ternary composition algebras and hurwitz s theorem axioms are proposed for a certain alternative kind of ternary composition algebra termed a 3cn algebra the axioms are shown to be for n 2 in a simple correspondence with the axioms for a ternary vector cross product algebra the axioms imply that n 1 2 4 or 8 from which the usual hurwitz theorem is deduced the existence of 3c8 algebras is demonstrated by an explicit construction in four dimensional hilbert space without appeal to the properties of the algebra of octonions

a new view of d 7 clifford algebra a new view of d 7 clifford algebra is described in which starting from a septet of mutually anticommuting clifford operators one introduces a septet of mutually commuting involutions in so 8 these involutions can be seen to arise naturally from a study of ternary vector cross products in eight dimensions

some groups of transformations defined by jordan algebras ii groups of type f4 the principal objective of this paper is the study of one of the types of linear groups defined in i 15 the groups of automorphisms of central simple exceptional jordan algebras this can also be considered as the second step in a program of studying exceptional linear groups analogous to the exceptional simple lie groups via non associative algebras the first step in this program has been taken in another paper 13 in which we studied the counterparts of the lie groups of type g 2 as groups of automorphisms of cayley algebras in an analogous fashion the present paper deals with the counterparts of the lie group f 4 as groups of automorphisms of exceptional jordan algebras

self dual yang mills fields in pseudo euclidean spaces the self duality yang mills equations in pseudo euclidean spaces of dimensions d 8 are investigated new classes of solutions of the equations are found extended solutions to the d 10 n 1 supergravity and super yang mills equations are constructed from these solutions

ternary vector cross products ternary vector cross products are studied in their own right results include a new proof of hurwitz s theorem and a principle of duplicity upon breaking the symmetry in eight dimensions by choosing a preferred axis this last principle implies the well known triality principle for octonions and so 8 transformations in displaying canonical forms it helps to put the eight basis vectors in correspondence with the eight points of the three dimensional affine geometry over f2

composition algebras with large derivation algebras abstract the finite dimensional flexible composition algebras include the hurwitz algebras composition algebras with unit element but also other interesting classes of algebras the para hurwitz and the okubo algebras the above mentioned algebras present many symmetries and this is reflected in their large derivation algebras in the present paper we study the opposite question what can be said about the composition algebras if we have some information about their derivation algebras our main result is the classification of all the composition algebras with such large derivation algebras

on triality and automorphisms and derivations of composition algebras abstract the principle of triality for orthogonal transformations on eight dimensional spaces is used to obtain different expressions for the automorphisms derivations and orthogonal transformations of cayley dickson algebras and symmetric composition algebras over fields of arbitrary characteristic

triple products and yang baxter equation i octonionic and quaternionic triple systems the yang baxter equation can be recast as a triple product equation assuming the triple product to satisfy some algebraic relations new solutions of the yang baxter equation can be found this program has been completed here for the simplest triple systems which are called octonionic and quaternionic the solutions are of rational type

lie algebras under constraints and non bijective canonical transformations the concept of a lie algebra under constraints is developed in connection with the theory of non bijective canonical transformations a finite dimensional vector space m carrying a faithful linear representation of a lie algebra l is mapped into a lower dimensional space m in such a manner that a subalgebra l0 of l is mapped into d l0 0 the lie algebra l under the constraint d l0 0 is the largest subalgebra l1 of l that can be represented faithfully on m if l0 is semisimple then l1 is shown to be the centraliser cent ll0 if l is semisimple and l0 is a one dimensional diagonal subalgebra of a cartan subalgebra of l then l1 is shown to be the factor algebra cent ll0 l0 the later two results are applied to non bijective canonical transformations generalising the kustaanheimo stiefel transformation

a classification of hidden variable properties hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics we start with a quantum mechanical model and describe various properties that can be asked of a hidden variable model we present six such properties and a venn diagram of how they are related with two existence theorems and three no go theorems epr bell and kochen specker we show which properties of empirically equivalent hidden variable models are possible and which are not formally our treatment relies only on classical probability models and physical phenomena are used only to motivate which models to choose

ternary composition algebras i structure theorems definite and neutral signatures previous work on ternary composition algebras of definite signature is generalized to the only other case allowed by the axioms namely that of neutral signature one new feature is the presence in the neutral case of counter automorphisms9 the structure of the exceptional eight dimensional algebras is probed and in various guises a fundamental dichotomy is unearthed canonical forms are obtained and various further properties are derived

