The magnetohydrodynamic (MHD) equations of incompressible viscous fluids with finite electrical conductivity describe the motion of viscous electrically conducting fluids in a magnetic field. In this paper, we find eight families of solutions of these equations by Xu's asymmetric and moving frame methods. A family of singular solutions may reflect basic characteristics of vortices. The other solutions are globally analytic with respect to the spacial variables. Our solutions may help engineers to develop more effective algorithms to find physical numeric solutions to practical models.  

We study the well-posedness of the equations with fractional derivative D^αu(t) = Au(t)+f(t) (0 ≤ t ≤ 2π), where A is a closed operator in a Banach space X, 0 < α < 1 and D^α is the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in L^p(0, 2π;X) or periodic continuous function spaces C_per([0, 2π];X), we show by using the method of sum that it is well-posed in some subspaces of L^p(0, 2π;X) or C_per([0, 2π];X).  

In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^∂ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T^∂ are given. The results are an answer to the problem posed by Nashed and Votruba.  

For a Tychonoff space X, we use ↓USC_F (X) and ↓C_F (X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0, 1] with the subspace topologies of the hyperspace Cld_F (X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology. In this paper, we shall show that there exists a homeomorphism h : ↓USC_F (X) → Q = [-1, 1]^ω such that h(↓C_F (X)) = c₀ = {(χ_n) ∈ Q| limn→∞ χ_n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.  

In this paper, the surplus of an insurance company is modeled by a Markovian regimeswitching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-switching economy consists of a fixed interest security and several risky shares. The optimal proportional reinsurance and investment strategies with no short-selling constraints for maximizing an exponential utility on terminal wealth are obtained.  

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.  

The learning approach of empirical risk minimization (ERM) is taken for the regression problem in the least square framework. A standard assumption for the error analysis in the literature is the uniform boundedness of the output sampling process. In this paper we abandon this boundedness assumption and conduct error analysis for the ERM learning algorithm with unbounded sampling processes satisfying an increment condition for the moments of the output. The key novelty of our analysis is a covering number argument for estimating the sample error.  

The V-system is a complete orthogonal system of functions defined on the interval [0, 1], generated by finite Legendre polynomials and the dilation and translation of a function generator, which consists of a finite number of continuous and discontinuous functions. The V-system has interesting properties, such as orthogonality, symmetry, completeness and short compact support. It is shown in this paper that the V-system is essentially a special multi-wavelet basis. As a result, some basic properties of the V-system are established through the well-developed theory of multi-wavelets. From this point of view, more other V-systems are constructed.  

We generalize Bangert's non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic R²ⁿ to asymptotically standard symplectic manifolds.  

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface. Some rather weaker size conditions, which imply the L^p-boundedness of such operators for certain fixed p (1 < p < ∞), are given.  

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter λ > 0 has at least two solutions (u_λ¹, u_λ²) for λ sufficiently large, which satisfy that u_λ¹ → -u₀ almost everywhere as λ → ∞, and that u_λ² → -∞ almost everywhere as λ → ∞, where u0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ→∞, and prove that u_λ² - u_λ² converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary ∂M is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.  

Given a vertex v of a graph G the second order degree of v denoted as d₂(v) is defined as the number of vertices at distance 2 from v. In this paper we address the following question: What are the sufficient conditions for a graph to have a vertex v such that d₂(v) ≥ d(v), where d(v) denotes the degree of v? Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K₄^--free graph or every maximal planar graph is shown to have a vertex v such that d₂(v) ≥ d(v). Other sufficient conditions on graphs for guaranteeing this property are also proved.  

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.  

We focus our attention to the set Gr(C) of grouplike elements of a coring C over a ring A. We do some observations on the actions of the groups U(A) and Aut(C) of units of A and of automorphisms of corings of C, respectively, on Gr(C), and on the subset Gal(C) of all Galois grouplike elements. Among them, we give conditions on C under which (C) is a group, in such a way that there is an exact sequence of groups {1} → U(A^g) → U(A) → GalC) → {1}, where A^g is the subalgebra of coinvariants for some g ∈ Gal(C).  

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.  

In this paper, we discuss the rational maps
F_λ(z)=zⁿ+λ/zⁿ, n≥ 2
with the positive real parameter λ. It is shown that the immediately attracting basin B_λ of ∞ for F_λ is always a Jordan domain if the Julia set of F_λ is not a Cantor set. Furthermore, B_λ is a quasidisk if there is no parabolic fixed point on the boundary of B_λ. It is also shown that if the Julia set of F_λ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpiński curve is given.

In this paper, we study affine spheres which are isotropic and we obtain a complete classification. In particular, we show that all such affine spheres are hyperbolic affine spheres, isometric with SL(3,R)/SO(3), SL(3,C)/SU(3), SU^*(6)/Sp(3) or E₆(?26)/F₄.

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.

