SIAM Journal Research Articles

Special Section on the Fifty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)

This special section contains five selected papers from the 50th Annual Symposium on Foundations of Computer Science (FOCS 2010) sponsored by the IEEE Technical Committee on Mathematical Foundations of Computing. The conference was held in Las Vegas, Nevada, October 23-26, 2010. The conference program consisted of 81 papers, which the program committee selected from 270 submissions. The program committee was composed of Scott Aaronson, Dorit Aharonov, Eli Ben-Sasson, Julia Chuzhoy, Ryan O'Donnell, Roberto Grossi, Nick Harvey, Adam Kalai, Nicole Immorlica, Yuval Ishai, Lap Chi Lau, James Lee, Tal Malkin, Joe Mitchell, Dana Moshkovitz, S. Muthukrishnan, Christos Papadimitriou, Sofya Raskhodnikova, Steve Skiena, Mikkel Thorup, Luca Trevisan, and Eric Vigoda. Each of the five papers appearing in this issue was subject to the standard refereeing process of the SIAM Journal on Computing. In “The Monotone Complexity of $k$-Clique on Random Graphs, Ben Rossman proves that monotone circuits that solve the $k$-cli...

GENERATING FUNCTIONS AND SHORT RECURSIONS, WITH APPLICATIONS TO THE MOMENTS OF QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS

Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962,Annals of Mathematical Statistics33, 542–570; Hillier, Kan, and Wang, 2009,Econometric Theory25, 211–242). Typically, in a recursion of this type thekth object of interest,dk, say, is expressed in terms of all lower orderdj’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, afixed lengthrecursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986,SIAM Journal on Scientific and Statistical Computing7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.

Optimal Execution and Price Manipulations in Time-varying Limit Order Books

This paper focuses on an extension of the limit order book (LOB) model with general shape introduced by Alfonsi, Fruth and Schied ((2010). Optimal execution strategies in limit order books with general shape functions. Quantitative Finance, 10(2), 143–157). Here, the additional feature allows a time-varying LOB depth. We solve the optimal execution problem in this framework for both discrete and continuous time strategies. This gives in particular sufficient conditions to exclude price manipulations in the sense of Huberman and Stanzl ((2004). Price manipulation and quasi-arbitrage. Econometrica, 72(4), 1247–1275) or transaction-triggered price manipulations (see Alfonsi, A., Schied, A., & Slynko, A. (2012). Order book resilience, prince manipulation, and the positive portfolio problem. SIAM Journal of Financial Mathematics, 3, 511–533.). These conditions give interesting qualitative insights on how market makers may create or not price manipulations.

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

Two Modified Hager and Zhang's Conjugate Gradient Algorithms For Solving Large-Scale Optimization Problems

At present, the conjugate gradient (CG) method of Hager and Zhang (Hager and Zhang, SIAM Journal on Optimization, 16(2005)) is regarded as one of the most effective CG methods for optimization problems. In order to further study the CG method, we develop the Hager and Zhang's CG method and present two modified CG formulas, where the given formulas possess the value information of not only the gradient but also the function. Moreover, the sufficient descent condition will be holden without any line search. The global convergence is established for nonconvex function under suitable conditions. Numerical results show that the proposed methods are competitive to the normal conjugate gradient method.

