Regularization with differential operators. I. General theory

The method of regularization is used to obtain least squares solutions of the linear equation Kx = y, where K is a bounded linear operator from one Hilbert space into another and the regularizing operator L is a closed densely defined linear operator. Existence, uniqueness, and convergence analyses are developed. An application is given to the special case when K is a first kind integral operator and L is an nth order differential operator in the Hilbert space L²[a, b].     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Differential operators on homogeneous spaces. I

Summary In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation of the enveloping algebra of by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI _x of _X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI _x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map of the cotangent-bundle ofX onto a nilpotent variety in , as studied e.g. in [BM1]. We prove, for instance, that grI _x is prime if and only if _X is birational with normal image. In general, we show that is prime. Equivalently, the associated variety ofI _x in is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI _y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

Generalized Watson transforms I: General theory

This paper introduces two main concepts, called a generalized Watson transform and a generalized skew-Watson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

     

Radial symmetry and symmetry breaking for some interpolation inequalities

We analyze the radial symmetry of extremals for a class of interpolation inequalities known as Caffarelli?CKohn?CNirenberg inequalities, and for a class of weighted logarithmic Hardy inequalities which appear as limiting cases of the first ones. In both classes we show that there exists a continuous surface that splits the set of admissible parameters into a region where extremals are symmetric and a region where symmetry breaking occurs. In previous results, the symmetry breaking region was identified by showing the linear instability of the radial extremals. Here we prove that symmetry can be broken even within the set of parameters where radial extremals correspond to local minima for the variational problem associated with the inequality. For interpolation inequalities, such a symmetry breaking phenomenon is entirely new.     

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the ?-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level ? and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.     

Universal deformation formulas and breaking symmetry

We show that an algebra with a non-nilpotent Lie group of automorphisms or “symmetries” (e.g., smooth functions on a manifold with such a group of diffeomorphisms) may generally be deformed (in the function case, “quantized”) in such a way that only a proper subgroup of the original group acts. This symmetry breaking is a consequence of the existence of certain “universal deformation formulas” which are elements, independent of the original algebra, in the tensor algebra of the enveloping algebra of the Lie algebra of the group.     

Elliptic theory of differential edge operators I

Examples of edge operators include Laplacians on asymptotically flat and asymptotically hyperbolic manifolds. Edge operators also arise in boundary problems around higher condimension boundaries. This paper is concerned with the analysis of general elliptic edge operators with constant indicide roots. We determine when such an operator has a distributional asymptotic expansion. Conditions are given to guarantee that the coefficients of this expansion are smooth. In Part I of this paper we only study the case when the operator is semi-Fredholm. Part II will examine edge operators with infinite dimensional kernel and cokernel, as well as develop the theory of Poisson edge operators.     

New perturbation theory representation of the conformal symmetry breaking effects in gauge quantum field theory models

We propose a hypothesis on the detailed structure for the representation of the conformal symmetry breaking term in the basic Crewther relation generalized in the perturbation theory framework in QCD renormalized in the [`(MS)]overline {MS} scheme. We establish the validity of this representation in the O(α _s ⁴ ) approximation. Using the variant of the generalized Crewther relation formulated here allows finding relations between specific contributions to the QCD perturbation series coefficients for the flavor nonsinglet part of the Adler function D _A ^ns for the electron-positron annihilation in hadrons and to the perturbation series coefficients for the Bjorken sum rule S _Bjp for the polarized deep-inelastic lepton-nucleon scattering. We find new relations between the α _s ⁴ coefficients of D _A ^ns and S _Bjp . Satisfaction of one of them serves as an additional theoretical verification of the recent computer analytic calculations of the terms of order α _s ⁴ in the expressions for these two quantities.     