In this paper, we study a class of stochastic processes, called evolving network Markov chains, in evolving networks. Our approach is to transform the degree distribution problem of an evolving network to a corresponding problem of evolving network Markov chains. We investigate the evolving network Markov chains, thereby obtaining some exact formulas as well as a precise criterion for determining whether the steady degree distribution of the evolving network is a power-law or not. With this new method, we finally obtain a rigorous, exact and unified solution of the steady degree distribution of the evolving network.

By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.

A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.

The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let  be the set of all trees of order n. In this paper, we will provide the ordering of trees in  up to the last eight trees according to their smallest algebraic connectivities when n≥13. This extends the result of Shao et al. [The ordering of trees and connected graphs by algebraic connectivity. Linear Algebra Appl., 428, 1421-1438 (2008)].

In this paper, we are concerned with the existence and uniqueness of global smooth solution for the Robin boundary value problem of Landau-Lifshitz equations in one dimension when the boundary value depends on time t. Furthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.

Let {u_s(x): s≥0, x∈R} be a random string taking values in R^d. The main goal of this paper is to discuss the characteristics of the polar functions of {u_s(x:s≥0, x∈R}. The relationship between a class of continuous functions satisfying the Hölder condition and a class of polar-functions of {u_s(x:s≥0, x∈R} is presented. The Hausdorff and packing dimensions of the set that the string intersects a given non-polar continuous function are determined. The upper and lower bounds are obtained for the probability that the string intersects a given function in terms of respectively Hausdorff measure and capacity.

In this paper, we firstly give the definition of meromorphic function element and algebroid mapping. We also construct the algebroid function family in which the arithmetic, differential operations are closed. On basis of these works, we firstly prove the Second Main Theorem concerning small algebroid functions for v-valued algebroid functions.

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 < ε < 1/2. Then for every bounded linear operator T: H → H and x₀∈ H with ‖T‖=1=‖x₀‖ such that ‖Tx₀‖ > 1?ε, there exist x_ε∈H and a bounded linear operator S:H→H with ‖S‖=1=‖x_ε‖ such that
‖Sx_ε‖=1,‖x_ε-x₀‖≤√2ε+⁴√2ε,‖S-T‖≤√2ε.

Here we introduce a subclass of the class of Ockham algebras (L; f) for which L satisfies the property that for every x ∈ L, there exists n ≥ 0 such that fⁿ(x) and fⁿ⁺¹(x) are complementary. We characterize the structure of the lattice of congruences on such an algebra (L; f). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.

A natural compactification of the virtual configuration space of N points on the Riemann sphere ? is constructed by using cross-ratios. We show that this compactification is homeomorphic to the Bers’ compactification of the virtual moduli space of a punctured Riemann sphere of type N. In particular, the system of global and explicit coordinates of this standard compactification is given by cross-ratios.

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation ∂_tf_ε+v·▽xf_ε=Q(f_ε, f_ε)+εΔ_vf_ε as ε→0⁺.We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L¹((0, T)×R^N×R^N). The proof is based on compactness analysis and velocity averaging theory.

We give a rigorous proof of the equivalence of Mañé’s supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather’s α- function is discussed.

A graph G satisfies the Ore-condition if d(x)+d(y) ≥ |V (G)| for any xy (2161) E(G). Luo et al. [European J. Combin., 2008] characterized the simple Z₃-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z₃-connected graphs G satisfying d(x) + d(y) ≥ |V (G)| - 1 for any xy  E(G), which improves the results of Luo et al.

A family (X, B₁), (X, B₂), . . . , (X, B_q) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTS_λ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTS_λ(v) if there does not exist an LSTS_λ' (v) contained in the collection for any λ' < λ. In this paper, we show that for λ = 5, 6, there is an IDLSTS_λ(v) for v ≡ 1 or 3 (mod 6) with the exception IDLSTS₆(7).

In this paper, we study the chromatic sum functions of rooted nonseparable neartriangulations on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of such maps are derived. Applying chromatic sum theory, the enumerating problem of different sorts maps can be studied, and a new method of enumeration can be obtained. Moreover, an asymptotic evaluation and some explicit expression of enumerating functions are also derived.

Kôpka’s D-poset is a very important notion in quantum structures. In this paper the conditional probability on the Kôpka’s D-posets is studied. The notion of conditional probability is introduced and the basic properties of conditional probability are proved.

We analyze the learning rates for the least square regression with data dependent hypothesis spaces and coefficient regularization algorithms based on general kernels. Under a very mild regularity condition on the regression function, we obtain a bound for the approximation error by estimating the corresponding K-functional. Combining this estimate with the previous result of the sample error, we derive a dimensional free learning rate by the proper choice of the regularization parameter.

The q-deformation of W(2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W(2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W(2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W(2, 2) Lie algebra in the q → 1 limit.

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrödinger equation 

where 2 < p < ∞, α₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^-α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H¹(R²)-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrödinger equation -Δu + V (ξ)u = K(ξ)|u|^p-2 ue^α0|u|γ has local minimum points. Furthermore, the concentration property of u_ε is also established as ε tends to zero.

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z₂-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z₂-equivariant non-analytic cubic differential systems.

A Hilbert transform for Hölder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in R²ⁿ, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Hölder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Hölder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.