Robust estimation in structural equation models using Bregman divergences

Structural equation models seek to find causal relationships between latent variables by analysing the mean and the covariance matrix of some observable indicators of the latent variables. Under a multivariate normality assumption on the distribution of the latent variables and of the errors, maximum likelihood estimators are asymptotically efficient. The estimators are significantly influenced by violation of the normality assumption and hence there is a need to robustify the inference procedures. We propose to minimise the Bregman divergence or its variant, the total Bregman divergence, between a robust estimator of the covariance matrix and the model covariance matrix, with respect to the parameters of interest. Our approach to robustification is different from the standard approaches in that we propose to achieve the robustification on two levels: firstly, choosing a robust estimator of the covariance matrix; and secondly, using a robust divergence measure between the model covariance matrix and its robust estimator. We focus on the (total) von Neumann divergence, a particular Bregman divergence, to estimate the parameters of the structural equation model. Our approach is tested in a simulation study and shows significant advantages in estimating the model parameters in contaminated data sets and seems to perform better than other well known robust inference approaches in structural equation models. References E. G. Baranoff, S. Papadopoulos and T. W. Seger. Capital and Risk Revisited: A Structural Equation Model Approach for Life Insurers. Journal of Risk and Insurance, 74:653–681, 2007. doi:10.1111/j.1539-6975.2007.00229.x K. Bollen. Structural Equations with Latent Variables . Wiley, New York, 1989. I. S. Dhillon and J. A. Tropp. Matrix Nearness Problems with Bregman Divergences. SIAM Journal on Matrix Analysis and Applications , 29:1120–1146, 2008. doi:10.1137/060649021 F. Nielsen and S. Boltz. The Burbea–Rao and Bhattacharyya Centroids. IEEE transactions in information theory , 57:5455–5466, 2011. doi:10.1109/TIT.2011.2159046 B. C. Vemuri, M. Liu, S.-I. Amari and F. Nielsen. Total Bregman Divergence and Its Applications to DTI Analysis. IEEE Transactions on medical imaging , 30:475–483, 2011. doi:10.1109/TMI.2010.2086464 S. Verboven and M. Hubert. LIBRA: a MATLAB library for robust analysis, Chemometrics and intelligent laboratory systems , 75:127–136, 2005. doi:10.1016/j.chemolab.2004.06.003 K.-H. Yuan and P. M. Bentler. Structural equation modeling with robust covariances. Sociological Methodology , 28:363–396, 1998. doi:10.1111/0081-1750.00052 K.-H. Yuan and P. M. Bentler. Robust mean and covariance structure analysis, British Journal of Mathematical and Statistical Psychology , 51:63–88, 1998. doi:10.1111/j.2044-8317.1998.tb00667.x K.-H. Yuan, P. M. Bentler and W. Chan. Structural Equation Modeling with Heavy Tailed Distributions, Psychometrika , 69:421–436, 2004. doi:10.1007/BF02295644 X. Zhong and K.-H. Yuan. Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood Versus Robust Methods. Multivariate Behavioral Research, 46:229–265, 2011. doi:10.1080/00273171.2011.558736 LIBRA: a Matlab Library for Robust Analysis. http://wis.kuleuven.be/stat/robust/LIBRA/LIBRA-home

