applications of vector bundles to factorization of rational matrices abstract we show that wiener hopf s and forney s factorization theorems are immediate consequences of grothendieck s theorem on vector bundles

simple diagonal locally finite lie algebras the ground field f is an algebraically closed field of zero characteristic and bbb n is the set of natural numbers let l be a locally finite dimensional or em locally finite for brevity lie algebra assume for simplicity that l has countable dimension then one can choose finite dimensional subalgebras l i i in bbb n of l in such a way that l i subset l i 1 for all i and l bigcup i in bbb n l i the set l i mid i in bbb n is called a em local system of l assume that l is simple then all l i can be chosen to be perfect that is l i l i l i we shall call such local systems em perfect let s i s i 1 oplus ldots oplus s i n i be a levi subalgebra of l i where s i 1 ldots s i n i are simple components of s i and let v i k be the standard s i k module since l i is perfect for every k there exists a unique irreducible l i module cal v i k such that the restriction cal v i k downarrow s i k is isomorphic to v i k an embedding l i to l i 1 is called em diagonal if for each l any non trivial composition factor of the restriction cal v i 1 l downarrow l i is isomorphic to cal v i k or its dual cal v i k for some k we prove that a simple locally finite lie algebra l can be embedded into a locally finite associative algebra if and only if there exists a perfect local system l i i in bbb n of l such that all embeddings l i to l i 1 are diagonal note that this result can be considered as a version of ado s theorem for simple locally finite lie algebras let v be a vector space an element x in frak gl v is called em finitary if dim xv infty the finitary transformations of v form an ideal frak fgl v of frak gl v and any subalgebra of frak fgl v is called a em finitary lie algebra we classify all finitary simple lie algebras of countable dimension it turns out that there are only three frak sl infty f frak so infty f and frak sp infty f the author has classified all finitary simple lie algebras and will describe this in a subsequent paper the analogous problem in group theory has been recently solved by j i hall he classified simple locally finite groups of finitary linear transformations 1991 mathematics subject classification 17b35 17b65

realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

algebras of quotients of graded lie algebras abstract in this paper we explore graded algebras of quotients of lie algebras with special emphasis on the 3 graded case and answer some natural questions concerning its relation to maximal jordan systems of quotients

classification of the direct limits of involution simple associative algebras and the corresponding dimension groups abstract a classification of the countable direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic is obtained this also classifies the corresponding dimension groups the set of invariants consists of two supernatural numbers and two real parameters

a controllability test for general first order representations we derive a new controllability rank test for general first order representations the criterion generalizes the well known controllability rank test for linear input state systems as well as a controllability rank test by mertzios et al for descriptor systems

quasiclassical lie algebras abstract in this paper we study finite dimensional non semisimple lie algebras that can be obtained as lie algebras of skew symmetric elements of associative algebras with involution we call such algebras quasiclassical and characterize them in terms of existence of so called plain representations we show that the theory of plain representations for quasiclassical lie algebras is almost equivalent to representation theory of associative algebras with involution

maximal ideals in modular group algebras of the finitary symmetric and alternating groups the main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups provided the characteristic p of the ground field is greater than 2 for the symmetric group there are exactly p 1 such ideals and for the alternating group there are p 1 2 of them the description is obtained in terms of the annihilators of certain systems of the completely splittable irreducible modular representations of the finite symmetric and alternating groups the main tools used in the proofs are the modular branching rules obtained earlier by the second author and the mullineux conjecture proved recently by fordkleshchev and bessenrodt olsson the results obtained are relevant to the theory of pi algebras they are used in a later paper by the authors and a e zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups

a bott borel weil theory for direct limits of algebraic groups we develop a bott borel weil theory for direct limits of algebraic groups some of our results apply to locally reductive ind groups g in general i e to arbitrary direct limits of connected reductive linear algebraic groups our most explicit results concern root reductive ind groups g the locally reductive ind groups whose lie algebras admit root decomposition given a parabolic subgroup p of g and a rational irreducible p module we consider the irreducible g sheaves inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i and their duals inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i these sheaves are locally free in general of infinite rank we prove a general analog of the bott borel weil theorem for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i namely that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i is nonzero for at most one index q q 0 and that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i is isomorphic to the dual of a rational irreducible g module v for q 0 0 we show that in contrast to the finite dimensional case v need not admit an irreducible p submodule there however one has a larger parabolic subgroup w p p constructed from p and a weyl group element w of length q 0 such that v is generated by an irreducible w p submodule consequently certain g modules v can appear only for q 0 0 never for q 0 0 for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i we show that there is no analog of bott s vanishing theorem more precisely that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i can have arbitrarily many nonzero cohomology groups finally we give an explicit criterion for the projectivity of the ind variety g p showing that g p is in general not projective

