an axiomatic approach to proportionality between matrices given a matrix p 0 what does it mean to say that a matrix f of the same dimension whose row and column sums must fall between specific limits is proportional to p this paper gives an axiomatic solution to this question in two distinct contexts first for any real allocation matrix f second for any integer constrained apportionment matrix f in the case of f real the solution turns out to coincide with what has been variously called biproportional scaling and diagonal equivalence and has been much used in econometrics and statistics in the case of f integer the problem arises in the simultaneous apportionment of seats to regions and to parties and also in the rounding of tables of census data

two phase free surface fluid dynamics on moving domains we develop in this paper a model for the numerical simulation of free surface debris and mudflows described as a mixture of two viscous phases a lagrangian approach with finite element approximation and regular remeshing is adopted in this paper the proposed model solves the equations of motion in both phases coupled with conservation of volume of the mixture implicitly volume fractions are computed explicitly at the end of each iteration a one step trapezoidal scheme is used for time integration the resulting model combines simplicity robustness and versatility and provides a novel and unified algorithmic framework for a large spectrum of applications it is also easily extensible to three spatial dimensions more complex constitutive models and coupling with geomechanical models application to sedimentation and to impact on an obstacle shows the effectiveness of the approach

large time step explicit integration method for solving problems with dominant convection abstract an explicit time integrator without the cfl the idea is to use the information existing at time t t n in the velocity streamlines as well as in the acceleration streamlines to update the particle position as well as the velocity in an updated lagrangian frame the method may be used with moving or fixed meshes

an ale pfem method for the numerical simulation of two phase mixture flow abstract a new finite element method is proposed to simulate two phase mixture problems with free surfaces and or large moving boundaries the two phases are treated as interpenetrating continua within a two fluid model in contrast to the conventional two fluid model formulations in which the governing equations are written in the eulerian eulerian description the modified two fluid model has the governing equations of one fluid phase written in the lagrangian description while the respective equations of the other phase is formulated in the arbitrary lagrangian eulerian ale description with relative velocities defined with respect to the moving mesh associated with the first fluid phase the finite element method fem together with stabilization techniques the characteristic based split cbs and the algebraic flux correction afc method is employed to solve the equations numerically in space and time the nodes of the computational mesh are updated together with the flow of the phase described in the lagrangian description the computational mesh is updated by applying techniques used in the so called particle finite element method pfem characterized by regenerating the mesh in each computational step according to the updated nodal positions while the current boundary of the domain is identified using shape method the performance of the method is demonstrated by means of three 2d and 3d examples including a mixing process of two fluids in a stirred tank a free surface flow a 3d two phase channel flow a phase separation problem and the flow of two immiscible fluids

some computational results on a problem concerning powerful numbers let d be a positive square free integer and let x y j d be the fundamental unit in the order with z basis 1 vd an algorithm which is of time complexity 0 d1 4 for any positive e is developed for determining whether or not d y results are presented for a computer run of this algorithm on all d 108 the conjecture of ankeny artin and chowla is verified for all primes 1 mod 4 less than 109

divisor based biproportional apportionment in electoral systems a real life benchmark study biproportional apportionment methods provide two way proportionality in electoral systems where the electoral region is subdivided into electoral districts the problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix and such that a set of row and column sum requirements are fulfilled in a divisor based method for biproportional apportionment the problem is solved by computing appropriate row and column divisors and by rounding the quotients we present a comprehensive experimental evaluation of divisor based biproportional apportionment in an electoral system context by performing experiments on real life benchmark instances election data with multimember districts we evaluate the general quality of divisor based apportionments with respect to e g deviation from quota reversal orderings and occurrences of ties for example we studied the frequency in which a party with a higher vote count in a district ended up with fewer seats in that district

on the parallel generation of the residues for the continued fraction factoring algorithm etude des proprietes du developpement en fraction continue d un irrationnel quadratique application au developpement d un algorithme efficace de factorisation

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

the continued fraction algorithm and regulator for quadratic function fields of characteristic 2 the continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied recently these ideas have even been used to produce cryptosystems much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic we will explore these ideas and show that the situation is very similar to the odd characteristic case this exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2

algorithms for proportional matrices in reals and integers letr be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh 0 closely related axiomatic approaches have been developed to ascribe meanings to the statements the real matrixf r and the integer matrixa r are proportional to a given matrixp 0

computer verification of the ankeny artin chowla conjecture for all primes less than 100 000 000 000 let p be a prime congruent to 1 modulo 4 and let t u be rational integers such that t u p 2 is the fundamental unit of the real quadratic field q p the ankeny artin chowla conjecture aac conjecture asserts that p will not divide u this is equivalent to the assertion that p will not divide b p 1 2 where b n denotes the nth bernoulli number although first published in 1952 this conjecture still remains unproved today indeed it appears to be most difficult to prove even testing the conjecture can be quite challenging because of the size of the numbers t u for example when p 40094470441 then both t and u exceed 10 330 000 in 1988 the aac conjecture was verified by computer for all p 10 9 in this paper we describe a new technique for testing the aac conjecture and we provide some results of a computer run of the method for all primes p up to 10 11

a numerical investigation into the length of the period of the continued fraction expansion of p d cd 1 21ogd o d l 2 where c 7 22t2 2 3546 the brief numerical study on the size of p d in beach and williams 1 seems to indicate that these bounds are not very good for larger values of d the purpose of this note is to suggest why this should be the case and to provide numerical evidence for the belief that p d can be bounded by an expression of the form cd 1 21og log d where c is a constant