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models are successfully used to model anomalous diffusion and spatial heterogeneity in different physical contexts. We develop a fractional model based on the Fisher--Kolmogoroff equation, analyse it for its wavespeed properties, and relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments. References R. J. Adler, R. E. Feldman and M. S. Taqqu. A practical guide to heavy tails: Statistical techniques and applications. Birkauser, 1998. I. J. Allan and D. F. Newgreen. The origin and differentiation of enteric neurons of the intestine of the fowl embryo. The American Journal of Anatomy , 157, 137--154, 1980. doi:10.1002/aja.1001570203 B. J. Binder, K. A. Landman, M. J. Simpson, M. Mariani and D. F. Newgreen. Modeling proliferative tissue growth: A general approach and an avian case study. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) , 78(3), 1--13, 2008. doi:10.1103/PhysRevE.78.031912 M. A. Breau, A. Dahmani, F. Broders--Bondon, J. P. Thiery and S. Dufour. $\beta 1$ integrins are required for the invasion of the caecum and proximal hindgut by enteric neural crest cells. Development , 136, 2791--2801, 2009. doi:10.1242/dev.031419 K. Burrage, N. Hale and D. Kay. An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM Journal on Scientific Computing , 34(4), A2145--A2172. doi:10.1137/110847007 N. R. Druckenbrod and M. L. Epstein. The patterns of neural crest advance in the cecum and colon. Developmental Biology , 287, 125--133, 2005. doi:10.1016/j.ydbio.2005.08.040 H. Engler. On the speed of spread for fractional reaction-diffusion equations. International Journal of Differential Equations , 2010, Article ID 315421, 2010. doi:10.1155/2010/315421 M. Ilic, F. Liu, I. Turner and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)--with nonhomogeneous boundary conditions. Fractional Calculus and Applied Analysis , 9, 333--349, 2006. http://eprints.qut.edu.au/23835/ P. K. Maini, D. L. S. McElwain and D. Leavesley. Travelling waves in a wound healing assay. Applied Mathematics Letters , 17, 575--580, 2004. doi:10.1016/S0893-9659(04)90128-0 R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini and P. M. Kulesa. Multiscale mechanisms of cell migration during development: theory and experiment. Development , 139, 2935--2944, 2012. doi:10.1242/dev.081471 J. D. Murray. Mathematical Biology I and II. Springer Verlag, 2003. M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen. Cell proliferation drives neural crest cell invasion of the intestine. Developmental Biology , 302, 553--568, 2007. doi:10.1016/j.ydbio.2006.10.017 H. M. Young, A. J. Bergner, R. B. Anderson, H. Enomoto, J. Milbrandt, D. F. Newgreen and P. M. Whitington. Dynamics of neural crest-derived cell migration in the embryonic mouse gut. Developmental Biology , 270, 455--473, 2004. doi:10.1016/j.ydbio.2004.03.015 P. Zhuang, F. Liu, V. Anh and I. Turner. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis , 47(3), 1760--1781, 2009. doi:10.1137/080730597

A CONIC AFFINE SCALING DOGLEG METHOD FOR NONLINEAR OPTIMIZATION WITH BOUND CONSTRAINTS

The conic model proposed by Davidon (1980) [Davidon, WC (1980). Conic approximations and collinear scalings for optimizers. SIAM Journal on Numerical Analysis, 17, 268–281.] is an extension of quadratic model incorporating more information than quadratic model. In this paper, we propose a conic affine scaling dogleg method for bound constrained optimization. Under some mild conditions, the sequence generated by our algorithm converges to a stationary point. Numerical tests on a series of standard testing problems confirm the efficiency of the method.

On the approximability of two capacitated vehicle routing problems

Abstract Vehicle routing is an important and active research topic in computer science and operations research. In this paper, we give some approximation results for two well-known capacitated vehicle routing problems. Our first result concerns the Capacitated Orienteering problem in Euclidean graphs. We are here given an Euclidean graph G, where each node has a profit value and a demand value, starting and end nodes s, t, a length bound D and a capacity bound C. The goal is to find an s-t-path of length at most D that collects maximum profit from nodes whose total demand does not exceed the capacity bound C. We give a PTAS for this problem, extending the corresponding known result given by Chen and Har-Peled [Chen, K., and S. Har-Peled, The Euclidean orienteering problem revisited. SIAM Journal on Computing, 2007] for the uncapacitated version. Our second result concerns the School Bus problem with regret minimization, where we are given a general metric graph, and the task is to design the routes for a given set of buses of limited capacity to transport a set of children to a school, while minimizing a certain regret threshold. Under the standard hypothesis P ≠ N P , we show that this problem cannot be approximated.