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

a six variable identity for a ternary vector cross product in eight dimensional space let x a b c be a ternary vector cross product for eight dimensional euclidean space e an identity is derived which expresses x a b c x u v w in terms of the spin 7 invariant scalar quadruple product phi a b c d a x b c d the proof of the identity is coordinate free and starts out from an explicit expression for x with e viewed as complex four dimensional hilbert space

hopf bifurcation subject to a large delay in a laser system hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system the problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks most of these instabilities are starting from a single hopf bifurcation because of the large delay a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi periodic states we determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double hopf bifurcation points we study numerically different cases of bistability between steady periodic and quasi periodic regimes finally the validity of the hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation

ternary composition algebras ii automorphism groups and subgroups we study an eight dimensional ternary composition algebra e e of signature 8 0 or 4 4 such an algebra is associated with an involutary outer automorphism m of the lie algebra so e of the automorphisms and in the 4 4 case counter automorphisms of e are obtained in terms of the d 7 spin group determined by m using m we derive a principle of duplicity for the ternary multiplication related to the well known principle of triality for the associated octonionic binary multiplication however rather than construct e out of the octonions as done in effect by previous authors we give a four dimensional complex construction of e this non octonionic view of e highlights certain 15 dimensional d 6 spin groups rather than the customary 14 dimensional octonionic automorphism groups in the 4 4 case we can choose to interpret complex in terms of the split complex numbers and are thereby led to consider the subgroup chain so 4 4 spin 3 4 sl 4 spin 3 3 instead of the chain so 4 4 spin 3 4 su 2 2 spin 2 4

quantum chains with a catalan tree pattern of conserved charges the δ 1 xxz model and the isotropic octonionic chain a class of quantum chains possessing a family of local conserved charges with a catalan tree pattern is studied recently we have identified such a structure in the integrable su n invariant chains in the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra two additional systems with a catalan tree structure of conserved charges are found one is the spin 1 2 xxz model with 1 the other is a new octonionic isotropic chain generalizing the heisenberg model this system provides an interesting example of an infinite family of noncommuting local conserved quantities

notes on a cross product of vectors the relation between quaternionlike algebras and cross products of vectors is demonstrated a classification of all cross products of vectors is given

finite geometries and clifford algebras ii the pleasant incidence properties of m dimensional projective geometry enable us upon employing the finite geometry over the field rm f rm 2 to deal nicely with certain commutativity anti commutativity aspects of clifford algebras cl o d in dimensions d 2m 1 1 m 2 3

two simple proofs of the kochen specker theorem a new proof of the kochen specker theorem uses 33 rays instead of 117 in the original proof if the number of dimensions is increased from 3 to 4 only 24 rays are needed

yang mills like instantons in eight and seven dimensions we consider a gauge theory in which a nonassociative moufang loop takes the place of a structure group we construct belavin polyakov schwartz tyupkin bpst and t hooft like instanton solutions of the gauge theory in seven and eight dimensions

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

clifford algebras and finite groups the structure of finite groups associated with clifford algebras of signatures 0 d d identical to 3 mod 4 is outlined a speculation by another author concerning the possible exceptional nature of the group associated with d 7 is seen to be false

