Diffusion approximation of the two-type Galton-Watson process with mean matrix close to the identity

In this paper a diffusion approximation to the two-type Galton-Watson branching processes with mean matrix close to the identity is given in the form of Berstein stochastic differentials. An associated diffusion equation is found using an extension of the one-dimensional Bernstein technique. Expressions for the mean vector and covariance matrix of the diffusion approximation are derived.     

On the Teichmüller theorem and the heights theorem for quadratic differentials

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmüller differential associated with the Teichmüller mapping between compact or finitely punctured Riemann surfaces.

     

Affine hypersurfaces with Gorenstein singular loci

The main result of this paper presents necessary and sufficient criteria for the torsion module of differentials of an affine hypersurface with isolated singularities to be cyclic.

     

Hamilton sequences for extremal quasiconformal mappings in the unit disk

By studying the mapping by heights for quadratic differentials introduced by Strebel, some relations have been established between the maximal norm sequence for quasisymmetric functions and the Hamilton sequence for extremal quasiconformal mappings in the unit disk. Consequently it is proved that a Hamilton sequence is only determined by e quasisymmetric function. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

A new coordinate of teichmüller space

By using the theory of quadratic differentials, we give a new coordinate to the Teichmüller space as well as the trajectory structures of a special class of Jenkins-Strebel quadratic differentials.     

The Codazzi equation for surfaces

In this paper we study the classical Codazzi equation in space forms from an abstract point of view, and use it as an analytic tool to derive global results for surfaces in different ambient spaces. In particular, we study the existence of holomorphic quadratic differentials, the uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.     

On holomorphic polydifferentials in positive characteristic

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m‐(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.     

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.     

Prym differentials as solutions to boundary value problems on Riemann surfaces

Construction of multiplicative functions and Prym differentials, including the case of characters with branch points, reduces to solving a homogeneous boundary value problem on the Riemann surface. The use of the well-established theory of boundary value problems creates additional possibilities for studying Prym differentials and related bundles. Basing on the theory of boundary value problems, we fully describe the class of divisors of Prym differentials and obtain new integral expressions for Prym differentials, which enable us to study them directly and, in particular, to study their dependence on the point of the Teichmüller space and characters. Relying on this, we obtain and generalize certain available results on Prym differentials by a new method.     

Fleurs, arbres et cellules: un invariant galoisien pour une famille d'arbres

This paper is principally concerned with the action of the absolute Galois group on a family of dessins d'enfants i.e. isomorphism classes of coverings of the projective line unramified outside three points. More precisely, we prove a generalisation of a conjecture proposed by Yu. Kotchetkov in 1997. The main tool used in this work is a correspondence between dessins d'enfants and ribbon graphs arising from the theory of Strebel differentials.     

On the dependence of the functions of state of a deformable body on the measure of rotation

We show that the measures of strain and initial values of vector and tensor state parameters are divided into subjective nonrotational and objective rotational. Representations of the functions of state are divided in a similar way, and only objective ones do not depend explicitly on the measure of rotation of material axes. We have constructed relations for the reduction of rotation-free differentials of the functions of state to expressions in terms of the tensors of infinitesimal strain and rotation. On this basis, we have obtained objective representations for stress tensor in terms of the derivatives of the state potential of an anisotropic material. The results obtained concern the nonlinear mechanics of initially stressed bodies.     

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

We provide a Boseck‐type basis of the space of holomorphic differentials for a large class of solvable covers of the projective line with perfect field of constants of characteristic . Within this class, we also describe the Galois module structure of holomorphic differentials for abelian covers.     

Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control.The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods.Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.Communicated by H. J. Pesch     

The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.     

Derivatives and differentials for multiplicative intuitionistic fuzzy information

By using the unsymmetrical scale instead of the symmetrical scale, the multiplicative intuitionistic fuzzy sets (MIFSs) reflect our intuition more objectively. Each element in a MIFS is expressed by an ordered pair which is called a multiplicative intuitionistic fuzzy number (MIFN) and is based on the unbalanced scale (i.e., Saaty’s 1-9 scale). In order to describe the derivatives and differentials for multiplicative intuitionistic fuzzy information more comprehensively, in this paper, we firstly propose two new basic operational laws for MIFNs, which are the subtraction law and the division law. Secondly, we describe the change values of MIFNs when considering them as variables, classify these change values based on the basic operational laws for MIFNs, and depict the convergences of sequences of MIFNs by the subtraction and division laws. Finally, we focus on the multiplicative intuitionistic fuzzy functions and derive some basic results related to their continuities, derivatives and differentials, and also give their application in selecting the configuration of a computer.     

On Some Concepts of Generalized Differentials

In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.      

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012)  [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.     

Impossible differential cryptanalysis using matrix method

The general strategy of impossible differential cryptanalysis is to first find impossible differentials and then exploit them for retrieving subkey material from the outer rounds of block ciphers. Thus, impossible differentials are one of the crucial factors to see how much the underlying block ciphers are resistant to impossible differential cryptanalysis. In this article, we introduce a widely applicable matrix method to find impossible differentials of block cipher structures whose round functions are bijective. Using this method, we find various impossible differentials of known block cipher structures: Nyberg’s generalized Feistel network, a generalized CAST256-like structure, a generalized MARS-like structure, a generalized RC6-like structure, Rijndael structures and generalized Skipjack-like structures. We expect that the matrix method developed in this article will be useful for evaluating the security of block ciphers against impossible differential cryptanalysis, especially when one tries to design a block cipher with a secure structure.     

VARIABILITY SETS AND HAMILTON SEQUENCES

This paper studies extremal quasiconformal mappings. Some properties of the variability set are obtained and the Hamilton sequences which are induced by point shift differentials are also discussed.