THE GROWTH OF SOLUTIONS OF SYSTEMS OF COMPLEX NONLINEAR ALGEBRAIC DIFFERENTIAL EQUATIONS

We investigate the problem of growth order of solutions of a type of systems of non-linear algebraic differential equations, and extend some results of the growth order of solutions of algebraic differential equations to systems of algebraic differential equations.     

拟线性可化约的双曲型方程组的奇性分析

In this paper we investigate the formation of singularities of hyperbolic systems. Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.     

ESTIMATES OF N-FUNCTION AND m-FUNCTION OF MEROMORPHIC SOLUTIONS OF SYSTEMS OF COMPLEX DIFFERENCE EQUATIONS

We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.     

Bifurcation of Equilibria in a Class of Planar Piecewise Smooth Systems with 3 Parameters

The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.     

WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

In this paper,we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces.Then we study the weak convergence of the two iterative sequences.Our results improve and extend the results announced by many others.     

Lagrange插值逼近导数的平均收敛

<正>We consider the rate of mean convergence of derivatives by Lagrange interpolation operators L_n(f,x) based on the zeros of Chebyshev polynomials of the first kind.A sharp estimate of the derivative of L_n(f,x)—f(x) in terms of the error of best approximation by polynomials of degree n is derived.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

WEIGHTED NONLINEAR REGRESSION

Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.     

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs （W,x）, where ∈ X and W is a closed downward subset of X     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Nonexistence in power and gamma density regression, sum of nonnegative terms

When we use the power function α(c x)^b and gamma density αx^be^-cx to fit the data by the least squares method, we have to address the question of existence. The closure of the set of each type of these functions defined on a finite domain is determined. We derive a way to determine the closure of a sum of nonnegative functions if the closures of the summands are available.     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Schreier rewriting beyond the classical setting Dedicated to our teacher Alfred Lvovich Shmelkin on his 70th birthday

Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras.     

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τ_t(ω)}_ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τ_t(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.     

Absolute Continuity of Measures in the Class of Markov and Semi-Markov Processes of Diffusion Type

The property of absolute continuity of measures in the class of one-dimensional semi-Markov processes of diffusion type is investigated. The measure of such a process can be composed of two measures. The first one is a distribution of a random track, and the second one is a conditional distribution of a time run along the track. The desired density is represented in the form of product of two corresponding densities.     

Toward eudaimonology: notes on a quantitative framework for the study of happiness

After noting factors (concern for others, ignorance, irrationality) accounting for the divergences between preference and happiness, the question of representing the preference of an individual by a utility function is discussed, taking account of lexicographic ordering, imperfect discrimination and the corresponding concepts of semiorder and sub-semiorder. Methods to improve upon the interpersonal comparability of measures of happiness such as pinning down the dividing line of zero happiness and the use of a just perceivable increment of happiness are discussed. The relation of social welfare to individual welfare (i.e. happiness) is then considered. Some reasonable set of axioms ensuring that social welfare is a separable function of and indeed an unweighted sum of individual welfares are reviewed. Finally, happiness is regarded as a function of objective, institutional and subjective factors; an interdisciplinary approach is needed even for an incomplete analysis.     

Clustering Methods in Microarrays

Summary DCT Given a finite set of points in an Euclidean space the \emph{spanning tree} is a tree of minimal length having the given points as vertices. The length of the tree is the sum of the distances of all connected point pairs of the tree. The clustering tree with a given length of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be column monotone orderable if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. Decomposition clusters of monotone orderings of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be column partitionable if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the partition sum of squares of the matrix is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. Decomposition of the rows of a matrix into clusters of partitions is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.