generalized state space realizations of non proper rational transfer functions in this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series k z 1 existence and uniqueness of minimal realizations are proved and their properties are described the dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized hankel matrix finally a correspondence between realizations and fractional representations over k z 1 of polynomial matrices is established and some properties of such representations are investigated

unitary highest weight modules of inductive limit lie algebras and groups abstract representations of inductive limits of finite dimensional lie groups and algebras am our main objects of study such representations can be constructed by taking inductive limits of the representations of the finite dimensional groups or algebras for finite dimensional lie groups and algebras of hermitian symmetric type enright howe wallach classified those highest weight representations which are unitarizable in this paper we consider the limits of the algebras u p n so 2n and sp n r the limits of the corresponding maximal compact subgroup and the limits of the corresponding generalized verma modules we give necessary and sufficient conditions for the inductive limit representation to be unitarizable in all three cases we show that a necessary condition for unitarizability is that the highest weight is eventually constant we also give sufficient conditions depending on the constant which will ensure that the limit is unitarizable

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

branching laws for tensor modules over classical locally finite lie algebras abstract let g and g be isomorphic to any two of the lie algebras gl sl sp and so let m be a simple tensor g module we introduce the notion of an embedding g g of general tensor type and derive branching laws for triples g g m where g g is an embedding of general tensor type more precisely since m is in general not semisimple as a g module we determine the socle filtration of m over g due to the description of embeddings of classical locally finite lie algebras given by dimitrov and penkov in 2009 our results hold for all possible embeddings g g unless g gl

finitary lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple and finitary irreducible lie algebras over an algebraically closed field of characteristic 2 3

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

asymptotic results on modular representations of symmetric groups and almost simple modular group algebras in fact this is a question about infinite simple groups because it is easy for g finite and because a non trivial normal subgroup gives rise to a non trivial ideal different from the augmentation ideal also note that the problem reduces easily to the question on when the augmentation ideal is simple as a ring the first interesting class of groups with almost simple fg was discovered in 3 this class is rather exotic and contains groups like the universal hall group and algebraically closed groups for locally finite groups g a substantial progress was achieved recently by using representation theory of finite groups see 18 19 16 and others the representation theory approach transforms the problems on ideals to certain problems on asymptotic behavior of representations of finite groups which are often of independent interest the method is more effective over fields of characteristic zero as the theory of ordinary representations is much better elaborated than the modular theory this paper is devoted to the modular aspect of the theory one of our main results deals with the case where g is a direct limit of alternating groups and f is any field of characteristic p 2 we note that the case where charf 0 was considered in 18 let n be the set of natural numbers denote by alt and sym or simply an and n the alternating and symmetric groups respectively on a set with n let alt 1 alt 2 alt i 1

simple locally finite dimensional lie algebras in positive characteristic abstract we prove two structure theorems for simple locally finite dimensional lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form r r z r r r or k r k r for a simple locally finite dimensional associative algebra r with involution the first proves that a condition we introduce known as locally nondegenerate along with the existence of an ad nilpotent element suffice the second proves that an ad integrable lie algebra is of this type if the characteristic of the ground field is sufficiently large lastly we construct a simple locally finite dimensional associative algebra r with involution such that k r k r k r to demonstrate the necessity of considering the commutator in the first two theorems

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

jordan algebras at jordan elements of semiprime rings with involution abstract in this paper we determine the jordan algebras associated to ad nilpotent elements of index at most 3 in lie algebras r and skew r for semiprime rings r without or with involution to do so we first characterize these ad nilpotent elements

output feedback invariants the paper is concerned with the problem of determining a complete set of invariants for output feedback using tools from geometric invariant theory it is shown that there exists a quasi projective variety whose points parameterize the output feedback orbits in a unique way if the mcmillan degree n mp the product of the number of inputs and number of outputs then it is shown that in the closure of every feedback orbit there is exactly one nondegenerate system

