AN ASYMPTOTIC ESTIMATE FOR THE CHARACTERISTIC AND NUMBER OF FIXED POINTS OF THE RIEMANN ZETA FUNCTION

Quadrature rules for singular integrals with application to Schwarz-Christoffel mappings

On the Atkinson formula for the ζ function

We consider the problem of approximately feedback linearizing a multi-input nonlinear system around the equilibrium manifold ε while making the error terms be of highest order on ε. Necessary and sufficient conditions are given for approximately feedback linearizing the system around ε with error terms of order ρ on ε.

Higher Order Approximate Feedback Linearization About a Manifold

Considers the problem of approximately feedback linearizing a multi-input nonlinear system around the equilibrium manifold /spl epsi/ while making the error terms be of highest order on /spl epsi/. Necessary and sufficient conditions are given for approximately feedback linearizing the system around /spl epsi/ with error terms of order /spl rho/ on /spl epsi/. A simple multi-input example is given to illustrate the higher order approximate feedback linearization procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

HIGHER ORDER APPROXIMATE FEEDBACK LINEARIZATION ABOUT A MANIFOLD

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N −2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N −γ , whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schr\"odinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

For divergence-form second-order elliptic operators with measurable $\varepsilon$-periodic coefficients in $\mathbb{R}^d$ resolvent approximations with error term of order $\varepsilon^2$ as $\varepsilon\to 0$ in the operator norm $\|\cdot\|_{H^1{\to}H^1}$ are constructed. The method of two-scale expansions in powers of $\varepsilon$ up to order two inclusive is used. The lack of smoothness in the data of the problem is overcome by use of Steklov smoothing or its iterates. First scalar differential operators with real matrix of coefficients which act on functions $u\colon \mathbb{R}^d\to \mathbb{R}$, and then matrix differential operators with complex-valued tensor of order four which act on functions $u\colon \mathbb{R}^d\to \mathbb{C}^n$ are considered. Bibliography: 20 titles.

Well-behaved Beurling primes and integers

For independent nearest-neighbor bond percolation on Zd with d≫6, we prove that the incipient infinite cluster’s two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n−3/2, plus an error term of order n−3/2−ε with ε&amp;gt;0. This is a strong version of the statement that the critical exponent δ is given by δ=2.

In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken into account. The results are applied to homogenization of a periodic parabolic Cauchy problem.

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of second and fourth order, requiring two and six substeps per time step, respectively. The number of substeps increases rapidly with order in time step, rendering higher order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per time step. This is achieved by only considering error terms of order and neglecting those of order 2, 3, etc. Here we devise symplectic algorithms with four and six substeps per step which effectively behave as fourth- and sixth-order integrators when is small. These algorithms are more efficient than the usual second- and fourth-order methods when applied to planetary systems.

On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

A homogenization problem is considered for the periodic elliptic differential operators on $L_2(\Pi )$, $\Pi =\mathbb {R} \times (0, a)$, defined by the differential expression \begin{align*} \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D}_j g_j(x_1/\varepsilon , x_2)\mathrm {D}_j + \sum _{j=1}^2 \bigl ( h_{j}^{*}(x_1/\varepsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )& \\ + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon , x_2)& \end{align*} with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $\mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $\mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $\mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $\varepsilon$ limit with error terms of order $\varepsilon$.