stable quartic spline integration method for solving stiff ordinary differential equations the convergence and stability analysis of a quartic c 2 spline collocation method for solving initial value problems including stiff differential equations will be considered letting the interior collocation point x i 1 b x i email protected be dependent on some parameter email protected 0 1 it will be shown that the method possesses convergence rate of order 4 if email protected 1 2 1 and is strongly unstable if b 1 0

stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

quasi valuations extending a valuation abstract suppose f is a field with valuation v and valuation ring o v e is a finite field extension and w is a quasi valuation on e extending v we study quasi valuations on e that extend v in particular their corresponding rings and their prime spectra we prove that these ring extensions satisfy inc incomparability lo lying over and gd going down over o v in particular they have the same krull dimension we also prove that every such quasi valuation is dominated by some valuation extending v under the assumption that the value monoid of the quasi valuation is a group we prove that these ring extensions satisfy gu going up over o v and a bound on the size of the prime spectrum is given in addition a one to one correspondence is obtained between exponential quasi valuations and integrally closed quasi valuation rings given r an algebra over o v we construct a quasi valuation on r we also construct a quasi valuation on r o v f which helps us prove our main theorem the main theorem states that if r e satisfies r f o v and e is the field of fractions of r then r and v induce a quasi valuation w on e such that r o w and w extends v thus r satisfies the properties of a quasi valuation ring

valuations on algebras with involution let a be a central simple algebra with involution of the first or second kind let v be a valuation on the fixed part f of z a a special v gauge g on a is a kind of value function on a extending v on f such that g x x 2g x for all x in a it is shown under certain restrictions if the residue characteristic is 2 that if v is henselian then there is a special v gauge g if and only if is anisotropic and g is unique if v is not henselian it is shown that there is a special v gauge g if and only if remains anisotropic after scalar extension from f to the henselization of f with respect to v when this occurs g is the unique invariant v gauge on a

linear stable unity feedback system necessary and sufficient conditions for stability under nonlinear plant perturbations the authors consider a linear not necessarily time invariant stable unity feedback system where the plant and the compensator have normalized right coprime factorizations they study two cases of nonlinear plant perturbations additive and feedback with four subcases resulting from 1 allowing exogenous input to delta p or not 2 allowing the observation of the output of delta p or not the plant perturbation delta p is not required to be stable using the factorization approach the authors obtain necessary and sufficient conditions for all cases in terms of two pairs of nonlinear pseudostate maps simple physical considerations explain the form of these necessary and sufficient conditions finally the authors obtain the characterization of all perturbations delta p for which the perturbed system remain stable

absolute monotonicity of rational functions occurring in the numerical solution of initial value problems this paper deals with rational functions o z approximating the exponential function exp z related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative numberr denoted byr o such that o is absolutely monotonic on r 0 an algorithm for the computation ofr o is presented application of this algorithm yields the valuer o for the well known pade approximations to exp z for some specific values ofm n andp we determine the maximum ofr o when o varies over the class of all rational functions o with degree of the numerator m degree of the denominator n and o z exp z z p 1 forz 0

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

the defektsatz for central simple algebras let q be a central simple algebra finite dimensional over its center f and let v be a valuation ring of f the v has en extension to q i e there exists a dubrovin valuation ring b of q satisfying v f b generally the number of extensions of v to q is not finite and therefore the so called intersection property of dubrovin valuation rings b 1 b n is introduced this property is defined in terms of the prime ideals and the valuation overrings of the intersection b b n it is shown that there exists a uniquely determined natural number n depending only on v and having the following property if b 1 b k are extensions of v having the intersection property then k n and k n holds if and only if b 1 b k is integral over v

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

on the dec group of finite abelian galois extensions over global fields abstract if k f is a finite abelian galois extension of global fields whose galois group has exponent t we prove that there exists a short exact sequence 0 dec k f br t k f q p r q z z 0 where r q q and p is a finite set of primes of f that is empty if t is square free in particular we obtain that if t is square free then dec k f br t k f which we use to show that prime exponent division algebras over henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras finally we construct a counterexample to the result for higher exponent division algebras

embedding henselian fields into power series abstract every henselian field of residue characteristic 0 admits a truncation closed embedding in a field of generalised power series possibly with a factor set as corollaries we obtain the ax kochen ershov theorem and an extension of mourgues and ressayre s theorem every ordered field which is henselian in its natural valuation has an integer part we also give some results for the mixed and the finite characteristic cases

