A New Kind of Trivariate C 1 Macro-element

We propose a construction of a trivariate C ¹ macro-element over a special tetrahedral partition and compare our construction with known C ¹ macro-elements which are summarized in this paper. Also, we propose an improvement of the Alfeld construction of a C ¹ quintic macro-element such that the new scheme is able to reproduce all polynomials of total degree ≤5.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Robust rigidity for circle diffeomorphisms with singularities

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C ¹-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C ¹-smoothly conjugate.     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Function algebras on disks II

This work is a continuation of [7]. In that paper, a sufficient condition was given on a real analytic fmlction g defined near 0 in C so that the algebra generated by z² and g² is dense in the space of continuous functions on D for all disks D close enough to the origin in C. By using the same methods and some ideas taken from the first named author's thesis we deal with the case where g is only of class C¹ near 0.     

Sur les algèbres absolument valuées qui vérifient l'identité (x, x, x) = 0

Let A denote a prehilbert absolute valued real algebra such that (x, x, x) = 0 for all x ε A; for this algebra we obtain the same results we have previously obtained for the flexible absolute valued algebra. Our main theorem is: A has a finite dimension 1, 2, 4 or 8, and is isotopic to or C. One of the results concerning the isomorphism between A and , C^*, or C shows that if for every two idempotents e₁ and e₂ in , then A is isomorphic to , C^*, or C. The example of infinite dimensional Hilbert absolute valued algebra given by Urbanik and Wright indicates that the assumption, (x, x, x) = 0 for all x ε A, is essential.     

The relative isoperimetric inequality outside convex domains in R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space R ⁿ is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we show that equality holds if and only if Σ is a hemisphere which meets ∂C orthogonally.     

ASYMPTOTICS OF THE NUMBER OF RESONANCES IN THE TRANSMISSION PROBLEM

We consider the resonances for the transmission problem associated with a strictly convex transparent obstacle. Under some natural assumptions we show that there is a free of resonances region in the complex upper half plane given by {C ≤ Im λ ≤ C ₁|λ|^1/3 ? C ₂}, where C, C ₁ and C ₂ are positive constants. Moreover, we obtain asymptotics for the number of resonances counted with multiplicities in the region {0 < Im λ ≤ C, 0 < Re λ ≤ r} as r → ∞, where C > 0 is the same constant as above.

     

Examples of C1-smoothly conjugate diffeomorphisms of the circle with break that are not C1+γ-smoothly conjugate

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C ^1+γ -smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C ¹-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.     

Conditions for a diffeomorghism to be embedded in aC r flow

We will consider aC ^r diffeomorphism of the real lineR, and give a necessary and sufficient condition for aC ^r diffeomorphism ofR to be embedded (uniquely) in aC ^r flow. As an application, we do the same for diffeomorphisms of the circleS ¹ and a class of analytic diffeomorphisms of the planeR ².     

Existence of codimension one foliations with minimal leaves

Every smooth closed manifold of dimension 4 or greater that has a smooth codimension one foliation, has such aC ¹ foliation whose leaves are minimal hypersurfaces for someC ¹ Riemannian metric.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

On Ω-blow-ups in the simplest <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth skew products of interval mappings

The authors prove a criterion (necessary and sufficient condition) for the emergence of the C ⁰-Ω-blow-up for C ¹-smooth skew products of interval mappings with closed set of periodic points. An example of the mapping with given properties that admits the C ⁰-Ω-blow-up is presented. It is proved that the C ¹-Ω-blow-up is impossible for mappings of such a type (in the space of C ¹-smooth skew products of interval mappings). It is proved that there is no one-parameter family of C ¹-smooth skew products of interval mappings with closed set of periodic points C ¹-smoothly depending on the parameter in which from one fixed point, periodic orbits with periods 2 and 4 simultaneously arise. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Dynamics on Ahlfors quasi-circles

The celebrated theory of Denjoy introduced a topological invariant distinguishingC ¹ andC ² diffeomorphisms of the circle. AC ² diffeomorphism of the circle cannot have an infinite minimal set other than the circle itself. However, this is possible forC ¹ diffeomorphisms. In dimension two there is a related invariant distinguishingC ² andC ³ diffeomorphisms. Partially supported by NSF grant No. MCS-83202062.     

C ∞ regularity of solutions of an equation of Levi’s type in R 2 n +1

We study the regularity of the solutions u of a class of P.D.E., whose prototype is the prescribed Levi curvature equation in ℝ^{2 n +1}. It is a second-order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every function u∈C ². If the Levi curvature never vanishes, we represent the operator ℒ associated with the Levi equation as a sum of squares of non-linear vector fields which are linearly independent at every point. By using a freezing method we first study the regularity properties of the solutions of a linear operator, which has the same structure as ℒ. Then we apply these results to the classical solutions of the equation, and prove their C ^∞ regularity. Received: October 10, 1998; in final form: March 5, 1999?Published online: May 10, 2001     

Symbolic extensions and smooth dynamical systems

Let f:XX be a homeomorphism of the compact metric space X. A symbolic extension of (f,X) is a subshift on a finite alphabet (g,Y) which has f as a topological factor. We show that a generic C¹ non-hyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r>1, we exhibit a residual subset of an open set of C^r diffeomorphisms of a compact surface such that if , then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C diffeomorphism has symbolic extensions with the same entropy. We also show that C^r generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.     

Intersecting self-similar Cantor sets

We define a self-similar set as the (unique) invariant set of an iterated function system of certain contracting affine functions. A topology on them is obtained (essentially) by inducing theC ¹-topology of the function space. We prove that the measure function is upper semi-continuous and give examples of discontinuities. We also show that the dimension is not upper semicontinuous. We exhibit a class of examples of self-similar sets of positive measure containing an open set. IfC ₁ andC ₂ are two self-similar setsC ₁ andC ₂ such that the sum of their dimensionsd(C ₁)+d(C ₂) is greater than one, it is known that the measure of the intersection setC ₂–C ₁ has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such thatC ₂–C ₁ has arbitrarily small measure.