complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

invariance of simultaneous similarity and equivalence of matrices under extension of the ground field in this work we give a new and elementary proof that simultaneous similarity and simultaneous equivalence of families of matrices are invariant under extension of the ground field a result which is non trivial for finite fields and first appeared in an earlier paper of klinger and levy

strong stability preserving explicit peer methods in this paper we study explicit peer methods up to order p 13 which have the strong stability preserving ssp property this class of general linear methods has the favourable property of a high stage order the effective ssp coefficient is maximized by solving a nonlinear constraint optimization problem numerically to high precision the coefficient matrices of the optimized methods are sparse in a very structured way linear multistep methods are obtained as a special case of only one stage

contractivity preserving implicit linear multistep methods we investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on nonlinear and linear systems d u t f t u t where f satisfies a so called circle condition in an arbitrary norm the results for the two types of systems turn out to be closely related we construct optimal multistep methods of given order and stepnumber which allow the use of a maximal stepsize

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

strong stability for additive runge kutta methods space discretization of some time dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties in these situ

poset block equivalence of integral matrices given square matrices b and b with a poset indexed block structure for which an ij block is zero unless i j when are there invertible matrices u and v with this required zero block structure such that ubv b we give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain r as one application when r is a field we classify such matrices up to similarity by matrices respecting the block structure we also give complete invariants for equivalence under the additional requirement that the diagonal blocks of u and v have determinant 1 the invariants involve an associated diagram the k web of r module homomorphisms the study is motivated by applications to symbolic dynamics and c algebras

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

on unitary similarity of matrices abstract we obtain a set of polynomials in a a a an n n matrix whose traces completely characterize the unitary similarity classes of a generic set of matrices the number of invariants required is an order of magnitude smaller than earlier examples of such sets of invariants

polynomial identity rings as rings of functions ii abstract in characteristic zero zinovy reichstein and the author generalized the usual relationship between irreducible zariski closed subsets of the affine space their defining ideals coordinate rings and function fields to a non commutative setting where varieties carry a pgl n action regular and rational functions on them are matrix valued coordinate rings are prime polynomial identity algebras and function fields are central simple algebras of degree n in the present paper much of this is extended to prime characteristic in addition a mistake in the earlier paper is corrected one of the results is that the finitely generated prime pi algebras of degree n are precisely the rings that arise as coordinate rings of n varieties in this setting for n 1 the definitions and results reduce to those of classical affine algebraic geometry

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

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices a b under a simultaneous similarity roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set then we construct a finite number of rational functions in the entries of a and b whose values are constant on all pairs similar to a b the values of the functions phi sub i a b i equals 1 s determine a finite number of similarity classes

an improvement on the positivity results for 2 stage explicit runge kutta methods in this paper we investigate the positivity property for a class of 2 stage explicit runge kutta rk2 methods of order two when applied to the numerical solution of special nonlinear initial value problems ivps for ordinary differential equations odes we also pay particular attention to monotonicity property we obtain new results for positivity which are important in practical applications we provide some numerical examples to illustrate our results

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

a structure theorem for some matrix algebras abstract let f be a field and m n f the algebra of n n matrices over f it is in general a very difficult and tedious problem to determine the structure of the subalgebra x of m n f generated by a given subset s of m n f we show that if x contains a matrix a with n distinct eigenvalues in f then x can be determined up to similarity very quickly by a graph theoretic method as a consequence we show that any such x can be generated by a pair of elements one of which can be taken to be a and the other a 0 1 matrix as an application we obtain a specht type similarity theorem

representations of runge kutta methods and strong stability preserving methods over the last few years a great effort has been made to develop monotone high order explicit runge kutta methods by means of their shu osher representations in this context the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation in this paper we extend the shu osher representations for any runge kutta method giving sufficient conditions for monotonicity we show how optimal shu osher representations can be constructed from the butcher tableau of a runge kutta method the optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the runge kutta method l ferracina and m n spijker siam j numer anal 42 2004 pp 1073 1093 and hence if this radius is zero the method is not monotone in the shu osher representation methods with zero radius require negative coefficients and to deal with them an extra associate problem is considered in this paper we interpret these schemes as representations of perturbed runge kutta methods we extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity optimal representations can be constructed from the butcher tableau of a perturbed runge kutta method

on invertibility preserving linear mappings simultaneous triangularization and property l abstract let a be a unital matrix algebra a m n c a unital linear mapping and b the algebra generated by a the mapping is a homomorphism modulo the jacobson radical in b if and only if for k dim b dim a 3 the mapping id a m k c b m k c preserves invertibility this result is closely related to the question on the relationship between property l and simultaneous triangularization we introduce a property kl and give criteria in terms of conditions on matrix pencils with coefficients in m k c rather than in c which ensure simultaneous triangularization