the bott borel weil theorem for direct limit groups we show how highest weight representations of certain infinite dimensional lie groups can be realized on cohomology spaces of holomorphic vector bundles this extends the classical bott borel weil theorem for finite dimensional compact and complex lie groups our approach is geometric in nature in the spirit of bott s original generalization of the borel weil theorem the groups for which we prove this theorem are strict direct limits of compact lie groups or their complexifications we previously proved that such groups have an analytic structure our result applies to most of the familiar examples of direct limits of classical groups we also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases one of the technical problems here is that in general the limit lie algebras will have root systems but need not have root spaces so we need to develop machinery to handle this somewhat delicate situation

finitary simple lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple lie algebras over a field of characteristic 0 we also describe finitary irreducible lie algebras

geometric representation theory for unitary groups of operator algebras geometric realizations for the restrictions of gns representations to unitary groups of c algebras are constructed these geometric realizations use an appropriate concept of reproducing kernels on vector bundles to build such realizations in spaces of holomorphic sections a class of complex coadjoint orbits of the corresponding real banach lie groups are described and some homogeneous holomorphic hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed

simple lie subalgebras of locally finite associative algebras abstract we prove that any simple lie subalgebra of a locally finite associative algebra is either finite dimensional or isomorphic to the commutator algebra of the lie algebra of skew symmetric elements of some involution simple locally finite associative algebra the ground field is assumed to be algebraically closed of characteristic 0 this result can be viewed as a classification theorem for simple lie algebras that can be embedded in locally finite associative algebras we also establish a link between this class of lie algebras and that of lie algebras graded by finite root systems

universal central extensions of direct limits of lie superalgebras abstract we show that the universal central extension of a direct limit of perfect lie superalgebras l i is isomorphic to the direct limit of the universal central extensions of l i as an application we describe the universal central extensions of some infinite rank lie superalgebras

representations of the symmetric group which are irreducible over subgroups let f be an algebraically closed field of characteristic p and n be the symmetric group on n letters in this paper we classify all pairs g d where d is an irreducible f n module of dimension greater than 1 and g is a proper subgroup of n such that the restriction d g is irreducible provided p 3

inner ideals of simple locally finite lie algebras abstract inner ideals of simple locally finite dimensional lie algebras over an algebraically closed field of characteristic 0 are described in particular it is shown that a simple locally finite dimensional lie algebra has a non zero proper inner ideal if and only if it is of diagonal type regular inner ideals of diagonal type lie algebras are characterized in terms of left and right ideals of the enveloping algebra regular inner ideals of finitary simple lie algebras are described

the classification of locally finite split simple lie algebras one of the great early achievements in lie theory is the classification of the finite dimensional simple complex lie algebras by w killing and e cartan more than 50 years later new aspects were added to this classification by the theory of coxeter groups and the visualization of the classification in terms of dynkin diagrams furthermore serre s description of the simple lie algebras by generators and relations provided a direct way to construct the lie algebras from the cartan matrix corresponding to the choice of a root base i e a system of simple roots

support varieties for modules over symmetric groups 1 1 in the late 1970s alperin a defined an invariant called the complexity of a module as a way to relate the modules with the complexes and resolutions that they admit several years later carlson ca1 ca2 defined affine algebraic varieties corresponding to modules over group algebras these varieties are subvarieties of the spectrum of the cohomology ring which was earlier described by quillen q they are known in present day language as support varieties it was discovered early on that the complexity of a module is equal to the dimension of the support variety of the module geometric methods involving support varieties have played a fundamental role in understanding the interplay between the modular representation theory and cohomology for finite groups despite substantial progress in this direction there have been few explicit computations of support varieties for important classes of modules over certain groups the goal of this paper is to introduce methods and techniques for computing support varieties for modules over the symmetric group d in the process we will provide explicit computations of support varieties for certain classes

a koszul category of representations of finitary lie algebras abstract we find for each simple finitary lie algebra g a category t g of integrable modules in which the tensor product of copies of the natural and conatural modules is injective the objects in t g can be defined as the finite length absolute weight modules where by absolute weight module we mean a module which is a weight module for every splitting cartan subalgebra of g the category t g is koszul in the sense that it is antiequivalent to the category of locally unitary finite dimensional modules over a certain direct limit of finite dimensional koszul algebras we describe these finite dimensional algebras explicitly we also prove an equivalence of the categories t o and t sp corresponding respectively to the orthogonal and symplectic finitary lie algebras o sp

on cartan subalgebras abstract in this paper we study the concept of a cartan subalgebra in the context of locally finite and borcherds kac moody lie algebras

