Collapse in f(R) gravity and the method of R matching

Collapsing solutions in f(R) gravity are restricted due to junction conditions that demand continuity of the Ricci scalar and its normal derivative across the time-like collapsing hypersurface. These are obtained via the method of R-matching, which is ubiquitous in f(R) collapse scenarios. In this paper, we study spherically symmetric collapse with the modification term $$\alpha R^2$$ α R 2 , and use R-matching to exemplify a class of new solutions. After discussing some mathematical preliminaries by which we obtain an algebraic relation between the shear and the anisotropy in these theories, we consider two metric ansatzes. In the first, the collapsing metric is considered to be a separable function of the co-moving radius and time, and the collapse is shear-free, and in the second, a non-separable interior solution is considered, that represents gravitational collapse with non-zero shear viscosity. We arrive at novel solutions that indicate the formation of black holes or locally naked singularities, while obeying all the necessary energy conditions. The separable case allows for a simple analytic expression of the energy-momentum tensor, that indicates the positivity of the pressures throughout collapse, and is further used to study the heat flux evolution of the collapsing matter, whose analytic solutions are presented under certain approximations. These clearly highlight the role of modified gravity in the examples that we consider.

BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

A SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS

We give a simple argument which shows that Gabor systems consisting of odd functions of$d$variables and symplectic lattices of density$2^{d}$cannot constitute a Gabor frame. In the one-dimensional, separable case, this follows from a more general result of Lyubarskii and Nes [‘Gabor frames with rational density’,Appl. Comput. Harmon. Anal.34(3) (2013), 488–494]. We use a different approach exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.

POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

ON λ-STRICT IDEALS IN BANACH SPACES

We define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE HELMHOLTZ EQUATION CONCENTRATED NEAR THE AXIS OF A DEEP-WATER WAVEGUIDE IN A RANGE-INDEPENDENT MEDIUM

To derive the integral representation for the axial wave describing the interference of near-axial waves in an arbitrary deep-water waveguide in a range-independent medium in long-range acoustic propagation in the ocean, it is necessary to construct the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a narrow strip having a width of order ω-1/2, where ω is a cyclic frequency. In a range-independent (separable) case the desired solutions coincide with the principal term of the uniform asymptotic expansion as ω→∞ of normal modes when a mode number is a value of order of unity. In this paper, the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a range-independent ocean and which decrease exponentially outside a strip containing the axis are constructed in the form admitting generalization to the case of a range-dependent medium. The solutions are represented as the product of exponentials and parabolic cylinder functions whose arguments are infinite series in powers of ω-1/2. Coefficients of these series are found from a recurrent system of partial differential equations up to terms allowing to get a residual (difference between the left-hand side of the Helmholtz equation and zero) of order ω-1/2. Numerical results are obtained for medium parameters corresponding to the Munk canonical sound-speed profile.