towers of complements to valuation rings and truncation closed embeddings of valued fields abstract we study necessary and sufficient conditions for a valued field k with value group g and residue field k with char k char k to admit a truncation closed embedding in the field of generalized power series k g f with factor set f we show that this is equivalent to the existence of a family tower of complements of k subspaces of k which are complements of the possibly fractional ideals of the valuation ring and satisfying certain natural conditions if k is a henselian field of characteristic 0 or more generally an algebraically maximal kaplansky field we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding we also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions

on dubrovin valuation rings in crossed product algebras let f be a field and let v be a valuation ring in f if a is a central simple f algebra then v can be extended to a dubrovin valuation ring in a in this paper we consider the structure of dubrovin valuation rings with center v in crossed product algebras k f g f where k f is a finite galois extension with galois group g unramified over v and f is a normalized two cocycle in the case where v is indecomposed in k we introduce a family of orders naturally associated to f examine their basic properties and determine which of these orders is dubrovin in the case where v is decomposed we determine the structure in the case of certain special discrete finite rank valuations

simultaneous stabilization of multiplicatively perturbed plants necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear time invariant multi input multi output nominal plant and a multiplicatively perturbed plant where the multiplicative perturbation is stable these conditions are derived from the general parity interlacing property applicable to any two arbitrary plants and are expressed explicitly in terms of the real axis poles and zeros of the nominal plant and the perturbation the class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions a special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

maximal orders over valuation rings abstract in this paper we study maximal orders over commutative valuation rings in central simple algebras we are particularly interested in maximal orders which are either bezout or semihereditary we construct a class of bezout maximal orders and a class of semihereditary maximal orders and show that for any valuation ring v resp v with value group z m any bezout resp semihereditary maximal order over v belongs to the class constructed furthermore we classify all maximal orders in m2 f over a valuation ring with value group z m and in mn f given a mild defectless assumption

a new construction of noncrossed product algebras new examples of noncrossed product division algebras are obtained using methods different from all previous noncrossed product constructions the examples are division algebras over intersections of henselian valued fields and they have schur index and exponent p for any prime number p and any integers m n 2 n 3if 2 the basic tools used in the construction are valuation theory and galois cohomology no generic methods are applied and there is no p i theory along the way local global principles are proved for central simple algebras over intersections of p henselian valued fields the first examples of central simple division algebras which are not crossed products were obtained by amitsur in am in 1972 amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years his examples were the generic division algebras ud q n of index n over q the rational numbers for any natural number n such that p2 n p an odd prime or 8 all subsequent constructions of noncrossed products in ss 83 883 r r an ti have heretofore been based on amitsur s specialization argument and they are all generic division algebras or extensions of generic division algebras the centers of these noncrossed product algebras are not known nor are the brauer groups nor the absolute galois groups of the centers we present here a new method of constructing noncrossed product algebras in our approach the noncrossed product is realized as the underlying division algebra d of a tensor product of suitably chosen cyclic algebras over a field f l n l2 where each l is a pth root henselian valued field we prove local global principles relating the splitting fields of d to those of d f lt i 1 2 it is shown that the part of the brauer group of f is completely determined by that of ll and l2 computations for central simple f algebras thus become very tractable for exam ple we not only show that d is not a crossed product but also calculate exactly how large r must be so that the matrix ring mr d is a crossed product and how large s must be so that ms d is a tensor product of cyclic algebras indeed the structure of the brauer group of f is so nice that we were somewhat surprised that noncrossed products could possibly exist over f

a necessary and sufficient condition for stability of a perturbed system the problem of determining the stability of a feedback system in the presence of perturbation is considered a necessary and sufficient condition under which the perturbed system remains stable is obtained this condition is very general in particular it does not require the perturbed plant to have the same number of poles as the nominal plant

