anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

primitive axial algebras of jordan type abstract an axial algebra over the field f is a commutative algebra generated by idempotents whose adjoint action has multiplicity free minimal polynomial for semisimple associative algebras this leads to sums of copies of f here we consider the first nonassociative case where adjoint minimal polynomials divide x 1 x x for fixed 0 1 jordan algebras arise when 1 2 but our motivating examples are certain griess algebras of vertex operator algebras and the related majorana algebras we study a class of algebras including these for which axial automorphisms like those defined by miyamoto exist and there classify the 2 generated examples always for 1 2 and in identifiable cases for 1 2 this implies that the miyamoto involutions are 3 transpositions leading to a classification

a brief history of the classification of the finite simple groups first published in bulletin of the american mathematical society in volume 38 issue 3 published by the american mathematical society

locally toroidal polytopes of rank 6 and sporadic groups abstract we augment the list of finite universal locally toroidal regular polytopes of type 3 3 4 3 3 due to p mcmullen and e schulte adding as well as removing entries this disproves a related long standing conjecture our new universal polytope is related to a well known y shaped presentation for the sporadic simple group fi 22 and admits s 4 o 8 2 s 3 as the automorphism group we also discuss further extensions of its quotients in the context of y shaped presentations as well we note that two known examples of finite universal polytopes of type 3 3 4 3 3 are related to y shaped presentations of orthogonal groups over f 2 mixing construction is used in a number of places to describe covers and 2 covers

associative subalgebras of low dimensional majorana algebras abstract a majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents called majorana axes that satisfy some of the properties of the 2a axes of the monster griess algebra the term was introduced by a a ivanov in 2009 inspired by the work of s sakuma and m miyamoto in the present paper we revisit mayer and neutsch s theorem on associative subalgebras of the griess algebra in the context of majorana theory we apply this result to determine all the maximal associative subalgebras of some low dimensional majorana algebras namely the majorana algebras generated by two majorana axes and the majorana representations of the symmetric group of degree 4 involving 3c algebras

constructing the monster amalgam abstract the monster group m contains a pair c n of subgroups where c 2 1 24 co 1 is the centralizer of an involution and n 2 2 11 2 11 m 24 sym 3 is the normalizer of an elementary subgroup of order four with n c having index three in n the amalgam c n plays an important role in the existing constructions and uniqueness proofs for the monster we suggest a transparent construction of this amalgam

majorana representations of a 5 the monster group m which is the largest among the 26 sporadic simple groups is the automorphism group of the 196 884 dimensional conway griess norton algebra simply called the monster algebra there is a remarkable correspondence between the so called 2a involutions in m and certain idempotents in the monster algebra we refer to these idempotents as majorana axes the isomorphism types of the subalgebras in the monster algebra generated by pairs of majorana axes were calculated by s norton a while ago there are precisely nine isomorphism types more recently these nine algebras were characterized by s sakuma in the context of vertex operator algebras relying on earlier work by m miyamoto the properties of monster algebras used in the proof of sakuma s theorem are rather elementary and they have been axiomatized under the name of majorana representations in this terminology sakuma s theorem amounts to classification of the majorana representations of the dihedral groups together with a remark that all the representations are based on embeddings into the monster in the present paper it is shown that the alternating group a 5 of degree 5 possesses precisely two majorana representations both based on embeddings into the monster the dimensions of the representations are 20 and 26 the scalar squares of their identities are 10 and 72 7 respectively in the vertex operator algebra context these numbers are doubled central charges

injective dg modules over non positive dg rings abstract let a be an associative non positive differential graded ring in this paper we make a detailed study of a category inj a of left dg modules over a which generalizes the category of injective modules over a ring we give many characterizations of this category generalizing the theory of injective modules and prove a derived version of the bass papp theorem the category inj a is closed in the derived category d a under arbitrary direct sums if and only if the ring h 0 a is left noetherian and for every i 0 the left h 0 a module h i a is finitely generated specializing further to the case of commutative noetherian dg rings we generalize the matlis structure theory of injectives to this context as an application we obtain a concrete version of grothendieck s local duality theorem over commutative noetherian local dg rings

