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 .
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$">.
We also show that factors through , where denotes the differential of . Therefore, the problem of determining should be of interest.
Problems of this kind arise, e.g., in hydrodynamics where the coefficients , , and are unbounded selfadjoint operators. It is assumed that is the dominating operator in the Cauchy problem above, i.e.,
We also suppose that and are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.
for all there exists a homomorphism such that
provided, in addition, that are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number and any 0$"> there is 0$"> such that in any unital amenable purely infinite simple -algebra if
for a pair of unitaries, then there exists a pair of unitaries and in such that
A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation
We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.
where , is a smooth bounded domain and . We assume that the nonlinear term
where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.
In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations
where the 1-periodic function is a smooth function and satisfies sublinearity:
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
for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .
Let be a unital Banach algebra and let be a closed two-sided ideal of . We prove that if any invertible element of has an invertible lifting in , then the quotient homomorphism is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foias, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna-Pick, and Carathéodory type interpolation for , the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra and , the algebra of bounded operators on a finite dimensional Hilbert space . A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for .
In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of , in which one bounds the spectral radius of the interpolant and not the norm.
where is some constant and some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.
In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category in terms of the geometry of and the Epstein filtration of the exceptional set . The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that
We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.
where 0$"> is a constant, 0$"> is a parameter, and is a continuous function on , 0$">, and 0$"> for . The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions of (H such that
It is shown that, if and if (H is strongly nonoscillatory, then there exists a sequence such that , as ; and with has exactly zeros in the interval and ; and with has exactly zeros in and . For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.
where is the Legendre symbol. For example for an odd prime,
where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,
where denotes the nearest integer, satisfies
where
Indeed we derive a closed form for the norm of all shifted Fekete polynomials
Namely
and if .
(1)
(2)
(3)
where is a bounded open set in with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and
The resulting problem is then reduced to the problem where the operator satisfies Condition This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.
Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .
Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.
The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.
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.
Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate
In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.