representations of finitary lie algebras abstract let f be an algebraically closed field of characteristic 2 3 w a f vector space and l gl w with nilwl 0 dimfl suppose l fgl w is a finitary subalgebra the faithful irreducible l modules are determined it is shown that l has minimal ideals if a minimal ideal s is infinite dimensional then sw is a completely reducible l module suppose l fgl w 0 w is l irreducible and char f 3 then l is classified in terms of l fgl w

an artinian theory for lie algebras abstract complemented lie algebras are introduced in this paper a notion similar to that studied by o loos and e neher in jordan pairs we prove that a lie algebra is complemented if and only if it is a direct sum of simple nondegenerate artinian lie algebras moreover we classify simple nondegenerate artinian lie algebras over a field of characteristic 0 or greater than 7 and describe the lie inner ideal structure of simple lie algebras arising from simple associative algebras with nonzero socle

hilbert schmidt groups as infinite dimensional lie groups and their riemannian geometry abstract we describe the exponential map from an infinite dimensional lie algebra to an infinite dimensional group of operators on a hilbert space notions of differential geometry are introduced for these groups in particular the ricci curvature which is understood as the limit of the ricci curvature of finite dimensional groups is calculated we show that for some of these groups the ricci curvature is

cartan subalgebras of root reductive lie algebras abstract root reductive lie algebras are direct limits of finite dimensional reductive lie algebras under injections which preserve the root spaces it is known that a root reductive lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of finite dimensional simple lie algebras as well as copies of the three simple infinite dimensional root reductive lie algebras sl so and sp as part of a structure theory program for root reductive lie algebras cartan subalgebras of the lie algebra gl were introduced and studied in k h neeb i penkov cartan subalgebras of gl canad math bull 46 2003 597 616 in the present paper we refine and extend the results of k h neeb i penkov cartan subalgebras of gl canad math bull 46 2003 597 616 to the case of a general root reductive lie algebra g we prove that the cartan subalgebras of g are the centralizers of maximal toral subalgebras and that they are nilpotent and self normalizing we also give an explicit description of all cartan subalgebras of the simple lie algebras sl so and sp we conclude the paper with a characterization of the set of conjugacy classes of cartan subalgebras of the lie algebras gl sl so and sp with respect to the group of automorphisms of the natural representation which preserve the lie algebra

the jordan algebras of a lie algebra abstract we attach a jordan algebra l x to any ad nilpotent element x of index of nilpotence at most 3 in a lie algebra l this jordan algebra has a behavior similar to that of the local algebra of a jordan system at an element thus l x inherits nice properties from l and keeps relevant information about the element x

simple module branching rules for k sp sp and ks2p abstract we study the branching rules between k s p s p and k s 2 p that is we determine in characteristic p for p odd the loewy structure of the principal block simple k s p s p modules induced to k s 2 p and the loewy structure of the principal block simple k s 2 p modules restricted to k s p s p in characteristic zero the answer follows easily from the littlewood richardson rule but in other characteristics the structure of the induced and restricted modules is not known kleshchev has obtained the simple module branching rules between k s n and k s n 1 for any n and any field characteristic we provide similar branching rules for our case

infinite kostant cascades and centrally generated primitive ideals of u n in types a c abstract we study the center of u n where n is the locally nilpotent radical of a splitting borel subalgebra of a simple complex lie algebra g sl c so c sp c there are infinitely many isomorphism classes of lie algebras n and we provide explicit generators of the center of u n in all cases we then fix n with largest possible center of u n and characterize the centrally generated primitive ideals of u n for g sl c sp c in terms of the above generators as a preliminary result we provide a characterization of the centrally generated primitive ideals in the enveloping algebra of the nilradical of a borel subalgebra of sl n c sp 2 n c

locally simple subalgebras of diagonal lie algebras abstract we describe up to isomorphism all locally simple subalgebras of any diagonal locally simple lie algebra

the socle of a nondegenerate lie algebra we define the socle of a nondegenerate lie algebra as the sum of all its minimal inner ideals the socle turns out to be an ideal which is a direct sum of simple ideals and satisfies the descending chain condition on principal inner ideals every classical finite dimensional lie algebra coincides with its socle while relevant examples of infinite dimensional lie algebras with nonzero socle are the simple finitary lie algebras and the classical banach lie algebras of compact operators on an infinite dimensional hilbert space this notion of socle for lie algebras is compatible with the previous ones for associative algebras and jordan systems we conclude with a structure theorem for simple nondegenerate lie algebras containing abelian minimal inner ideals and as a consequence we obtain that a simple lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank one element