on primitive 3 generated axial algebras of jordan type abstract axial algebras of jordan type are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing x 1 x x where 0 1 is fixed with restrictive multiplication rules these properties generalize the peirce decompositions for idempotents in jordan algebras where 1 2 is replaced with in particular jordan algebras generated by idempotents are axial algebras of jordan type 1 2 if 1 2 then it is known that axial algebras of jordan type are factors of the so called matsuo algebras corresponding to 3 transposition groups we call the generating idempotents axes and say that an axis is primitive if its adjoint operator has 1 dimensional 1 eigenspace it is known that a subalgebra generated by two primitive axes has dimension at most three the 3 generated case has been opened so far we prove that every axial algebra of jordan type generated by three primitive axes has dimension at most nine if the dimension is nine and 1 2 then we either show how to find a proper ideal in this algebra or prove that the algebra is isomorphic to certain jordan matrix algebras

the squaring operation for commutative dg rings abstract let a b be a homomorphism of commutative rings the squaring operation is a functor sq b a from the derived category d b of complexes of b modules into itself this operation is needed for the definition of rigid complexes in the sense of van den bergh that in turn leads to a new approach to grothendieck duality for rings schemes and even dm stacks in our paper with j j zhang from 2008 we introduced the squaring operation and explored some of its properties unfortunately some of the proofs in that paper had severe gaps in them in the present paper we reproduce the construction of the squaring operation this is done in a more general context than in the first paper here we consider a homomorphism a b of commutative dg rings our first main result is that the square sq b a m of a dg b module m is independent of the resolutions used to present it our second main result is on the trace functoriality of the squaring operation we give precise statements and complete correct proofs in a subsequent paper we will reproduce the remaining parts of the 2008 paper that require fixing this will allow us to proceed with the other papers mentioned in the bibliography on the rigid approach to grothendieck duality the proofs of the main results require a substantial amount of foundational work on commutative and noncommutative dg rings including a study of semi free dg rings their lifting properties and their homotopies this part of the paper could be of independent interest

the uniqueness of groups of lyons type we give the first computer free proof of the uniqueness of groups of type ly we also show that certain simpicial complexes associated to the lyons group ly have the same homotopy type as the quillen complex for ly at the prime 3 we show this complex is simply connected and we calculate its homology finally we supply simplified proofs of some properties of groups of type ly such as the group order a group of type ly is a finite group g possessing an involution t such that h cg t is the covering group of the alternating group a1l and t is not weakly closed in h with respect to g we prove

extensions of cocycles cohen macaulay geometries and a vanishing theorem for cohomology of alternating groups abstract a new approach to the computation of the cohomology of groups acting on buekenhout geometries is described it is shown that in many cases including bn pairs and groups acting on dimensional linear spaces to compute the kth cohomology group hk g m it is sufficient roughly speaking to consider only parabolic subgroups of rank k 1 as an application it is shown that h k a n f p 0 for k 1 p k 2 where an is the alternating group

an expansion algorithm for constructing axial algebras abstract an axial algebra a is a commutative non associative algebra generated by primitive idempotents called axes whose adjoint action on a is semisimple and multiplication of eigenvectors is controlled by a certain fusion law different fusion laws define different classes of axial algebras axial algebras are inherently related to groups namely when the fusion law is graded by an abelian group t every axis a leads to a subgroup of automorphisms t a of a the group generated by all t a is called the miyamoto group of the algebra we describe a new algorithm for constructing axial algebras with a given miyamoto group a key feature of the algorithm is the expansion step which allows us to overcome the 2 closedness restriction of seress s algorithm computing majorana algebras at the end we provide a list of examples for the monster fusion law computed using a magma implementation of our algorithm

classical locally finite tits chamber systems of rank 3 a chamber system v 9 j zi is over some index set z is a set of chambers together with partitions in z of v if ze z and c e v then bj is the join of all partitions fijzi e z and d c is the element of fij containing c d c is again a chamber system over j with fij restricted to z we say q is connected if d c furthermore 111 is the rank of v the most important examples of chamber systems are those obtained from groups if g is a group b a subgroup of g and xi ie z are subgroups of g containing b then 5 g b xi icl has as chambers the cosets bg g e g two such cosets bg bh being i adjacent if and only if xi g xih then g acts chamber transitively on with kernel b n bg and is connected if and only if g x 1 i e z furthermore each chamber system with transitive automorphism group g can be obtained in this way by setting b g and xi gd cr a rank 2 chamber system u hi fi is called a classical generalized ma gon mq 3 if is isomorphic to g b pi p where g is an essentially simple rank 2 group of lie type in the sense of cl b is a bore1 subgroup and p pj are the two maximal parabolic subgroups of g containing b here md is given by the relation vu wj wiw g 1 defined by the weyl group of g a rank 2 chamber system is a generalized digon if is isomorphic to v g pin p pi p where pi p are different proper subgroups of g satisfying g pipj pjpi a connected chamber system over z is called a classical tits chamber system if for each ce and i j c z di j c is either a generalized digon or a classical generalized m gon for some mi e n is focally finite if all d c are finite for motivation to consider such classical locally finite tits chamber systems see 15 and 16 if is a classical tits chamber system then the diagram d z is defined in 9 0021 8693 89 3 00

