Nonsingular solution with anisotropic fluid in mini bang cosmology

In this work, we study cosmological evolution in the mini creation event with anisotropic fluid under the Bianchi-I spacetime. After providing the basic mathematical formalism of the model, we find out the exact solutions in its particular form to the field equations. In order to get physical validity, we have presented elaborate discussions on the graphical results, especially nonsingular behaviour of the model. It is shown that the model under mini bang can successfully exhibit several interesting cosmological features which have specific signatures to tally with the observational evidences.

Invariant Bianchi type I models in f(R,T) gravity

In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of [Formula: see text] gravity with special case [Formula: see text]. The exact solution of the Einstein’s field equations are derived by using Lie point symmetry analysis method that yield two models of invariant universe for symmetries [Formula: see text] and [Formula: see text]. The model with symmetries [Formula: see text] begins with big bang singularity while the model with symmetries [Formula: see text] does not favor the big bang singularity. Under this specification, we find out at set of singular and nonsingular solution of Bianchi type I model which present several other physically valid features within the framework of [Formula: see text] gravity.

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's [Formula: see text] invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

The Real Solutions of Equation of in Control Theory

. The research of matrix equations is an active research field, matrix equations have applied in many physical applications in recent years. As one of them, the equation is applied more and more extensively, such as control theory, chemistry and chemical engineering and so on. In this paper, motivated by [1], we give two discriminations about the real solutions of equation . The matrix is proved that it is a nonsingular solution of equation whenever are nonsingular solutions of equations at last.

Localization of periodic orbits of autonomous systems based on high-order extremum conditions

This paper gives localization and nonexistence conditions of periodic orbits in some subsets of the state space. Mainly, our approach is based on high-order extremum conditions, on high-order tangency conditions of a nonsingular solution of a polynomial system with an algebraic surface, and on some ideas related to algebraically-dependent polynomials. Examples of the localization analysis of periodic orbits are presented including the Blasius equations, the generalized mass action (GMA) system, and the mathematical model of the chemical reaction with autocatalytic step.

PROBING THE GRAVITATIONAL GEON

The Brill-Hartle gravitational geon construct as a spherical shell of small amplitude, high frequency gravitational waves is reviewed and critically analyzed. The Regge-Wheeler formalism is used to represent the most general gravitational wave perturbations of the spherical background as a superposition of tensor spherical harmonics and an attempt is made to build a nonsingular solution to meet the requirements of a gravitational geon. The attempted constructs of gravitational and electromagnetic geons are contrasted. High-frequency waves are seen to be a necessary condition for the geon and the field equations are decomposed accordingly. It is shown that this leads to the impossibility of forming a spherical gravitational geon. The attempted constructs of gravitational and electromagnetic geons are constructed. The spherical shell in the proposed BrillHartle geon does not meet the regularity conditions required for a nonsingular source and hence cannot be regarded as an adequate geon construct. Since it is the high frequency attribute which is the essential cause of the geon nonviability, it is argued that a geon with less symmetry is an unlikely prospect. The broader implications of the result are discussed with particular reference to the problem of gravitational energy.

Singularity of Constraint Mechanical Systems

The solution set of nonlinear kinematic constraint equations can be divided into regular and critical solution subsets, according to linear dependency of gradient vectors of the constraint equations. However, relative to a given input coordinate set, the solution set can also be separated into singular and nonsingular solution subsets, according to whether the sub-Jacobian matrix with respect to the output and intermediate coordinates is rank deficient or not. By providing precise definitions and classifications of singular configurations, from both mathematical and physical points of view, a better understanding of the kinematic behavior of singularity is obtained. Moreover, by extending the definition of the singular solution set to the output space and exploring the mathematical meaning of it, the difficulty in formulating mathematical expressions for workspace problems is resolved.