two dimensional representations of finitely generated free group over an arbitrary field working over an arbitrary field we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible respectively also we classify all two dimensional representations of a finitely generated free group

strong stability preserving transformed dimsims abstract in this paper we investigate the strong stability preserving ssp property of transformed diagonally implicit multistage integration methods dimsims within this class examples of ssp methods of order p 1 2 3 and 4 and stage order q p have been determined and also suitable starting and finishing procedure have been constructed the numerical experiments performed on a set of test problems have shown that ssp transformed dimsims can be more accurate and competitive with ssp runge kutta methods of the same order

global optimization of explicit strong stability preserving runge kutta methods strong stability preserving runge kutta ssprk methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations pdes under a suitable stepsize restriction these methods share a desirable nonlinear stability property with the underlying pde e g positivity or stability with respect to total variation this is of particular interest when the solution exhibits shock like or other nonsmooth behaviour a variety of optimality results have been proven for simple ssprk methods however the scope of these results has been limited to low order methods due to the detailed nature of the proofs in this article global optimization software baron is applied to an appropriate mathematical formulation to obtain optimality results for general explicit ssprk methods up to fifth order and explicit low storage ssprk methods up to fourth order throughout our studies allow for the possibility of negative coefficients which correspond to downwind biased spatial discretizations guarantees of optimality are obtained for a variety of third and fourth order schemes where optimality is impractical to guarantee specifically for fifth order methods and certain low storage methods extensive numerical optimizations are carried out to derive numerically optimal schemes as a part of these studies several new schemes arise which have theoretically improved time stepping restrictions over schemes appearing in the recent literature

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

stepsize conditions for boundedness in numerical initial value problems for runge kutta methods rkms linear multistep methods lmms and classes of general linear methods glms much attention has been paid in the literature to special nonlinear stability requirements indicated by the terms total variation diminishing strong stability preserving and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many well known methods the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we focus on stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual rkms lmms and glms thereby generalizing results on the special stability properties mentioned above

optimal explicit strong stability preserving general linear methods this paper constructs strong stability preserving general linear time stepping methods that are well suited for hyperbolic pdes discretized by the method of lines these methods generalize both runge kutta rk and linear multistep schemes they have high stage orders and hence are less susceptible than rk methods to order reduction from source terms or nonhomogeneous boundary conditions a global optimization strategy is used to find the most efficient schemes that have low storage requirements numerical results illustrate the theoretical findings

a wild quiver in linear systems theory abstract we study the action t l a b t 1 at t 1 bl of the group gl n k gl m k of linear state and input transformations on the space u n m of control systems a b k n n k n m with n different eigenvalues k an algebraically closed field this action is of interest in systems theory as well as the representation theory of quivers we construct a canonical form for on u n m and determine completely its stabilizer subgroups the results are based on an analysis of the scaling action d e b d 1 be of the group of diagonal matrices d n k d m k on k n m we construct a canonical form for this action by graph theoretic methods and determine the adherence order of the underlying stratification of k n m

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

comparison of boundedness and monotonicity properties of one leg and linear multistep methods one leg multistep methods have some advantage over linear multistep methods with respect to storage of the past results in this paper boundedness and monotonicity properties with arbitrary semi norms or convex functionals are analyzed for such multistep methods the maximal stepsize coefficient for boundedness and monotonicity of a one leg method is the same as for the associated linear multistep method when arbitrary starting values are considered it will be shown however that combinations of one leg methods and runge kutta starting procedures may give very different stepsize coefficients for monotonicity than the linear multistep methods with the same starting procedures detailed results are presented for explicit two step methods

numerical integration of positive linear differential algebraic systems in the simulation of dynamical processes in economy social sciences biology or chemistry the analyzed values often represent non negative quantities like the amount of goods or individuals or the density of a chemical or biological species such systems are typically described by positive ordinary differential equations odes that have a non negative solution for every non negative initial value besides positivity these processes often are subject to algebraic constraints that result from conservation laws limitation of resources or balance conditions and thus the models are differential algebraic equations daes in this work we present conditions under which both these properties the positivity as well as the algebraic constraints are preserved in the numerical simulation by runge kutta or multistep discretization methods using a decomposition approach we separate the dynamic and the algebraic equations of a given linear positive dae to give positivity preserving conditions for each part separately for the dynamic part we generalize the results for positive odes to daes using the solution representation via drazin inverses for the algebraic part we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non negative we analyze these conditions for some common runge kutta and multistep methods and observe that for index 1 systems and stiffly accurate runge kutta methods positivity is conditionally preserved under similar conditions as for odes for higher index problems however none of the common methods is suitable

two dimensional representations of the free group in two generators over an arbitrary field abstract working over an arbitrary field we classify all two dimensional representations of the free group with two generators