code algebras axial algebras and voas abstract inspired by code vertex operator algebras voas and their representation theory we define code algebras a new class of commutative non associative algebras constructed from binary linear codes let c be a binary linear code of length n a basis for the code algebra a c consists of n idempotents and a vector for each non constant codeword of c we show that code algebras are almost always simple and under mild conditions on their structure constants admit an associating bilinear form we determine the peirce decomposition and the fusion law for the idempotents in the basis and we give a construction to find additional idempotents called the s map which comes from the code structure for a general code algebra we classify the eigenvalues and eigenvectors of the smallest examples of the s map construction and hence show that certain code algebras are axial algebras we give some examples including that for a hamming code h 8 where the code algebra a h 8 is an axial algebra and embeds in the code voa v h 8

weak proregularity weak stability and the noncommutative mgm equivalence abstract let a be a commutative ring and let a be a finitely generated ideal in it it is known that a necessary and sufficient condition for the derived a torsion and a adic completion functors to be nicely behaved is the weak proregularity of a in particular the mgm equivalence holds because weak proregularity is defined in terms of elements of the ring it involves limits of koszul complexes it is not suitable for noncommutative ring theory in this paper we introduce a new condition on a torsion class t in a module category weak stability our first main theorem says that in the commutative case the ideal a is weakly proregular if and only if the corresponding torsion class t is weakly stable we then study weak stability of torsion classes in module categories over noncommutative rings there are three main theorems in this context for a torsion class t that is weakly stable quasi compact and finite dimensional the right derived torsion functor is isomorphic to a left derived tensor functor the noncommutative mgm equivalence that holds under the same assumptions on t a theorem about symmetric derived torsion for complexes of bimodules this last theorem is a generalization of a result of van den bergh from 1997 and corrects an error in a paper of yekutieli zhang from 2003

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

the twisted inverse image pseudofunctor over commutative dg rings and perfect base change abstract let k be a gorenstein noetherian ring of finite krull dimension and consider the category of cohomologically noetherian commutative differential graded rings a over k such that h 0 a is essentially of finite type over k and a has finite flat dimension over k we extend grothendieck s twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup we prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps and prove a perfect base change result for f in this setting as application we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings our results generalize and solve some recent conjectures of yekutieli

reduction of derived hochschild functors over commutative algebras and schemes abstract we study functors underlying derived hochschild cohomology also called shukla cohomology of a commutative algebra s essentially of finite type and of finite flat dimension over a commutative noetherian ring k we construct a complex of s modules d and natural reduction isomorphisms ext s k l s s k m k l n ext s r hom s m d n for all complexes of s modules n and all complexes m of finite flat dimension over k whose homology h m is finitely generated over s such isomorphisms determine d up to derived isomorphism using grothendieck duality theory we establish analogous isomorphisms for any essentially finite type flat map f x y of noetherian schemes with f o y in place of d

signalizer functors proper 2 generated cores and nonconnected groups as has been well understood since its inception the signalizer functor method works most smoothly in finite groups satisfying some connectedness condition in particular in groups with a connected sylow 2 subgroup the original approach to nonconnectivity disposed of the problem once and for all by completely classifying all groups of sectional 2 rank at most 4 and then determining those with a nonconnected sylow 2 subgroup as a corollary by a theorem of macwilliams 141 every section of a nonconnected 2group can be generated by at most four elements even though this approach was very effective it was extremely elaborate for the actual proof of the classification of such groups runs to 464 pages l 11 and in addition depends upon a number of other classification theorems concerning groups with specified types of sylow 2 subgroups moreover although the sectional 2 rank 4 theorem has been invoked at many other points in the classification of finite simple groups the analysis of nonconnected groups has remained its primary application hence in attempting to simplify the existing classification of the finite simple groups one of the first natural questions to consider is whether there exists a shorter more direct way of treating nonconnectivity indeed harada has recently achieved just such a direct approach by a short elementary fusion argument 121 his result can be stated as follows here a k group is any finite group whose composition factors are among the known simple groups

