539 research outputs found
Matrix probing and its conditioning
When a matrix A with n columns is known to be well approximated by a linear
combination of basis matrices B_1,..., B_p, we can apply A to a random vector
and solve a linear system to recover this linear combination. The same
technique can be used to recover an approximation to A^-1. A basic question is
whether this linear system is invertible and well-conditioned. In this paper,
we show that if the Gram matrix of the B_j's is sufficiently well-conditioned
and each B_j has a high numerical rank, then n {proportional} p log^2 n will
ensure that the linear system is well-conditioned with high probability. Our
main application is probing linear operators with smooth pseudodifferential
symbols such as the wave equation Hessian in seismic imaging. We demonstrate
numerically that matrix probing can also produce good preconditioners for
inverting elliptic operators in variable media
Riesz transforms on generalized Heisenberg groups and Riesz transforms
Let 1 < q < \infty. We prove that the Riesz transforms on a generalized Heisenberg group satisfy where , are respectively the dimensions of the first and second layer of the Lie algebra of . We prove similar inequalities on Schatten spaces , with dimension free constants, for Riesz transforms associated to commuting inner -derivations and a suitable substitute of the square function. An example is given by the derivations associated to commuting pairs of operators on a Hilbert space satisfying the canonical commutation relations [P
Dark energy, Ricci-nonflat spaces, and the Swampland
It was recently pointed out that the existence of dark energy imposes highly
restrictive constraints on effective field theories that satisfy the Swampland
conjectures. We provide a critical confrontation of these constraints with the
cosmological framework emerging from the Salam-Sezgin model and its string
realization by Cvetic, Gibbons, and Pope. We also discuss the implication of
the constraints for string model building.Comment: Matching version to be published in PL
Structured Random Matrices
Random matrix theory is a well-developed area of probability theory that has
numerous connections with other areas of mathematics and its applications. Much
of the literature in this area is concerned with matrices that possess many
exact or approximate symmetries, such as matrices with i.i.d. entries, for
which precise analytic results and limit theorems are available. Much less well
understood are matrices that are endowed with an arbitrary structure, such as
sparse Wigner matrices or matrices whose entries possess a given variance
pattern. The challenge in investigating such structured random matrices is to
understand how the given structure of the matrix is reflected in its spectral
properties. This chapter reviews a number of recent results, methods, and open
problems in this direction, with a particular emphasis on sharp spectral norm
inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and
Applications" (Springer
Asymptotically (anti)-de Sitter solutions in Gauss-Bonnet gravity without a cosmological constant
In this paper we show that one can have asymptotically de Sitter (dS),
anti-de Sitter (AdS) and flat solutions in Gauss-Bonnet gravity without any
need to a cosmological constant term in field equations. First, we introduce
static solutions whose 3-surfaces at fixed and have constant positive
(), negative (), or zero () curvature. We show that for
, one can have asymptotically dS, AdS and flat spacetimes, while for
the case of , one has only asymptotically AdS solutions. Some of these
solutions present naked singularities, while some others are black hole or
topological black hole solutions. We also find that the geometrical mass of
these 5-dimensional spacetimes is , which is different from
the geometrical mass, , of the solutions of Einstein gravity. This feature
occurs only for the 5-dimensional solutions, and is not repeated for the
solutions of Gauss-Bonnet gravity in higher dimensions. We also add angular
momentum to the static solutions with , and introduce the asymptotically
AdS charged rotating solutions of Gauss-Bonnet gravity. Finally, we introduce a
class of solutions which yields an asymptotically AdS spacetime with a
longitudinal magnetic field which presents a naked singularity, and generalize
it to the case of magnetic rotating solutions with two rotation parameters.Comment: 13 pages, no figur
Governance in Service Delivery in the Middle East and North Africa. World Development Report Background Paper
This paper examines the clientelistic equilibrium that remains prevalent in much of the Middle East and North Africa (MENA) region during the post-independence period, undermining service delivery and creating inequality in access. Political institutions and social practices that shape incentives for policymakers, service providers, and citizens create what can be called a potentially tenuous, “clientelistic equilibrium.” Service delivery is influenced by political institutions that allow for the capture of public jobs and service networks, and by social institutions that call upon individuals to respond more readily to members of their social networks than to others. The result is poor quality service delivery (e.g., absenteeism, insufficient effort), difficulties in access (e.g., need for bribes, connections), and inequalities in the provision of services
Planet Formation in the Outer Solar System
This paper reviews coagulation models for planet formation in the Kuiper
Belt, emphasizing links to recent observations of our and other solar systems.
At heliocentric distances of 35-50 AU, single annulus and multiannulus
planetesimal accretion calculations produce several 1000 km or larger planets
and many 50-500 km objects on timescales of 10-30 Myr in a Minimum Mass Solar
Nebula. Planets form more rapidly in more massive nebulae. All models yield two
power law cumulative size distributions, N_C propto r^{-q} with q = 3.0-3.5 for
radii larger than 10 km and N_C propto r^{-2.5} for radii less than 1 km. These
size distributions are consistent with observations of Kuiper Belt objects
acquired during the past decade. Once large objects form at 35-50 AU,
gravitational stirring leads to a collisional cascade where 0.1-10 km objects
are ground to dust. The collisional cascade removes 80% to 90% of the initial
mass in the nebula in roughly 1 Gyr. This dust production rate is comparable to
rates inferred for alpha Lyr, beta Pic, and other extrasolar debris disk
systems.Comment: invited review for PASP, March 2002. 33 pages of text and 12 figure
- …