Local C1 stability versus global C1 unstability for iterative roots

Stability of iterative roots is important in the numerical computation of iterative roots. Known results show that under some conditions iterative roots of strictly monotonic self-mappings are $C^0$ stable in both the local sense and the global sense. In this paper we discuss the $C^1$ stability for iterative roots of strictly increasing self-mappings on a compact interval between two fixed points. We prove that those iterative roots are locally $C^1$ stable but globally $C^1$ unstable.     

C^1 C^2 INTERPOLATION OF SCATTERED DATA POINTS

In this paper an error in [4] is pointed out and a method for constructing surface interpolating scattered data points is presented, The main feature of the method in this paper is that the surface so constructed is polynomial, which makes the construction simple and the calculation easy.     

关于C1-积分

本文利用Riemann和的Moore-Smith极限来定义并研究C1-积分.利用C1-积分的有关性质,讨论了C1-积分与C-积分之间的关系,并给出了一个零测度集特征函数C1-可积的充要条件,从而推广了文献[5]的有关结论.     

On nuclear C1-algebras

A C¹-algebra is called nuclear if there is a unique way of forming its tensor product with any other C¹-algebra. Takesaki [17] showed that all C¹-algebras of type I and all inductive limits of such algebras are nuclear, but that the C¹-algebra $\text{C}{}_{\mathrm{r}}{}^1\text{(G)}$ generated by the left regular representation of G on l²(G) is nonnuclear, where G is the free group on two generators. In this paper an extension property for tensor products of C¹-algebras is introduced, and is characterized in terms of the existence of a certain family of weak expectations on the algebra. Nuclearity implies the extension property, and this is used to show that for a discrete group $\text{G, C}{}_{\mathrm{r}}{}^1\text{(G)}$ is nuclear if and only if G is amenable.An approximation property in the dual of a C¹-algebra is introduced, and shown to imply nuclearity. It is not clear whether the converse implication holds, but it is proved that the known nuclear C¹-algebras satisfy the approximation property.     

C1, 1 Regularity in semilinear elliptic problems

In this paper we give an astonishingly simple proof of C^{1, 1} regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optical results. The setting we'll consider is the following. Let u be a solution to where B, is the unit ball in ?ⁿ, f(x, t) is a bounded Lipschitz function in x, and f_t′ is bounded from below. Then we prove that u ? C^{1, 1} (B_1/2). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig. © 2002 Wiley Periodicals, Inc.     

C 1 surface interpolation with constraints

Givenn pairwise distinct and arbitrarily spaced pointsP _i in a domainD of thex–y plane andn real numbersf _i, consider the problem of computing a bivariate functionf(x, y) of classC ¹ inD whose values inP _i are exactlyf _i,i=1,,n, and whose first or second order partial derivatives satisfy appropriate equality and inequality constraints on a given set ofp pointsQ _l inD.In this paper we present a method for solving the above problem, which is designed for extremely large data sets. A step of this method requires the solution of a large scale quadratic programming (QP) problem.The main purpose of this work is to analyse an iterative method for determining the solution of this QP problem: such a method is very efficient and well suited for parallel implementation on a multiprocessor system.Work supported by MURST Project of Computational Mathematics, Italy.     

C1 monotone cubic Hermite interpolant

Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bézier cubic interpolant.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

DIFFEOMORPHISMS WITH VARIOUS C1 STABLE PROPERTIES

Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f|Λ satisfies a C1-genericstable shadowable property (resp.,C1-gene...