lie group structures on quotient groups and universal complexifications for infinite dimensional lie groups we characterize the existence of lie group structures on quotient groups and the existence of universal complexifications for the class of baker campbell hausdorff bch lie groups which subsumes all banach lie groups and linear direct limit lie groups as well as the mapping groups ckr m g cr m g m k 1 for every bch lie group g second countable finite dimensional smooth manifold m compact subset k of m and 0 r also the corresponding test function groups dr m g kckr m g are bch lie groups

iwasawa decompositions of some infinite dimensional lie groups we set up an abstract framework that allows the investigation of iwasawa decompositions for involutive infinite dimensional lie groups modeled on banach spaces this provides a method to construct iwasawa decompositions for classical real or complex banach lie groups associated with the schatten ideals o p h on a complex separable hilbert space h if 1 p

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

realization of autoregressive equations in pencil and descriptor form a linear system described by autoregressive equations with a given input output structure cannot be transformed to standard state space form if the implied input output relation is nonproper instead a realization in descriptor form must be used in this paper it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies and without going through a reduction process external equivalence is used so that even situations in which there is no transfer matrix can be considered the approach is based on the so called pencil representation of linear systems and it is shown that there is a natural realization of autoregressive equations in pencil form in this way the link between the realization theories of willems and fuhrmann can also be clarified

a class of simple matrix algebras abstract fix an infinite set i and consider the associative matrix algebra m i f where f is a base field with char f 2 for any couple of bijective maps i i such that and 2 2 we introduce a linear subspace of m i f we endow it with a structure of non associative algebra for a certain bilinear product and obtain a wide class of non associative algebras containing in particular the lie algebras lie m i f t we show that each algebra is simple

infinite dimensional irreducible lie algebras containing transformations of finite rank let mathbb f be a field of characteristic zero and let v be an infinite dimensional vector space over mathbb f a linear transformation x of v is called finitary if dim xv infty the aim of this paper is to describe irreducible lie subalgebras of mathfrak g mathfrak l v containing nonzero finitary transformations it turns out that any such algebra is a semidirect product of a finite dimensional lie algebra and a dense lie subalgebra of mathfrak g mathfrak l w for some vector space w

irreducible finitary lie algebras over fields of characteristic zero abstract a lie subalgebralof g l k v is said to befinitaryif it consists of elements of finite rank we show that if char k 0 if dim k vis infinite and iflacts irreducibly onv then the derived algebra oflis simple

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

conical representations for direct limits of symmetric spaces we extend the definition of conical representations for riemannian symmetric spaces to a certain class of infinite dimensional riemannian symmetric spaces using a certain infinite dimensional version of weyl s unitary trick we show that there is a correspondence between smooth representations of infinite dimensional noncompact type riemannian symmetric spaces and smooth representations of infinite dimensional compact type symmetric spaces we classify all smooth conical representations which are unitary on the compact type side finally a new class of non smooth unitary conical representations appears on the compact type side which has no analogue in the finite dimensional case we classify these representations and show how to decompose them into direct integrals of irreducible conical representations

second derivative of an iterative solver boosts its acceleration by kocak s method abstract many mathematical applications involve the solution of a nonlinear equation x g x the target fixed point z of g may be a zero of a related function f x or equivalently a root of the equation f x 0 it is an important and challenging task to develop practical efficient and robust solvers kocak s method achieves acceleration by generating a superior secondary solver gk via a transformation gk g mx 1 m where m is a linear combination of the first derivative of g at two different x values namely the current iterate and the target z gk is the result of a simple rearrangement of the equation x g after subtracting the product mx from both sides therefore gk z g z z the process is piecewise linearization where m is the slope of a straight line approximating g ideally m g z x z let n denote the convergence order of g the initial version employs w 1 2 irrespective of n its gk is of third order when n is 1 or 2 when n exceeds 2 the order remains the same the second version harnesses w 1 n when n exceeds 2 and obtains a gk of order n 1 this manuscript reports a third version involving the second derivative of g

kocak s method shows that an unwittingly exaggerated convergence order is in fact 2 iterative solution of a nonlinear equation f x 0 usually means a repetitive scheme to locate a fixed point of a related equation x g x kocak s acceleration method smoothly gears up iterations with the aid of a superior secondary solver g k x g x g x x g x m x x 1 m x where g x 1 1 m x is a gain and m x 1 1 g x is a straight line slope the accelerator shows that a previously published article a biazar a amirtemoori an improvement to the fixed point iterative method amc 182 2006 567 571 unwittingly exaggerated the convergence order of the solver it presented this solver boils down to an indirect application of newton s method solving g x x 0 which means that it is of second order hence their claim that it increases the order of convergence as much as desired is false the scheme wastes higher derivatives

