the factorization method for the numerical solution of two point boundary value problems for linear ode s the theoretical analysis and computational implementation of factorization based methods for the numerical solution of linear boundary value problems for ordinary differential equations are presented the methods are optimal with respect to certain clearly defined criteria numerical examples show the effectiveness of a general code based on the factorization method

a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

magnus integrators for solving linear quadratic differential games we consider magnus integrators to solve linear quadratic n player differential games these problems require to solve backward in time non autonomous matrix riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game to be integrated forward in time we analyze different magnus integrators which can provide either analytical or numerical approximations to the equations they can be considered as time averaging methods and frequently are used as exponential integrators we show that they preserve some of the most relevant qualitative properties of the solution for the matrix riccati differential equations as well as for the remaining equations the analytical approximations allow us to study the problem in terms of the parameters involved some numerical examples are also considered which show that exponential methods are in general superior to standard methods

cofinality spectrum theorems in model theory set theory and general topology we connect and solve two longstanding open problems in quite different areas the modeltheoretic question of whether sop2 is maximal in keisler s order and the question from general topology set theory of whether p t the oldest problem on cardinal invariants of the continuum we do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely using model theoretic methods by a cofinality spectrum problem s we essentially mean the data of a pair of models m m1 which code sufficient set theory possibly in an expanded language along with a distinguished set of formulas s which define linear orders in m1 let ts the treetops of s be the smallest regular cardinal such that one of a set of derived trees in m1 has a strictly increasing sequence with no upper bound let c s ts be the set of pairs of regular cardinals 1 2 such that 1 2 or what is equivalent such that d contains no cuts for cf is good we obtain several consequences notably existence of a minimum keisler class among the non simple theories

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures a stable approach for integrating the impedance matrix in cylindrical radial inhomogeneous structures is developed and studied a stroh like system using the time harmonic displacement traction state vector is used to derive the riccati matrix differential equation involving the impedance matrix it is found that the resulting equation is stiff and leads to exponential instabilities a stable scheme for integration is found in which a local expansion is performed by combining the matricant and impedance matrices this method offers a stable solution for fully anisotropic materials which was previously unavailable in the literature several approximation schemes for integrating the riccati equation in cylindrical coordinates are considered exponential magnus taylor series lagrange polynomials with numerical examples indicating that the exponential scheme performs best the impedance matrix is compared with solutions involving buchwald potentials in which the material is limited to piecewise constant transverse isotropy lastly a scattering example is considered and compared with the literature

wild pfister forms over henselian fields k theory and conic division algebras the epicenter of this paper concerns pfister quadratic forms over a field f with a henselian discrete valuation all characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but f does not we also prove results about round quadratic forms composition algebras generalizations of composition algebras we call conic algebras and central simple associative symbol algebras finally we give relationships between these objects and kato s filtration on the milnor k groups of f

on a theorem of tignol for defectless extensions and its converse abstract let k v be a henselian valued field of arbitrary rank in 1990 tignol proved that if k v k v is a finite separable defectless extension of degree a prime number then the set a k k v tr k k v k 0 has a minimum element this paper extends tignol s result to all finite separable extensions moreover a characterization of finite separable defectless extensions is given by showing that k v k v is a defectless extension if and only if the set a k k has a minimum element our proof also leads to a new proof of the well known result that each finite extension of a formally adic field or more generally of a finitely ramified valued field is defectless

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

general topology meets model theory on p and t cantor proved in 1874 cantor g 1874 j reine angew math 77 258 262 that the continuum is uncountable and hilbert s first problem asks whether it is the smallest uncountable cardinal a program arose to study cardinal invariants of the continuum which measure the size of the continuum in various ways by godel godel k 1939 proc natl acad sci usa 25 4 220 224 and cohen cohen p 1963 proc natl acad sci usa 50 6 1143 1148 hilbert s first problem is independent of zfc zermelo fraenkel set theory with the axiom of choice much work both before and since has been done on inequalities between these cardinal invariants but some basic questions have remained open despite cohen s introduction of forcing the oldest and perhaps most famous of these is whether which was proved in a special case by rothberger rothberger f 1948 fund math 35 29 46 building on hausdorff hausdorff 1936 fund math 26 241 255 in this paper we explain how our work on the structure of keisler s order a large scale classification problem in model theory led to the solution of this problem in zfc as well as of an a priori unrelated open question in model theory

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

model theory and the cardinal numbers p and t modern mathematical logic is a multifaceted subject which concerns itself with the strengths and limitations of formal proofs and algorithms and the relationship between language and mathematical structure modern mathematical logic also addresses foundational issues that arise in mathematics this commentary summarizes the groundbreaking results of malliaris and shelah 1 recently published in pnas 2 relating two branches of logic model theory and set theory

numerical solution of eigenvalue problems for linear boundary value odes abstract interrelationships between several popular approaches for solving eigenvalue problems for linear boundary value odes are given for linear eigenvalue problems the popular methods can be interpreted in a common framework this leads us to propose and justify alternative strategies the choice of numerical methods used here is motivated by the desire to solve eigenvalue problems for stiff odes in particular we consider a one step global method spline collocation and two initial value methods riccati and continuous orthonormalization to solve the orr sommerfeld equation a comparison of results for these methods using various implementation strategies is given

constructing regular ultrafilters from a model theoretic point of view this paper contributes to the set theoretic side of understanding keisler s order we consider properties of ultralters which aect saturation of unstable theories the lower conality lcf 0 d of 0 modulo d saturation of the minimum unstable theory the random graph exi bility goodness goodness for equality and realization of symmetric cuts we work in zfc except when noted as several constructions appeal to complete ultralters thus assume a measurable car dinal the main results are as follows first we investigate the strength of exibility known to be detected by non low theories assuming 0 is measurable we construct a regular ultralter on 2 which is exible but not good and which moreover has large lcf 0 but does not even saturate models of the random graph this implies a that exibility alone cannot characterize saturation of any theory however b by separating exibility from goodness we remove a main obstacle to proving non low does not imply maximal since exible is precisely ok this also shows that c from a set theoretic point of view consistently ok need not imply good addressing a prob lem from dow 1985 second under no additional assumptions we prove that there is a loss of saturation in regular ultrapowers of unstable theories and also give a new proof that there is a loss of saturation in ultrapowers of non simple theories more precisely for d regular onand m a model of an unstable theory m d is not 2 saturated and for m a model of a non simple theory and m d is not saturated in the third part of the paper we investigate realization and omission of symmetric cuts signicant both because of the maximality of the strict order property in keisler s order and by recent work of the authors on sop2 we prove that if d is a complete ultralter on any ultrapower of a suciently saturated model of linear or der will have no cuts and that ifd is also normal it will have a cut we apply this to prove that for any n assuming the existence of n measurable cardinals below there is a regular ultralter d onsuch that any d ultrapower of a model of linear order will haven alternations of cuts as dened below moreover d will saturate all stable theories but will not 2 saturate any unstable theory whereis the smallest measurable cardinal used in the construction

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

a formula for tignol s constant abstract let k v be a henselian valued field and l w be a finite separable extension of k v in 2004 it was proved that the set a l k defined by a l k v tr l k w l 0 has a minimum element if and only if l k e f where e f are the ramification index and the residual degree of w v i e l w k v is defectless the constant min a l k was first introduced by tignol and is referred to as tignol s constant in 2005 k ota and khanduja gave a formula for min a l k when l w k v is an extension of local fields in this paper we give this formula when l w is any finite separable defectless extension of a henselian valued field of arbitrary rank and thereby generalize some well known results of dedekind regarding different of extensions of algebraic number fields and ramification of prime ideals

some numerical considerations and newton s method revisited for solving algebraic riccati equations analyzes some of the numerical aspects of solving the algebraic riccati equation are this analysis applies to both the symmetric and unsymmetric cases the author reconsiders the numerically relevant problems of balancing the are and the conditioning properties of the are and shows how these can be exploited by a solution algorithm he proposes an estimator for the condition number of the sylvester equation ax xb c based on iterative refinement also he interprets newton s method as a sequence of similarity transformations on the underlying system matrix this closes the gap between so called global and iterative methods for solving the are and also suggests an altogether revised implementation of newton s method one of the advantages of this revised implementation is that in the case where newton s method converges to a solution different from the desired solution enough information emerges to allow a switch to the desired solution the author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

existence of optimal ultrafilters and the fundamental complexity of simple theories abstract in the first edition of classification theory the second author characterized the stable theories in terms of saturation of ultrapowers prior to this theorem stability had already been defined in terms of counting types and the unstable formula theorem was known a contribution of the ultrapower characterization was that it involved sorting out the global theory and introducing nonforking seminal for the development of stability theory prior to the present paper there had been no such ultrapower characterization of an unstable class in the present paper we first establish the existence of so called optimal ultrafilters on suitable boolean algebras which are to simple theories as keisler s good ultrafilters 15 are to all first order theories then assuming a supercompact cardinal we characterize the simple theories in terms of saturation of ultrapowers to do so we lay the groundwork for analyzing the global structure of simple theories in zfc via complexity of certain amalgamation patterns this brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability

stable parallel elimination for boundary value odes a parallelizable and vectorizable algorithm for solving linear algebraic systems arising from two point boundary value odes is described the method is equivalent to gaussian elimination with row partial pivoting applied to a certain column reordered version of the usual almost block diagonal coefficient matrix we present analytical and numerical evidence to show that the algorithm is stable in most circumstances results from implementation on a shared memory multiprocessor and a vector processor are given the approach can be extended to handle problems with multipoint and integral conditions or with algebraic parameters

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

staircase matrices and systems difference equations spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure a staircase system s variables can be partitioned into a natural sequence of periods in such a way that every equation involves at most the variables of two adjacent periods this paper surveys the numerous properties of staircase systems and of the staircase matrices that comprise their coefficients particular attention is given to reduced and sparse staircase forms and to the varieties of gaussian elimination for solving staircase systems

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough