The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on for the equation
where is the classical Laplacian operator, and denotes the open ball of center and radius in When is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.
Let denote the ring of integers of an algebraic number field of degree which is totally and tamely ramified at the prime . Write , where . We evaluate the twisted Kloosterman sum
where and denote trace and norm, and where is a Dirichlet character (mod ). This extends results of Salié for and of Yangbo Ye for prime dividing Our method is based upon our evaluation of the Gauss sum
which extends results of Mauclaire for .
The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding
where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .
Let be Singer's invariant-theoretic model of the dual of the lambda algebra with , where denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, , into is a chain-level representation of the Lannes-Zarati dual homomorphism
The Lannes-Zarati homomorphisms themselves, , correspond to an associated graded of the Hurewicz map
Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in , of positive degree represents the homology class in for 2$">.
We also show that factors through , where denotes the differential of . Therefore, the problem of determining should be of interest.
- (1)
- For all 0$">, all but finitely many integer quadratics satisfy
where is the height function. - (2)
- For all 0$"> there exists a sequence of integer quadratics such that
with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.
We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that
The constant is sharp.
The proof is based on a Lipschitz selection result which is interesting in its own right.
A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation
We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.
It is known that sometimes this number can be expressed in a natural way using the Newton polygon of . We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal is also analyzed.
We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form:
where goes to zero in a way related to the asymptotic mass of upper floors, is some Lipschitz norm and is some norm. The fact that the dependence on is given by an norm is useful to study asymptotic laws of successive entrance times.
We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition
then (R) holds with no growth assumptions on .
We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function defined in a domain and such that
We also assume that the interior boundary of the positivity set, \nobreak 0\}$">, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:
Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).
The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.
Erdos and Straus conjectured that for any positive integer the equation has a solution in positive integers , and . Let k \geq 3$"> and
We show that parametric solutions can be used to find upper bounds on where the number of parameters increases exponentially with . This enables us to prove
This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization' of the work of Vaughan (1970), who considered the case .
are studied when may grow linearly with respect to the highest-order derivative, and admissible sequences converge strongly in It is shown that under certain continuity assumptions on convexity, -quasiconvexity or -polyconvexity of
ensures lower semicontinuity. The case where is -quasiconvex remains open except in some very particular cases, such as when
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test' polynomials (those tangent from above or below to the graph of at a point ) satisfy the correct inequality only if . That is, we simply disregard those test polynomials for which .
Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for , (0$">) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.
where () is an appropriately smooth bounded domain and 0$"> is a small parameter. It is known that under some conditions on , the solution corresponding to develops boundary layers when . We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.
where , is a smooth bounded domain and . We assume that the nonlinear term
where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.