Verification of quantum programs

This paper develops verification methodology for quantum programs, and the contribution of the paper is two-fold. •Sharir, Pnueli and Hart [M. Sharir, A. Pnueli, S. Hart, Verification of probabilistic programs, SIAM Journal of Computing 13 (1984) 292–314] presented a general method for proving properties of probabilistic programs, in which a probabilistic program is modeled by a Markov chain and an assertion on the output distribution is extended to an invariant assertion on all intermediate distributions. Their method is essentially a probabilistic generalization of the classical Floyd inductive assertion method. In this paper, we consider quantum programs modeled by quantum Markov chains which are defined by super-operators. It is shown that the Sharir–Pnueli–Hart method can be elegantly generalized to quantum programs by exploiting the Schrödinger–Heisenberg duality between quantum states and observables. In particular, a completeness theorem for the Sharir–Pnueli–Hart verification method of quantum programs is established.•As indicated by the completeness theorem, the Sharir–Pnueli–Hart method is in principle effective for verifying all properties of quantum programs that can be expressed in terms of Hermitian operators (observables). But it is not feasible for many practical applications because of the complicated calculation involved in the verification. For the case of finite-dimensional state spaces, we find a method for verification of quantum programs much simpler than the Sharir–Pnueli–Hart method by employing the matrix representation of super-operators and Jordan decomposition of matrices. In particular, this method enables us to compute easily the average running time and to analyze some interesting long-run behaviors of quantum programs in a finite-dimensional state space.

A Family of Maximum Entropy Densities Matching Call Option Prices

We investigate the position of the Buchen–Kelly density (Peter W. Buchen and Michael Kelly. The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis, 31(1), 143–159, March 1996.) in the family of entropy maximizing densities from Neri and Schneider (Maximum entropy distributions inferred from option portfolios on an asset. Finance and Stochastics, 16(2), 293–318, April 2012.), which all match European call option prices for a given maturity observed in the market. Using the Legendre transform, which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen–Kelly density and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen–Kelly density in the sense of relative entropy. The method presented here can also be used to interpolate between call option prices, and we compare it to a method proposed by Kahalé (An arbitrage-free interpolation of volatilities. Risk, 17(5), 102–106, May 2004). Orozco Rodriguez and Santosa (Estimation of asset distributions from option prices: Analysis and regularization. SIAM Journal on Financial Mathematics, 3(1), 374–401, 2012.) have produced examples in which the Buchen–Kelly algorithm becomes numerically unstable, and we use these as test cases to show that the algorithm given here remains stable and leads to good results.

An improved bound on acyclic chromatic index of planar graphs

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χa′(G), is the least number of colors k such that G has an acyclic k-edge-coloring. Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet and T. Müller, Acyclic edge-coloring of planar graphs, SIAM Journal of Discrete Mathematics 25 (2) (2011) 463–478] showed that χa′(G)≤Δ(G)+12 for any planar graph G with maximum degree Δ(G). In this paper, the bound is improved to Δ(G)+10.

André Nies. Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305. - Bjørn Kjos-Hanssen, André Nies, and Frank Stephan. Lowness for the class of Schnorr random reals. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647–657. - Noam Greenberg and Joseph S. Miller. Lowness for Kurtz randomness. The Journal of Symbolic Logic, vol.

André Nies. Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305. - Bjørn Kjos-Hanssen, André Nies, and Frank Stephan. Lowness for the class of Schnorr random reals. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647–657. - Noam Greenberg and Joseph S. Miller. Lowness for Kurtz randomness. The Journal of Symbolic Logic, vol. 74 (2009), no. 2, pp. 665–678. - Laurent Bienvenu and Joseph S. Miller. Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506–518. - Johanna N. Y. Franklin, Frank Stephan, and Liang. Yu Relativizations of randomness and genericity notions. The Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 721–733. - George Barmpalias, Joseph S. Miller, and André Nies. Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791–816. - Volume 19 Issue 1