rigid complexes via dg algebras let a be a commutative ring b a commutative a algebra and m a complex of b modules we begin by constructing the square sq b a m which is also a complex of b modules the squaring operation is a quadratic functor and its construction requires differential graded dg algebras if there exists an isomorphism p m sq b a m then the pair m p is called a rigid complex over b relative to a there are some finiteness conditions there is an obvious notion of rigid morphism between rigid complexes we establish several properties of rigid complexes including their uniqueness existence under some extra hypothesis and formation of pullbacks f b m p resp f m p along a finite resp essentially smooth ring homomorphism f b c in the subsequent paper rigid dualizing complexes over commutative rings we consider rigid dualizing complexes over commutative rings building on the results of the present paper the project culminates in our paper rigid dualizing complexes and perverse sheaves on schemes where we give a comprehensive version of grothendieck duality for schemes the idea of rigid complexes originates in noncommutative algebraic geometry and is due to van den bergh 1997

a 1d exact treatment of shock waves within spectral methods in plane geometry abstract we present a very exact numerical technique for solving 1d euler equations coupled with the transport equations for the entropy and the chemical abundances with or without shock formation two moving grids are used before and after the shock formation quantities are expanded on both sides of the matching point in chebychev polynomials series after the shock is formed rankine hugoniot conditions are used to determine the velocity of the shock and the matching conditions across the shock typical results are presented

a correction of the decomposability result in a paper by meyer neutsch abstract in their paper of 1993 meyer and neutsch established the existence of a 48 dimensional associative subalgebra in the griess algebra g by exhibiting an explicit counter example the present paper shows a gap in the proof one of the key results in meyer and neutsch s paper which states that an idempotent a in the griess algebra is indecomposable if and only its peirce 1 eigenspace i e the 1 eigenspace of the linear transformation l a x a x is one dimensional the present paper fixes this gap and shows a more general result let v be a real commutative nonassociative algebra with an associative inner product and let c be a nonzero idempotent of v such that its peirce 1 eigenspace is a subalgebra then c is indecomposable if and only if its peirce 1 eigenspace is one dimensional the proof of this result is based on a general variational argument for real commutative metrised algebras with inner product

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

commuting involution graphs for sporadic simple groups let kless than or equals slantgless than or equals slantaut k where k is one of the 26 sporadic finite simple groups and let tset membership variantg be an involution and x tg the commuting involution graph view the mathml source has x as its vertex set with two distinct elements of x joined by an edge whenever they commute in g for most of the sporadic simple groups we compute the diameter of view the mathml source and give detailed information about the elements at a given distance from a fixed involution t

the universal nonabelian representation of the petersen type geometry related toj4 abstract let g be a geometry with three points on each line the universal representation groupr g of g is generated by a set of involutions one for each point subject to the relations that for every line the product of the three generators corresponding to the points on the line is the identity letj4be the fourth sporadic simple group of janko and let g j4 be the petersen type geometry ofj4whose points and lines are subgroups of order 2 and 22inj4with normalizers containing sylow 2 subgroups the incidence relation is given by inclusion we show thatr g j4 is exactlyj4

natural representations of the p geometries of co2 type abstract we consider flag transitive p geometries that are geometries belonging to the diagrams formula where the rightmost edge means the geometry of the edges and the vertices of the petersen graph with the natural incidence relation the main result of the paper together with some earlier results provides the classification of the flag transitive p geometries possessing natural representations together with a description of all such representations

presenting the baby monster abstract we prove that y 433 is isomorphic to 2 2 f 2 where f 2 is fischer s baby monster group this answers the last open problem concerning the description of y groups the result implies in particular that y 553 and y 555 are fabulous systems of generators for the monster and bimonster respectively in a sense introduced by j conway we also give a new independent proof that y 555 is the bimonster

a presentation for j4 let g g j 4 be the rank 4 p geometry with janko s fourth group j 4 as a flag transitive automorphism group let a be the amalgam of maximal parabolics corresponding to the action of j 4 on g we prove that g is simply connected i e that it has no proper coverings this result is equivalent to the fact that j 4 coincides with the universal closure of the amalgam a so any presentation for a is a presentation for j 4 this implies in particular that a nice set of generators and relations given by g stroth and r weiss is a presentation for j 4 in addition we obtain a uniqueness proof for j 4

