an adaptive conservative 0d 2v multispecies rosenbluth fokker planck solver for arbitrarily disparate mass and temperature regimes in this study we propose an adaptive velocity space discretization scheme for the multi species multidimensional rosenbluth fokker planck rfp equation which is exactly mass momentum and energy conserving unlike most earlier studies our approach normalizes the velocity space coordinate to the temporally varying individual plasma species local thermal velocity v t h t and explicitly considers the resulting inertial terms in the fokker planck equation our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the fokker planck collision operator to deal with situations of extreme thermal velocity disparities among different species we employ an asymptotic v t h ratio based expansion of the rosenbluth potentials that only requires the computation of several velocity space integrals numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme in particular we show that the combined use of the velocity grid adaptivity and asymptotic expansions delivers many orders of magnitude savings in mesh resolution requirements compared to a single static uniform mesh

dressing cmc n noids the purpose of this paper is to construct cmc n noids with bubbletons we recall a specific class of dressing matrices which we will refer to as simple factor dressing matrices which in many known cases add bubbletons to a cmc surface our new surfaces are obtained by dressing known n noids with embedded delaunay ends by well chosen simple factor dressing matrices in such a way that the dressed surface is also a cmc n noid

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

the eckmann hilton argument and higher operads abstract the classical eckmann hilton argument shows that two monoid structures on a set such that one is a homomorphism for the other coincide and moreover the resulting monoid is commutative this argument immediately gives a proof of the commutativity of the higher homotopy groups a reformulation of this argument in the language of higher categories is suppose we have a one object one arrow 2 category then its hom set is a commutative monoid a similar argument due to a joyal and r street shows that a one object one arrow tricategory is the same as a braided monoidal category in this paper we begin to investigate how one can extend this argument to arbitrary dimension we provide a simple categorical scheme which allows us to formalise the eckmann hilton type argument in terms of the calculation of left kan extensions in an appropriate 2 category then we apply this scheme to the case of n operads in the author s sense and classical symmetric operads we demonstrate that there exists a functor of symmetrisation sym n from a certain subcategory of n operads to the category of symmetric operads such that the category of one object one arrow one n 1 arrow algebras of a is isomorphic to the category of algebras of sym n a under some mild conditions we present an explicit formula for sym n a which involves taking the colimit over a remarkable categorical symmetric operad we will consider some applications of the methods developed to the theory of n fold loop spaces in the second paper of this series

backlund transformations ward solitons and unitons the ward equation also called the modified 2 1 chiral model is obtained by a dimension reduction and a gauge fixing from the self dual yang mills field equation on r it has a lax pair and is an integrable system ward constructed solitons whose extended solutions have distinct simple poles he also used a limiting method to construct 2 solitons whose extended solutions have a double pole ioannidou and zakrzewski and anand constructed more soliton solutions whose extended solutions have a double or triple pole some of the main results of this paper are i we construct algebraic backlund transformations bts that generate new solutions of the ward equation from a given one by an algebraic method ii we use an order k limiting method and algebraic bts to construct explicit ward solitons whose extended solutions have arbitrary poles and multiplicities iii we prove that our construction gives all solitons of the ward equation explicitly and the entries of ward solitons must be rational functions in x y and t iv since stationary ward solitons are unitons our method also gives an explicit construction of all k unitons from finitely many rational maps from c to c

a cellular nerve for higher categories abstract we realise joyal cell category as a dense subcategory of the category of categories the associated cellular nerve of an category extends the well known simplicial nerve of a small category cellular sets like simplicial sets carry a closed model structure in quillen s sense with weak equivalences induced by a geometric realisation functor more generally there exists a dense subcategory a of the category of a algebras for each operad a in batanin s sense whenever a is contractible the resulting homotopy category of a algebras i e weak categories is equivalent to the homotopy category of compactly generated spaces

multi soliton multi breather and higher order rogue wave solutions to the complex short pulse equation abstract in the present paper we are concerned with the general analytic solutions to the complex short pulse csp equation including soliton breather and rogue wave solutions with the aid of a generalized darboux transformation we construct the n bright soliton solution in a compact determinant form the n breather solution including the akhmediev breather and a general higher order rogue wave solution the first and second order rogue wave solutions are given explicitly and analyzed the asymptotic analysis is performed rigorously for both the n soliton and the n breather solutions all three forms of the analytical solutions admit either smoothed cusped or looped type ones for the csp equation depending on the parameters it is noted that due to the reciprocal hodograph transformation the rogue wave solution to the csp equation can be a smoothed cusponed or a looped one which is different from the rogue wave solution found so far

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

flows of constant mean curvature tori in the 3 sphere the equivariant case we present a deformation for constant mean curvature tori in the 3 sphere we show that the moduli space of equivariant constant mean curvature tori in the 3 sphere is connected and we classify the minimal the embedded and the alexandrov embedded tori therein

on autoequivalences of the infty 1 category of infty operads we study the infty 1 category of autoequivalences of infty operads using techniques introduced by toen lurie and barwick and schommer pries we prove that this infty 1 category is a contractible infty groupoid our calculation is based on the model of complete dendroidal segal spaces introduced by cisinski and moerdijk similarly we prove that the infty 1 category of autoequivalences of non symmetric infty operads is the discrete monoidal category associated to mathbf z 2 mathbf z we also include a computation of the infty 1 category of autoequivalences of infty n categories based on rezk s theta n spaces

monoidal globular categories as a natural environment for the theory of weakn categories we present a definition of weak categories based on a higher order generalization of apparatus of operads

simplicial structures on model categories and functors we produce a highly structured way of associating a simplicial category to a model category which improves on work of dwyer and kan and answers a question of hovey we show that model categories satisfying a certain axiom are quillen equivalent to simplicial model categories a simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits we also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial or continuous functors this is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial quillen equivalence

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

a uniqueness theorem for stable homotopy theory roughly speaking the stable homotopy category of algebraic topology is obtained from the homotopy category of topological spaces by inverting the suspension functor yielding a linear approximation to the homotopy category of spaces the isomorphism classes of objects in the stable homotopy category represent the generalized cohomology theories defined by the eilenberg steenrod axioms es45 without the dimension axiom which distinguishes ordinary from generalized cohomology theories the first construction of the full stable homotopy category was given by boardman b69 nowadays manymodels for the stable homotopy category are known most of which have the additional structure of a closed model category in the sense of quillen q67 in m83 h r margolis introduced a short list of axioms and conjectured that they characterize the stable homotopy category up to an equivalence of categories in theorem 3 2 we prove that these axioms do uniquely specify the stable homotopy category whenever there is some underlying quillen model category we also prove amore structured version theuniqueness theorembelow which states that the model category of spectra itself is uniquely determined by certain equivalent conditions up to so called quillen equivalence of

harmonic maps backlund darboux transformations and line solution analogues harmonic maps from or one connected domain into and u m are treated the gbdt version of the b cklund darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given using some methods from system theory a new general formula on the gbdt transformations of the sym tafel immersions is derived a new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions similar in certain ways to line solutions is obtained explicitly and studied

an efficient explicit numerical scheme for diffusion type equations with a highly inhomogeneous and highly anisotropic diffusion tensor abstract a locally split step explicit lsse algorithm was developed for efficiently solving a multi dimensional advection diffusion type equation involving a highly inhomogeneous and highly anisotropic diffusion tensor which makes the problem very ill conditioned for standard implicit methods involving the iterative solution of large linear systems the need for such an optimized algorithm arises in particular in the frame of thermonuclear fusion applications for the purpose of simulating fast charged particle slowing down with an ion fokker planck code the lsse algorithm is presented in this paper along with the results of a model slowing down problem to which it has been applied

inner horns for 2 quasi categories abstract dimitri ara s 2 quasi categories which are certain presheaves over andre joyal s 2 cell category 2 are an example of a concrete model that realises the abstract notion of 2 category in this paper we prove that the 2 quasi categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by david oury these maps are more tractable than the maps that ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2 quasi categories

topological hypercovers and 1 realizations we show that if u is a hypercover of a topological space x then the natural map hocolim u x is a weak equivalence this fact is used to construct topological realization functors for the 1 homotopy theory of schemes over real and complex fields in an appendix we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant

iterated wreath product of the simplex category and iterated loop spaces generalising segal s approach to 1 fold loop spaces the homotopy theory of n fold loop spaces is shown to be equivalent to the homotopy theory of reduced tn spaces where tn is an iterated wreath product of the simplex category a sequence of functors from tn to g allows for an alternative description of the segal spectrum associated to a g space this yields a canonical reduced tn set model for each eilenberg maclane space

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

isothermic submanifolds of symmetric r spaces we extend the classical theory of isothermic surfaces in conformal 3 space due to bour christoffel darboux bianchi and others to the more general context of submanifolds of symmetric r spaces with essentially no loss of integrable structure

hamiltonian evolution of curves in classical affine geometries abstract in this paper we study geometric poisson brackets and we show that if m g r n g endowed with an affine geometry in the klein sense and if g is a classical lie group then the geometric poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves except for the case g gl n r this trivial extension does not exist in other nonaffine cases projective conformal etc

a homotopy coherent cellular nerve for bicategories abstract the subject of this paper is a nerve construction for bicategories introduced by leinster which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over joyal s category 2 we prove that the nerve of a bicategory is a 2 quasi category a model for 2 categories due to ara and moreover that the nerve functor restricts to the right part of a quillen equivalence between lack s model structure for bicategories and a bousfield localisation of ara s model structure for 2 quasi categories we deduce that lack s model structure for bicategories is quillen equivalent to rezk s model structure for 2 2 spaces on the category of simplicial presheaves over 2 to this end we construct the homotopy bicategory of a 2 quasi category and prove that a morphism of 2 quasi categories is an equivalence if and only if it is essentially surjective on objects and fully faithful we also prove a quillen equivalence between ara s model structure for 2 quasi categories and the hirschowitz simpson pellissier model structure for quasi category enriched segal categories from which we deduce a few more results about 2 quasi categories including a conjecture of ara concerning weak equivalences of 2 categories

rigidity and exotic models for v1 local g equivariant stable homotopy we prove that the v 1 local g equivariant stable homotopy category for g a finite group has a unique g equivariant model at p 2 this means that at the prime 2 the homotopy theory of g spectra up to fixed point equivalences on k theory is uniquely determined by its triangulated homotopy category and basic mackey structure the result combines the rigidity result for k local spectra of the second author with the equivariant rigidity result for g spectra of the first author further when the prime p is at least 5 and does not divide the order of g we provide an algebraic exotic model as well as a g equivariant exotic model for the v 1 local g equivariant stable homotopy category showing that for primes p ge 5 equivariant rigidity fails in general

reedy categories and the vartheta construction we use the notion of multi reedy category to prove that if mathcal c is a reedy category then vartheta mathcal c is also a reedy category this result gives a new proof that the categories vartheta n are reedy categories we then define elegant reedy categories for which we prove that the reedy and injective model structures coincide

on an extension of the notion of reedy category we extend the classical notion of a reedy category so as to allow non trivial automorphisms our extension includes many important examples occurring in topology such as segal s category or the total category of a crossed simplicial group such as connes cyclic category for any generalized reedy category mathbb r and any cofibrantly generated model category mathcal e the functor category mathcal e mathbb r is shown to carry a canonical model structure of reedy type

simple determinant representation for rogue waves of the nonlinear schrodinger equation we present a simple representation for arbitrary order rogue wave solution and a study on the trajectories of them explicitly we find that the trajectories of two valleys on whole temporal spatial distribution all look x shaped for rogue waves additionally we present different types of high order rogue wave structures which could be helpful towards realizing the complex dynamics of rogue waves

the vortex filament in the minkowski 3 space and generalized bi schrodinger maps by extending the concept of generalized bi schrodinger maps to the case that the target manifold is a para kahler manifold we show that the third order timelike and spacelike correction models of the vortex filament in the minkowski 3 space are equivalent to the generalized bi schrodinger maps from to the hyperbolic 2 space and the de sitter 2 space respectively as a consequence all three typical second to fourth order integrable systems of the akns hierarchy are interpreted in a unified way in terms of generalized bi schrodinger maps based on this exploitation and a general discussion of the generalized bi schodinger maps from into an arbitrary riemannian surface we reveal a property of generalized bi schrodinger maps that is not admitted for schrodinger maps

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

on the modified constrained kadomtsev petviashvili equations three reduction sub hierarchies in constrained kp hierarchies are studied the corresponding matrix integrable systems are presented furthermore backlund transformations bi hamiltonian structures and functions are formulated

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

homotopical and operator algebraic twisted k theory using the framework for multiplicative parametrized homotopy theory introduced in joint work with c schlichtkrull we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted k theory both in the real and complex case we also improve several comparison results about twisted k theory of c algebras to include multiplicative structures our results can also be interpreted in the infty categorical setup for parametrized spectra

combinatorial model categories have presentations abstract we show that every combinatorial model category is quillen equivalent to a localization of a diagram category where diagram category means diagrams of simplicial sets this says that every combinatorial model category can be built from a category of generators and a set of relations

modules over motivic cohomology abstract for fields of characteristic zero we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to voevodsky s big category of motives the proof makes use of some highly structured models for motivic stable homotopy theory motivic spanier whitehead duality the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper working with rational coefficients we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory

universal homotopy theories abstract begin with a small category c the goal of this short note is to point out that there is such a thing as a universal model category built from c we describe applications of this to the study of homotopy colimits the dwyer kan theory of framings sheaf theory and the homotopy theory of schemes

a discrete generalized coupled dispersionless integrable system and its multisoliton solutions abstract an integrable discretization of generalized coupled dispersionless dgcd integrable system via lax pair is presented a lax pair for the dgcd system is defined a darboux transformation is used on the lax pair to obtain multi soliton solutions of the dgcd system the solutions are expressed in terms of quasideterminants explicit expressions of discrete one and two soliton solutions are obtained for the s u 2 case by using properties of quasideterminants we also study continuous analogue of the dgcd system by applying continuum limit

breathers as homoclinic geometric wave maps a classical breather solution of the sine gordon equation can be viewed as a homoclinic wave map from periodic minkowski space mp2 into s2 this wave map is an unstable orbit of a simple steady state wave map homoclinic wave maps are also constructed into non round spheres

generalized weierstrass representations of surfaces with the constant gauss curvature in pseudo riemannian three dimensional space forms we know that there exists the correspondence between one of the nonumbilical riemannian surface with constant gauss curvature k in m 3 k k k the nonumbilical spacelike or timelike surfaces with constant gauss curvature k in m 13 k k k and a solution of one of the following differential equations the sine gordon sinh gordon sine laplace sinh laplace and cosh gordon equations in this paper we consider the initial value problems of these equations and give the generalized weierstrass representations of these surfaces that depend only on the initial values of these equations

monads and theories abstract given a locally presentable enriched category e together with a small dense full subcategory a of arities we study the relationship between monads on e and identity on objects functors out of a which we call a pretheories we show that the natural constructions relating these two kinds of structure form an adjoint pair the fixpoints of the adjunction are characterised on the one side as the a nervous monads those for which the conclusions of weber s nerve theorem hold and on the other as the a theories which we introduce here the resulting equivalence between a nervous monads and a theories is best possible in a precise sense and extends almost all previously known monad theory correspondences it also establishes some completely new correspondences including one which captures the globular theories defining grothendieck weak groupoids besides establishing our general correspondence and illustrating its reach we study good properties of a nervous monads and a theories that allow us to recognise and construct them with ease we also compare them with the monads with arities and theories with arities introduced and studied by berger mellies and weber

decomposition spaces incidence algebras and mobius inversion i basic theory abstract this is the first in a series of papers devoted to the theory of decomposition spaces a general framework for incidence algebras and mobius inversion where algebraic identities are realised by taking homotopy cardinality of equivalences of groupoids a decomposition space is a simplicial groupoid satisfying an exactness condition weaker than the segal condition expressed in terms of active and inert maps in just as the segal condition expresses composition the new exactness condition expresses decomposition and there is an abundance of examples in combinatorics after establishing some basic properties of decomposition spaces the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra spanned by the space of 1 simplices and with coefficients in groupoids we take a functorial viewpoint throughout emphasising conservative ulf functors these induce coalgebra homomorphisms reduction procedures in the classical theory of incidence coalgebras are examples of this notion and many are examples of decalage of decomposition spaces an interesting class of examples of decomposition spaces beyond segal spaces is provided by hall algebras the waldhausen s construction of an abelian or stable infinity category is shown to be a decomposition space in the second paper in this series we impose further conditions on decomposition spaces to obtain a general mobius inversion principle and to ensure that the various constructions and results admit a homotopy cardinality in the third paper we show that the lawvere menni hopf algebra of mobius intervals is the homotopy cardinality of a certain universal decomposition space two further sequel papers deal with numerous examples from combinatorics note the notion of decomposition space was arrived at independently by dyckerhoff and kapranov 17 who call them unital 2 segal spaces our theory is quite orthogonal to theirs the definitions are different in spirit and appearance and the theories differ in terms of motivation examples and directions

gauge theory of ba cklund transformations ii abstract a class of gauge transformations is constructed for hamiltonian hierarchies of completely integrable systems on semi simple lie algebras these transformations create or annihilate poles in the scattering data hence create or annihilate solitons in the potential q in that sense they generalize backlund transformations to which they reduce precisely for the 2 2 akns systems to each backlund gauge transformation is associated an automorphism in the weyl group of the lie algebra two proofs are given to show that these backlund gauge transformations preserve the nonlinear evolution equation one proof uses the transformation of the scattering data under the backlund transformation the second proof is purely gauge theoretic

tzitzeica transformation is a dressing action we classify the simplest rational elements in a twisted loop group and prove that dressing actions of them on proper indefinite affine spheres give the classical tzitzeica transformation and its dual we also give the group point of view of the permutability theorem construct complex tzitzeica transformations and discuss the group structure for these transformations

generators for rational loop groups uhlenbeck proved that a set of simple elements generates the group of rational loops in gl n that satisfy the u n reality condition for an arbitrary complex reductive group a choice of representation defines a notion of rationality and enables us to write a natural set of simple elements using these simple elements we prove generator theorems for the fundamental representations of the remaining neo classical groups and most of their symmetric spaces we also obtain explicit dressing and permutability formulae

a twisted integrable hierarchy with d2 symmetry a loop algebra approach to the gerdjikov mikhailov valchev gmv equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered using the twisted loop algebra structure we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the gmv equation

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

complete integrability of shock clustering and burgers turbulence we consider scalar conservation laws with convex flux and random initial data the hopf lax formula induces a deterministic evolution of the law of the initial data in a recent article we derived a kinetic theory and lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative markov process here we show that i the lax equations are hamiltonian and describe a principle of least action on the markov group ii the lax equations are completely integrable and linearized via a loop group factorization of operators iii the associated zero curvature equations can be solved via inverse scattering our results are rigorous for n dimensional approximations of the lax equations and yield formulas for the limit n the main observation is that the lax equations and zero curvature equations are a markovian analog of known integrable systems geodesic flow on lie groups and the n wave model respectively this allows us to introduce a variety of methods from the theory of integrable systems

the cauchy problem of the ward equation with mixed scattering data we solve the cauchy problem of the ward equation with both continuous and discrete scattering data

flat time like submanifolds in s2n 12q 1 in this paper it is shown that the gq qn q n q ii system gives the gauss codazzi ricci equations of a class of flat time like n submanifolds with index q in s2n 12q 1 where gq qn q n q o 2n 2q 2q o n q q o n q q and 0 q n moreover we construct a dressing action on the gq qn q n q system space of solutions of the gq qn q n q system which gives rise to backlund transformations for flat time like n submanifolds with index q in s2n 12q 1

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

diagram spaces and symmetric spectra abstract we present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra the main motivating examples are the i spaces which are diagrams indexed by finite sets and injections and j spaces which are diagrams indexed by quillen s localization construction 1 on the category of finite sets and bijections we show that the category of i spaces provides a convenient model for the homotopy category of spaces in which every e space can be rectified to a strictly commutative monoid similarly the commutative monoids in the category of j spaces model graded e spaces using the theory of j spaces we introduce the graded units of a symmetric ring spectrum the graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures

singular chains on topological stacks i abstract we extend the functor sing of singular chains to the category of topological stacks and establish its main properties we prove that sing respects weak equivalences and takes a morphism of topological stacks that is both a serre and a reedy fibration to a kan fibration of simplicial sets when restricted to the category of topological spaces sing coincides with the usual singular functor

orbispaces orthogonal spaces and the universal compact lie group this paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory at the same time we provide a new perspective by interpreting it as the homotopy theory of spaces with an action of the universal compact lie group the upshot is a novel way to construct and study genuine cohomology theories on stacks orbifolds and orbispaces defined from stable global homotopy types represented by orthogonal spectra the universal compact lie group which is neither compact nor a lie group is a well known object namely the topological monoid mathcal l of linear isometric self embeddings of mathbb r infty the underlying space of mathcal l is contractible and the homotopy theory of mathcal l spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces however the monoid mathcal l contains copies of all compact lie groups in a specific way and we define global equivalences of mathcal l spaces by testing on corresponding fixed points we establish a global model structure on the category of mathcal l spaces and prove it to be quillen equivalent to the global model category of orthogonal spaces and to the category of orbispaces i e presheaves of spaces on the global orbit category

left fibrations and homotopy colimits for a small category mathbf a we prove that the homotopy colimit functor from the category of simplicial diagrams on mathbf a to the category of simplicial sets over the nerve of mathbf a establishes a left quillen equivalence between the projective or reedy model structure on the former category and the covariant model structure on the latter we compare this equivalence to a quillen equivalence in the opposite direction previously established by lurie from our results we deduce that a categorical equivalence of simplicial sets induces a quillen equivalence on the corresponding over categories equipped with the covariant model structures also we show that a version of quillen s theorem a for infty categories easily follows

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

an adaptive implicit conservative 1d 2v multi species vlasov fokker planck multi scale solver in planar geometry abstract we consider a 1d 2v vlasov fokker planck multi species ionic description coupled to fluid electrons we address temporal stiffness with implicit time stepping suitably preconditioned to address temperature disparity in time and space we extend the conservative adaptive velocity space discretization scheme proposed in taitano et al j comput phys 318 391 420 2016 to a spatially inhomogeneous system in this approach we normalize the velocity space coordinate to a temporally and spatially varying local characteristic speed per species we explicitly consider the resulting inertial terms in the vlasov equation and derive a discrete formulation that conserves mass momentum and energy up to a prescribed nonlinear tolerance upon convergence our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the vlasov operator and the fokker planck collision operator numerical examples of varying degrees of complexity including shock wave propagation demonstrate the favorable efficiency and accuracy properties of the scheme

a spectral curve approach to lawson symmetric cmc surfaces of genus 2 minimal and cmc surfaces in s 3 can be treated via their associated family of flat sl 2 mathbb c connections in this the paper we parametrize the moduli space of flat sl 2 mathbb c connections on the lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of lawson s geometric construction the parametrization uses hitchin s abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1 dimensional torus this description is used to develop a spectral curve theory for the lawson surface this theory applies as well to other cmc and minimal surfaces with the same holomorphic symmetries as the lawson surface but different riemann surface structure additionally we study the space of isospectral deformations of compact minimal surface of genus g ge 2 and prove that it is generated by simple factor dressing