linear inequality scaling problems let a in r m b in r m and c in r m times n be given where a is strictly positive a c scaling of the vector a is defined to be a vector a in r n with a j a j pi k 1 m u k c kj for some strictly positive vector u in r m the problem of finding a c scaling of the vector a that satisfies the linear system cx b is called the linear equality scaling problem lesp the current paper considers the linear inequality scaling problem lisp which concerns the identification of a c scaling of a which satisfies the linear inequality system cx leqq b where it is required that u leqq 1 and that u i 1 for each i with cx i b i it is shown that lisp generalizes lesp and that it unifies a number of matrix scaling problems that have been studied recently further it is shown that lisp can be reduced to one of two convex optimization problems and these reductions are used to characterize solutions to lisp and to derive necessary and sufficient condition

improving mass conservation in simulation of incompressible flows summary we propose a technique for improving mass conservation features of fractional step schemes applied to incompressible flows the method is illustrated by using a lagrangian fluid formulation where the mass loss effects are particularly apparent however the methodology is general and could be used for fixed grid approaches the idea consists in reflecting the incompressibility condition already in the intermediate velocity this is achieved by predicting the end of step pressure and using this prediction in the fractional momentum equation the resulting intermediate velocity field is thus much closer to the final incompressible one than that of the standard fractional step scheme in turn the predicted pressure can be used as the boundary condition necessary for the solution of the pressure poisson equation in case a continuous laplacian matrix is employed using this approximation of the end of step incompressible pressure as the essential boundary condition considerably improves the conservation of mass specially for the free surface flows of fluids with low viscosity the pressure prediction does not require the resolution of any additional equations system the efficiency of the method is shown in two examples the first one shows the performance of the method with respect to mass conservation the second one tests the method in a challenging fluid structure interaction benchmark which can be naturally resolved by using the presented approach copyright 2012 john wiley sons ltd

computation of real quadratic fields with class number one a rapid method for determining whether the real quadratic field si s d has class number one is described the method makes use of the infrastructure idea of shanks to determine the regulator of w and then uses the generalized riemann hypothesis to rapidly estimate l l x to the accuracy needed for determining whether or not the class number of 3 is one the results of running this algorithm on a computer for all prime values of d up to 109 are also presented together with further results on runs on intervals of size 107 starting at 10 t 9 10 16

on the complexity of general matrix scaling and entropy minimization via the ras algorithm given an n m nonnegative matrix a a ij and positive integral vectors r e n and c e m having a common one norm h the r c scaling problem is to obtain positive diagonal matrices x and y if they exist such that xay has row and column sums equal to r and c respectively the entropy minimization problem corresponding to a is to find an n m matrix z z ij having the same zero pattern as a the sum of whose entries is a given number h its row and column sums are within given integral vectors of lower and upper bounds and such that the entropy function consisting of the sum of the terms z ij ln z ij a ij is minimized when the lower and upper bounds coincide matrix scaling and entropy minimization are closely related in this paper we present several complexity bounds for the e approximate r c scaling problem polynomial in n m h 1 e and ln v v where v and v are the largest and the smallest positive entries of a respectively these bounds although not polynomial in ln 1 1 e not only provide alternative complexities for the polynomial time algorithms but could result in better overall complexities in particular our theoretical analysis supports the practicality of the well known ras algorithm in our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities but also a complexity for square nonnegative matrices having positive permanent in particular our results extend nontrivially many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by kalantari and khachiyan to the case of general r c scaling finally we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals and the sum of all entries is also prescribed balinski and demange described an ras type algorithm for its continuous version but did not analyze its complexity we show that this algorithm produces an 1 e approximate solution within complexity polynomial in n m h v v and 1 1 e

on real quadratic fields of class number two it is the primary purpose of the paper to determine all real quadratic fields q v d of class number h d 2 when k 24 with one possible exception here k is the period length of the continued fraction expansion of either co v 7 in the case d 2 or 3 mod 4 or of co 1 d 2 in the case d 1 mod 4

matrix scaling entropy minimization and conjugate duality ii the dual problem we present a matrix scaling problem calledtruncated scaling and describe applications arising in economics urban planning and statistics we associate a dual pair of convex optimization problems to the scaling problem and prove that the existence of a solution for the truncated scaling problem is characterized by the attainment of the infimum in the dual optimization problem we show that optimization problems used by bacharach 1970 bachem and korte 1979 eaves et al 1985 marshall and olkin 1968 and rothblum and schneider 1989 to study scaling problems can be derived as special cases of the dual problem for truncated scaling we present computational results for solving truncated scaling problems using dual coordinate descent thereby showing that truncated scaling provides a framework for modeling and solving large scale matrix scaling problems

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

interval constrained matrix balancing abstract we define two interval constrained matrix balancing problems the models encompass several known formulations of problems that appear in regional input output analysis estimation of traffic over transportation and telecommunications networks and estimation of social accounting matrices we develop primal dual row action algorithms for the solution of these models and establish their convergence both algorithms generalize existing scaling algorithms for equality constrained matrix balancing problems one of the algorithms is a generalization of the well known procedure for matrix balancing called ras that can handle the range constraints and is thus called range ras the structure of the problem makes the algorithm suitable for implementation on massively parallel computers details of our implementation on a connection machine cm 2 are given numerical results for the solution of problems of size up to 500 500 are given and the performance of the algorithm is compared with that of ras