A decomposition theory for matroids. I. General results

A new matroid decomposition with several attractive properties leads to a new theorem of alternatives for matroids. A strengthened version of this theorem for binary matroids says roughly that to any binary matroid at least one of the following statements must apply: (1) the matroid is decomposable, (2) several elements can be removed (in any order) without destroying 3-connectivity, (3) the matroid belongs to one of 2 well-specified classes or has 10 elements or less. The latter theorem is easily specialized to graphic matroids. These theorems seem particularly useful for the determination of minimal violation matroids, a subject discussed in part II.     

Topological geometrodynamics,Part I: General theory

The recent status of topological geometrodynamics (TGD) is reviewed. One can end up with TGD either by starting from the energy problem of general relativity or from the need to generalize hadronic or superstring models. The basic principle of the theory is `Do not quantize!' meaning that quantum physics is reduced to Kähler geometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H=M⁴₊×CP₂ with physical states represented by classical spinor fields. General coordinate invariance implies that classical theory becomes an exact part of the quantum theory and configuration space geometry and that space-time surfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional Kähler geometric existence fixes imbedding space and the dimension of the space-time highly uniquely and implies that superconformal and supercanonical symmetries acting on the lightcone boundary δM⁴₊×CP₂ are cosmologies symmetries.The work with the p-adic aspects of TGD, the realization of the possible role of quaternions and octonions in the formulation of quantum TGD, the discovery of infinite primes, and TGD inspired theory of consciousness encouraged the vision about TGD as a generalized number theory. The vision leads to a considerable generalization of TGD and to an extension of the symmetries of the theory to include superconformal and Super-Kac-Moody symmetries associated with the group P×SU(3)×U(2)_ew (P denotes the Poincaré group) acting as the local symmetries of the theory. Quantum criticality, which can be seen as a prediction of the theory, fixes the value spectrum for the coupling constants of the theory.The proper mathematical and physical interpretation of the p-adic numbers has remained a long-lasting challenge. Both TGD inspired theory of consciousness and the vision about physics as a generalized number theory suggest that p-adic space-time regions obeying p-adic counterparts of the field equations are geometric correlates of mind in the sense that they provide cognitive representations for the physics in the real space-time regions representing matter. Evolution identified as a gradual increase of the infinite p-adic prime characterizing the entire Universe is basic prediction of the theory.S-matrix elements can be identified as Glebsch–Gordan coefficients between interacting and free Super-Kac-Moody algebra representations and it is now possible to give Feynmann rules for the S-matrix in the approximation that elementary particles correspond to the so-called CP₂ type extremals.     

General theory of wavemaker,I. theoretical results

A general theory of the wavemaker is presented based on a recent formulation of the water wave equations by Hui and Tenti. It exploits the fact that the free surface is a surface of constant pressure in order to make the surface boundary conditions linear and to be evaluated at a fixed boundary. The main features of the present theory are as follows: First, it applies to any weakly nonlinear wavemaker. Second, the full initial-boundary value problem is solved, thus including the transient effects in contrast to the classical approaches. Third, the finite amplitude (weakly nonlinear) effects are explicitly calculated. Finally, it is notable from a mathematical standpoint that the complicated second-order problem can be transformed to the form of the linear problem, and can therefore be solved by identical techniques.

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Résumé Nous présentons une étude théorique générale des batteurs à houle fondée sur une formulation récente des équations d'un fluide parfait, pesant et incompressible, par Hui and Tenti, dans laquelle la pression est une variable indépendante et la condition aux limites à la surface libre devient une condition linéaire portant sur une frontière fixe. Cette théorie est applicable à un batteur quelconque, et nous résolvons le problème de Cauchy en incorporant les effets transitoires, contrairement aux travaux classiques. En particulier nous calculons explicitement les corrections nonlinéaires du second ordre, ce qui est notable du point de vue mathématique parce que nous pouvons ramener la forme du problème du second ordre à celle du premier ordre.

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This work was supported in part by the Natural Sciences and Engineering Research Council of Canada through grants to G. Tenti and W. H. Hui.