a general theory of canonical induction formulae abstract so far there exist three independent constructions of two different canonical versions of brauer s induction theorem for complex characters due to v snaith p symonds and the author canonical in this context means functorial with respect to restrictions along group homomorphisms in this article we axiomatize the situation in which the above canonical induction formulae are constructed mackey functors and related structures arise in this way naturally as a convenient language this approach allows us to construct canonical induction formulae for arbitrary mackey functors in particular we obtain canonical induction formulae for the brauer character ring the group of projective characters the ring of trivial source modules and the ring of linear source modules in most cases it is not difficult to construct such formulae over the rational numbers a much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers we give a sufficient condition in the general framework of mackey functors for a canonical induction formula to be integral as an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings as for example the span of linear characters in the case of the canonical brauer induction formula this will be used in a subsequent article in the case of adams operations and chern classes

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

the inverse of a singular linear transformation and some applications in constrained mechanics we extend the notion of the moore penrose inverse of a matrix to the case of a morphism between vector bundles and a more general concept is used to develop the geometric theory of udwadia and kalaba s approach to constrained lagrangian mechanics

singular points of harmonic maps from 4 dimensional domains into 3 spheres in this article we show that if a stable stationary harmonic map u from a domain in r4 into s3 has an isolated singular point then the mapping degree deg u of u at is 1 or 1 furthermore the complete characterization of stable stationary tangent maps from b4 to s3 is given

propagating sharp group homology decompositions a collection c of subgroups of a finite group g can give rise to three different standard formulas for the cohomology of g in terms of either the subgroups in c or their centralizers or their normalizers we give a short but systematic study of the relationship among such formulas for nine standard collections c of p subgroups obtaining some new formulas in the process to do this we exhibit some sufficient conditions on the poset c which imply comparison results

the helmholtz conditions revisited a new approach to the inverse problem of lagrangian dynamics deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second order ordinary equations the structure of euler lagrange equations the approach is based on a generalisation of previous studies on linear systems the main result concerns a set of necessary and sufficient conditions for the existence of a multiplier which contains an infinite set of algebraic equations the coefficients of which can be used to derive necessary conditions involving only the given right hand sides of the differential equations an outline is given of interesting points for future studies and an example is presented for which all multipliers are explicitly constructed

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

first order equivalent lagrangians and conservation laws we present a theorem for first order lagrangian theories which associates several conserved quantities to one s equivalence symmetry transformation

lax formalism for a family of integrable toda related n particle systems this article can be divided in two parts first a lax representation is obtained for a family of integrable three particle toda related systems the second part contains the generalization to the case of n particles

brauer relations for finite groups in the ring of semisimplified modular representations abstract let g be a finite group and p be a prime we study the kernel of the map between the burnside ring of g and the grothendieck ring of f p g modules taking a g set to its associated permutation module we are able for all finite groups to classify the primitive quotient of the kernel that is for each g the kernel modulo elements coming from the kernel for proper subquotients of g we are able to identify exactly which groups have non trivial primitive quotient and we give generators for the primitive quotient in the soluble case

non noether constants of motion mathematical tools of modern differential geometry are used to derive in an intrinsic formulation more general results about non noether constants of motion a relation between two different ways of obtaining such constants is found by making use of leverrier s method of determining the characteristics polynomial of a matrix in terms of the traces of its increasing powers

le complexe de chaines d un g complexe simplicial acyclique resume let g be a finite group the main result of this note shows that the reduced chain complex of an acyclic finite dimensional simplicial g complex is a split acyclic complex of z g modules the proof requires the extension of well known results on finitely generated p permutation modules to arbitrary p permutation modules the key is a theorem stating that a bounded complex of p permutation modules is split acyclic if and only if all its brauer quotients are acyclic

equivalent lagrangians multidimensional case we generalize a theorem known for one dimensional nonsingular equivalent lagrangians l and to the multidimensional case in particular we prove that the matrix which relates the left hand sides of the euler lagrange equations obtained from l and is such that the trace of all its integer powers are constants of the motion we construct several multidimensional examples in which the elements of are functions of position velocity and time and prove that in some cases equivalence prevails even if det 0

lagrangians for differential equations of any order in this work the inverse problem of the variational calculus for systems of differential equations of any order is analyzed it is shown that if a lagrangian exists for a given regular system of differential equations then it can be written as a linear combination of the equations of motion the conditions that these coefficients must satisfy for the existence of an s equivalent lagrangian are also exhibited a generalization is also made of the concept of lagrangian symmetries and they are related with constants of motion

