We develop c-supplemented subalgebras, E-algebras and Frattini theory of Lie algebras for Lie-Rinehart algebras, obtain its some important properties and give a necessary conditions for solvable Lie-Rinehart algebras. Moreover, we obtain a necessary and sufficient conditions for E-Lie-Rinehart algebras and c-supplemented Lie-Rinehart algebras, respectively.
On the basis of the Prüfer-Baer theorem of the bounded module over a principal ideal domain, this paper study several basic problems about the algebraic linear transformation of some vector space (infinite dimensional). Let V be a vector space (infinite dimensional) over a field F, A be an algebraic linear transformation of V:
(1) Suppose any linear transformation commuting with A commutes also with a linear transformation B, then B=f(A), where f is a polynomial over F.
(2) There exists a basis for V such that the matrix of A relative to this basis has the rational canonical form (classical canonical form). Moreover the classical canonical form becomes the Jordan canonical form when F is algebraic closed.
(3) There exists the Jordan-Chevalley decomposition of A when F is algebraically closed.
This result prevails for the perfect field in general. These results extend some theorems of finite dimensional vector spaces to infinite dimensional vector spaces.
We study the properties of the category of the Yetter-Drinfeld modules over a weak Hopf algebra, and give sufficient condition for the Yetter-Drinfeld category to be semisimple.
We establish the fundamental W1,p estimate for the weak solution of a system in a bounded domain Ω in R3. The system is related to the steady-state of Maxwell's equations for the magnetic field. The inverse of the principle coefficient matrix is assumed to be in the VMO space. We transform the system to scalar elliptic equations by using the properties of curl and divergence of vector fields in R3. By the regularity theory of elliptic equations, we get the W1,p estimate for 1 < p < ∞.
Let dwI denote the class of #-injective complexes of left R-modules (i.e., complexes of injective left R-modules). We prove that over left noetherian rings R, the pair (⊥(dwI), dwI) is a perfect injective cotorsion pair. In particular, we get that every complex of left R-modules has a #-injective envelope. As an application, we prove that over left noetherian rings R, every complex of left R-modules has a special Etac (I)-preenvelope, where Etac (I) is the class of complete acyclic complexes of injective left R-modules.
We study the conformal transformations between two (α, β)-metrics. We prove that, if F is a locally dually flat regular (α, β)-metric and is conformally related to F, that is, F=eσ(x)F, then F is also a locally dually flat (α, β)-metric if and only if the conformal transformation is a homothety. Further, in the case with singularity, we prove that any conformal transformation between two locally dually flat general Kropina metrics must be a homothety.
We consider two Toeplitz operators Tu and Tv on the generalized Fock space over the complex plane C. Let's assume that u is a radial function and the two operators commute. Under certain growth condition at infinity of u and v, we prove that v must be a radial function as well. Finally, we also construct a Sp class of Toeplitz operators on the generalized Fock space with symbols which are essentially unbounded on any point of the complex plane C.
Fixed point theorems for several cyclic contractive mappings are established in dislocated quasi-b-metric spaces. The results obtained in this paper improve and unify some previous results in the existing literature. Moreover, several nontrivial examples are given to illustrate the superiority of the main results.
We study the existence of solutions to nonlinear impulsive boundary value problems with small non-autonomous perturbations on the half-line. We show the existence of at least three distinct classical solutions by using variational methods and a three critical points theorem.
We study preliminary properties and algebraic properties of Bergman-type Toeplitz operators which are induced by harmonic symbols on the Dirichlet space, including self-adjointness, products, commutativity and invertibility. Moreover, the spectra of the Toeplitz operator are calculated.
The main purpose of this paper is to study the growth and approximation on entire functions represented by Laplace-Stieltjes transforms of finite logarithmic order convergent on the whole complex plane, and obtain some results about the logarithmic order, the logarithmic type, the error, and the coefficients of Laplace-Stieltjes transforms which are generalization and improvement of the previous results given by Luo and Kong, Singhal and Srivastava.
Hamiltonian of a classical quantum system is a self-adjoint operator. The self-adjoint property of a Hamiltonian not only ensures that the system follows unitary evolution, but also ensures that it has real energy eigenvalues. However, there exist indeed some physical systems, their Hamiltonians are nonself-adjoint, but also have real energy eigenvalues. The systems with nonself-adjoint Hamiltonians are called nonself-adjoint quantum systems (NSAQSs). The systems with pseudo self-adjoint Hamiltonians are a special class of NSAQSs, their Hamiltonians are similar to selfadjoint operators. In this paper, we discuss unitary evolution and adiabatic theorem of pseudo self-adjoint quantum systems (PSAQSs). First, pseudo self-adjoint operators are defined and characterized. Second, a given time-dependent pseudo self-adjoint Hamiltonian H(t) is self-adjoint with respect to a new time-dependent inner product, and then a condition for the system to be unitary evolving under the new inner product is derived. Last, adiabatic evolving theorem and adiabatic approximation theorem for a PSAQS are proved.
This paper is concerned with traveling waves for the nonlocal dispersal equation with the state-dependent delay. If the birth function is monotone, then the existence and nonexistence of monotone traveling waves are established. By a prior estimate and Ikehara's Theorem, we obtain the asymptotic behavior of critical traveling wave fronts. Finally, by introducing two auxiliary quasi-monotone equations, we improve our results of existence to the non-quasi-monotone equation.
The aim of this paper is to use the definition of the regular integers modulo a positive integer q and the analytic method to study the computational problem of one kind sums related to Dedekind sums, and give some interesting identities for the sums at some special integer points.
We obtain some boundedness results for the θ-type Calderón-Zygmund operators Tθ under natural regularity assumptions on a class of generalized Lebesgue spaces with weight and variable exponent. Furthermore, the boundedness of Tθ is established on the weighted variable Herz and Herz-Morrey spaces based on the above conclusions. We also prove the boundedness of the corresponding commutator[b, Tθ] in the generalized weighted Morrey spaces with variable exponent.
Let r:D → R3 be a minimal surface M with isothermal parameter, where D is a domain in R2, then the Gauss map of minimal surface is a meromorphic function on D. Supposing g(z) to be an arbitrary meromorphic function on D ⊂ C, whether there exists a complete minimal surface M, such that g(z) is the Gauss map of M, which is proposed by Xavier and Chao is still to be unsolved. In this paper, we prove that for the meromorphic function g(z) on C with the property that either the exponent of convergence of zeros or the exponent of convergence of poles is less than 1/2, g(z) must be the Gauss map of some complete minimal surface.
Let R be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category introduced in (see[Models for singularity categories, Adv. Math., 2014, 254:187-232]). As an application, we explicitly give an equivalence Kex(P)?Kex(I) for the homotopy categories of exact complexes of projective and injective modules.