totally ramified valuations on finite dimensional division algebras division algebras d with valuation v are studied where d is lsnite dimensional and totally ramified over its center f i e the ramification index of v over wlf equals d f such division algebras have appeared in some important constructions but the structure of these algebras has not been systematically analyzed before when vlf is henselian a full classification of the f subalgebras of d is given when f has a henselian valuation v with separably closed residue field and a is any tame central simple f algebra an algorithm is given for computing the underlying division algebra of a from a suitable subgroup of a f some examples are constructed using this valuation theory including the first example of finitbdimensional f central division algebras d1 and d2 with d1 xf d2 not a division ring but d1 and d2 having no common subfield k d f valuation theory long a basic tool in commutative algebra has been relatively neglected in the study of division algebras until quite recently nontheless valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties particularly algebras over iterated laurent power series fields for example such division algebras have been key ingredients in amitsur s noncrossed product construction am and in platonov s construction p of division algebras d with sk1 d 7 1 valuations are not so prevalent on division algebras as on fields but if a division algebra d does have a valuation this structure contains a substantial amount of information about d which would scarcely be accessible otherwise we consider here valued divisioll algebras d for which d is totqlly ramified and tame over its center f i e for which the ramification index ird rfi equals the dimension d f of d over f and the characteristic char d does not divide d f here rd is the value group of the valuation on d and d is the residue division algebra we assume throughout that d f oo valued division algebras of this type appear e g in amitsur s noncrossed product paper am 2 in saltman s work on indecomposable division algebras sa in certain of the malcevneumann division algebras considered by the first author and amitsur ta2 4 etc however the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before this may be because most past work on valued division algebras has concentrated on discrete valuations when rd z for such a valuation d is never totally ramified over its center f unless d f cf 3 2 below received by the editors june 26 1986 1980 mathematics subject clmsification 1985 reon primary 16a39 2supported in part by f n r s 2supported in part by the national science foundation 1987 american mathematlcal society 0002 9947 87 00 25 per page

non linear systems stabilization and coprimeness a theory of coprimeness is developed for non linear systems with the intention of constructing analytic tools for the solution of the problem of stabilizing a non linear system through the application of additive non linear feedback the qualitative features of the resulting theory are strikingly similar to the situation in the linear case

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

strong stabilizability of systems with multiaffine uncertainties and numerator denominator coupling this paper considers a set of proper transfer functions whose numerator and denominator polynomial coefficients display dependent multiaffine parametric uncertainties it is shown that provided no member transfer function has positive real pole zero cancellations all members satisfy the parity interlacing property iff all corner members do the same notice that while this implies that each member is strongly stabilizable it does not imply the existence of a single stable controller that stabilizes the whole set the paper also shows that whenever the numerator and denominator polynomials lie in dependent polytopes the task of verifying the absence of positive real pole zero cancellations can be accomplished by checking the edges

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

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

stability and boundedness in the numerical solution of initial value problems abstract this paper concerns the theoretical analysis of step by step meth ods for solving initial value problems in ordinary and partial differential equa tions the main theorem of the paper answers a natural question arising in thelinear stability analysis of such methods it guarantees a strong version ofnumerical stability under a stepsize restriction related to the stability regionof the numerical method and to a circle condition on the differential equation the theorem also settles an open question related to the properties total variation diminishing strong stability preserving monotonic and total variation bounded under a monotonicity condition on the forward eulermethod the theorem specifies a stepsize condition guaranteeing boundednessfor linear problems the main theorem is illustrated by applying it to linear multistep methods for important classes of these methods conclusions are thus obtained whichsupplement earlier results in the literature

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

regions of stability equivalence theorems and courant friedichs lewy condition the celebrated cfl condition for discretizations of hyperbolic pdes is shown to be equivalent to some results of jeltsch and nevanlinna concerning regions of stability ofk step m stage linear methods for the integration of odes we characterize the methods for the numerical integration of the model equation ut ux which are weakly stable when the mesh ratio takes the maximum value allowed by the cfl condition we provide new equivalence theorems between stability and convergence which improve on the classical results

norm bounds for rational matrix functions if the field of values of a matrixa is contained in the left complex halfplaneh and a functionf mapsh into the unit disc then f a 2 1 by a theorem of j v neumann we prove a theorem of this type only the field of values ofa is used for functions which are absolutely bounded by one in only part ofh an extension can be used to show norm stability of single step methods for stiff differential equations the results are applicable among others to several subdiagonal pade approximations which are nota stable

simple riesz groups of rank one having wild intervals abstract we prove that every partially ordered simple group of rank one which is not riesz embeds into a simple riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals using this result we construct examples of simple riesz groups of rank one g containing unbounded intervals d n n 1 and d that satisfy a for each n 1 t d n g for every t q n but q n d n g where q n is a sequence of relatively prime integers b for every n 1 n d g we sketch some potential applications of these results in the context of k theory

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

on the use of stability regions in the numerical analysis of initial value problems this paper deals with the stability analysis of one step methods in the numerical solution of initial boundary value problems for linear ordinary and partial differential equations restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size these restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration the errors in the one step methods are measured in arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of the heat equation and some other differential equations where the error growth is measured in the maximum norm

