a new aural drawing system called, "Auditory Sketching" as a way for design students to visualize the auditory fields that structure one's perception of architectural space. The Auditory Sketching procedure requires that the design student first mediate aural perceptions linguistically through meticulous observation during which they are encouraged to define aural qualities within the space such as notes, tones, clusters, bands, and pitches that would be difficult to capture using a graphic system of notation. Using this written language as a guide for graphic expression, they explore different ways of expressing these time and space notations with ink pens; creating an auditory system of 'tones' by building up marks in different ways: hatching, layering up, scribbling, etc. The eventual translation of aural perception into graphic patterns is crucial because designers need to visualize the sound in order to consider it a significant medium of architecture that they can define and shape.

Fisher and Inference for Scores

Summary This paper examines the work of Fisher and Bartlett on discriminant analysis, ordinal response regression and correspondence analysis. Placing these methods with canonical correlation analysis in the context of the singular value decomposition of particular matrices, we use explicit models and vector space notation to unify these methods, understand Fisher's approach, understand Bartlett's criticisms of Fisher and relate both to modern thinking. We consider in particular the formulation of certain hypotheses and Fisher's arguments to obtain approximate distributions for tests of these hypotheses (without assuming multivariate normality) and put these in modern notation. Using perturbation techniques pioneered by G.S. Watson, we give an asymptotic justification for Fisher's test for assigned scores and thereby resolve a long standing conflict between Fisher and Bartlett.

Conformally invariant wave equations and massless fields in de Sitter spacetime

Conformally invariant wave equations in de Sitter space, for scalar and vector fields, are introduced in the present paper. Solutions of their wave equations and the related two-point functions, in the ambient space notation, have been calculated. The Hilbert space structure and the field operator, in terms of coordinate independent de Sitter plane waves, have been defined. The construction of the paper is based on the analyticity in the complexified pseudo-Riemannian manifold, presented first by Bros et al. Minkowskian limits of these functions are analyzed. The relation between the ambient space notation and the intrinsic coordinates is then studied in the final stage.

ABELIAN GAUGE THEORY IN DE SITTER SPACE

Quantization of spinor and vector free fields in four-dimensional de Sitter spacetime, in the ambient space notation, has been studied in the previous works. Various two-point functions for the above fields are presented in this paper. The interaction between the spinor field and the vector field is then studied by the Abelian gauge theory. The U (1) gauge invariant spinor field equation is obtained in a coordinate independent way notation and their corresponding conserved currents are computed. The solution of the field equation is obtained by the use of the perturbation method in terms of the Green's function. The null curvature limit is discussed in the final stage.

Formula omitted] de Sitter supersymmetry algebra

Recalling the universal covering group of de Sitter universe, the transformation properties of the spinor fields ψ(x) and ψ¯(x), in the ambient space notation, are presented in this Letter. The charge conjugation symmetry of the de Sitter spinor field is then discussed in the above notation. Using this spinor field and charge conjugation, de Sitter supersymmetry algebra in the ambient space notation has been established. It is shown that a novel de Sitter superalgebra can be attained by the use of spinor field and charge conjugation in the ambient space notation.