Approximation of Hypersingular Integral Operators With Cauchy Kernel

In the present article, the hypersingular integral operator with Cauchy kernel H is approximated by a sequence of operators of a special form, and it is proved that the approximating operators H_n strongly converge to the operator H and for an algebraic polynomial of degree not higher than n the operators H_n and H coincide. Therefore, the estimate established in this article yields more exact results in terms of the convergence rate than traditional methods. At the end we give the approximate solution of the hypersingular integral equation of the first kind.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil III

Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A ₀-stability of the method as well as the influence of rounding errors are investigated.

     

Locally nilpotent derivations and Nagata-type utomorphisms of a polynomial algebra

We study locally nilpotent derivations belonging to a Lie algebra sa _n of a special affine Cremona group in connection with the root decompositions of sa _n relative to the maximum standard torus. It is proved that all root locally nilpotent derivations are elementary. As a continuation of this research, we describe two- and three-root derivations. By using the results obtained by Shestakov and Umirbaev, it is shown that the exponents of almost all obtained three-root derivations are wild automorphisms of a polynomial algebra in three variables.     

L∞-boundedness of L2-projections on splines for a geometric mesh

In this paper we complete the investigations started by [14.]. We study C. de Boor's conjecture of the L_∞-boundedness of the L₂-projection P on smooth splines in the special case of a geometric mesh x = {q^v}. A connection to the interpolation projection is established and the uniform boundedness of ¦|P¦|_π with respect to q is proved.     

Convergence and Error Bound of a Method for Solving Variational Inequality Problems via the Generalized D-Gap Function

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. Recently, a hybrid Newton-type method was proposed by Peng and Fukushima for minimizing a special form of the generalized D-gap function. In this paper, the hybrid Newton-type algorithm is extended to minimize the general form g of the generalized D-gap function. It is shown that the algorithm has nice convergence properties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, it is proved that the function g has bounded level sets for strongly monotone VIP. An error bound of the algorithm is obtained.     

Special cohomology classes for modular Galois representations

Building on ideas of Vatsal [Uniform distribution of Heegner points, Invent. Math. 148(1) (2002) 1-46], Cornut [Mazur's conjecture on higher Heegner points, Invent. Math. 148(3) (2002) 495-523] proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E_/Q as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article, Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's two-dimensional ?-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.     

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R_ and let K be the cokernel of the canonical ring homomorphism from R to R_. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter _K of ideals of R, which is a subset of , such that all _K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and _K is investigated.     

Classification of cocycles over ergodic automorphisms with values in the Lorentz group and recurrence of cocycles

It is proved that any SO₀(1, d)-valued cocycle over an ergodic (probability) measurepreserving automorphism is cohomologous to a cocycle having one of three special forms; the recurrence property of such cocycles is also studied.     

Homogenization of the Neumann problem in thick multi‐structures of type 3 : 2 : 2

Using some special extension operator, a convergence theorem is proved for the solution to the Neumann boundary value problem for the Ukawa equation in a junction Ω_ε, which is the union of a domain Ω₀ and a large number N of ε‐periodically situated thin annular disks with variable thickness of order ε=??(N^‐1), as ε → 0. Copyright © 2004 John Wiley & Sons, Ltd.     

On the relation of Cesàro summability to generalized (F α, q) summability

Standard special cases of the sequence-to-functionF (a, q)-transform of Meir4 admit of a more general, essentially more refined, characterization as theF ^k (a, q)-transform of the sequel, adapted from one of Faulhaber’s.1 The theorem proved,viz., that (F _{a, q} ^k) summability for sequences, corresponding to the latter transform, includes Cesàro summability of a positive integer order with a certain rapidity, applies to the standard special cases of (F _{α, q}) and (F _{a, q} ^k) summabilities which are, in famiiar notation, summabilities(E _p), (T _α), (S _β), (V _a) and (B _{α, γ}) whose further special case (B_{1, 1}) is Borel summability. The special cases of (V_1/2) and Borel summabilities go back to Hyslop.3     

Well-posedness of an infinite system of partial differential equations modelling parasitic infection in an age-structured host

We study a deterministic model for the dynamics of a population infected by macroparasites. The model consists of an infinite system of partial differential equations, with initial and boundary conditions; the system is transformed in an abstract Cauchy problem on a suitable Banach space, and existence and uniqueness of the solution are obtained through multiplicative perturbation of a linear C₀-semigroup. Positivity and boundedness are proved using the specific form of the equations.     

Lp – Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper L_p – L_q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.     

A New Derivation of the Inner Product Formula for the Macdonald Symmetric Polynomials

We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type A_n-1. As a special case, the corresponding constant term conjecture is also proved.     

Über die verallgemeinerte Form der Lagrangeschen Gleichungen,welche auch die Behandlung von nicht-holonomen mechanischen Systemen gestattet

Summary A generalized form of the equations of motion of a rheonomic-holonomic mechanical system is proved. As special cases the equations ofNielsen andTzenoff are obtained. By a method of the last author the form ofAppell's equations is derived, in which the kinetic energy, but not the energy of acceleration, appears. The generalized equations can be extended to the case of non-holonomical systems.     

Some projectively flat (α, β)-metrics

In this paper, we study a class of Finsler metrics in the form $F = \alpha + \varepsilon \beta + 2k\tfrac{{\beta ^2 }}{\alpha } - \tfrac{{k^2 \beta ^4 }}{{3\alpha ^3 }}$ , where $\alpha = \sqrt {\alpha _{ij} y^i y^j } $ is a Riemannian metric, β = b _i y ⁱ is a 1-form, and ε and k ≠ 0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.     

On an Inverse Problem for a Parabolic Equation

In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u _t and u depend on u _(x,t) , with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

Novikov’s inversion formula for the attenuated Radon transform—A new approach

We study the inversion of weighted Radon transforms in two dimensions, R_ρƒ(L)=ƒ_L =ƒ(·), where the weight function ρ(L, x), L a line and x ∈ L, has a special form. It was an important breakthrough when R.G. Novikov recently gave an explicit formula for the inverse of R_ρ when ρ has the form(1.2); in this case R_ρ is called the attenuated Radon transform. Here we prove similar results for a somewhat larger class of ρ using completely different and quite elementary methods.