Characterization of Bounded Symmetric Domain by Carathèodory and Eisenmann-Kobayashi Volume Elements

1 IntroductionClassification of bounded domain is an important problem in several complex variables.The best result was got by E. Cartan. He proved that any boundd symmetric domain must bebiholimorphic to one of the following domains or their topological …     

On the Stability of Eaton’s Characterization by the Properties of Linear Forms

We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics $(X_{1}+\cdots +X_{k_{1}})/k_{1}^{1/\alpha}We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics and have the same distribution as the monomial X ₁, then this monomial has a symmetric stable distribution of order α. The stability estimation in this theorem is investigated in the λ ₀-metric.      

Explosion of Jump-Type Symmetric Dirichlet Forms on ℝ d

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? ^d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? ^d .     

Equivalent Semi-Norms of Non-Local Dirichlet Forms on the Sierpiński Gasket and Applications

We construct equivalent semi-norms of non-local Dirichlet forms on the Sierpiński gasket and apply these semi-norms to a convergence problem and a trace problem. We also construct explicitly a sequence of non-local Dirichlet forms with jumping kernels equivalent to |x ? y|^?α?β that converges exactly to local Dirichlet form.     

Variational Formula for Dirichlet Forms and Estimates of Principal Eigenvalues for Symmetric α-stable Processes

In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure charging no set of zero capacity, we give a new proof of an L²()-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed -stable processes and obtain conditions for some measures being gaugeable. Mathematics Subject Classifications (2000) Primary 31C25; Secondary 34L15, 60G52.     

Stability-enhancing control of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement

We investigate the problem of enhancing the stability of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement. In the recent paper [H. Sano, Neumann boundary control of a coupled transport–diffusion system with boundary observation, J. Math. Anal. Appl. 377 (2011) 807–816], we treated the stabilization problem for the case with Neumann actuation and Dirichlet measurement, where the variable transformation of the state is performed by using the fractional power of an unbounded operator. However, we cannot use the similar transformation for the case with Dirichlet actuation and Dirichlet measurement, since it brings an ill-posed expression of the system. So, we use an algebraic approach for the formulation of the system. In this paper, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. The fact enables us to construct a finite-dimensional stability-enhancing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach. The novelty of this paper is the structure that the controller contains the dynamics with respect to the control variable. As a result, the state vector of the resulting closed-loop system includes the control variable as its entry.     

Characterization of Prüfer-Like Monoids and Domains by GCD-Theories

We combine ideal-theoretic and divisor-theoretic methods to characterize various classes of Prüfer-like monoids and domains by the gcd-properties of certain semigroups of invertible ideals.     

Characterization of Day–James Spaces and Generalized Day–James Spaces

In this paper, we introduce the definition of generalized Day–James space on R~n(n ≥2) and give a characterization of it, which extend some known results. In addition, we provide a sufficient and necessary condition for Day–James space, which reappeared Day's construction for any two-dimensional normed space to make Birkhoff orthogonality symmetry.     

Approximation of conjugate differentiable functions by their Abel–Poisson integrals

We obtain the exact values of upper bounds of approximations of classes of periodic conjugate differentiable functions by their Abel–Poisson integrals in uniform and integral metrics. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 73–82, January, 2009.     

Complex Dirichlet Forms: Non Symmetric Diffusion Processes and Schrödinger Operators

The concept of complex Dirichlet forms ^c resp. operators L _c in complex weighted L ²-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.Complex Dirichlet operators L _c are unitarily equivalent with (a family of) Schrödinger operators with electromagnetic potentials.To ^c there is associated a pair of real-valued non symmetric Dirichlet forms on the corresponding real weighted L ²-spaces, which in turn are associated with (non-symmetric) diffusion processes.Results by Stannat on non symmetric Dirichlet forms and their perturbations can be used for discussing the essential self-adjointness of L _c .New closability criteria for (perturbation of) non symmetric Dirichlet forms are obtained.     

Pointwise Properties of Eigenfunctions and Heat Kernels of Dirichlet–Schrödinger Operators

Let D be an open domain, a second order elliptic operator with continuous coefficients, and let be a Schrödinger operator associated with H₀, acting on L²(D), with Dirichlet boundary conditions. We provide in this paper both L² and pointwise bounds for the eigenfunctions of H, in term of the Agmon's metric of q and of the quasi-hyperbolic geometry of D. At least when H₀=-, we show that the pointwise bounds obtained for the ground state eigenfunction of H are qualitatively sharp either when q diverges sufficiently fast at the boundary, or in planar regular domains. We also give applications to the intrinsic ultracontractivity of H. Finally, we prove a result concerning the pointwise decay at the boundary of the heat kernel of H in -regular domains.     

Balayages on Excessive Measures,their Representation and the Quasi-Lindelöf Property

If Exc is the set of all excessive measures associated with a submarkovian resolvent on a Lusin measurable space and B is a balayage on Exc then we show that for any mExc there exists a basic set A (determined up to a m-polar set) such that B=(B^A)* for any Exc, m. The m-quasi-Lindelöf property (for the fine topology) holds iff for any B there exists the smallest basic set A as above. We characterize the case when any B is representable i.e. there exists a basic set such that B=(B^A)* on Exc.     

Characterization of Levi-Civita and Newton–Cartan connections in dimension 2

In dimension 2, we give a local characterization of Levi-Civita connections, improving a result by G. Thompson (1991), and of Newton–Cartan connections (connections which are limit of Levi-Civita connections).     

An Integrated Version of Varadhan’s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.