Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation

The Lagrangian formulation of the class of general second-order ordinary differential equations invariant under translation in the independent variable and rescaling is presented. The differential equations arising from this analysis are analysed using the Painlevé test. The well-known differential equation, y^″+yy^′+ky³=0, is a unique member of this class when k=3 since it is linearisable by a point transformation. A wider subset is shown to be linearisable by means of a nonlocal transformation.     

Automorphism groups of second order differential equations

The automorphism group of a system ofn second order differential equations is shown to be a Lie group of dimension at mostn ²+n in the autonomous case, and of dimension at most (n+1)² in the heteronomous case. The equations whose automorphism groups have this maximum dimension are classified.     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

On the focus order of planar polynomial differential equations

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime pk, and show that these systems can have a focus order n²−n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n²−1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.     

Derivations for a class of matrix function algebras

We study a class of matrix function algebras, here denoted T⁺(C_n). We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of T⁺(C_n). We use point derivations and information about n×n matrices to show that every T⁺(C_n)-valued derivation on T⁺(C_n) is inner.     

Smooth hypersurfaces with a CR contraction

In this paper we show that a C^∞ real hypersurface in Cn+1 of finite D'Angelo type admitting a weakly contracting local CR automorphism is CR equivalent to a weighted homogeneous hypersurface. As an application, we show that a bounded pseudoconvex domain in Cn+1 with C^∞ boundary of finite D'Angelo type with a hyperbolic orbit accumulation point is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.     

Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x _n ⁰(ε) satisfying this equation within a residual error of O(ε ⁿ⁺¹) is found. A modified Newton’s sequence starting from the element x _n ⁰(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x _n ⁰(ε) has the order of O(ε ⁿ⁺¹), which proves the asymptotic character of the approximation x _n ⁰(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.     

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ⁰(n)⁺, for n=2,3,5,7,11,13 and 17.     

Long time <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-behavior for the incompressible magneto-hydrodynamic equations in a half-space

The large time L ¹-behavior of the strong solution (including the first and second order spacial derivatives) to the incompressible magneto-hydrodynamic (MHD) equations is given in a half-space. The main tool employed in this article is a new weighted estimate for the Stokes flow in L ¹(R₊ ⁿ), such a study is of independent interest.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Tactical decompositions and orbits of projective groups

We consider decompositions of the incidence structure of points and lines of PG(n, q) (n?3) with equally many point and line classes. Such a decomposition, if line-tactical, must also be point-tactical. (This holds more generally in any 2-design.) We conjecture that such a tactical decomposition with more than one class has either a singleton point class, or just two point classes, one of which is a hyperplane. Using the previously mentioned result, we reduce the conjecture to the case n=3, and prove it when q²+q+1 is prime and for very small values of q. The truth of the conjecture would imply that an irreducible collineation group of PG(n, q) (n?3) with equally many point and line orbits is line-transitive (and hence known).     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→⁰⁺ to the classical Filippov solution (Filippov, 1964, 1988 [13] and [14]). Several properties of the solutions corresponding to small ε>0 are presented.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Generalized Wald equations in discrete time

Let S = {S_n, n ? 1} be a martingale. Expectations of m^th order quantities associated with S are related by two forms of Wald-type identity, called Generalized Wald equations. The previously known sufficient conditions for the validity of Wald equations are shown to be of a set of three equivalent conditions, each of which is necessary as well as sufficient for the validity of both types of Generalized Wald Equation.     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.     

Necessary conditions for the stability of difference schemes

For difference equations of the form uⁿ⁺=Auⁿ+f ⁿh, n ≤ T/h the necessary condition for stability due to von Neumann is well known; this condition is expressed in terms of the spectrum of the operator A: r(A)≤1+ch. In this note, for a certain class of difference equations, we express this condition in terms of the spectral radius of the symbol of the operator A.     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

Improving an estimate of the convergence rate of the seidel method by selecting the optimal order of equations in the system of linear algebraic equations

The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs’ book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is 2n ² additions and (12)n ² divisions. Numerical results for random matrices of order 100 are presented that confirm the proposed improvement.     

Infinitesimal bendings of high orders for homogeneous surfaces with positive curvature and a flat point

For any given n∈Z⁺ and k∈Z⁺, infinitesimal bendings of order n and of class Ck are constructed for a class of surfaces in R³. These surfaces are given as graphs of homogeneous polynomials P(x,y) and they have positive curvature except at the origin.