Optimal Asymptotic Estimates Research Articles

On unimodular multilinear forms with small norms on sequence spaces

The Kahane–Salem–Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries 1 and −1 generating unimodular m-linear forms Am,n:ℓp1n×⋯×ℓpmn⟶R (or C) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of Am,n when {p1,…,pm}⊂[2,∞] and when {p1,…,pm}⊂[1,2) are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: {p1,…,pm} intercepts both [2,∞] and [1,2). In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.

Asymptotically Optimal Multi-Paving

Abstract Anderson’s paving conjecture, now known to be true [14], asserts that every zero-diagonal matrix admits a nontrivial paving with dimension independent bounds. We study this problem for a collection of matrices and show that given $k$ zero-diagonal $n\times n$ Hermitian matrices $A_1,\ldots ,A_k$ and $\epsilon>0$ there are diagonal projections $P_1,\ldots ,P_r$ with $\sum _{j\le r} P_j=I$ such that $||P_{j}A_iP_{j}||\le \epsilon ||A_i||$ for $i\le k, j\le r$, that is, a simultaneous paving of the matrices, with $r\le 18k/\epsilon ^2$. As a consequence, we get the optimal asymptotic estimates for paving a single zero-diagonal (not necessarily Hermitian) matrix: every square zero-diagonal complex matrix can be $\epsilon -$paved using $O(\epsilon ^{-2})$ blocks, improving the previously known bound of $O(\epsilon ^{-8})$. We use our result to strengthen a result of Johnson–Ozawa–Schechtman on commutator representations of zero trace matrices.

Geodesic distance Riesz energy on the sphere

We study energy integrals and discrete energies on the sphere, in particular, analogues of the Riesz energy with the geodesic distance in place of the Euclidean, and we determine that the range of exponents for which uniform distribution optimizes such energies is different from the classical case. We also obtain a very general form of the Stolarsky principle, which relates discrete energies to certain L 2 L^2 discrepancies, and prove optimal asymptotic estimates for both objects. This leads to sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case, as well as new proofs of discrepancy estimates.

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work [17]. We study the following quasilinear elliptic equations -Δpu-μ|x|p|u|p-2u=Q(x)|u|NpN-p-2u, x∈ℝN, where 1 < p < N,0 ≤ μ < ((N-p)/p)p and Q ɛ L∞ ℝN. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.

On the stability of some imaging functionals

This work is devoted to the stability/resolution analysis of several imaging functionals in complex environments. We consider both linear functionals in the wavefield as well as quadratic functionals based on wavefield correlations. Using simplified measurement settings and reduced functionals that retain the main features of functionals used in practice, we obtain optimal asymptotic estimates of the signal-to-noise ratios depending on the main physical parameters of the problem. We consider random media with possibly long-range dependence and with a correlation length that is less than or equal to the central wavelength of the source we aim to reconstruct. This corresponds to the wave propagation regimes of radiative transfer or homogenization.

Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

AbstractA method for constructing the exact quadratures for Müntz and Müntz‐logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz‐logarithmic polynomials in terms of the number of Gauss–Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end‐point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run‐time evaluation than the existing methods for Müntz polynomials. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Estimates of constants in right-hand sides of Marcinkiewicz’s and Rosenthal’s inequalities

Some optimal asymptotic estimates of constants for the right-hand inequalities of Marcinkiewicz and Rosenthal are derived. These estimates imply some new inequalities for the rate of increase of sums and optimal right-hand estimates for the law of the iterated logarithm. Similar estimates are derived for self-normalized sums. Bibliography: 12 titles.

Higher order boundary estimates for blow-up solutions of elliptic equations

We investigate blow-up solutions of the equation $\Delta u=u^p+g(u)$ in a bounded smooth domain $\Omega$. If $p>1$ and if $g$ satisfies appropriate growth conditions (compared with the growth of $t^p$) as $t$ goes to infinity we find optimal asymptotic estimates of the solution $u(x)$ in terms of the distance of $x$ from the boundary $\partial\Omega$.

Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements

We recently derived a very general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction ( cf. Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single measurement) estimates, even for moderate volume fractions.

Optimal bounds and blow up phenomena for parabolic problems in narrowing domains

We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.