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

symmetries separability and volume forms in order to clarify certain misconceptions in the literature we discuss the details of the way determining equations for general symmetries of a lagrangian system differ from determining equations for noether symmetries and establish the minimal set of extra conditions which have to be imposed in practical situations for the former to reduce to the latter we further derive properties by which in situations where the system is integrable through quadratic integrals in involution the components of the corresponding noether symmetries themselves can be used to compute the separation variables if they exist for the hamilton jacobi equation

alternative lagrangians for spherically symmetric potentials a new geometrical account of the existence of alternative s equivalent lagrangians is given for a particle moving in three dimensional euclidean space under the influence of a central force the fact that the lagrangian for this familiar problem is not unique was pointed out by henneaux and shepley j math phys 23 2101 1982 the present analysis makes the reasons for the phenomenon more transparent than theirs

the action principle for a system of differential equations we consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with euler lagrange equations for an action principle two ways of constructing the action principle are presented from simple consideration we derive the necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second order differential equations with the structure of the euler lagrange equations an explicit form of the action is constructed if such a multiplier exists if a given set of differential equations cannot be derived from an action principle one can reformulate such a set in an equivalent first order form which can always be treated as the euler lagrange equations of a certain action we construct such an action explicitly there exists an ambiguity not reduced to a total time derivative in associating a lagrange function with a given set of equations we present a complete description of this ambiguity the general procedure is illustrated by several examples

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

some recent results on symmetries of lagrangian systems re examined hojman s results 1984 on alternative lagrangians and first integrals related to symmetries of lagrangian systems are shown to be covered by previous treatments in a more geometrical setting it is argued that the use of lagrangians linear in the accelerations for second order systems does not widen the spectrum of results

density of lipschitz mappings in the class of sobolev mappings between metric spaces we prove that lipschitz mappings are dense in the newtonian sobolev classes n1 p x y of mappings from spaces x supporting p poincare inequalities into a finite lipschitz polyhedron y if and only if y is p connected 1 y 2 y p y 0 where p is the largest integer less than or equal to p

a separability theorem for dynamical systems admitting alternative lagrangian descriptions by generalising earlier results the authors prove that whenever a lagrangian dynamical system can be associated with a 1 1 type tensor field which is left invariant by the dynamics satisfies the nijenhuis condition and is compatible with the tangent bundle structure the system necessarily decomposes into a collection of lower dimensional mutually non interacting lagrangian subsystems

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

removability of singular sets of harmonic maps it is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density moreover the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability this work generalizes the results in cl df lg12 which were proved under the assumption that the singular sets are isolated points or smooth submanifolds

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

geometric theory of the equivalence of lagrangians for constrained systems the lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular lagrangians generalising that of the regular case geometric and gauge equivalence of singular lagrangians are studied and the authors also give some conditions in which second order differential equations exist satisfying the dynamical equation on the final constraint submanifold

some remarks on the generalized noether theory of point symmetry transformations of the lagrangian the noether theory of infinitesimal point symmetry transformations is revisited and generalized under the broad scope of the general theory of transformations of lagrangian mechanics the basic generalized point symmetries of a lagrangian function are thus obtained and briefly discussed it is proved that under very general and physically reasonable provisions the point symmetry group of the lagrangian is necessarily a finite lie group and a rather simple technique is then introduced for the explicit calculation of the associated lie algebra next the algebra obeyed by the set of basic noether quantities which also includes the traditional noether constants of motion is examined in this way it is shown that not all the generalized point symmetries of a lagrangian yield an associated conservation law the author presents three simple examples of this not so well known aspect of the noether theory

new classes of conversed quantities associated with non noether symmetries a result of hojman and harleston 1981 has been used to extend the known relationship between conserved quantities and non noether symmetries it is shown that n constants of the motion may be derived from each non noether symmetry of an n dimensional lagrangian system this applies whether the symmetry is continuous or discrete in addition the properties of the cartan form are utilised to to give a new proof of the hojman harleston theorem

