A quasi‐symmetric 2‐(49,9,6) design

Huffman and Tonchev discovered four non‐isomorphic quasi‐symmetric 2‐(49,9,6) designs. They arise from extremal self‐dual [50,25,10] codes with a certain weight enumerator. In this note, a new quasi‐symmetric 2‐(49,9,6) design is constructed. This is established by finding a new extremal self‐dual [50,25,10] code as a neighbor of one of the four extremal codes discovered by Huffman and Tonchev. A number of new extremal self‐dual [50,25,10] codes with other weight enumerators are also found. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 173–179, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10007     

Quadrature rules for qualocation

Qualocation is a method for the numerical treatment of boundary integral equations on smooth curves which was developed by Chandler, Sloan and Wendland (1988‐2000) [1,2]. They showed that the method needs symmetric J–point–quadrature rules on [0, 1] that are exact for a maximum number of 1–periodic functions The existence of 2–point–rules of that type was proven by Chandler and Sloan. For J ∈ {3, 4} such formulas have been calculated numerically in [2]. We show that the functions G_α form a Chebyshev–system on [0, 1/2] for arbitrary indices á and thus prove the existence of such quadrature rules for any J.     

Consistency adjustments for pairwise comparison matrices

This paper is concerned with the development of a ‘best’ rank one transitive approximation to a general paired comparison matrix in a least‐squares sense. A direct attack on the non‐linear problem is frequently replaced by a sub‐optimal linear problem and, here, the best procedure of this kind is obtained. The Newton–Kantorovich method for the solution of the non‐linear problem is also studied, including the role of the best linear approximation as a starting point for this method. Numerical examples are included. Copyright © 2002 John Wiley & Sons, Ltd.     

Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures

Suppose that there are n jobs and n machines and it costs c_ij to execute job i on machine j. The assignment problem concerns the determination of a one‐to‐one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. When the c_ij are independent and identically distributed exponentials of mean 1, Parisi [Technical Report cond‐mat/9801176, xxx LANL Archive, 1998] made the beautiful conjecture that the expected cost of the minimum assignment equals . Coppersmith and Sorkin [Random Structures Algorithms 15 ( 6 ), 113–144] generalized Parisi's conjecture to the average value of the smallest k‐assignment when there are n jobs and m machines. Building on the previous work of Sharma and Prabhakar [Proc 40th Annu Allerton Conf Communication Control and Computing, 20 , 657–666] and Nair [Proc 40th Annu Allerton Conf Communication Control and Computing, 17 , 667–673], we resolve the Parisi and Coppersmith‐Sorkin conjectures. In the process we obtain a number of combinatorial results which may be of general interest.© 2005 Wiley Periodicals, Inc. Random Struct. Alg. 2005     

A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems

We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

New upper bounds on the decomposability of planar graphs

It is known that a planar graph on n vertices has branch‐width/tree‐width bounded by . In many algorithmic applications, it is useful to have a small bound on the constant α. We give a proof of the best, so far, upper bound for the constant α. In particular, for the case of tree‐width, α < 3.182 and for the case of branch‐width, α < 2.122. Our proof is based on the planar separation theorem of Alon, Seymour, and Thomas and some min–max theorems of Robertson and Seymour from the graph minors series. We also discuss some algorithmic consequences of this result. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two‐group transport operators with general boundary conditions in the Banach space L_p ([–a, a ] × [–1, 1]) × L_p ([–a, a ] × [–1, 1]), a > 0, p ≥ 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

High‐girth cubic graphs are homomorphic to the Clebsch graph

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5‐colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1 ). Hopkins and Staton [J Graph Theory 6(2) (1982), 115–121] and Bondy and Locke [J Graph Theory 10(4) (1986), 477–504] proved that every (sub)cubic graph of girth at least 4 has an edge‐cut containing at least of the edges. The existence of such an edge‐cut follows immediately from the existence of a 5‐edge‐coloring as described above; so our theorem may be viewed as a coloring extension of their result (under a stronger girth assumption). Every graph which has a homomorphism to a cycle of length five has an above‐described 5‐edge‐coloring; hence our theorem may also be viewed as a weak version of Ne?et?il's Pentagon Problem (which asks whether every cubic graph of sufficiently high girth is homomorphic to C₅). Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 241—259, 2011     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Subspace methods for three‐parameter eigenvalue problems

We propose subspace methods for three‐parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for two‐parameter eigenvalue problems exist, their extensions to a three‐parameter setting seem challenging. An inherent difficulty is that, while for two‐parameter eigenvalue problems, we can exploit a relation to Sylvester equations to obtain a fast Arnoldi‐type method, such a relation does not seem to exist when there are three or more parameters. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi–Davidson‐type method for three or more parameters, which we generalize from its two‐parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi–Davidson approach is devised to locate eigenvalues close to a prescribed target; yet, it often also performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. The proposed approaches are suitable especially for problems where the computation of several eigenvalues is required with high accuracy. MATLAB implementations of both methods have been made available in the package MultiParEig (see http://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig ).     