SIGEST

The SIGEST paper in this issue, “Nonlocal Aggregation Models: A Primer of Swarm Equilibria” by Andrew J. Bernoff and Chad M. Topaz, is from the SIAM Journal on Applied Dynamical Systems. One of the many remarkable things in this paper is the authors' clear account of the very direct connections between first-kind Fredholm integral equations and apocalyptic plagues of locusts. Those of you who solve integral equations for a living will not be surprised that a first-kind Fredholm equation can be apocalyptic. While the results in the paper apply to many types of swarms, the authors focus on the particular behavior of Schistocerca gregaria, the swarming desert locust in Africa. These locusts self-organize into bubble-like swarms having a dense group of locusts on the ground, a flying group overhead, and an empty gap between them. The swarm moves with the wind with a rolling motion (see Figure 1 in the paper). This type of swarming is observed in nature and captured by the model in the paper. The paper looks at the relation between two models of swarming behavior. The discrete model is a system of ODEs for the time-dependent path of each individual bug. The forces on the locust are both exogenous (gravity, air resistance, food, predators) and endogenous (the preference to avoid collisions and stay with the group). If the swarm is at all large, the system of ODEs can become intractable, hence motivating a continuum approach. The continuum model, which the authors derive from the discrete model, is a variational principle for the density of locusts. The necessary conditions for stationarity of the energy functional can be expressed as a first-kind Fredholm equation. The analysis in the paper is for the continuous model. The paper looks deeply into the stability of the swarm configurations and the types of forces on the locusts, finds exact solutions for swarm equilibria in many parameter regimes, and carefully examines the regularity of the solutions of the continuous model. They find parameters for which the bubble-like swarms are stable solutions for both models. This is a satisfying paper with deep analysis, a great application, and a discrete formulation that is simple enough for a reader to play with individually or share with a class.

SIGEST

The SIGEST paper in this issue, “Global Convergence of Radial Basis Function Trust-Region Algorithms for Derivative-Free Optimization,” by Stefan M. Wild and Christine Shoemaker, is an expanded version of “Global Convergence of Radial Basis Function Trust Region Derivative-Free Algorithms,” which appeared in the SIAM Journal on Optimization in 2011. The theoretical contribution of this paper is a convergence proof for a derivative-free optimization method. There are many classes of derivative-free methods. The methods of interest in this paper are for smooth problems where derivatives of the objective function (the function to be minimized) are not available. Such problems are common, arising, for example, from applications where source code is not available. These methods approximate the functions with interpolation, and the challenge for a particular method is using as few interpolatory nodes as possible while simultaneously producing an interpolant which is a useful surrogate for the objective function. The algorithm in this paper uses radial basis functions (RBFs) for the interpolation and manages the convergence of the nonlinear iteration with a trust-region approach. The authors do a fine job of explaining the technical aspects of the algorithm, the nodes for the interpolation, and the convergence analysis. The paper concludes with a compelling application to a subsurface contamination problem. The site is the Blaine Naval Ammunition Depot in Nebraska, and the authors collaborated with a large interdisciplinary team to design a remediation strategy. Three dimensional hydrology codes are used to evaluate the objective function. Because of the internal simulations, the objective function for this problem has noise which arises from internal iterations in the simulators and truncation errors in the discretizations. The RBF trust-region algorithm solved this large and complex design problem very nicely.

Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems

The aim of this paper is to obtain, in a Hilbert space $H$, the weak and strong convergence of a penalty proximal algorithm and a splitting one for a bilevel equilibrium problem: find $ x\in S_F $ such that $\ G(x,y)\geq 0\ $ for all $ \ y\in S_F$, where $S_F :=\lbrace y\in K\; :\; F(y,u)\geq 0\;\; \forall u\in K \rbrace$, and $F,G:K\times K\longrightarrow \mathbb{R}$ are two bifunctions with $K$ a nonempty closed convex subset of $H$. In our framework, results of convergence generalize those recently obtained by Attouch et al. (SIAM Journal on Optimization 21, 149-173 (2011)). We show in particular that for the strong convergence of the penalty algorithm, the geometrical condition they impose is not required. We also give applications of the iterative schemes to fixed point problems and variational inequalities.

