algebras of quotients of jordan algebras abstract we define a jordan analogue of lambek and utumi s associative algebra of quotients and we construct the maximal algebra of quotients for nondegenerate jordan algebras we apply those results to other classes of algebras of quotients appearing in the literature

discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

optimal bi level quantization of i i d sensor observations for binary hypothesis testing we consider the problem of binary hypothesis testing using binary decisions from independent and identically distributed i i d sensors identical likelihood ratio quantizers with threshold spl lambda are used at the sensors to obtain sensor decisions under this condition the optimal fusion rule is known to be a k out of n rule with threshold k for the bayesian detection problem we show that given k the probability of error is a quasi convex function of spl lambda and has a single minimum that is achieved by the unique optimal spl lambda sub opt except for the trivial situation where one hypothesis is always decided we obtain a sufficient and necessary condition on spl lambda sub opt and show that spl lambda sub opt can be efficiently obtained via the secant algorithm the overall optimal solution is obtained by optimizing every pair of k spl lambda for the neyman pearson detection problem we show that the use of the lagrange multiplier method is justified for a given fixed k since the objective function is a quasi convex function of spl lambda we further show that the receiver operating characteristic roc for a fixed k is concave downward

coordinate algebras of extended affine lie algebras of type a1 abstract the cores of extended affine lie algebras of reduced types were classified except for type a1 in this paper we determine the coordinate algebra of extended affine lie algebras of type a1 it turns out that such an algebra is a unital z n graded jordan algebra of a certain type called a jordan torus we classify jordan tori and get five types of jordan tori

strong primeness of hermitian jordan systems this paper establishes the strong primeness of all jordan systemsjof hermitian type trapped between ample hermitian elements of a prime associative systemrand its martindale system of symmetric quotientsq r h0 r j h q r this completes the converse of zelmanov s classification of strongly prime jordan systems providing if as well as only if classifications of strongly prime and primitive jordan systems

minimal ideals of jordan systems we show that minimal ideals of jordan systems algebras triple systems and pairs are either simple or trivial so answering the question posed by nam and mccrimmon in 1983

quasi convexity and optimal binary fusion for distributed detection with identical sensors in generalized gaussian noise we present a technique to find the optimal threshold spl tau for the binary hypothesis detection problem with n identical and independent sensors the sensors all use an identical and single threshold spl tau to make local decisions and the fusion center makes a global decision based on the n local binary decisions for generalized gaussian noise and some non gaussian noise distributions we show that for any admissible fusion rule the probability of error is a quasi convex function of threshold spl tau hence the problem decomposes into a series of n quasi convex optimization problems that may be solved using well known techniques assuming equal a priori probability we give a sufficient condition of the non gaussian noise distribution g x for the probability of error to be quasi convex furthermore this technique is extended to bayes risk and neyman pearson criteria we also demonstrate that in practice it takes fewer than twice as many binary sensors to give the performance of infinite precision sensors in our scenario

the suboptimality of randomized tests in distributed and quantized detection systems the design of decentralized or quantized detection systems requires simultaneous optimization of quantizer mappings and ultimate fusion rule it is shown that if the likelihood ratios of the unquantized or raw observations are independent and contain no point masses of probability the optimal test does not randomize this is so despite the fact that the data to be fused can be considered discrete the result is a considerable simplification in the design search

explicit runge kutta schemes and finite elements with symmetric stabilization for first order linear pde systems we analyze explicit runge kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first order linear differential operator in space of friedrichs type for the time discretization we consider explicit second and third order runge kutta schemes we identify a general set of properties on the space stabilization encompassing continuous and discontinuous finite elements under which we prove stability estimates using energy arguments then we establish l 2 norm error estimates with quasi optimal convergence rates for smooth solutions in space and time these results hold under the usual cfl condition for third order runge kutta schemes and any polynomial degree in space and for second order runge kutta schemes and first order polynomials in space for second order runge kutta schemes and higher polynomial degrees in space a tightened 4 3 cfl condition is required numerical results are presented for smooth and rough solutions the case of finite volumes is briefly discussed

discontinuous galerkin methods for friedrichs systems part iii multifield theories with partial coercivity this paper is the third and last part of a work attempting to give a unified analysis of discontinuous galerkin methods the purpose of this paper is to extend the framework that has been developed in part ii for two field friedrichs systems associated with second order pdes we now consider two field friedrichs systems with partial l 2 coercivity and three field friedrichs systems with an even weaker l 2 coercivity hypothesis in particular this work generalizes the discontinuous galerkin methods of part ii to compressible and incompressible linear continuum mechanics we also show how the stabilizing parameters of the method must be set when the two field friedrichs system is composed of terms that may be of different magnitude thus accounting for instance for advection diffusion equations at high peclet numbers

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

new asymptotic results in parallel distributed detection the performance of a parallel distributed detection system is investigated as the number of sensors tends to infinity it is assumed that the i i d sensor data are quantized locally into m ary messages and transmitted to the fusion center for binary hypothesis testing the boundedness of the second moment of the postquantization log likelihood ratio is examined in relation to the asymptotic error exponent it is found that when that second moment is unbounded the neyman pearson error exponent can become a function of the test level whereas the bayes error exponent remains as previously conjectured by j n tsitsiklis 1986 unaffected large deviations techniques are also used to show that in bayes testing the equivalence of absolutely optimal and best identical quantizer systems is not limited to error exponents but extends to the actual bayes error probabilities up to a multiplicative constant

jordan centers and martindale like covers abstract in this paper we show that the scalar center of a nondegenerate quadratic jordan algebra is contained in the scalar center of any of its martindale like covers

counterexamples in distributed detection specific counterexamples violating three very plausible conjectures about optimal design of distributed detection and decision systems are reported the numerical magnitude of the violation is small in each case

friedrichs systems in a hilbert space framework solvability and multiplicity abstract the friedrichs 1958 theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics irrespective of their type this theory was recast in an abstract hilbert space setting by ern guermond and caplain 2007 and by antonic and burazin 2010 in this work we make a further step presenting a purely operator theoretic description of abstract friedrichs systems and proving that any pair of abstract friedrichs operators admits bijective extensions with a signed boundary map moreover we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces and by the universal operator extension theory grubb 1968 we get a complete identification of all such pairs which we illustrate on two concrete one dimensional examples

encoding of correlated observations an important class of engineering problems involves sensing an environment and making estimates based on the phenomena sensed in the traditional model of this problem the sensors observations are available to the estimator without alteration there is growing interest in em distributed sensing systems in which several observations are communicated to the estimator over channels of limited capacity the observations must be separately encoded so that the target can be estimated with minimum distortion two questions are addressed for a special case of this problem wherein there are two sensors which observe noisy data and communicate with a single estimator 1 if the encoder is unlimited in complexity what communication rates and distortions can be achieved 2 if the encoder must be a quantizer a mapping of a single observation sample into a digital output how can it be designed for good performance the first question is treated by the techniques of information theory it is proved that a given combination of rates and distortion is achievable if there exist degraded versions of the observations that satisfy certain formulas the second question is treated by two approaches in the first the outputs of the quantizers undergo a second stage of encoding which exploits their correlation to reduce the output rate algorithms which design the second stage are presented and tested the second approach is based on the em distributional distance a measure of dissimilarity between two probability distributions an algorithm to modify a quantizer for increased distributional distance is derived and tested

quadratic jordan systems of hermitian type abstract in this paper we extend zel manov s classification of linear jordan systems triple systems and pairs of hermitian type to quadratic jordan systems over an arbitrary ring of scalars using associative triple systems of the first kind for what we believe to be a more natural tool to describe special jordan triples we show that an i special prime quadratic jordan triple system the ideals of which remain semiprime with a nonzero hermitian part lies between an ample subspace of hermitian elements in a prime associative triple system and those in its martindale system of symmetric quotients modulo some extra definitions the structure of strongly prime jordan pairs of hermitian type follows almost immediately from the jordan triple result

