SIAM Journal Research Articles

Quantum computation, entanglement and state reduction

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

The writhing of circular cross–section rods: undersea cables to DNA supercoils

The large deflection theory circular cross–section elastic rods is used to consider the writhing of long straight rods subjected to tension and torque, such as undersea cables, and to closed loops with inserted twist, such as DNA supercoils. The writhed shape of the long straight rod under tension and torque is easily generated by twisting a piece of string with the fingers and consists of three separate parts: a balanced ply region, a free end loop, and two tail regions. The solution for the rod shape in each of the regions is found. The results are then joined together to ensure continuity of the position and tangent vectors of the strand centre line through the introduction of point forces and moments at the points where the strands enter and exit the balanced ply. The results of the model are consistent with simple experiments on long braided rope. The writhed shape of the closed loop with twist inserted between the ends prior to closure is modelled as a balanced ply joined to two end loops. The analysis combines the mechanics solution with the conservation of topological link to provide a simple formula which quantitatively predicts the approximate shape and helix angle of the supercoil. The results are in good agreement with simple experiments on rope and with available data on DNA supercoils.

Fully copositive matrices

The class of fully copositive (C 0 f ) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268–1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class ofQ 0-matrices areP 0-matrices. As a corollary of this main result, we establish that a bisymmetricQ 0-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization ofQ 0-matrices within the class ofC 0 f . While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable toC 0 f ∩Q 0 class. As a byproduct of this characterization, we observe that aC 0 f -matrix is inQ 0 if, and only if, it is completelyQ 0. Also, from Aganagic and Cottle's [M. Aganagic, R.W. Cottle, Mathematical Programming 37 (1987) 223–231] result, it is observed that LCPs arising fromC 0 f ∩Q 0 class can be processed by Lemke's algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An efficient approximation algorithm for the survivable network design problem

The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph ofn vertices and connectivity requirements that are at mostk. Both algorithms find a solution that is within a factor 2k − 1 of optimal fork ⩾ 2 and a factor 2 of optimal fork = 1. Our algorithm improves the time from O(k 3n4) to O $$(k^2 n^2 + kn^2 \sqrt {\log \log n} )$$ ). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the “redundant” edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (s, t) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Introduction to Special Section on Probabilistic Proof Systems

The study of probabilistically verifiable proofs originated in the mid 1980s with the introduction of Interactive Proof Systems (IPs). The primary focus of research in this area in the '80s has been twofold: the role of zero-knowledge interactive proofs within cryptographic protocols, and characterizing which languages are efficiently interactively provable. In the 1990s, the focus of research on the topic shifted. Extensions of the interactive proof model, such as Multiprover Interactive Proofs (MIPs) and Probabilistically Checkable Proofs (PCPs), were considered with the intention of expanding our notion of what should be considered efficiently verifiable. In addition, researchers have taken a closer look at the exact resources (and tradeoffs amongst them) needed to verify a proof using various proof systems. This culminated in the important discovery that it is possible to verify NP statements (with a constant error probability) by only examining a constant number of bits of a PCP and using logarithmic amount of randomness. Perhaps, however, the most dramatic development has been the connection which was found between probabilistically verifiable proofs and proving hardness of approximation for optimization problems. It has been shown that a large variety of optimization versions of NP-hard problems (e.g., the maximum size of a clique in a graph, the minimum number of colors necessary to color a graph, and the maximum number of clauses satisfiable in a CNF formula) are not only NP-hard to solve exactly but also NP-hard to approximate in a very strong sense. The tools to establish hardness of approximation came directly from results on MIPs and PCPs. Indeed, almost every improvement in the efficiency of these proof systems translates directly into showing larger factors within which these optimization problems are hard to approximate. In 1994--1995 two exciting workshops were held at the Weizmann Institute in Israel on the new developments in probabilistically verifiable proofs and their applications to approximation problems, cryptography, program checking, and complexity theory at large. Over 60 papers were presented in the workshop series, and we are proud to include three of them in this special section. "On the Power of Finite Automata with Both Nondeterministic and Probabilistic States" by Anne Condon, Lisa Hellerstein, Samuel Pottle, and Avi Wigderson, considers constant round interactive proof systems where the verifier is restricted to use constant space and public coins. An equivalent characterization is finite automata with both nondeterministic and random states (npfa's), which accept their languages with a small probability of error. The paper shows that npfa's restricted to run in polynomial expected time accept only the regular languages in the case of npfa with 1-way input head, and that if Lis a nonregular language, then either L or its complement is not accepted by any npfa with a 2-way input head. "A Parallel Repetition Theorem" by Ran Raz, addresses and resolves the Parallel Repetition Conjecture which has eluded researchers for some time. The broader topic is what happens to the error probability of proof systems when they are composed. It has been known for awhile that sequential composition of proof systems (both single and multiprover interactive proofs) reduces the error exponentially, but this increases the number of rounds. For interactive proof systems, parallel repetition is known to reduce the error exponentially, and the Parallel Repetition Conjecture asserts that the same holds in a one-round two-prover proof system. Raz proves a constructive bound on the probability of error which indeed reduces at an exponential rate. The constant in the exponent is logarithmic in the total number of possible answers of the two provers, which means one can achieve two-prover one-round MIPs for NP statements with arbitrarily small constant error probability. This, in turn, has played a crucial role in further developments in the area and in particular in those reported in the next paper. "Free Bits, PCPs, and Nonapproximability---Towards Tight Results" by Mihir Bellare, Oded Goldreich, and Madhu Sudan, continues the investigation of PCPs and nonapproximability with emphasis on trying to get closer to tight results. The work consists of three parts. The first part presents several PCP proof systems for NP, based on a new error-correcting code called the Long Code. The second part shows that the connection between PCPs and hardness of approximation is not accidental. In particular, it shows that the transformation of a PCP for NP into hardness results for MaxClique can be reversed. Finally, the third part initiates a systematic investigation of the properties of PCPs as a function of the various parameters: randomness, query complexity, free-bit complexity, amortized free-bit complexity, proof size, etc. Two more papers submitted for this special section were not ready at this time for publication. They will appear in future issues of the SIAM Journal on Computing.

Empirical determinants of measles metapopulation dynamics in England and Wales.

A key issue in metapopulation dynamics is the relative impact of internal patch dynamics and coupling between patches. This problem can be addressed by analysing large spatiotemporal data sets, recording the local and global dynamics of metapopulations. In this paper, we analyse the dynamics of measles meta-populations in a large spatiotemporal case notification data set, collected during the pre-vaccination era in England and Wales. Specifically, we use generalized linear statistical models to quantify the relative importance of local influences (birth rate and population size) and regional coupling on local epidemic dynamics. Apart from the proportional effect of local population size on case totals, the models indicate patterns of local and regional dynamic influences which depend on the current state of epidemics. Birth rate and geographic coupling are not associated with the size of major epidemics. By contrast, minor epidemics--and especially the incidence of local extinction of infection--are influenced both by birth rate and geographical coupling. Birth rate at a lag of four years provides the best fit, reflecting the delayed recruitment of susceptibles to school cohorts. A hierarchical index of spatial coupling to large centres provides the best spatial model. The model also indicates that minor epidemics and extinction patterns are more strongly influenced by this regional effect than the local impact of birth rate.

Introduction to the Special Issue on Iterative Methods for Solving Systems of Algebraic Equations

This is a special issue of SIAM Journal on Scientific Computing devoted to iterative methods for solving systems of algebraic equations. The development of efficient methods is of fundamental impor...

Introduction to the Special Issue on Iterative Methods for Solving Systems of Algebraic Equations

This is a special issue of SIAM Journal on Scientific Computing devoted to iterative methods for solving systems of algebraic equations. The development of efficient methods is of fundamental importance in modern computational modeling, as the solution of large systems of equations often represents the dominant computational cost in mathematical simulations. In contrast to direct methods based on Gaussian elimination, iterative methods tend to have storage requirements proportional to the size of the problem. Consequently, they have the potential for handling significantly larger problems than direct methods. Iterative methods are the only option when fine resolution is required, particularly for three-dimensional models. Similarly, they are the only choice for nonlinear problems, and such solvers in the nonlinear case are frequently combined with inner iterations for linear subproblems. The papers in this volume represent a snapshot of the current state of the art in iterative solution algorithms. The results in these papers were originally presented at the Copper Mountain meeting on iterative methods held in April 1996. This was the eleventh annual meeting in the Copper Mountain series and the fourth devoted exclusively to iterative methods. There were 200 attendees at the conference, and more than 150 presentations were made. The papers included here (identified by first author) can be loosely grouped into five categories: This collection points to some significant trends in the study of iterative methods, including the maturation of algorithm technology for new "model problems" such as saddle-point and fourth-order problems; the increasing complexity of problems, such as the full potential equation in fluid dynamics, being handled by iterative methods; and the integration of parallel computation and software development into many of the research efforts. I wish to thank the following members of the Copper Mountain program committee who acted as special editors for this issue: Through their efforts, submissions were carefully refereed and brought to print on schedule. I also want to thank Tom Manteuffel and Steve McCormick, the organizers of the Copper Mountain conferences, for setting a consistently high standard for the meetings. Special thanks are due to Fred Howes of the Department of Energy, Michael Steuerwalt of the National Science Foundation, and Eric Pritcher from Cray Research for generous support of the meeting. Our hope is that the diverse topics covered in this issue make it clear that the development and analysis of iterative methods continues to be a challenging and fruitful area of research. Howard ElmanUniversity of Maryland, College Park

Classical Solutions of Multidimensional Hele--Shaw Models

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

On the non-universality of the error function in the smoothing of stokes discontinuities

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Chapman S. J. 1996On the non-universality of the error function in the smoothing of stokes discontinuitiesProc. R. Soc. Lond. A.4522225–2230http://doi.org/10.1098/rspa.1996.0118SectionRestricted accessArticleOn the non-universality of the error function in the smoothing of stokes discontinuities S. J. Chapman Google Scholar Find this author on PubMed Search for more papers by this author S. J. Chapman Google Scholar Find this author on PubMed Search for more papers by this author Published:01 January 1996https://doi.org/10.1098/rspa.1996.0118AbstractThe non-universality of the error function in the smoothing of Stokes lines is demonstrated by means of an example with smoothing function ∫ ϕ-∞ e -u2m+g(u) du, where m is any integer greater than 2 and g is any polynomical of degree less than or equal to 2m - 1.FootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Say F (2020) Optimal successive complementary expansion for singular differential equations, Mathematical Methods in the Applied Sciences, 10.1002/mma.6228, 44:9, (7423-7432), Online publication date: 1-Jun-2021. Lustri C, Pethiyagoda R and Chapman S (2019) Three-dimensional capillary waves due to a submerged source with small surface tension, Journal of Fluid Mechanics, 10.1017/jfm.2018.1030, 863, (670-701), Online publication date: 25-Mar-2019. Dean A, Matthews P, Cox S and King J (2015) Orientation-Dependent Pinning and Homoclinic Snaking on a Planar Lattice, SIAM Journal on Applied Dynamical Systems, 10.1137/140966897, 14:1, (481-521), Online publication date: 1-Jan-2015. Chu Y, Li Y, Xia W and Zhang X (2014) Best possible inequalities for the harmonic mean of error function, Journal of Inequalities and Applications, 10.1186/1029-242X-2014-525, 2014:1, (525), . Pasquetti S and Schiappa R (2010) Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models, Annales Henri Poincaré, 10.1007/s00023-010-0044-5, 11:3, (351-431), Online publication date: 1-Aug-2010. Chapman S, King J and Adams K (1998) Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454:1978, (2733-2755), Online publication date: 8-Oct-1998. Xia W (2012) Two Optimal Double Inequalities for Error Function, Advanced Materials Research, 10.4028/www.scientific.net/AMR.557-559.2092, 557-559, (2092-2095) This Issue08 October 1996Volume 452Issue 1953 Article InformationDOI:https://doi.org/10.1098/rspa.1996.0118Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Manuscript received24/08/1995Manuscript accepted11/12/1995Published online01/01/1997Published in print01/01/1996 License:Scanned images copyright © 2017, Royal Society Citations and impact

Star-shapedness of the kernel for homogeneous games

Homogeneous games and weighted majority games were introduced by von Neumann-Morgenstern ( Theory of Games and Economic Behavior, 1944) in the constant-sum case. Peleg ( Illinois Journal of Mathematics, 1966, 10, 39–48; SIAM Journal of Applied Mathematics, 1968, 16, 527–532) studied the kernel and nucleolus for these classes of games. The general theory of homogeneous, not necessarily constant-sum, games was developed by Ostmann ( International Journal of Game Theory, 1987a, 16, 69–81), Rosenmüller (On homogeneous weights for simple games, Working Paper 115, 1982; Zeitschrift für Operations Research, 1984, 12, 123–141; Mathematics of Operations Research, 1987a, 12, 309–330), and Sudhölter ( International Journal of Game Theory, 1989, 18, 433–469). Peleg-Rosenmüller ( Games and Economic Behaviour, 1992, 4, 588–605) used it to discuss several solution concepts for homogeneous games without steps. A reduction theorem for the nucleolus and kernel of homogeneous games with steps was proved by Rosenmüller-Sudhölter ( Discrete Applied Mathematics, 1994, 50, 53–76) and Peleg et al. (The kernel of homogeneous games with steps, Essays in Game Theory in Honor of Michael Maschler, 1994), respectively. On the basis of these results, this paper shows that the kernel of each homogeneous game without winning players is star-shaped. Moreover, each of these games possesses a truncated game that is uniquely determined and homogeneous itself. The normalized vector of weights of the minimal representation of the truncated game is a center of both of the kernels of the original game and the truncated game. Moreover, this preimputation is Lorenz maximal within the kernel and can be characterized as the unique minimizer of a social welfare ordering in the kernel. To be more precise, the center solution uniquely minimizes a weighted Gini coefficient. Every weighted majority game occurs as a reduced game of certain homogeneous games called homogeneous extensions. The kernel of a weighted majority game arises from that of each of its homogeneous extensions in a very simple way. Moreover, the kernels of partition games are shown to be singletons.