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

hecke algebras and cohomotopical mackey functors in this paper we define the concept of the cohomotopical mackey functor which is more general than the usual cohomological mackey functor and show that hecke algebra techniques are applicable to cohomotopical mackey functors our theory is valid for any possibly infinite discrete group some applications to topology are also given

blow up regularization of singular lagrangians we propose a method for regularizing singular lagrangians by adding new degrees of freedom we illustrate this regularization method with some particular examples this procedure is shown to be very useful when the lagrangian is homogeneous of degree one in the velocities giving rise to an identically vanishing hamiltonian

equivalent lagrangians in field theory a theorem by hojman and harleston 1981 for multidimensional equivalent lagrangians l and l in the realm of newtonian mechanics is generalised to field theory equivalent lagrangians for some set of field equations are found

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

a note on green functors with inflation abstract this note is motivated by the problem to understand given a commutative ring f which g sets x y give rise to isomorphic f g representations f x f y a typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of g in the present paper we axiomatise the situation and prove such a result in the generality of mackey functors and green functors with inflation our result includes as special cases a result of deligne on monomial relations a result of the first author and tim dokchitser on brauer relations in characteristic 0 and a new result on brauer relations in characteristic p 0 we will need the new result in a forthcoming paper on brauer relations in positive characteristic

deformations of annuli with smallest mean distortion we determine the extremal mappings with smallest mean distortion for mappings of annuli as a corollary we find that the nitsche harmonic maps are dirichlet energy minimizers among all homeomorphisms h mathbb a r r to mathbb a r r however outside the nitsche range of the modulus of the annuli within the class of homeomorphisms no such energy minimizers exist in this case we identify the bv limits of minimizers

the nitsche conjecture the nitsche conjecture is deeply rooted in the theory of doubly connected minimal surfaces however it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli h a a r r onto a r r a in the early 1960s while attempting to describe all doubly connected minimal graphs over a given annulus a j c c nitsche observed that their conformal modulus cannot be too large then he conjectured in terms of isothermal coordinates even more a harmonic homeomorphism h a onto a exists if an only if r r 1 2 r r r r this fascinating and engaging problem remained open for almost a half of a century in the present paper we provide among further generalizations an affirmative answer to his conjecture

note on equivalent lagrangians and symmetries shows that every ordered pair of equivalent lagrangians determines an equivalence class of dynamical symmetries of non noether type its explicit construction merely requires a particular solution of a single partial differential equations various related aspects are discussed

symmetries of lagrangians and of their equations of motion a new kind of lagrangian symmetry is defined in such a way that the resulting set of lagrangian symmetries coincides with the set of symmetries of its equations of motion several constants of motion may be associated to each of the new symmetry transformations one example is presented

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

symmetries of space time and geodesic symmetries it is shown how to construct symmetries of the geodesic equation starting from space time symmetries the constants of motion associated with space time symmetries are recovered and a few new ones are found using non noetherian conservation theorems explicit examples are presented

another demonstration of the theorem by hojman and harleston another demonstration of a theorem on the calculus of variations derived recently by hojman and harleston 1981 using helmholtz conditions is presented

quantization of non standard hamiltonian systems the quantization of classical theories that admit more than one hamiltonian description is considered this is done from a geometrical viewpoint both at the quantization level geometric quantization and at the level of the dynamics of the quantum theory a spin system is taken as an example in which all the steps can be completed it is shown that the geometry of the quantum theory imposes restrictions on the physically allowed non standard quantum theories