associative subalgebras of the griess algebra abstract the structure of the griess algebra g whose automorphism group is the fischer griess monster f1 is investigated we show the existence of 48 dimensional associative subalgebras in g and furthermore demonstrate that they are not contained in strictly larger ones this is an immediate consequence of a more general necessary and sufficient criterion for the maximality of associative subalgebras of g it is conjectured that the given explicit examples are of maximal possible dimension among all associative subalgebras in g this depends on the validity of a certain inequality

universal axial algebras and a theorem of sakuma abstract in the first half of this paper we define axial algebras nonassociative commutative algebras generated by axes that is semisimple idempotents the prototypical example of which is griess algebra 2 for the monster group when multiplication of eigenspaces of axes is controlled by fusion rules the structure of the axial algebra is determined to a large degree we give a construction of the universal frobenius axial algebra on n generators with specified fusion rules of which all n generated frobenius axial algebras with the same fusion rules are quotients in the second half we realise this construction in the majorana ising vir 4 3 case on 2 generators and deduce a result generalising sakuma s theorem in voas 13

y groups via transitive extension abstract motivated by an earlier observation by j h fischer around 1980 b conway conjectured that the coxeter diagram y555 on fig 1 together with a single additional so called spider relation ab1c1ab2c2ab3c3 10 1 form a presentation for the wreath product of the monster group m and a group of order 2 this conjecture was proved by s p norton and the author in 1990 the original proof was rather involved relying on simple connectedness results for certain diagram geometries on numerous data obtained by coset enumeration on a computer and on some delicate calculations with subgroups coming from the 26 node theorem in the present work we follow an inductive approach to the identification of y groups by considering larger y groups as transitive extensions of smaller ones along these lines we obtain an alternative identification proof for the y groups which is almost computer free we refer to only one result of double coset enumeration our approach provides a uniform understanding of the y groups particularly of features such as centres and redundant generators

absolute embeddings of point line geometries abstract a method is given for showing that an embeddable point line geometry possesses an absolutely universal projective embedding this method is applied to show that virtually every embeddable lie incidence geometry possesses an absolutely universal embedding

on the uniqueness of the harada norton group the definition of a group 2 8 a 5 wr2 type is given in 2 2 1 see also 2 3 1 and the notation in 2 1 4 and 2 1 5 for 2 hs 2 a group g satisfying the hypotheses of the main theorem exists its existence is a consequence of the existence of the monster constructed by r griess in 151 we know of no published work proving the uniqueness of f5 of course many properties of f were derived in harada s major work on f5 7 a group g satisfying the hypotheses of the main theorem will be said to be of f type let g be a group of f type and let a z be involutions in g as in the main theorem we consider the graph f defined as follows its vertex set is the conjugacy class a g two involutions x y e ag form an edge iff xy a we show that the hypotheses of the main theorem imply the uniqueness of r and that g aut t aut t and hence g is unique our paper relies on some properties of 2 hs 2 see 2 3 and some properties of co z g z as in the main theorem but we do not assume but rather derive the isomorphism type of c otherwise our paper is independent of 7 and we prove all properties of g which we require after some preliminary and important work in section 2 we start in section 3 to obtain properties of g and r r as above perhaps the most important result of section 3 is that c a has 8 nontrivial orbits on ag as given in the following table 261 0021 8693192 55 00

non abelian representations of some sporadic geometries for a point line incidence system s p l with three points per line we define the universal representation group of s as formula we prove that if g is the 2 local parabolic geometry of the sporadic simple groupf1 the monster orf2 the baby monster thenr g f1or 2 f2 respectively

a moonshine path for 5a and associated lattices of ranks 8 and 16 abstract we continue the program begun in griess and lam in press 18 to make a moonshine path between a node of the extended e 8 diagram and the monster our goal is to provide a context for observations of mckay glauberman and norton by realizing their theories in a more concrete form in this article we treat the 5a node most work in this article is a study of certain lattices of ranks 8 16 and 24 whose determinants are powers of 5