Accuracy and stability of numerical algorithms

From the Publisher: What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included here. This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis. Software practicalities are emphasized throughout, with particular reference to LAPACK and MATLAB. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon. Because of its central role in revealing problem sensitivity and providing error bounds, perturbation theory is treated in detail. Historical perspective and insight are given, with particular reference to the fundamental work of Wilkinson and Turing, and the many quotations provide further information in an accessible format. The book is unique in that algorithmic developments and motivations are given succinctly and implementation details minimized, so that attention can be concentrated on accuracy and stability results. Here, in one place and in a unified notation, is error analysis for most of the standard algorithms in matrix computations. Not since Wilkinson's Rounding Errors in Algebraic Processes (1963) and The Algebraic Eigenvalue Problem (1965) has any volume treated this subject in such depth. A number of topics are treated that are not usually covered in numerical analysis textbooks, including floating point summation, block LU factorization, condition number estimation, the Sylvester equation, powers of matrices, finite precision behavior of stationary iterative methods, Vandermonde systems, and fast matrix multiplication. Although not designed specifically as a textbook, this volume is a suitable reference for an advanced course, and could be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises (many of which have never before appeared in textbooks). The book is designed to be a comprehensive reference and its bibliography contains more than 1100 references from the research literature. Audience Specialists in numerical analysis as well as computational scientists and engineers concerned about the accuracy of their results will benefit from this book. Much of the book can be understood with only a basic grounding in numerical analysis and linear algebra. About the Author Nicholas J. Higham is a Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 40 publications and is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications and the IMA Journal of Numerical Analysis. His book Handbook of Writing for the Mathematical Sciences was published by SIAM in 1993.

Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Bridges Thomas J. 1996Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface wavesPhil. Trans. R. Soc. A.354533–574http://doi.org/10.1098/rsta.1996.0019SectionRestricted accessArticlePeriodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Published:15 March 1996https://doi.org/10.1098/rsta.1996.0019AbstractSpace and time periodic waves at the two-dimensional surface of an irrotational inviscid fluid of finite depth are considered. The governing equations are shown to have a new formulation as a generalized Hamiltonian system on a multisymplectic structure where there is a distinct symplectic operator corresponding to each unbounded space direction and time. The wave-generated mean flow in this framework has an interesting characterization as drift along a group orbit. The theory has interesting connections with, and generalizations of, the concepts of action, action flux, pseudofrequency and pseudowavenumber of the Whitham theory. The multisymplectic structure and novel characterization of mean flow lead to a new constrained variational principle for all space and time periodic patterns on the surface of a finite-depth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for three-dimensional travelling wavesFootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Hu W, Xiao C and Deng Z (2023) Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System Geometric Mechanics and Its Applications, 10.1007/978-981-19-7435-9_3, (89-201), . Clamond D and Dutykh D (2012) Practical use of variational principles for modeling water waves, Physica D: Nonlinear Phenomena, 10.1016/j.physd.2011.09.015, 241:1, (25-36), Online publication date: 1-Jan-2012. Bridges T and Donaldson N (2010) VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME‐DEPENDENT MOVING MESH, Mathematika, 10.1112/S0025579310001233, 57:1, (147-173), Online publication date: 1-Jan-2011. Sedletsky Y (2006) A new type of modulation instability of Stokes waves in the framework of an extended NSE system with mean flow, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/39/31/L03, 39:31, (L529-L537), Online publication date: 4-Aug-2006. Sedletsky Y (2005) The modulational instability of Stokes waves on the surface of finite-depth fluid, Physics Letters A, 10.1016/j.physleta.2005.04.076, 343:4, (293-299), Online publication date: 1-Aug-2005. Laine-Pearson F and Bridges T (2004) Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves, SIAM Journal on Applied Mathematics, 10.1137/S0036139903423753, 64:6, (2096-2120), Online publication date: 1-Jan-2004. Bridges T (2003) Benjamin Memorial Lecture: Stability of Solitary Waves: Geometry, Symplecticity and Three-Dimensionality Wind over Waves II, 10.1533/9780857099532.178, (178-188), . Bridges T (1999) A new framework for studying the stability of genus-1 and genus-2 KP patterns, European Journal of Mechanics - B/Fluids, 10.1016/S0997-7546(99)80044-2, 18:3, (493-500), Online publication date: 1-May-1999. Dias F and Kharif C (1999) NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES, Annual Review of Fluid Mechanics, 10.1146/annurev.fluid.31.1.301, 31:1, (301-346), Online publication date: 1-Jan-1999. Bridges T (1999) Toral-equivariant partial differential equations and quasiperiodic patterns, Nonlinearity, 10.1088/0951-7715/11/3/005, 11:3, (467-500), Online publication date: 1-May-1998. Knobloch E and Pierce R (2009) On mean flows associated with traveling water waves, Fluid Dynamics Research, 10.1016/S0169-5983(97)00030-0, 22:2, (61-71), Online publication date: 1-Feb-1998. Bridges T (1997) A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453:1962, (1365-1395), Online publication date: 8-Jul-1997. This Issue15 March 1996Volume 354Issue 1707 Article InformationDOI:https://doi.org/10.1098/rsta.1996.0019Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Manuscript received20/09/1994Manuscript accepted07/02/1995Published online01/01/1997Published in print15/03/1996 License:Scanned images copyright © 2017, Royal Society Citations and impact

