Sharp determinants

We introduce a sharp trace Tr^# and a sharp determinant Det^#(1-z) for an algebra of operators acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of . The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators.Oblatum 8-V-1995 & IX-1995On leave from CNRS, UMR 128, ENS Lyon, France     

Sharp Morawetz estimates

We prove sharp Morawetz estimates — global in time with a singular weight in the spatial variables — for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Sharp Log-Sobolev Inequalities

We show existence of a wide variety of Log-Sobolev inequalities in which the constant is exactly that required by the Poincaré inequality which may be inferred from the Log-Sobolev.

     

Sharp local embedding inequalities

In this paper we establish local versions of the Onofri and sharp Sobolev inequalities. Such local inequalities enable us to give a more direct and simpler proof of the Onofri inequality on ??², as well as an alternative proof of sharp Sobolev inequalities on ??ⁿ (for n ≥ 3). © 2005 Wiley Periodicals, Inc.     

Sharp inequalities via truncation

We show that Sobolev-Poincaré and Trudinger inequalities improve to inequalities on Lorentz-type scales provided they are stable under truncations.     

Sharp de Rham realization

We introduce the sharp (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the sharp de Rham realization T by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. Thus we show how to provide one-dimensional sharp de Rham cohomology of algebraic varieties.     

Sharp Berezin Lipschitz estimates

F.A. Berezin introduced a general ``symbol calculus" for linear operators on reproducing kernel Hilbert spaces. For the Segal-Bargmann space of Gaussian square-integrable entire functions on complex -space, , or for the Bergman spaces of Euclidean volume square-integrable holomorphic functions on bounded domains in , we show here that earlier Lipschitz estimates for Berezin symbols of arbitrary bounded operators are sharp.

     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp asymptotics for isotonic regression

 The asymptotic behavior of the isotonic estimator of a monotone regression function (that is the least-squares estimator under monotonicity restriction) is investigated. In particular it is proved that the ?₁-distance between the isotonic estimator and the true function is of magnitude n ^-1/3. Moreover, it is proved that a centered version of this ?₁-distance converges at the n ^1/2 rate to a Gaussian variable with fixed variance. Received: 20 September 1999 / Revised version: 10 May 2001 / Published online: 19 December 2001     

Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.     

Sharp bounds on Castelnuovo-Mumford regularity

The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity of a nondegenerate projective variety , , provided is -Buchsbaum for , and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.

     

Sharp bounds for NBUE distributions

Let F be a new better than used in expectation (NBUE) distribution function with mean μ. In a previous paper (Brown in Probab. Eng. Inf. Sci. 20:195–230, 2006), the author derived the following bound. For any t≥μ, $$\overline{F}(t) = \mathit{Pr}(X \ge t) \le e^{-[{t\over\mu}-1]}. $$ The main result of this paper is to show that this bound is sharp. Other sharp bounds for NBUE distributions are also derived.     

Note on Sharp Hardy-Type Inequality

We prove Hardy-type inequality with weight singular at ${0 \in \Omega}$ in the class of functions which are not zero on the boundary ${\partial \Omega}$ . The Hardy constant is optimal and the inequality is sharp due to the additional boundary term.     

Sharp inequalities and geometric manifolds

Sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6]. From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups. Among the principal tools used in this analysis are the linear and multilinear operators mapping L^p(M) to L^q(M) defined by the Stein-Weiss integral kernels which extend the Hardy-Littlewood-Sobolev fractional integrals\(\mathcal{H}^1 (\mathbb{R}^d )\) conformal geometry, and the notion of equimeasurable geodesic radial decreasing rearrangement. To illustrate these ideas, four model problems will be examined here: (1) logarithmic Sobolev inequality and the uncertainty principle, (2) SL(2,R) and axial symmetry in fluid dynamics, (3) Stein-Weiss integrals on the Heisenberg group, and (4) Morpurgo’s work on zeta functions and trace inequalities of conformally invariant operators.     

Sharp concentration of random polytopes

We prove that key functionals (such as the volume and the number of vertices) of a random polytope is strongly concentrated, using a martingale method. As applications, we derive new estimates for high moments and the speed of convergence of these functionals. Received: September 2004 Revision: November 2004 Accepted: November 2004     

Sharp bounds for harmonic numbers

In the paper, we collect some inequalities and establish a sharp double inequality for bounding the n-th harmonic number.