Smoothed Analysis of Belief Propagation for Minimum-Cost Flow and Matching

Belief propagation (BP) is a message-passing heuristic for statistical inference in graphical models such as Bayesian networks and Markov random fields. BP is used to compute marginal distributions or maximum likelihood assignments and has applications in many areas, including machine learning, image processing, and computer vision. However, the theoretical understanding of the performance of BP remains limited. Recently, BP has been applied to combinatorial optimization problems. It has been proved that BP can be used to compute maximum-weight matchings and minimum-cost flows for instances with a unique optimum. The number of iterations needed for this is pseudo-polynomial and hence BP is not efficient in general. We study BP in the framework of smoothed analysis and prove that with high probability the number of iterations needed to compute maximum-weight matchings and minimum-cost flows is bounded by a polynomial if the weights/costs of the edges are randomly perturbed. To prove our upper bounds, we use an isolation lemma by Beier and Vocking (SIAM Journal on Computing, 2006) for the matching problem and we generalize an isolation lemma by Gamarnik, Shah, and Wei (Operations Research, 2012) for the min-cost flow problem. We also prove lower tail bounds for the number of iterations that BP needs to converge that almost match our upper bounds.

SIGEST

The SIGEST paper in this issue, “The Validity of Johnson--Nédélec's BEM--FEM Coupling on Polygonal Interfaces” by Francisco-Javier Sayas, is from the SIAM Journal on Numerical Analysis. The paper resolves a thirty-year old question about the coupling of the boundary element method (BEM) with the finite element method (FEM) for problems that need both. For example, a problem which naturally contains both an exterior (or infinite) domain and an interior (or bounded) domain might best be solved by discretizing the exterior and interior with FEM and the interface or boundary with BEM. The Johnson--Nédélec coupling was the earliest such method. While this coupling has been successful in practice, the assumptions for the underlying theory did not hold in many situations, including problems with nonsmooth boundaries and linear elasticity. Hence, the theory did not apply to the polygonal interfaces that arise in many finite element discretizations. This led to a perception by some that the Johnson--Nédélec coupling was not sufficient for such problems. In this paper the author shows that in some situations the Johnson--Nédélec coupling works for any Lipschitz boundary and any choice of discrete spaces, thereby putting an old question to rest.

Special Section on the Fiftieth Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)

This section of SIAM Journal on Computing contains extended versions of selected papers from the 50th Annual Symposium on Foundations of Computer Science, sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing. The conference was held in Atlanta, Georgia, October 24--27, 2009, at the Renaissance Atlanta Hotel Downtown. The program committee consisted of Sanjeev Arora, Maria Florina Balcan, Boaz Barak, Mark Braverman, Amit Chakrabarti, Ken Clarkson, Alon Efrat, Martin Furer, Anna Gilbert, Phil Klein, Ming Li, Mihai Patrascu, Dana Ron, Tim Roughgarden, Daniel Spielman, Mario Szegedy, Kunal Talwar, Eli Upfal, Umesh Vazirani, Vijay Vazirani, and Berthold Vocking. They accepted 75 papers from 249 submissions. We briefly describe the papers that appear here. In “On the Power of Randomization in Algorithmic Mechanism Design,” Shahar Dobzinski and Shaddin Dughmi analyze multi-unit auctions to show that truthfulness in expectation is more powerful than universal truthfu...

Junction point on partially singular trajectories

In recent times, several works have been performed on the design of fuel optimal trajectories for space navigation. These works show the possibility of the existence of partially singular trajectories for systems that are linear analytic (Park et al. 2010). Linear analytic systems may show the existence of partially singular subarcs, and the point where these subarcs meet is called a junction point. Thus, knowledge about junction conditions became necessary when solving the optimal control problem for such systems. This led to the development of two ‘theorems’ on junction conditions, given by McDanell and Powers (McDanell, J.P. and Powers W.F. (1971), ‘Necessary Conditions for Joining Optimal Singular and Nonsingular Sub Arcs’, SIAM Journal of Control, 9, 161–173). However, the second ‘theorem’, which is now known as a conjecture, could not satisfy all classes of linear analytic system. Therefore, the aim of this study was to detect and correct the errors in the derivation of the McDanell and Powers conjecture. The error in their derivations was corrected and then tested on two newly mathematically constructed systems. The results of these tests were found to be satisfactory. This implies that by making the necessary corrections, the conjecture can still be useful in generating a general theorem for all classes of systems.