Random Utility Representation of Binary Choice Probabilities: Critical Graphs Yielding Critical Necessary Conditions

Representing binary choice probabilities by random utility variables is known to amount to a representation by a probability distribution on the collection of rankings of the alternatives and the representation problem consists in finding a minimal complete set of linear inequalities defining the convex polytope spanned by these rankings or linear orderings. A technique for this problem, developed in the field of operations research, is first described and next applied to determine which of the inequalities derived in Koppen ( 1991, in J. P. Doignon & J.-C. Falmagne (Eds.), Mathematical psychology: Current developments (pp. 181-201), New York: Springer-Verlag) define facets of this polytope. Then, by the same technique, a considerable generalization is obtained, leading to a graph-theoretic formulation of a wide class of facet-defining inequalities, including not only Koppen's, but also Cohen and Falmagne's (1990, Journal of Mathematical Psychology34, 88-94) and Fishburn's (1990, SIAM Journal of Discrete Mathematics3, 478-488) inequalities as specific subschemes. Apart from these, many more infinite schemes of facet-defining inequalities are shown to be implied.

Some properties of globally δ-convex functions∗

The notions of δ-convex and midpoint δ-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27 1989). It is known that such functions have some important optimization properties: each r-local minimum is a global minimum, and if they assume their global maximum on a bounded convex domain of a Hilbert space then they do so at least at some r-extreme points of this domain. In this paper some analytical properties of δ-convex and midpoint δ-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, δ-convex functions defined on the entire real line is always locally bounded, and midpoint δ-convex function on the real line is either locally bounded or totally unbounded. Further on, it is proved that there are totally discontinuous (i.e., nowhere differentiable) δ-convex and midpoint δ-convex functions on the real line

Erratum: Generalized Jacobi Weights, Christoffel Functions, and Jacobi Polynomials