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

subcritical hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation a subcritical hopf bifurcation in a dynamical system modeled by a scalar nonlinear delay differential equation is studied theoretically and experimentally the subcritical hopf bifurcation leads to a significant domain of bistability where stable steady and time periodic states coexist

isotopes of octonion algebras g2 torsors and triality octonion algebras over rings are in contrast to those over fields not determined by their norm forms octonion algebras whose norm is isometric to the norm q of a given algebra c are twisted forms of c by means of the aut c torsor o q o q aut c we show that over any commutative unital ring these twisted forms are precisely the isotopes c a b of c with multiplication given by x y xa by for unit norm octonions a b of c the link is provided by the triality phenomenon which we study from new and classical perspectives we then study these twisted forms using the interplay thus obtained between torsor geometry and isotope computations thus obtaining new results on octonion algebras over e g rings of laurent polynomials

gradings on symmetric composition algebras abstract the group gradings on the symmetric composition algebras over arbitrary fields are classified applications of this result to gradings on the exceptional simple lie algebras are considered too

division composition algebras through their derivation algebras abstract composition algebras are algebras with a nondegenerate quadratic multiplicative form in this paper we construct five families of division composition algebras and prove that any division composition algebra with nonabelian derivation algebra belongs to one of these families as a by product we obtain new examples of real division algebras

non resonant double hopf bifurcation of a class b laser system we investigate the double hopf bifurcation at positive equilibrium point of a class b laser system firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for a class b laser system and derive the universal unfolding of the double hopf bifurcation thirdly some dynamic behaviors such as periodic and quasi periodic solutions are observed using numerical simulation

octonionic representations of so 8 and its subgroups and cosets the generators of the subgroups so 7 l r and the cosets s7l r so 8 so 7 l r and 7l r so 7 l r g2 of so 8 are constructed out of the generators of so 8 using octonionic structure tensors and the representations of these generators are given as products of octonions

simple unified form for the major no hidden variables theorems two examples are given that substantially simplify the no hidden variables theorem of kochen and specker greatly reducing the number of observables considered and requiring no intricate geometric argument while one of the examples also obeys a more powerful version of bell s theorem the other does not the examples provide a new perspective on both of these fundamental theorems and on the relation between them

on the moyal deformation of nahm equations in seven dimensions we show how the reduced anti self dual yang mills equations to seven dimensions described by the nahm equations can be carried over to the weyl wigner moyal formalism in the process some new solutions for the cases of gauge groups su 2 and sl 2 r are explicitly obtained

two graded absolute valued algebras abstract we study two graded absolute valued algebras these are two graded algebras satisfying the absolute value multiplicative property only on homogeneous elements thus hexagonions also called sedenions and other sixteen dimensional algebras arise as examples of these algebras the even parts of two graded absolute valued algebras are the absolute valued algebras while the odd parts are exactly the absolute valued triple systems so in a way these two graded algebras give a unifying viewpoint of both structures we also study the simplicity and give several ways to construct two graded absolute valued algebras we also provide a description of isomorphism classes for two graded absolute valued algebras of dimensions 1 2 and 4

clifford algebra parametrized octonions and generalizations abstract introducing products between multivectors of c l 0 7 the clifford algebra over the metric vector space r 0 7 and octonions resulting in an octonion and leading to the non associative standard octonionic product in a particular case we generalize the octonionic x product associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7 sphere s 7 and the xy product this generalization is accomplished in the u and u v products where u v c l 0 7 are fixed but arbitrary moreover we extend these original products in order to encompass the most general non associative products r r 0 7 c l 0 7 r r 0 7 c l 0 7 r r 0 7 r r 0 7 and c l 0 7 c l 0 7 r r 0 7 we also present the formalism necessary to construct clifford algebra parametrized octonions which provides the structure to present the o 1 u algebra finally we introduce a method to construct o algebras endowed with the u v product from o algebras endowed with the u product these algebras are called o like algebras and their octonionic units are parametrized by arbitrary clifford multivectors when u is restricted to the underlying paravector space r r 0 7 c l 0 7 of the octonion algebra o these algebras are shown to be isomorphic the products between clifford multivectors and octonions leading to an octonion are shown to share graded associative supersymmetric properties we also investigate the generalization of moufang identities for each one of the products introduced

multiple scale analysis of delayed dynamical systems abstract the model equations of a semiconductor laser with delayed feeback are studied as a prototype of general delayed systems in the case of long delay thermodynamic limit it is shown that its behaviour in the vicinity of the hopf bifurcation can be described by suitable amplitude equations the derivation of the final model follows naturally from the space time representation of the evolution and by the corresponding re formulation of the linear stability problem regions characterized by both direct and inverse hopf bifurcations are found and in particular parameter values are identified for which a chaotic evolution is expected

bifurcation analysis on a turning system with large and state dependent time delay abstract stability and bifurcation analyses were performed in this study on the turning process with a state dependent and large time delay using the method of multiple scales mms the turning system tool was modeled as an oscillator with two degrees of freedom and both the cubic nonlinear stiffness and the nonlinear cutting force were considered the nonlinear cutting force was appropriately expanded in a taylor series considering the state dependency of the time delay the time delay and parameters were scaled through the proper ordering process to reflect the large delay effect on an asymptotic formulation of the mms asymptotic solutions were then obtained by the mms in the large delay regime and used to calculate the linear stability boundaries i e hopf bifurcation points and coexisting one period periodic solutions i e limit cycles of the turning system to investigate the local and global behaviors of the tool chatter bifurcation diagrams were obtained at various workpiece rotating speeds the validity of the results was examined by comparison with those obtained through the method of harmonic balance and direct numerical integration additionally using the bifurcation diagrams the effects of the state dependent time delay and nonlinear stiffness on the chatter vibration behaviors were examined

generalized vector products duality and octonionic identities in d 8 geometry in an explicit unified and covariant formulation we study and generalize the exceptional vector products in r8 of zvengrowski gray and kleinfeld we derive the associated general quadratic cubic and quartic g2 invariant algebraic identities and uncover an octonionic counterpart to the d 4 quaternionic duality when restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on s7 we further link up with the ogievetski tzeitlin vector product and obtain explicit tensor forms of the so 7 and g2 structure constants so 8 transformations related by triality are characterized by means of invariant tensors possible physical applications are discussed

domain wall and periodic solutions of coupled asymmetric double well models coupled asymmetric double well a 2 b 3 c 4 one dimensional potentials arise in the context of first order phase transitions both in condensed matter physics and field theory here we provide an exhaustive set of exact periodic solutions of such a coupled asymmetric model in terms of elliptic functions domain wall arrays and obtain single domain wall solutions in specific limits we also calculate the energy and interaction between solitons for various solutions both topological kinklike at t tc and nontopological pulselike for t tc domain wall solutions are obtained we relate some of these solutions to domain walls in hydrogen bonded materials and also in the field theory context as a by product we also obtain a new one parameter family of kink solutions of the uncoupled asymmetric double well model

loop algebras and superalgebras based on s 7 loop algebras of parallel transformations are defined on seven spheres these algebras can be extended with eight fermionic charges and the virasoro algebra to a soft superalgebra using the geometry of cosets of supergroups or to a nonassociative infinite dimensional n 8 d 2 superconformal algebra

joint distributions quantum correlations and commuting observables we provide necessary and sufficient conditions for several observables to have a joint distribution when applied to the bivalent observables of a quantum correlation experiment we show that these conditions are equivalent to the bell inequalities and also to the existence of deterministic hidden variables we connect the no hidden variables theorem of kochen and specker to these conditions for joint distributions we conclude with a new theorem linking joint distributions and commuting observables and show how violations of the bell inequalities correspond to violations of commutativity as in the theorem

the ten classical types of group representations each irreducible n dimensional complex representation of an arbitrary group g is shown to be associated with a unique classical group such representations are thereby classified into ten classical types associated with the respective classical groups gl n c o n c sp n c u r s gl n r gl n 2 h o r s sp n r o n 2 h sp r 2 s 2 where r s n

mermin pentagrams arising from veldkamp lines for three qubits we study the geometry of the space of mermin pentagrams objects that are used to rule out the existence of noncontextual hidden variable theories as alternatives to quantum theory it is shown that this space of 12 096 possible pentagrams is organized into 1008 families with each family containing a double six of pentagrams the 1008 families are connected to a special set of veldkamp lines in the veldkamp space for three qubits an object well known to finite geometers but has only been introduced to physics recently due to the transitive action of the symplectic group on this set of veldkamp lines it is enough to study only one canonical double six configuration of pentagrams we prove that the geometry of this double six configuration is encapsulated in the weight diagram of the 20 dimensional irreducible representation of the group as an interesting by product of our approach we show that mermin pentagrams and a class of mermin squares labelled by three qubit pauli operators are inherently related we conjecture that by studying the representation theoretic content of other veldkamp lines of the veldkamp space for n qubits makes it possible to find new contextual configurations in a systematic manner

finite lorentz transformations automorphisms and division algebras an explicit algebraic description of finite lorentz transformations of vectors in ten dimensional minkowski space is given by means of a parametrization in terms of the octonions the possible utility of these results for superstring theory is mentioned along the way automorphisms of the two highest dimensional normed division algebras namely the quaternions and the octonions are described in terms of conjugation maps similar techniques are used to define so 3 and so 7 via conjugation so 4 via symmetric multiplication and so 8 via both symmetric multiplication and one sided multiplication the noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions

exceptional lie algebras su 3 and jordan pairs part 2 zorn type representations a representation of the exceptional lie algebras reflecting a simple unifying view based on realizations in terms of zorn type matrices is presented the role of the underlying jordan pair and jordan algebra content is crucial in the development of the structure each algebra contains three jordan pairs sharing the same lie algebra of automorphisms and the same external su 3 symmetry the applications in physics are outlined

moyal nahm equations various aspects of the nahm equations in three and seven dimensions are investigated the residues of the variables at simple poles in the seven dimensional case form an algebra a large class of matrix representations of this algebra is constructed the large n limit of these equations is taken by replacing the commutators by moyal brackets and a set of nontrivial solutions in a generalized form of wigner distribution functions is obtained

coupling induced resonance in two mutually and asymmetrically coupled oscillators we consider two mutually coupled oscillators where we have independent control over the magnitude sign and delay of the coupling signal for appropriate tuning of the coupling constants there is a coupling induced resonance where the amplitude becomes large we investigate the role of nonlinear dissipation and amplitude dependent frequency correction on the coupling resonance with delayed coupling we track the deformation of the resonant bifurcation equation through imperfect bifurcations and the generation of isolas which generate intervals of multistability between oscillations of different amplitudes the resonance should be observable in coupled oscillator systems where the amplitude remains an important dynamical variable but not in limit cycle oscillators modeled by phase only descriptions

local triality and some related algebras abstract eight dimensional symmetric composition algebras give very nice formulae for triality and local triality these latter formulae will be extended to any algebra in a suitable variety the simple algebras in this variety will be shown to be precisely the symmetric composition algebras over fields while the prime algebras are central orders in the simple ones

relaxation oscillations in a class of delay differential equations we study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics under certain circumstances the model exhibits self sustained oscillations with periods which can be significantly longer than the basic cell cycle time we show that the long periods in the oscillations occur when the cell generation rate is small and we provide an asymptotic analysis of the model in this case this analysis bears a close resemblance to the analysis of relaxation oscillators such as the van der pol oscillator except that in our case the slow manifold is infinite dimensional despite this a fairly complete analysis of the problem is possible

hidden variables joint probability and the bell inequalities it is shown that the following statements about a quantum correlation experiment are mutually equivalent 1 there is a deterministic hidden variables model for the experiment 2 there is a factorizable stochastic model 3 there is one joint distribution for all observables of the experiment returning the experimental probabilities 4 there are well defined compatible joint distributions for all pairs and triples of commuting and noncommuting observables 5 the bell inequalities hold