183 research outputs found
Generalized Dirac operators on Lorentzian manifolds and propagation of singularities
We survey the correct definition of a generalized Dirac operator on a
Space--Time and the classical result about propagation of singularities. This
says that light travels along light--like geodesics. Finally we show this is
also true for generalized Dirac operators
A Note on BIBO Stability
The statements on the BIBO stability of continuous-time convolution systems
found in engineering textbooks are often either too vague (because of lack of
hypotheses) or mathematically incorrect. What is more troubling is that they
usually exclude the identity operator. The purpose of this note is to clarify
the issue while presenting some fixes. In particular, we show that a linear
shift-invariant system is BIBO-stable in the -sense if and only if
its impulse response is included in the space of bounded Radon measures, which
is a superset of (Lebesgue's space of absolutely integrable
functions). As we restrict the scope of this characterization to the
convolution operators whose impulse response is a measurable function, we
recover the classical statement
A calculus for magnetic pseudodifferential super operators
This work develops a magnetic pseudodifferential calculus for super operatorsOpA(F); these map operators onto operators (as opposed to Lp functions onto Lqfunctions). Here, F could be a tempered distribution or a H\"ormander symbol.An important example is Liouville super operators defined in terms of amagnetic pseudodifferential operator. Our work combines ideas from magneticWeyl calculus developed in [MP04, IMP07, Lei11] and the pseudodifferentialcalculus on the non-commutative torus from [HLP18a, HLP18b]. Thus, our calculusis inherently gauge-covariant, which means all essential properties of OpA(F)are determined by properties of the magnetic field B = dA rather than thevector potential A. There are conceptual differences to ordinary pseudodifferential theory. Forexample, in addition to an analog of the (magnetic) Weyl product that emulatesthe composition of two magnetic pseudodifferential super operators on the levelof functions, the so-called semi-super product describes the action of apseudodifferential super operator on a pseudodifferential operator.<br
An Urban Economic Model over a Continuous Plane with Spatial Characteristic Vector Field - Consideration of Heterogeneous Geographical Conditions -
Among others Beckmann (1952) firstly introduced the concept of a two dimensional continuous space into economics. This great step had unfortunately not shown further expansion in economics. Through several papers related to Beckmann's initiation, Beckmann and Puu (1985) at last reached a systematic treatment of the continuous spatial economics. Although their achievement is fascinated by employing a partial differential equations approach, Beckmann's original philosophy, that is, the gradient law has still been inherited. Following their achievement, Puu (2003) alone developed their theory by using many computer simulations to visually show the significance of their theory. Beckmann and Puu's book (1985) aims to study formation of urban configuration in a two dimensional continuous space, focusing on flows of commodities. However, consideration of households and firms location is not necessarily sufficient, resulting in reconsideration from a new urban economics point of view. As another exceptional urban economic study of a plane city, Lucus and Rossi-Hansberg (2002) is pointed out. They were inspired by Fujita and Ogawa (1982), and indicate endogenous land use pattern over a plane city. However they neglect commodity market to simplify the analysis. Well discussion about formation of urban configuration is summarized in Anas, Arnott, and Small (1998). Differing from Beckmann and Puu's studies, Miyata (2010) introduces bid rent functions (Fujita (1989)), which are familiar in the new urban economics, for land of households and firms, and then it studies how the results of Beckmann and Puu are rigorously modified by using the theory of partial differential equations (Courant and Hilbert (1953, 1962). However Miyata (2010) deals with a symmetric equilibrium which seems to be a little unrealistic. This article extends the author's previous study introducing spatial characteristic vector field in the model which stands for heterogeneity in geographical conditions in a city, and try to show asymmetry in land use pattern and endogenous formation of transport networks in a two dimensional continuous space.
On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus
Polyhomogeneous symbols, defined by Kohn-Nirenberg and H\"ormander in the
60's, play a central role in the symbolic calculus of most pseudodifferential
calculi. We prove a simple characterisation of polyhomogeneous functions which
avoids the use of asymptotic expansions. Specifically, if is open subset of
, then a polyhomogeneous symbol on is
precisely the restriction to of a function on
which is homogeneous for the dilations of modulo Schwartz
class functions. This result holds for arbitrary graded dilations on the vector
space . As an application, using the generalisation of A.~Connes'
tangent groupoid for a filtered manifold, we show that the Heisenberg calculus
of Beals and Greiner on a contact manifold or a codimension 1 foliation
coincides with the groupoid calculus of Van Erp and the second author
Conformal scattering for a nonlinear wave equation on a curved background
The purpose of this paper is to establish a geometric scattering result for a
conformally invariant nonlinear wave equation on an asymptotically simple
spacetime. The scattering operator is obtained via trace operators at null
infinities. The proof is achieved in three steps. A priori linear estimates are
obtained via an adaptation of the Morawetz vector field in the Schwarzschild
spacetime and a method used by H\"ormander for the Goursat problem. A
well-posedness result for the characteristic Cauchy problem on a light cone at
infinity is then obtained. This requires a control of the nonlinearity uniform
in time which comes from an estimates of the Sobolev constant and a decay
assumption on the nonlinearity of the equation. Finally, the trace operators on
conformal infinities are built and used to define the conformal scattering
operator
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