on majorana representations of a 6 and a 7 the majorana representations of groups were introduced in ivanov the monster group and majorana involutions 2009 by axiomatising some properties of the 2a axial vectors of the 196 884 dimensional monster algebra inspired by the sensational classification of such representations for the dihedral groups achieved by sakuma int math res notes 2007 this classification took place in the heart of the theory of vertex operator algebras and expanded earlier results by miyamoto j alg 268 653 671 2003 every subgroup g of the monster which is generated by its intersection with the conjugacy class of 2a involutions possesses the possibly unfaithful majorana representation obtained by restricting to g the action of the monster on its algebra this representation of g is said to be based on an embedding of g in the monster so far the majorana representations have been classified for the groups g isomorphic to the symmetric group s 4 of degree 4 ivanov et al in j alg 324 2432 2463 2010 the alternating group a 5 of degree 5 ivanov aa seress a in majorana representations of a 5 2010 and the general linear group gl 3 2 in dimension 3 over the field of two elements ivanov aa shpectorov s in majorana representations of l 3 2 2010 all these representations are based on embeddings in the monster of either the group g itself or of its direct product with a cyclic group of order 2 the dimensions and shapes of these representations are given in the following table

spectral methods for the euler equations ii chebyshev methods and shock fitting on exprime chaque composante de la vitesse dans les equations d euler par une somme de polynomes de tchebycheff qui constitue une fonction d interpolation on utilise les expressions spectrales ainsi obtenues pour calculer plusieurs ecoulements gazeux bidimensionnels par la methode d ajustement d ondes de choc selon cette methode les conditions initiales correspondent a une onde de choc plane normale a un ecoulement supersonique et on determine la forme veritable de l onde a l aide des formules de rankine hugoniot qui expriment les conditions aux limites

flux corrected pseudospectral method for scalar hyperbolic conservation laws abstract the pseudospectral method has under used advantages in problems involving shocks and discontinuities these emerge from superior accuracy in phase and group velocities as compared to finite difference schemes of all orders dispersion curves for finite difference schemes suggest that group velocity error typically outranks gibbs error as a cause of numerical oscillation a flux conservative form of the pseudospectral method is derived for compatibility with flux limiters used to preserve monotonicity the resulting scheme gives high quality results in linear advection and shock formation propagation examples

moonshine paths for 3a and 6a nodes of the extended e8 diagram abstract we continue the program to make a moonshine path between a node of the extended e 8 diagram and the monster our theory is a concrete model expressing some of the mysterious connections identified by john mckay george glauberman and simon norton in this article we treat the 3a and 6a nodes we determine the orbits of triples x y z in the monster where z 2 b x y 2 a c z and x y 3 a 6 a such x y correspond to a rootless ee 8 pair in the leech lattice for the 3a and 6a cases we shall say something about the half weyl groups which are proposed in the glauberman norton theory most work in this article is with lattices due to their connection with dihedral subgroups of the monster these lattices are m n where m n is the relevant pair of ee 8 sublattices and their annihilators in the leech lattice the isometry groups of these four lattices are analyzed

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

majorana representations of the symmetric group of degree 4 the monster group m acts on a real vector space vm of dimension 196 884 which is the sum of a trivial 1 dimensional module and a minimal faithful m module there is an m invariant scalar product on vm an m invariant bilinear commutative non associative algebra product on vm commonly known as the conway griess norton algebra and a subset a of vm 0 indexed by the 2a involutions in m certain properties of the quintet m m vm a have been axiomatized in chapter 8 of ivanov 2009 iv09 under the name of majorana representation of m the axiomatization enables one to study majorana representations of an arbitrary group g generated by its involutions a representation might or might not exist but it always exists whenever g is a subgroup in m generated by the 2a involutions contained in g we say that thus obtained representation is based on an embedding of g in the monster the essential motivation for introducing the majorana terminology was the most remarkable result by s sakuma 2007 sak07 which gave a classification of the majorana representations of the dihedral groups there are nine such representations and every single one is based on an embedding in the monster of the relevant dihedral group it is a fundamental property of the monster that its 2a involutions form a class of 6 transpositions and that there are precisely nine m orbits on the pairs of 2a involutions and also on the set of 2a generated dihedral subgroups in m in the present paper we are making a further step in building up the majorana theory by classifying the majorana representations of the symmetric group s4 of degree 4 we prove that s4 possesses precisely four majorana representations the monster is known to contain four classes of 2a generated s4 subgroups so each of the four representations is based on an embedding of s4 in the monster the classification of 2a generated s4 subgroups in the monster relies on calculations with the character table of the monster our elementary treatment shows that there are at most four isomorphism types of subalgebras in the conway griess norton algebra of the monster generated by six majorana axial vectors canonically indexed by the transpositions of s4 two of these subalgebras are 13 dimensional the other two have dimensions 9 and 6 these dimensions not to mention the isomorphism type of the subalgebras were not known before