neohookean deformations of annuli existence uniqueness and radial symmetry let mathbb x x in mathbb r 2 r x r and mathbb y y in mathbb r 2 r ast y r ast be annuli in the plane we consider the class fancyscript f mathbb x mathbb y of all orientation preserving homeomorphisms h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev space fancyscript w 1 2 mathbb x mathbb y which keep the boundary circles in the same order this means that lim x searrow r h x r ast and lim x nearrow r h x r ast we study the neohookean energy integral mathcal e h int limits mathbb x dh 2 phi rm det dh quad rm for h in fancyscript f mathbb x mathbb y quad quad quad 1 where phi in fancyscript c infty 0 infty is positive and strictly convex we assume in addition that the function psi z 1 ddot phi z and its derivative extend continuously to 0 with 0 0 then we prove

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

equivalence and s equivalence of vector tensor lagrangians it will be proven that if a gauge invariant lagrangian density having the local form l l gij ai ai j is such that its euler lagrange equations ei l 0 have the same set of solutions as ei l0 0 where l0 g1 2fijfij then l and cl0 are equivalent for same constant c i e ei l ei cl0 from a previous result it follows that l cl0 d eg1 2 where d is a divergence and e is a constant

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

algebraicisation of explicit brauer induction this paper deals with canonical forms for induction theorems in the complex finite dimensional representation theory of a finite group g the first example of an explicit form for an induction theorem is due to r brauer br who in 1951 gave a canonical form for artin s induction theorem in 1946 r brauer had also proved that each virtual character of g may be expressed as an integral linear combination of characters which are induced from linear i e one dimensional characters of subgroups of g one may reformulate brauer s induction theorem in the following manner define r g to be the free abelian group on the g conjugacy classes h 5 s of linear characters of subgroups define 6 r g r g by b h s indz rp brauer s theorem states that 6 is onto by an explicit brauer induction formula we will mean a section for the map 6 given such a section one may use it to obtain a presentation for r g in terms of generators h s c this problem is mentioned in ser p 71 footnote in 1986 snaith produced such a section t by means of the algebraic topology of smooth group actions related to g representations details appear in sl see also s2 s3 s4 in his thesis boltje algebraically constructed another such section a in fact it is possible to give a topological construction of a similar to that of sl for t this construction is given in 2 26 see theorem 2 27 in this paper we derive algebraic formulae for t and its topological ancestor ro in a manner similar to that used for a in bl these results appear in theorems 3 23 and 3 29 in section 4 we derive a formula which relates ac and t directly rather than by means of the algebraic formulae of section 3 the prerequisites for the understanding of a t and l the explicit

an induction principle for the weighted p energy minimality of x x abstract in this paper we investigate minimizing properties of the map x x from the euclidean unit ball b n to its boundary s n 1 for the weighted energy functionals e p n u b n x u p d x where p 2 we establish the following induction principle if the map x x b n 1 s n minimizes e p n 1 among maps u b n 1 s n satisfying u x x on s n then the map y y b n s n 1 minimizes e p 1 n among maps v b n s n 1 satisfying v y y on s n 1 this result enables us to enlarge the range of values of p and for which x x minimizes e p n

the helmholtz conditions in terms of constants of motion in classical mechanics the helmholtz conditions are the necessary and sufficient conditions for a set of second order differential equations to be equivalent to a variational principle in this work an alternative approach to the inverse problem in classical mechanics is described it is proven that the helmholtz conditions can be transformed into a set of conditions for a nonsingular antisymmetric matrix whose entries are constants of motion of the problem in question

regularity of the inverse of a planar sobolev homeomorphism let be a domain suppose that f w1 1loc r2 is a homeomorphism such that df x vanishes almost everywhere in the zero set of j f we show that f 1 w1 1loc f r2 and that df 1 y vanishes almost everywhere in the zero set of sharp conditions to quarantee that f 1 w1 q f r2 for some 1 q 2 are also given

on the inverse problem of the calculus of variations in field theory the inverse problem of the calculus of variations is investigated in the case of field theory uniqueness of the action principle is demonstrated for the vector laplace equation in a non decomposable riemannian space as well as for the harmonic map equation