derivations and automorphisms of jordan algebras in characteristic two a jordan algebra j over a field k of characteristic 2 becomes a 2 lie algebra l j with lie product x y x y and squaring x 2 x 2 we determine the precise ideal structure of l j in case j is simple finite dimensional and k is algebraically closed we also decide which of these algebras have smooth automorphism groups finally we study the derivation algebra of a reduced albert algebra j h3 o k and show that der j has a unique proper nonzero ideal vj isomorphic to l j k 1j with quotient der j v j independent of o on the group level this gives rise to a special isogeny between the automorphism group of j and that of the split albert algebra whose kernel is the infinitesimal group determined by vj

a posteriori error analysis for numerical approximations of friedrichs systems the global error of numerical approximations for symmetric positive systems in the sense of friedrichs is decomposed into a locally created part and a propagating component residual based two sided local a posteriori error bounds are derived for the locally created part of the global error these suggest taking the l 2 norm as well as weaker dual norms of the computable residual as local error indicators the dual graph norm of the residual vec r h is further bounded from above and below in terms of the l 2 norm of h vec r h where h is the local mesh size the theoretical results are illustrated by a series of numerical experiments

boundary operator from matrix field formulation of boundary conditions for friedrichs systems abstract following the recent progress in understanding the abstract setting for friedrichs symmetric positive systems by ern guermond and caplain 2007 8 as well as antonic and burazin 2010 3 an attempt is made to relate these results to the classical friedrichs theory a comparison of two approaches via the trace operator and the boundary operator has been made favouring the latter finally a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given and the applicability of this procedure in realistic situations is shown by examples

zelmanov s prime theorem for quadratic jordan algebras e i zelmanov has recently shown that any prime linear jordan algebra without nil ideals but no finiteness conditions imposed is either a homomorphic image of a special algebra or is a form of the 27 dimensional exceptional algebra in particular all division algebras and all simple unital algebras are either derived from associative algebras or are 27 dimensional over their centers this effectively ends the search for infinite dimensional exceptional algebras in the present paper zelmanov s result is extended to quadratic jordan algebras jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics an algebraic system which behaved like the usual algebra of quantum mechanical observables hermitian operators on hilbert space but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra in their pioneering paper of 1933 91 jordan von neumann and wigner classified the finite dimensional formally real linear jordan algebras and a a albert showed 11 that the only simple algebra in the list which was exceptional was a certain 27 dimensional algebra of 3 x 3 hermitian matrices with entries from an gdimensional cayley algebra all subsequent investigations of exceptional jordan algebras led back to this same albert algebra the next breakthrough in the structure of jordan algebras was n jacobson s 1965 introduction 6 j of inner ideals and the resultant quadratilication of the theory in terms of the quadratic product xyx instead of the linear product xy vx the final breakthrough was the astounding 1979 result of a young russian mathematician e i zelmanov that the only simple exceptional linear jordan are albert algebras thus the search for an infinite dimensional setting for exceptional quantum mechanics is doomed to failure the only simple exceptional structure allotted to mortals is the 27 dimensional albert algebra this came as a complete surprise to western researchers who hoped the free jordan algebra might lead to an 291 002 i 8693 82 060297 30 02 00 0

a unified discontinuous petrov galerkin method and its analysis for friedrichs systems we propose a unified discontinuous petrov galerkin dpg framework with optimal test functions for friedrichs like systems which embrace a large class of elliptic parabolic and hyperbolic partial differential equations pdes the well posedness i e existence uniqueness and stability of the dpg solution is established on a single abstract dpg formulation and two abstract dpg methods corresponding to two different but equivalent norms are devised we then apply the single dpg framework to several linear ized pdes including but not limited to scalar transport laplace diffusion convection diffusion convection diffusion reaction linear ized continuum mechanics e g linear ized elasticity a version of linearized navier stokes equations etc time domain acoustics and a version of the maxwell s equations the results show that we not only recover several existing dpg methods but also discover new dpg methods for both pdes currently considered in the dpg community and new ones as

tits kantor koecher algebras of strongly prime hermitian jordan pairs abstract in this paper we describe the tits kantor koecher algebras of strongly prime hermitian jordan pairs in terms of prime associative pairs and their martindale pairs of symmetric quotients