simultaneous similarity and triangularization of sets of 2 by 2 matrices abstract let a a 1 a n be a finite or infinite sequence of 2 2 matrices with entries in an integral domain we show that except in a very special case a is simultaneously triangularizable if and only if all pairs a j a k are triangularizable for 1 j k we also provide a simple numerical criterion for triangularization using constructive methods in invariant theory we define a map with the minimal number of invariants that distinguishes simultaneous similarity classes for non commutative sequences over a field of characteristic 2 we also describe canonical forms for sequences of 2 2 matrices over algebraically closed fields and give a method for finding sequences with a given set of invariants

quaternionic matrices unitary similarity simultaneous triangularization and some trace identities abstract we construct six unitary trace invariants for 2 2 quaternionic matrices which separate the unitary similarity classes of such matrices and show that this set is minimal we have discovered a curious trace identity for two unit speed one parameter subgroups of sp 1 a modification gives an infinite family of trace identities for quaternions as well as for 2 2 complex matrices we were not able to locate these identities in the literature we prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field finally we consider the problem of describing the semi algebraic set of pairs x y of quaternionic n n matrices which are simultaneously triangularizable even the case n 2 which we analyze in more detail remains unsolved

versal deformations of invariant subspaces we describe a miniversal deformation of invariant subspaces with regard to a fixed endomorphism by means of a technique which can be applied to obtain explicitly miniversal deformations in a general orbit space in addition we present an application to the problem of classifying invariant subspaces

the codimension of degenerate pencils abstract let dn dn r denote the codimension of the set of pairs of n n hermitian really symmetric matrices a b for which det i a xb p x is a reducible polynomial we prove that dn r n 1 dn n 1 n odd dn n n even we conjecture that the equality holds in all three inequalities we prove this conjecture for n 2 3

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

unitary and orthogonal equivalence of sets of matrices abstract two matrices a and b are called unitary resp orthogonal equivalent if a u v b for two unitary resp orthogonal matrices u and v using trace identities criteria are given for simultaneous unitary orthogonal or complex orthogonal equivalence between two sets of matrices

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

monotonicity preserving linear multistep methods in this paper we provide an analysis of monotonicity properties for linear multistep methods these monotonicity properties include positivity and the diminishing of total variation we also pay particular attention to related boundedness properties such as the total variation bounded tvb property in the analysis the multistep methods are considered in combination with suitable starting procedures this allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results

boundedness and strong stability of runge kutta methods in the literature much attention has been paid to runge kutta methods rkms satisfying special nonlinear stability requirements indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many rkms the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we study stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we focus on stepsize conditions which are optimal in that they are not unnecessarily restrictive we nd that in situations where the special stability properties mentioned above are violated boundedness can be present only within a class of very special rkms as a byproduct our analysis sheds new light on the known theory of monotonicity for rkms we obtain separate results for internal and external monotonicity as well as a new proof of the fundamental relation between monotonicity and kraaijevanger s coecient this proof distinguishes itself from older ones in that it is shorter and more transparent while it requires more simple assumptions on the rkms under consideration

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

on simultaneous similarity of matrices and related questions abstract let f be a field and f x the ring of polynomials in an indeterminate x over f let m n f m n f x denote the algebras of n n matrices over f f x respectively and gl n f gl n f x their corresponding groups of units given a x b x m n f x we say that a x b x are ps equivalent polynomial scalar if there exist p x gl n f x q gl n f with b x p x a x q we consider the problem of determining whether a x and b x are ps equivalent in other words we wish to classify the orbits of m n f x under the action of gl n f x gl n f acting via t x q a x t x 1 a x q we observe that the classical problems of determining the simultaneous equivalence of two k tuples of elements of m n f and the simultaneous similarity of two k tuples of elements of m n f are special cases of this problem we observe that the smith invariants of a x and b x that is invariants for the action of gl n f x gl n f x on m n f x via t x s x a x t x 1 a x s x must be equal if a x b x are ps equivalent based on this we present a near canonical form for ps equivalence and an algorithm for determining whether two matrices in near canonical form are ps equivalent we examine in detail the generic case in which a x has a single smith invariant different from 1 and obtain a further set of invariants in this case and based on these we present an improved algorithm to determine ps equivalence in this situation while the main emphasis in the paper is on finding a reasonably good algorithm in the generic ease we also discuss the question of finding a complete set of invariants for ps equivalence especially in the case n 2 where connections with linear fractional transformations arise a much more comprehensive account of the invariants in the simultaneous similarity problem can be found in friedland s paper adv in math 50 1983 189 265

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn

the matrizant and saxon hutner theorem relying on the one to one correspondence between real localized potentials and transfer matrices well known from inverse scattering theory the saxon hutner conjecture is reformulated initially as a group theoretical and subsequently as a lie algebraic problem a very basic lie theory in conjunction with time reversal symmetry of the time independent schrodinger equation leads to a fairly general condition which ensures the validity of the saxon hutner theorem