In this paper, the ill-posed Cauchy problem for two-dimensional Helmholtz equation with mixed boundary is investigated. To obtain stable numerical solution, a mollification regularization method with the de la Vallée Poussin operator is proposed. Error estimate between the exact solution and its approximation is given under the proper choice of a priori parameter. A numerical experiment shows that our procedure is effective and stable with respect to perturbations of noise in the data.
Hamiltonian of a classical quantum system is a self-adjoint operator which ensures that the energy eigenvalues are real and the eigenstates (unit eigenvectors) form an orthonormal basis for the state space. However, there exists the parity-time-reversal (PT) symmetric physical system, the Hermiticity (transpose and complex conjugate) of a Hamiltonian is replaced by the physically transparent condition of PT-symmetry. If a Hamiltonian has an unbroken PT-symmetry, then the spectrum is real and further more one can construct a CPT inner product with a positive-definite inner product. In this paper, we discuss the PT-symmetric operator in the system. First, given the fixed time reversal operator T as the complex conjugation, the matrix representations of both the parity operator P and PT-symmetric Hamiltonian H are obtained. Then all possible concrete forms of P and the corresponding forms of H are expressed. Next, as an application, it is established that PT-symmetric quantum theory for realizing the discrimination of two quantum states which are not orthogonal in the conventional quantum mechanics.
In this paper, we discuss several basic properties of a class of quasiconformal close-to-convex harmonic mappings with starlike analytic part, such results as coefficient inequalities, an integral representation, a growth theorem, an area theorem, and radii of close-to-convexity of partial sums of the class, are derived.
The Pang-Zalcman lemma is an important tool to study normal families of meromorphic functions. In this paper, we extend Pang-Zalcman lemma to the case of holomorphic functions of several complex variables and establish some normality criteria as applications.
This paper is concerned with a non-uniform flexible structure with thermal effect governed by Coleman-Gurtin law. By using semi-group method, we establish the global well-posedness of the system. The main result is the long-time dynamics of the system. We prove the quasi-stability property of the system and obtain the existence of a global attractor, and the global attractor has finite fractal dimension. The existence of exponential attractors is also proved.
We propose a new multidimensional filter algorithm with a nonmonotone trust-region strategy for linear inequality constrained optimization. The objective function and the components of its projection gradient constitute a new multidimensional filter which is related to the trust-region radius. When the trust-region radius is small enough, the new filter can accept the trial point to avoid the infinite cycle of the algorithm. The nonmonotone trust-region strategy maintains global convergence of the new algorithm. The analysis of local convergence on multidimensional filter algorithms is a problem that has not been solved so far. We analyze the local superlinear convergence of the new algorithm under some suitable conditions. Numerical results show that the new approach is efficient.
In this paper, sets of the form[0,1]×Z in Rd will be called sets of Tyson type, where d > 1 and Z ⊂ Rd-1. It is known that every set E of Tyson type is quasisymmetrically minimal, provided that Z is compact in Rd-1. We prove that this result holds by only assuming that Z is Borel in Rd-1. As applications, we prove that the quasisymmetric minimality of three variants of sets of Tyson type. One of the results is that the graph G(h) of h is quasisymmetrically minimal, provided that Z is a Borel set in Rd-1 and h:Z → R1 is a Borel function satisfying the condition dimH({h ≠ 0} ∩ Z)=dimH Z, where G(h)={(z, y):z ∈ Z, y ∈[0, h(z)]}.
Let X and Y be normed space. We say that a map f:X→Y is a phaseisometry if it satisfies {||f(x) + f(y)||,||f(x)-f(y)||}={||x + y||,||x-y||} (x, y ∈ X). Suppose that g, f:X→Y are maps. If there is a phase function ε:X→{-1, 1} such that ε·f=g, then we say that f is phase-equivalent to g. We shall prove that every phase-isometry between two modified Tsirelson spaces TM is phase-equivalent to a linear isometry. This can be considered as a new version of the famous Wigner's theorem for the modified Tsirelson space TM.
In this note, we establish some new p-adic Hardy-Littlewood-Pólya-type inequalities with the best constant factors. The equivalent forms of these inequalities and some particular cases are also considered.
The theory of operators on Hilbert spaces is one of fundamental frameworks of quantum mechanics. Hilbert space effect algebra, which is the convex set of positive operators between 0 and the identity, is one of important aspects in quantum mechanics. In the paper, we introduce a kind of sub-sequential effect algebra and explore some algebraic properties of the sequential product on it. We show that these properties on the sub-sequential effect algebra is different from those of existing ones.
In this paper, a parametered Jordan-von Neumann type constant is introduced and estimated. We also give some sufficient conditions of which a Banach space has the normal structure by discussing the relationship among the parametrized Jordan-von Neumann constant, the weak orthogonality coefficient and the Domínguez Benavides coefficient, respectively. These results further improve some results in the previous literatures.
In this paper, we first introduce the definition of a Hom-Poisson bialgebra and give an equivalent description via the Manin triple of Hom-Poisson algebras. Also we introduce the notion of coboundary Hom-Poisson bialgebras, and construct solutions of Hom-Possion Yang-Baxter equations.