asymptotic expansions for large deviation probabilities in the strong law of large numbers letx1 x2 be a sequence of independent random variables with common distribution functionf having zero mean and let sn be the random walk of partial sums the weak and strong laws of large numbers respectively imply that for any and e 0 the probabilitiesp sm em and pm p s n alpha varepsilon n text for some n geqq m tend to 0 asm tends to building upon work of bahadur and ranga rao ann math stat 31 1015 1027 1960 siegmund z wahrscheinlichkeitstheor verw geb 31 107 113 1975 and fill and wichura probab th rel fields81 189 212 1989 we produce complete asymptotic expansions for the probabilitiesp sm em andpm

moufang sets from groups of mixed type abstract we consider the existence of moufang sets related to certain groups of mixed type this way we obtain new examples of moufang sets and new constructions of known classes the most interesting class of new examples is related to the moufang quadrangles of type f 4 and to the ree tits octagon over a nonperfect field and the root groups of each member have nilpotency class three

outer fractions in quadratic jordan algebras abstract using new techniques of zelmanov c martinez improved on work of jacobson mccrimmon and parvathi to give a necessary and sufficient ore type condition for an arbitrary linear jordan algebra with no 2 or 3 torsion to have an algebra of fractions in this paper we extend to quadratic algebras the concept of algebras of outer fractions with respect to an ore monad and describe necessary and sufficient ore type conditions for the embedding in such an algebra of fractions the details of the actual embedding will appear in a subsequent paper

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

the ring of fractions of a jordan algebra abstract we derive a necessary and sufficient ore type condition for a jordan algebra to have a ring of fractions

fine quantization in signal detection and estimation the performance lost to data quantization in signal detection and estimation procedures is considered the performance is measured by f divergences which are useful indices of discrimination information between statistical hypotheses some properties of the f divergences are briefly discussed and a general result for the loss in divergence due to uniform data quantization is obtained several applications of this result in specific problems of signal detection and estimation being developed and some numerical results that illustrate the asymptotic behavior of the divergence in these applications are given the divergence lost to finite nonuniform quantization through the use of a companding model is considered and the asymptotically optimum nonuniform quantizer within this class is derived some interesting problems that remain open are discussed

jordan quadruple systems abstract we define jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the jordan tetrad a b c d a b c d d c b a as a quadrilinear operation on associative algebras we find further identities in degree 10 which are not consequences of the defining identities we introduce four infinite families of finite dimensional jordan quadruple systems and construct the universal associative envelope for a small system in each family we obtain analogous results for the anti tetrad a b c d a b c d d c b a our methods rely on computer algebra especially linear algebra on large matrices the lll algorithm for lattice basis reduction representation theory of the symmetric group noncommutative grobner bases and wedderburn decompositions of associative algebras

general formulas for the neyman pearson type ii error exponent subject to fixed and exponential type i error bounds the general formulas for the neyman pearson type ii error exponent subject to two different type i error constraints as indicated in the title of the correspondence are established as revealed in the formulas the type ii error exponents are fully determined by the ultimate statistical characteristic of the normalized log likelihood ratio evaluated under the null hypothesis distribution applications of the general formulas to distributed neyman pearson detection and the channel reliability function are also demonstrated

heat equation as a friedrichs system abstract inspired by recent advances in the theory of friedrichs symmetric positive systems we apply newly developed results to the heat equation by showing how the intrinsic theory of ern guermond and caplain 2007 can be used in order to get a well posedness result for the dirichlet initial boundary value problem we also demonstrate the application of the two field theory with partial coercivity of ern and guermond 2008 originally developed for elliptic problems and also discuss different possibilities for the construction of the appropriate boundary operator

asymptotic locally optimal detector for large scale sensor networks under the poisson regime we consider distributed detection with a large number of identical binary sensors deployed over a region where the phenomenon of interest poi has spatially varying signal strength each sensor makes a binary decision based on its own measurement and the local decision of each sensor is sent to a fusion center using a random access protocol the fusion center decides whether the event has occurred under a global size constraint in the neyman pearson formulation assuming homogeneous poisson distributed sensors we show that the distribution of alarmed sensors satisfies the local asymptotic normality lan we then derive an asymptotically locally most powerful almp detector optimized jointly over the fusion form and the local sensor threshold under the poisson regime we establish conditions on the spatial signal shape that ensure the existence of the almp detector we show that the almp test statistic is a weighted sum of local decisions the optimal weights being the shape of the spatial signal the exact value of the signal strength is not required we also derive the optimal threshold for each sensor for the case of independent identically distributed iid sensor observations we show that the counting based detector is also almp under the poisson regime the performance of the proposed detector is evaluated through analytic results and monte carlo simulations and compared with that of the counting based detector the effect of mismatched signal shapes is also investigated