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

stability and singularities of harmonic maps into 3 spheres the behavior of harmonic maps from a domain in r4 into s3 is discussed in this paper we show that if u is a strictly stable stationary harmonic map b4 s3 such that the singular set sing u of u consists of 0 then the degree deg u 0 of u at 0 is zero and that if u is a weakly stable harmonic map b4 s3 with sing u 0 then deg u 0 0 or 1 furthermore the characterization of homogeneous stable stationary harmonic maps from b4 into s3 is given

on the inverse problem of the calculus of variations we consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first order form we provide a way of constructing infinitely many lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach we present examples of first order lagrangians for systems for which no second order lagrangians exist the hamiltonian theory for first order degenerate lagrangians is constructed using dirac s method for singular lagrangians

construction of alternative hamiltonian structures for field equations we use symmetry vectors of nonlinear field equations to build alternative hamiltonian structures we construct such structures even for equations which are usually believed to be non hamiltonian such as heat burger and potential burger equations we improve on a previous version of the approach using recursion operators to increase the rank of the poisson bracket matrices cole hopf and miura type transformations allow the mapping of these structures from one equation to another

categorifying induction formulae via divergent series abstract we show how to get explicit induction formulae for finite group representations and more generally for rational green functors by summing a divergent series over dwyer s subgroup and centralizer decomposition spaces this results in formulae with rational coefficients the former space yields a well known induction formula the latter yields a new one as essentially immediate corollaries of the existing literature we get similar formulae in group cohomology and stable splittings of classifying spaces

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

higher order noether symmetries and constants of the motion the authors identify a class of symmetries for lagrangian systems called higher order noether symmetries which yield a corresponding first integral without further integrations first a version of noether s theorem is recalled in which noether symmetries are considered to be symmetries of the two form d theta theta being the cartan form nth order noether symmetries are then defined by the requirement lynd theta 0 the picture is further generalised by exploring various ways in which the computation of repeated lie derivatives of d theta can lead to the identification of first integrals a number of illustrative examples are discussed

generalized lie symmetries and complete integrability of certain nonlinear hamiltonian systems with three degrees of freedom a systematic method for investigating the existence of nontrivial generalized lie symmetries is presented and the associated integrals of motion for nonlinear oscillator systems with three degrees of freedom defined in terms of the lagrangian by l 1 2 x2 y2 z2 v x y z are constructed then the method is applied to study the integrability properties of quartically and cubically coupled nonlinear oscillators with three degrees of freedom compatibility with the painleve property is also investigated

new derivation of a conserved quantity for lagrangian systems a new derivation of a recently discussed conserved quantity is given using a general procedure which determines n constants of the motion for any n dimensional system possessing a non noether symmetry

a class of recursion operators on a tangent bundle we generalize the construction of a class of type 1 1 tensor fields r on a tangent bundle which was introduced in a preceding paper the generalization comes from the fact that apart from a given lagrangian the further data consist of a type 1 1 tensor j along the tangent bundle projection tq q rather than a tensor on q the main features under investigation are two kinds of recursion properties of r namely its potential invariance under the flow of the given dynamics and the property of having vanishing nijenhuis torsion the theory is applied in particular to the case of second order dynamics coming from a finsler metric

sl 3 r as the group of symmetry transformations for all one dimensional linear systems iii equivalent lagrangian formalisms the sl 3 r theory of projective transformations of the plane is applied to the lagrangians of all one dimensional newtonian linear systems noether and non noether equivalent lagrangians as well as the associated noether and non noether constants of motion are thus obtained in a completely general and systematic way complete unification is achieved by this group theoretic approach to lagrangians of one dimensional linear systems

canonoid transformations from a geometric perspective the concept of canonoid transformation for a locally hamiltonian vector field is introduced and its relation with the existence of non noether constants of the motion is shown from a geometrical viewpoint the equations determining generating functions for such canonoid transformations are obtained and applications to some particular problems given

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians