Some Results on Ricci Almost Solitons

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

Weak cosmic censorship with self-interacting scalar and bound on charge to mass ratio

We study the model of four-dimensional Einstein-Maxwell-Λ theory minimally coupled to a massive charged self-interacting scalar field, parameterized by the quartic and hexic couplings, labelled by λ and β, respectively. In the absence of scalar field, there is a class of counterexamples to cosmic censorship. Moreover, we investigate the full nonlinear solution with nonzero scalar field included, and argue that these counterexamples can be removed by assuming charged self-interacting scalar field with sufficiently large charge not lower than a certain bound. In particular, this bound on charge required to preserve cosmic censorship is no longer precisely the weak gravity bound for the free scalar theory. For the quartic coupling, for λ < 0 the bound is below the one for the free scalar fields, whereas for λ > 0 it is above. Meanwhile, for the hexic coupling the bound is always above the one for the free scalar fields, irrespective of the sign of β.

Indefinite Einstein metrics on nice Lie groups

We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension \geq 8.

Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F. Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d+1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d≥3.

Nonlinear preservers of group invertible operators

Let [Formula: see text] be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space [Formula: see text]. We prove that a bijective bicontinuous map [Formula: see text] on [Formula: see text] preserves the difference of group invertible operators in both directions if and only if [Formula: see text] is either of the form [Formula: see text] or of the form [Formula: see text], where [Formula: see text] is a nonzero scalar, [Formula: see text] and [Formula: see text] are two bounded invertible linear or conjugate linear operators.

Maps preserving strong 2-Jordan product on some algebras

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

On Ulam Stability of a Functional Equation in Banach Modules

Let X and Y be Banach spaces and let f : X → Y be an odd mapping. For any rational number r≠2, C. Baak, D. H. Boo, and Th. M. Rassias proved the Hyers–Ulam stability of the functional equationwhere d and l are positive integers so that In this note we solve this equation for arbitrary nonzero scalar r and show that it is actually Hyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered.

SOME ASPECTS OF SPHERICAL SYMMETRIC EXTREMAL DYONIC BLACK HOLES IN 4D N = 1 SUPERGRAVITY

In this paper, we study several aspects of extremal spherical symmetric black hole solutions of four-dimensional N = 1 supergravity coupled to vector and chiral multiplets with the scalar potential turned on. In the asymptotic region, the complex scalars are fixed and regular which can be viewed as the critical points of the black hole and the scalar potentials with vanishing scalar charges. It follows that the asymptotic geometries are of a constant and nonzero scalar curvature which are generally not Einstein. These spaces could also correspond to the near horizon geometries which are the product spaces of a two anti-de Sitter surface and the two sphere if the value of the scalars in both regions coincide. In addition, we prove the local existence of nontrivial radius dependent complex scalar fields which interpolate between the horizon and the asymptotic region. We finally give some simple ℂn-models with both linear superpotential and gauge couplings.

EFFECTIVE POTENTIAL FOR LIFSHITZ-TYPE z = 3 GAUGE THEORIES

We consider the one-loop effective potential at zero temperature in Lifshitz-type field theories with anisotropic space–time scaling, with critical exponent z = 3, including scalar, fermion and gauge fields. The fermion determinant generates a symmetry breaking term at one loop in the effective potential and a local minimum appears, for nonzero scalar field, for every value of the Yukawa coupling. Depending on the relative strength of the coupling constants for the scalar and the gauge field, we find a second symmetry breaking local minimum in the effective potential for a bigger value of the scalar field.

FRW MODELS WITH PERFECT FLUID AND SCALAR FIELD IN HIGHER DERIVATIVE THEORY

In this paper we study the dynamics of the universe in Friedmann–Robertson–Walker models including perfect fluid and coupled scalar field with nonzero scalar potential in higher derivative theory of gravitation. We study the evolution of the universe by assuming the scalar potential and scale factor as functions of the scalar field. Exact cosmological solutions are obtained for flat, closed and open models, which are physically interesting for the description of the present-day universe. The properties of scalar field and other physical parameters are discussed in detail. Some special cases have been studied by imposing certain constraints on constants to discuss the decelerated and accelerated phases of the universe.