the maximal symmetric ring of quotients abstract the maximal symmetric ring of quotients q r as defined by utumi is a symmetric version of the maximal ring of quotients of r for the most part we study this ring when r k g is a group algebra for example we show that if g is a free product of groups and if r k g is a domain then q r is usually equal to r on the other hand there are certainly groups for which q r is properly larger than r and we construct a number of such examples

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

primitive jordan pairs and triple systems abstract in this paper we give a characterization of primitivity of jordan pairs and triple systems in terms of their local algebras as a consequence of that local characterization we extend to jordan pairs and triple systems most of the known results about primitive jordan algebras in particular we describe primitive jordan pairs and triple systems over an arbitrary ring of scalars in the sense of the structure of primitive quadratic jordan algebras by j a anquela t cortes and f montaner 1995 j algebra172 530 553 5 1

the structure of primitive quadratic jordan algebras in this paper we give a complete description of primitive quadratic jordan algebras following the classification of strongly prime quadratic jordan algebras k mccrimmon and e i zel manov adv math 69 no 2 1988 133 222 the proof is based in two results of independent interest we show that every p i primitive jordan algebra is simple and unital with nonzero socle and prove that associative tight tight envelopes of special primitive jordan algebras are also primitive primitive as a consequence of the latter fact we see that an associative algebra a is one sided primitive if and only if a is primitive and an associative algebra a with involution is primitive if and only if an ample subspace h 0 a is primitive

asymptotic optimality of likelihood ratio threshold tests in decentralized detection two distributed systems are considered for discriminating between two finite alphabet bivariate memoryless sources and for detecting a known signal in stationary bivariate additive gaussian noise each system comprises two sensors m ary local quantizers and a fusion center which makes decisions based on quantized source observations the problem of asymptotically optimal quantization is considered in detail for the binary m 2 case it is shown that optimality is achieved by quantizing a locally computed likelihood ratio wherein one distribution is in general different from the appropriate source marginal for the problem of detection in gaussian noise it is further demonstrated that the optimal distributed system attains the same asymptotic performance as the optimal centralized system for appropriate choice of m

zel manov polynomials in quadratic jordan triple systems abstract zel manov polynomials are elements of a free special jordan system which are both hermitian their values look like hermitian elements and clifford they do not vanish on systems t containing h 3 h m 3 t these polynomials decide the classification of strongly prime special jordan systems t is either clifford or hermitian according as some zel manov polynomial does or does not vanish on t the existence of such polynomials has been established by mccrimmon and zel manov for quadratic jordan algebras and by zel manov for linear jordan triple systems and pairs in this paper we carry out the construction for quadratic jordan triple systems and pairs

algebras of quotients of nonsingular jordan algebras abstract we define a jordan analogue of johnson s associative algebra of quotients and study it under a suitable condition of nonsingularity of the jordan algebra which we call strong nonsingularity in particular we prove the existence and describe the maximal algebras of quotients of prime strongly nonsingular jordan algebras

goldie theory for jordan algebras it is shown that zelmanov s version of goldie s conditions still characterizes quadratic jordan algebras having an artinian algebra of quotients which is nondegenerate at the same time jordan versions of the main notions of the associative theory such as those of the uniform ideal uniform element singular ideal and uniform dimension are studied moreover it is proved that the nondegenerate unital jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element 2002 elsevier science usa

the etale tits process of jordan algebras revisited the etale tits process which was called the toral tits process by petersson racine 12 may be viewed as a jordan theoretical method to construct associative algebras with involution more specifically starting from a cubic etale algebra e and a quadratic etale algebra l both over an arbitrary base field as well as from invertible elements u e b l having the same norms the etale tits process produces an absolutely simple jordan algebra j j e l u b of degree 3 and dimension 9 which by structure theory must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind in addition e identifies canonically with a subalgebra of j

