Kähler Metrics on the Projective Bundle of a Holomorphic Finsler Vector Bundle

Let (M, g) be a compact Kähler manifold and (E, F) be a holomorphic Finsler vector bundle of rank r ≥ 2 over M. In this paper, we prove that there exists a Kähler metric φ defined on the projective bundle P (E) of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for φ having positive scalar curvature is obtained, and a sufficient condition for φ having positive Ricci curvature is established.     

Dual nature of Ricci scalar and creation of spinless particles

Manifestation of Ricci scalar like a matter field as well as a geometrical field, at high energy, has been noted earlier [9]. Here, its interaction with another scalar field is considered in four-dimensional curved space-time. This interaction leads to the production of a large number of pairs of spinless particle-antiparticle due to expansion of the early universe in the vacuum state (provided by temperature dependent Coleman-Weinberg like potential for Ricci field), where spontaneous symmetry breaking takes place.     

Set inertia-preserving matrices

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

Algebraically Decaying Modes in Many Dimensions: Application of Fubini’s Identity to Guided-Wave Optics

We show analytically that application of an identity in conformal invariant field theories to multidimensional guided-wave optics predicts a generic family of modes with algebraic (power-law) tails in the evanescent field.     

On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms

Gram–Schmidt-like orthogonalization process with respect to a given skew-symmetric scalar product is a key step in model reduction methods, structure-preserving, for large sparse Hamiltonian eigenvalue problem. Theoretical as well as numerical aspects of this step do not benefit of enough attention, compared to the one allowed to the classical Gram–Schmidt algorithm and its modified version. The aim of this paper is to revisit the symplectic Gram–Schmidt algorithms, to built some modified versions and to deal with their theoretical and numerical features. AMS subject classification 65F15, 65F50     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

Dark Energy Model with Canonical Scalar Field and Non-Linear Born–Infeld Type Scalar Field

In this paper, we propose the non-linear Born–Infeld scalar field and canonical scalar field dark energy models with the potential , which admits late time de Sitter attractor solution. The attractor solution corresponds to an equation of state ω_φ → − 1 and a cosmic density parameter Ω_φ → 1, which are important features for a dark energy model that can meet the current observations. dark energy; canonical scalar field, non-linear Born–Infeld type scalar field, attractor solution. PACS number(s):98.80.-k; 98.80.Cq; 98.80.Es.     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if is a complete noncompact gradient steady Ricci soliton satisfying , and if the scalar curvature attains its maximum at some point in the interior of M, then either is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

Spatially almost periodic solutions for active scalar equations

We are concerned with a family of dissipative active scalar equation on . By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution θ of 1.1 satisfying 1.11 . In this paper, we consider some countable sum‐closed frequency sets (see Remark 1.1 ). We show that the property of the solution is rather different from Chae et al 1 and obtain that with some initial data θ₀ for all t≥0, and 0≤α≤ω, where ω is a fixed constant. Furthermore, arranging the elements of a countable sum‐closed frequency set F_δ as in Remark 1.3 , we have for any 0≤α≤ω that belongs to , where F_δ is defined in 1.4 or 1.5 .