二次有限元解的渐近展开与外推

在Poisson方程的求解域Ω存在一致的三角剖分,并且相邻两初始单元构成平行四边形的假设下,证明了若Poisson方程的解u属于H6(Ω),那么二次有限元的误差有h4的渐近展开.基于误差的渐近展开,可以利用h4-Richardson外推进一步提高数值解的精度阶,并且能够得到一个后验误差估计.最后,一个数值算例验证了理论分析.     

Imprimitive matrices of zeros and ones with extremal numbers of ones

In [X. Zhan, Extremal numbers of positive entries of imprimitive nonnegative matrices, Linear Algebra Appl. 424 (2007) 132–138], Zhan determined the maximum and minimum numbers of positive entries of irreducible nonnegative matrices. In this paper, we characterize the irreducible (0,1) matrices with the maximum and minimum numbers of positive entries.     

Change of variable formulas for non-anticipative functionals on path space

We derive a change of variable formula for non-anticipative functionals defined on the space of Rd-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations.     

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.     

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software.     

Bipower-type estimation in a noisy diffusion setting

We consider a new class of estimators for volatility functionals in the setting of frequently observed Itō diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general Itō semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale.     

On the theory of convolution equations of the third kind, II

Existence results of Part I of the paper are generalized to two types of autoconvolution equations of the third kind having free terms with nonzero values at x=0 like the well-known Bernstein-Doetsch equation for the Jacobian theta zero functions. Also uniqueness results for the linear convolution equations in Part I of the paper are extended to more general function spaces. Further, a special class of integro-differential equations with autoconvolution integral and two classes of the linear singular Abel-Volterra equations are dealt with.     

Optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations

In this work, we take advantage of the powerful quadratic programming theory to obtain optimal solutions of scheduling problems. We apply a methodology that starts, in contrast to more classical approaches, by formulating three unrelated parallel machine scheduling problems as 0–1 quadratic programs under linear constraints. By construction, these quadratic programs are non-convex. Therefore, before submitting them to a branch-and-bound procedure, we reformulate them in such a way that we can ensure convexity and a high-quality continuous lower bound. Experimental results show that this methodology is interesting by obtaining the best results in literature for two of the three studied scheduling problems.     

On local convergence of sequential quadratically-constrained quadratic-programming type methods,with an extension to variational problems

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098–1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-Moré type condition. Research of the second author is partially supported by CNPq Grants 300734/95-6 and 471780/2003-0, by PRONEX–Optimization, and by FAPERJ.     

Proficiency testing scheme for the harmonization and comparability of analytical measurements

According to the experience of the successful implementation of proficiency tests (PT) by using the certified reference value as the assigned value, a new scheme of evaluation is presented by suggesting the use of the uncertainty associated with the certified value. The technical performance of laboratories is evaluated by the parameter quadratic mean error (QME), which is the square root of the sum of the square of the bias and that of the standard deviation of the laboratory. This parameter is considered as the estimate of the measurement uncertainty of the laboratory and is compared to the uncertainty (U) associated with the certified value provided by an NMI. Considering that the calibration and measurement capability, known as the CMC, is recognized among NMIs, the ratio QME/U enables us to compare the PT relative to the CMC of an NMI, and, consequently, to any other comparison results based on the CMC of signatories of the mutual recognition arrangement (MRA) of the International Committee of Weights and Measure (CIPM). Presented at BERM-11, October 2007, Tsukuba, Japan.