185 research outputs found
Renormalization and blow up for charge one equivariant critical wave maps
We prove the existence of equivariant finite time blow up solutions for the
wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the
sum of a dynamically rescaled ground-state harmonic map plus a radiation term.
The local energy of the latter tends to zero as time approaches blow up time.
This is accomplished by first "renormalizing" the rescaled ground state
harmonic map profile by solving an elliptic equation, followed by a
perturbative analysis
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
A new approach to hyperbolic inverse problems
We present a modification of the BC-method in the inverse hyperbolic
problems. The main novelty is the study of the restrictions of the solutions to
the characteristic surfaces instead of the fixed time hyperplanes. The main
result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a
part of the boundary uniquely determines the coefficients of the self-adjoint
hyperbolic operator up to a diffeomorphism and a gauge transformation. In this
paper we prove the crucial local step. The global step of the proof will be
presented in the forthcoming paper.Comment: We corrected the proof of the main Lemma 2.1 by assuming that
potentials A(x),V(x) are real value
Concerning the Wave equation on Asymptotically Euclidean Manifolds
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the
wave equation on , , when metric
is non-trapping and approaches the Euclidean metric like with
. Using the KSS estimate, we prove almost global existence for
quadratically semilinear wave equations with small initial data for
and . Also, we establish the Strauss conjecture when the metric is radial
with for .Comment: Final version. To appear in Journal d'Analyse Mathematiqu
Conormal distributions in the Shubin calculus of pseudodifferential operators
We characterize the Schwartz kernels of pseudodifferential operators of
Shubin type by means of an FBI transform. Based on this we introduce as a
generalization a new class of tempered distributions called Shubin conormal
distributions. We study their transformation behavior, normal forms and
microlocal properties.Comment: 23 page
Strichartz Estimates for the Vibrating Plate Equation
We study the dispersive properties of the linear vibrating plate (LVP)
equation. Splitting it into two Schr\"odinger-type equations we show its close
relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces
appear to be the natural setting to show Strichartz-type estimates for the LVP
equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we
prove the well-posedness of the Cauchy problem for the LVP equation with
time-dependent potentials. Finally, we exhibit the sharpness of our results.
This is achieved by finding a suitable solution for the stationary homogeneous
vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte
Spectral Analysis for Matrix Hamiltonian Operators
In this work, we study the spectral properties of matrix Hamiltonians
generated by linearizing the nonlinear Schr\"odinger equation about soliton
solutions. By a numerically assisted proof, we show that there are no embedded
eigenvalues for the three dimensional cubic equation. Though we focus on a
proof of the 3d cubic problem, this work presents a new algorithm for verifying
certain spectral properties needed to study soliton stability. Source code for
verification of our comptuations, and for further experimentation, are
available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe
A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem
We consider a transmission wave equation in two embedded domains in ,
where the speed is in the inner domain and in the outer
domain. We prove a global Carleman inequality for this problem under the
hypothesis that the inner domain is strictly convex and . As a
consequence of this inequality, uniqueness and Lip- schitz stability are
obtained for the inverse problem of retrieving a stationary potential for the
wave equation with Dirichlet data and discontinuous principal coefficient from
a single time-dependent Neumann boundary measurement
On the uniqueness and global dynamics of AdS spacetimes
We study global aspects of complete, non-singular asymptotically locally AdS
spacetimes solving the vacuum Einstein equations whose conformal infinity is an
arbitrary globally stationary spacetime. It is proved that any such solution
which is asymptotically stationary to the past and future is itself globally
stationary.
This gives certain rigidity or uniqueness results for exact AdS and related
spacetimes.Comment: 18pp, significant revision of v
Primary liver cancer in the UK: Incidence, incidence-based mortality, and survival by subtype, sex, and nation
Background & Aims: The incidence of primary liver cancer (PLC) is increasing in Western Europe. To understand trends over time and the current burden in the UK, a detailed analysis of the epidemiology of PLC and its subtypes was conducted. Methods: Data on PLCs diagnosed during 1997-2017 were obtained from population-based, nationwide registries in the UK. European age-standardised incidence (ASR) and incidence-based mortality rates (ASMR) per 100,000 person-years were calculated overall and by sex and UK-nation. Annual percentage change in rates was estimated using Joinpoint regression. One-, 2-, and 5-year age-standardised net survival was estimated. Results: A total of 82,024 PLCs were diagnosed. Both hepatocellular carcinoma (HCC) incidence and mortality rates trebled (ASR 1.8-5.5 per 100,000, ASMR 1.3-4.0). The rate of increase appeared to plateau around 2014/2015. Scottish men consistently had the highest HCC incidence rates. PLC survival increased, driven by a substantial increase in the proportion that are HCC (as prognosis is better than other PLCs) and in HCC survival (change in 1-year survival 24-47%). Intrahepatic cholangiocarcinoma was the most common PLC in women and 1-year survival improved from 22.6% to 30.5%. Conclusions: PLC incidence has been increasing rapidly but, as most risk factors are modifiable, it is largely a preventable cancer. This rate of increase has slowed in recent years, possibly attributable to effective treatment for hepatitis C. As other risk factors such as obesity and diabetes remain prevalent in the UK, it is unlikely the considerable burden of this disease will abate. While improvements in survival have been made, over half of patients are not alive after 1 year, therefore further progress in prevention, early detection, and treatment innovation are needed. Lay summary: Many more people are getting liver cancer, particularly the subtype hepatocellular carcinoma, than 20 years ago. Men in Scotland are most likely to get liver cancer and to die from it. Survival after liver cancer diagnosis is getting longer but still less than half are alive after 1 year
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