Numerical simulation of generalized newtonian blood flow past a couple of irregular arterial stenoses

This paper looked at the numerical investigations of the generalized Newtonian blood flow through a couple of irregular arterial stenoses. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mild stenosed artery, so that the flow effectively becomes two‐dimensional. The Marker and Cell (MAC) method is developed for the governing unsteady generalized Newtonian equations in staggered grid for viscous incompressible flow in the cylindrical polar co‐ordinates system. The derived pressure‐Poisson equation was solved using Successive‐Over‐Relaxation (S.O.R.) method and the pressure‐velocity correction formulae have been derived. Computations are performed for the pressure drop, the wall shear stress distribution and the separation region. The presented computations show that in comparison to the corresponding Newtonian model the generalized Newtonian fluid experiences higher pressure drop, lower peak wall shear stress and smaller separation region. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 960–981, 2011     

Singular Sturmian separation theorems for nonoscillatory symplectic difference systems

ABSTRACT

In this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445–459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431–444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Do?lý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268–1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. ?epitka and R. ?imon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243–275; P. ?epitka and R. ?imon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657–698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Computational fluid dynamics simulation of aerodynamic performance and flow separation by single element and slatted airfoils under rainfall conditions

This study investigated the effects of rainfall on flow separation and the aerodynamic performance of single element and slatted NACA 0012 airfoils by using a mathematical model developed with the commercial computational fluid dynamics solver ANSYS FLUENT 18.2. A two-way momentum coupled Eulerian–Lagrangian multiphase approach was used to simulate the formation of the water film layer on the airfoil's surface. According to the results, very low values of the lift-to-drag ratio at low angles of attack reflected severe degradation of the aerodynamic performance of the airfoil in the presence of water accumulated on its surface. The impact of rain droplets on the leading-edge slat surface led to less water accumulating on the main section of the airfoil. In particular, the maximum water film mass concentrated on the airfoil surface decreased from 15 g to 1 g compared with the single element airfoil. Hence, the thickness of the water film layer was not sufficiently large to significantly affect the aerodynamic coefficients of the slatted airfoil, especially the maximum lift coefficient, compared with the thicker water film layer on the single element airfoil. In addition, the use of slats clearly enhanced the aerodynamic coefficients and increased the stall angle from 13° to 22° in dry conditions, and from 16° to 24° in rainy conditions. Slats also significantly decreased the boundary layer thickness and delayed the separation at higher angles of attack.     

Steady Groundwater Flow with Steep Gradients: An Analytic Approach to the 2D–Problem with a Vertical Pressure Boundary Condition and Steady Recharge/Drainage

Steady groundwater flow with steep gradients in a vertical plane due to superficial recharge/drainage, inner sources/sinks and a one‐sided pressure boundary condition can be described by a 2D Poisson equation with a nonlinear free surface boundary condition. By means of conformal mapping techniques Schmitz and Edenhofer [1] derived the exact explicit solution of this problem in a horizontally infinite aquifer. Their results are extended to problems with a one‐sided vertical pressure boundary condition, modelling f. ex. the boundary between an aquifer and an adjacent free water body. According to ist simple parametrization, this approach can be applied on one hand to model various real world phenomena like river–aquifer–systems. It may on the other hand serve as a tool for investigating the exactness of numerical solutions and the range of validity of simplifying assumptions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Membrane fission: A computational complexity perspective

Membrane fission is a process by which a biological membrane is split into two new ones in the manner that the content of the initial membrane is separated and distributed between the new membranes. Inspired by this biological phenomenon, membrane separation rules were considered in membrane computing. In this work, we investigate cell‐like P systems with symport/antiport rules and membrane separation rules from a computational complexity perspective. Specifically, we establish a limit on the efficiency of such P systems which use communication rules of length at most two, and we prove the computational efficiency of this kind of models when using communication rules of length at most three. Hence, a sharp borderline between tractability and NP –hardness is provided in terms of the length of communication rules. © 2015 Wiley Periodicals, Inc. Complexity 21: 321–334, 2016