complex friedrichs systems and applications recently there has been a significant development of the abstract theory of friedrichs systems in hilbert spaces ern a guermond j l and caplain g commun partial differ equations 32 317 341 2007 and antonic n and burazin k commun partial differ equations 35 1690 1715 2010 and its applications to specific problems in mathematical physics however these applications were essentially restricted to real systems we check that the already developed theory of abstract friedrichs systems can be adjusted to the complex setting with some necessary modifications which allows for applications to partial differential equations with complex coefficients we also provide examples where the involved hilbert space is not the space of square integrable functions as it was the case in previous studies but rather its closed subspace or the space hs rd cr for real s this setting appears to be suitable for particular systems of partial differential equations such as the dirac system the di

discontinuous galerkin methods for friedrichs systems part ii second order elliptic pdes this paper is the second part of a work attempting to give a unified analysis of discontinuous galerkin methods the setting under scrutiny is that of friedrichs systems endowed with a particular 2 times 2 structure in which one unknown can be eliminated to yield a system of second order elliptic like pdes for the remaining unknown a general discontinuous galerkin method for approximating such systems is proposed and analyzed the key feature is that the unknown that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems all the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated examples are given for advection diffusion reaction linear continuum mechanics and a simplified version of the magnetohydrodynamics equations comparisons with well known discontinuous galerkin approximations for the poisson equation are presented

the convergence rate for the strong law of large numbers general lattice distributions letx1 x2 be a sequence of independent random variables with common lattice distribution functionf having zero mean and let sn be the random walk of partial sums the strong law of large numbers slln implies that for any and e 0 pm p s n alpha varepsilon n text for some n geqq m decreases to 0 asm increases to under conditions on the moment generating function off we obtain the convergence rate by determiningpm up to asymptotic equivalence when 0 and e is a point in the lattice forf the result is due to siegmund z wahrscheinlichkeitstheor verw geb 31 107 113 1975 but this restriction on e precludes all small values of e and these values are the most interesting vis a vis the slln even when 0 our result handles important distributionsf for which siegmund s result is vacuous for example the two point distributionf giving rise to simple symmetric random walk on the integers we also identify for both lattice and non lattice distributions the behavior of certain quantities in the asymptotic expression forpm as e decreases to 0

primitivity in jordan systems is ubiquitous abstract the main result of the paper asserts that if a jordan pair v v v is primitive at some 0 b v then it is primitive at any 0 b v also if a jordan triple system t is primitive at some 0 b t then it is primitive at any 0 b t as a tool similar results concerning one sided primitivity and primitivity of associative pairs and triple systems are established equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence

maximal algebras of martindale like quotients of strongly prime linear jordan algebras abstract in this paper we prove the existence and give precise descriptions of maximal algebras of martindale like quotients for arbitrary strongly prime linear jordan algebras as a consequence we show that zelmanov s classification of strongly prime jordan algebras can be viewed exactly as the description of their maximal algebras of martindale like quotients as a side result we show that the martindale associative algebra of symmetric quotients can be expressed in terms of the symmetrized product i e in purely jordan terms

the zelmanov approach to jordan homomorphisms of associative algebras abstract zelmanov s work on prime jordan algebras leads to an idempotent free version of martindale s theorem on the extension of jordan homomorphisms and derivations from the hermitian elements h r of an associative algebra of degree 2 with involution to associative homomorphisms and derivations on r the condition that j h r be of degree 2 is replaced by the intrinsic condition that j z j h a consist entirely of values of zelmanov polynomials

nondegenerate jordan algebras satisfying local goldie conditions abstract a structure theorem is given for nondegenerate jordan algebras j satisfying the ascending chain condition on annihilators of a single element and such that j contains no infinite direct sum of inner deals inside the inner ideal generated by each element x j as a consequence of this theorem and of the main results of a previous paper by the authors j algebra 174 1995 1024 1048 it is obtained that such jordan algebras j are precisely the local orders in nondegenerate jordan algebras satisfying dcc on principal inner ideals and without non artinian quadratic ideals which extends to local orders the zel manov theorem for goldie jordan algebras