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

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

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

convergence analysis of one step schemes in the method of lines we present an expository account of some fundamental results concerning the analysis of one step schemes for semidiscretizations of evolutionary problems in partial differential equations in the paper the emphasis lies in the interplay between the stability and convergence properties of the fully discrete scheme and those of the ordinary differential equations solver much attention is paid to the phenomenon of order reduction

monoids of intervals of ordered abelian groups abstract for any partially ordered abelian group g we relate the structure of the ordered monoid g of intervals of g i e nonempty upward directed lower subsets of g to various properties of g as for example interpolation properties or topological properties of the state space when g has an order unit this allows us to solve a problem by k r goodearl by proving that even in most natural cases multiplier groups of dimension groups often fail to be interpolation groups furthermore the study of monoids of intervals in the totally ordered case yields a characterization of hahn powers of the real line by a first order sentence on the positive interval monoid

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

value functions and associated graded rings for semisimple algebras we introduce a type of value function y called a gauge on a finite dimensional semisimple algebra a over a field f with valuation v the filtration on a induced by y yields an associated graded ring gr y a which is a graded algebra over the graded field gr v f key requirements for y to be a gauge are that gr y a be graded semisimple and that dim gr v f gr y a dim f a it is shown that gauges behave well with respect to scalar extensions and tensor products when v is henselian and a is central simple over f it is shown that gr y a is simple and graded brauer equivalent to gr w d where d is the division algebra brauer equivalent to a and w is the valuation on d extending v on f the utility of having a good notion of value function for central simple algebras not just division algebras and with good functorial properties is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras

discriminants of symplectic graded involutions on graded simple algebras abstract in this article we define and study the discriminant of symplectic graded involutions on non inertially split graded simple algebras with simple 0 component in particular we show that if f is a graded field of characteristic different from 2 d is a graded central division algebra over f with exp d 2 and ker d 4 see the preliminaries below a m n d and is a graded involution of symplectic type on a then there is only a finite number of values for the discriminants where describes all graded involutions of symplectic type on a see proposition 2 11 consequently for any graded central simple algebra c over f with c 0 simple non split exp c 2 ker c 4 and deg c ind c even we have 0 for any graded involutions of symplectic type and on c see corollary 2 12 we prove also that if e is a henselian valued field with residue characteristic different from 2 d is a central division algebra of exponent 2 over e with ker d 4 and b m n d with n even then for any symplectic involutions on b preserving a tame gauge defined on b we have 0 see corollary 3 5

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

robust stability under additive perturbations we consider a mimo linear time invariant feedback system1 s p c which is assumed to be u stable the plant p is subjected to an additive perturbation delta p which is proper but not necessarily stable we prove that the perturbed system is u stable if and only if delta p i q delta p 1 is u stable here q c i pc 1

simultaneous stabilizability of p and kp it is shown that there exists a common controller which simultaneously stabilizes any given nominal plant p and perturbed plant kp for any given positive real constant k in the standard linear time invariant multi input multi output unity feedback system the class of plants such that p and kp can be simultaneously stabilized for negative k is also determined

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

springer s theorem for tame quadratic forms over henselian fields a quadratic form over a henselian valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension the witt group of tame quadratic forms is shown to be canonically isomorphic to the witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation hence also isomorphic to a direct sum of copies of the witt group of the residue field indexed by the value group modulo 2

absolute monotonicity of polynomials occurring in the numerical solution of initial value problems this paper deals with polynomial approximations o x to the exponential function exp x related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument denoted by r o at which o is absolutely monotonic for given integersp 1 m 1 we determine the maximum ofr o when o varies over the class of all polynomials of a degree m with phi left x right exp left x right mathcal o left x p 1 right left for x to 0 right forx 0

hardy type derivations on fields of exponential logarithmic series abstract we consider the valued field k r of formal series with real coefficients and monomials in a totally ordered multiplicative group we investigate how to endow k with a logarithm l which satisfies some natural properties such as commuting with infinite products of monomials we studied derivations on k kuhlmann and matusinski in press km10 here we investigate compatibility conditions between the logarithm and the derivation i e when the logarithmic derivative is the derivative of the logarithm we analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives in kuhlmann 2000 kuh00 the first author described the exponential closure k el of k l here we show how to extend such a log compatible derivation on k to k el