Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

Resolvent estimates and smoothing for homogeneous partial differential operators on graded Lie groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|^1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?^N some new classes of partial differential operators are treated.     

Approximations of spectra of Schrödinger operators with complex potentials on ℝd

We study spectral approximations of Schrödinger operators T = ?Δ+Q with complex potentials on Ω = ?^d, or exterior domains Ω??^d, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ω_n, and of boundary conditions on ?Ω_n such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators T_n without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1,2,3, illustrate our results.     

解析Sobolev型空间上的算子与算子代数

介绍Hardy-Sobolev空间和Fock空间及其算子与算子代数研究方面所做的工作,包括对这两类空间上几类特殊算子有界性、紧性、Fredholm性、指标理论、谱和本性谱、范数和本性范数、Schatten-p类的讨论,以及由它们所生成的C~*-代数的研究.     

保持拟可逆性或拟零因子的可加映射

设X和Y是维数大于1的复Banach空间,A和B分别是B(X)和B(Y)中包含有限秩算子的范数闭子代数.A,B∈A,定义A。B=A+B-AB,称。为A,B的拟积.刻画了从A到B的双边保持算子的(左,右)拟可逆性或(左,右,半)拟零因子的可加满射的结构.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.     

A self‐adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application

The self‐adaptive intelligence gray predictive model (SAIGM) has an alterable‐flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.