Global existence for a class of triangular parabolic systems on domains of arbitrary dimension

A class of triangular parabolic systems given on bounded domains of with arbitrary is investigated. Sufficient conditions on the structure of the systems are found to assure that weak solutions exist globally.

     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

On convergence of LAD estimates in autoregression with infinite variance

The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a₁, …, a_q)′ for a stationary autoregressive model, are shown to have the property that N^σ(L(N) ? a) converge to zero in probability, for $\text{σ <}\text{1}\text{α}$, where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions.     

Weak convergence of partial sums of absolutely regular sequences

We show that weak convergence results for partial sums of absolutely regular sequences can easily be derived from the corresponding convergence results for independent triangular arrays. The link to be used is a simple lemma on the total variation norm.     

A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

Partial regularity of solutions to a class of degenerate systems

We consider the system , in coupled with suitable initial-boundary conditions, where is a bounded domain in with smooth boundary and is a continuous and positive function of . Our main result is that under some conditions on there exists a relatively open subset of such that is locally Hölder continuous on , the interior of is empty, and is essentially bounded on .

     

A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems

We study two-point Lagrange problems for integrands :

 

Under very weak regularity hypotheses [ is Hölder continuous and locally elliptic on each compact subset of ] we obtain, when is of superlinear growth in , a characterization of problems in which the minimizers of (P) are -regular for all boundary data. This characterization involves the behavior of the value function : defined by . Namely, all minimizers for (P) are -regular in neighborhoods of and if and only if is Lipschitz continuous at . Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where is nearly autonomous, nearly independent of , or jointly convex in are presented.

     

Rumely's local global principle for algebraic PC fields over rings

Let be a finite set of rational primes. We denote the maximal Galois extension of in which all totally decompose by . We also denote the fixed field in of elements in the absolute Galois group of by . We denote the ring of integers of a given algebraic extension of by . We also denote the set of all valuations of (resp., which lie over ) by (resp., ). If , then denotes the ring of integers of a Henselization of with respect to . We prove that for almost all , the field satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that for each and for each . Then . We also prove two approximation theorems for .

     

The Castelnuovo regularity of the Rees algebra and the associated graded ring

It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of -sequences.

     

Crystal structure and absolute configuration of the indole alkaloid arborescidine C

The structure and absolute configuration (3R, 17R) of the indole alkaloid arborescidine C were determined by x-ray diffraction. The six-membered ring assumes a half-chair conformation and the seven-membered ring has a twist-like conformation. The crystal packing is characterized by intermolecular hydrogen-bonding between the hydroxyl group and nitrogen atom N4 which leads to the formation of infinite chains of molecules along the a-axis of the crystal. The absolute configurations of two related indole alkaloids, arborescidine B and arborescidine D are inferred from the experimentally determined configuration of arborescidin C molecule. A comparison of the present structure with that of a related indole alkaloid akagerine showed significant conformational and configurational differences. Crystal data: C₁₆H₁₉N₂OBr, orthorhombic, P2₁2₁2, a = 10.3376(8), b = 15.461(4), c = 9.2094(9)Å, V = 1471.9(6)ų, Z = 4, D _calc = 1.510 g cm^–3, = 1.54178Å.