Previous article Full AccessErratum: Generalized Jacobi Weights, Christoffel Functions, and Jacobi PolynomialsTamás Erdélyi, Alphonse P. Magnus, and Paul NevaiTamás Erdélyi, Alphonse P. Magnus, and Paul Nevaihttps://doi.org/10.1137/0525082PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Erratum: Generalized Jacobi Weights, Christoffel Functions, and Jacobi Polynomials." SIAM Journal on Mathematical Analysis, 25(5), p. 1461[1] Paul Nevai, , Tamás Erdélyi and , Alphonse P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal., 25 (1994), 602–614 10.1137/S0036141092236863 95f:33011a 0805.33013 LinkISIGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails ASYMPTOTIC LINEAR PROGRAMMING LOWER BOUNDS FOR THE ENERGY OF MINIMIZING RIESZ AND GAUSS CONFIGURATIONS27 September 2018 | Mathematika, Vol. 65, No. 1 Cross Ref Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operatorAdvances in Mathematics, Vol. 333 Cross Ref Fractional Pearson diffusionsJournal of Mathematical Analysis and Applications, Vol. 403, No. 2 Cross Ref Decomposition of spaces of distributions induced by tensor product basesJournal of Functional Analysis, Vol. 263, No. 5 Cross Ref A Cohen type inequality for Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer-Sobolev inner product12 January 2011 | Mathematische Nachrichten, Vol. 284, No. 2-3 Cross Ref An upper bound on Jacobi polynomialsJournal of Approximation Theory, Vol. 149, No. 2 Cross Ref Inequalities for orthonormal Laguerre polynomialsJournal of Approximation Theory, Vol. 144, No. 1 Cross Ref On the maximum value of Jacobi polynomialsJournal of Approximation Theory, Vol. 136, No. 1 Cross Ref Inequalities for real-root polynomials and entire functionsAdvances in Applied Mathematics, Vol. 29, No. 1 Cross Ref Large Sieve Estimates on Arcs of a CircleJournal of Number Theory, Vol. 91, No. 2 Cross Ref A new bound for the Laguerre polynomialsJournal of Computational and Applied Mathematics, Vol. 133, No. 1-2 Cross Ref New Bounds for Hahn and Krawtchouk PolynomialsHolger Dette17 February 2012 | SIAM Journal on Mathematical Analysis, Vol. 26, No. 6AbstractPDF (1173 KB) Volume 25, Issue 5| 1994SIAM Journal on Mathematical Analysis History Submitted:13 April 1994Accepted:14 April 1994Published online:17 July 2006 InformationCopyright © 1994 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0525082Article page range:pp. 1461-1461ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Finding k Disjoint Paths in a Directed Planar Graph

It is shown that, for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.

New algorithms for generalized network flows

This paper, of which a preliminary version appeared in ISTCS'92, is concerned with generalized network flow problems. In a generalized network, each edgee = (u, v) has a positive ‘flow multiplier’a e associated with it. The interpretation is that if a flow ofx e enters the edge at nodeu, then a flow ofa e x e exits the edge atv. The uncapacitated generalized transshipment problem (UGT) is defined on a generalized network where demands and supplies (real numbers) are associated with the vertices and costs (real numbers) are associated with the edges. The goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand, and the sum over the edges of the product of the cost and the flow is minimized. Adler and Cosares [Operations Research 39 (1991) 955–960] reduced the restricted uncapacitated generalized transshipment problem, where only demand nodes are present, to a system of linear inequalities with two variables per inequality. The algorithms presented by the authors in [SIAM Journal on Computing, to appear result in a faster algorithm for restricted UGT. Generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges (i.e., upper bounds on the amount of flow). The goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized. We present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of UGT. The algorithm can be used to find an optimal flow or an approximation thereof. When used to find a constant factor approximation, the algorithm is not only more efficient than previous algorithms but also strongly polynomial. It is believed to be the first strongly polynomial approximation algorithm for generalized circulation. The existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm.

Special Section on Iterative Methods in Numerical Linear Algebra

Previous article Next article SPECIAL SECTIONSpecial Section on Iterative Methods in Numerical Linear AlgebraTom Manteuffel and Steve McCormickTom Manteuffel and Steve McCormickhttps://doi.org/10.1137/0915intro1PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Special Section on Iterative Methods in Numerical Linear Algebra." SIAM Journal on Scientific Computing, 15(2), pp. 295–296 Previous article Next article FiguresRelatedReferencesCited byDetails Efficient numerical integration of stiff differential equations in polymerisation reaction engineering: Computational aspects and applications17 February 2012 | The Canadian Journal of Chemical Engineering, Vol. 90, No. 4 Cross Ref Volume 15, Issue 2| 1994SIAM Journal on Scientific Computing† Special Section on Iterative Methods in Numerical Linear Algebra History Published online:13 July 2006 InformationCopyright © 1994 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0915intro1Article page range:pp. 295-296ISSN (print):1064-8275ISSN (online):1095-7197Publisher:Society for Industrial and Applied Mathematics

On the Controllability of Matrix Pairs $( A, K )$ with K Positive Semidefinite, II

The subject of the previous article by David Carlson, B. N. Datta, and Hans Schneider [SIAM Journal on Algebraic and Discrete Methods, 5 (1984), pp. 346--350] is revisited to improve and clarify results given there and elsewhere.