completely splittable representations of symmetric groups and affine hecke algebras abstract we study the class of completely splittable representations of the symmetric group and its associated degenerate affine hecke algebra in positive characteristic using techniques developed by okounkov and vershik for the symmetric group over the complex numbers the representations are classified and constructed and we provide a character formula in the symmetric group case

simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order direct substitution x k 1 g x k generally represents iterative techniques for locating a root z of a nonlinear equation f x at the solution f z 0 and g z z efforts continue worldwide both to improve old iterators and create new ones this is a study of convergence acceleration by generating secondary solvers through the transformation g m x g x m x x 1 m x or equivalently through partial substitution g m p s x x g x g x g x 1 1 m x as a matter of fact g m x g m p s x is the point of intersection of a linearised g with the g x line aitken s and wegstein s accelerators are special cases of g m simple geometry suggests that m x g x g z 2 is a good approximation for the ideal slope of the linearised g indeed this renders a third order g m the pertinent asymptotic error constant has been determined the theoretical background covers a critical review of several partial substitution variants of the well known newton s method including third order halley s and chebyshev s solvers the new technique is illustrated using first second and third order primaries a flexible algorithm is added to facilitate applications to any solver the transformed newton s method is identical to halley s the use of m x g x g z 2 thus obviates the requirement for the second derivative of f x comparison and combination with halley s and chebyshev s solvers are provided numerical results are from the square root and cube root examples

locally semisimple and maximal subalgebras of the finitary lie algebras gl sl so and sp abstract we describe all locally semisimple subalgebras and all maximal subalgebras of the finitary lie algebras gl sl so and sp for simple finite dimensional lie algebras these classes of subalgebras have been described in the classical works of a malcev and e dynkin

the jordan socle and finitary lie algebras in this paper we introduce the notion of jordan socle for nondegenerate lie algebras which extends the definition of socle given in a fernandez lopez et al 3 graded lie algebras with jordan finiteness conditions comm algebra in press for 3 graded lie algebras any nondegenerate lie algebra with essential jordan socle is an essential subdirect product of strongly prime ones having nonzero jordan socle these last algebras are described up to exceptional cases in terms of simple lie algebras of finite rank operators and their algebras of derivations when working with lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0 the exceptions disappear and the algebras of derivations are computed

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

impulsive smooth behavior in multimode systems part i state space and polynomial representations abstract a switched or multimode system is one that can switch between various modes of operation we consider here switched systems in which the modes of operation are characterized as linear finite dimensional systems not necessarily all of the same mcmillan degree when a switch occurs from one of the modes to another of lower mcmillan degree the state space collapses and an impulse may result followed by a smooth evolution under the new regime this paper is concerned with the description of such impulsive smooth behavior on a typical interval we propose an algebraic framework modeled on the class of impulsive smooth distributions as defined by hautus both state space and polynomial representations are considered and we discuss transformations between the two forms

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

on modular homology in the boolean algebra iii abstract let be a set ra ring of characteristicp 0 and denote bymkther module withk element subsets of as basis theset inclusion map mk mk 1is the homomorphism which associates to ak element subset the sum 1 2 kof all its k 1 element subsets i in this paper we study the chain formula arising from we introduce the notion ofp exactness for a sequence if is infinite we show that isp exact for all prime characteristicsp 0 this result can be extended to various submodules and quotient modules and we give general constructions arising from permutation groups with a finitary section two particular applications are the following the orbit module sequence of such a permutation group on isp exact for every primep and we give a formula for thep rank of the orbit inclusion matrix if the group has finitely many orbits onk element subsets

quivers geometric invariant theory and moduli of linear dynamical systems we use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems a general approach to this problem is presented and applied to two well known cases we show how both lomadze s and helmke s compactification arises naturally as a geometric invariant theory quotient both moduli spaces are proven to be smooth projective manifolds furthermore a description of lomadze s compactification as a quot scheme is given whereas helmke s compactification is shown to be an algebraic grassmann bundle over a quot scheme this gives an algebro geometric description of both compactifications as an application we determine the cohomology ring of helmke s compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive