fixed points of uniformly lipschitzian mappings abstract two fixed point theorems for uniformly lipschitzian mappings in metric spaces due respectively to e lifsic and to t c lim and h k xu are compared within the framework of the so called cat 0 spaces it is shown that both results apply in this setting and that lifsic s theorem gives a sharper result also a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces a class which includes those possessing property p of lim and xu

a simple method for group analysis and its application to a model of detonation a simple procedure is suggested to obtain extensions of the principal lie algebra for invariance transformations of a given family of equations by using an equivalence algebra an application to a qualitative model of detonation is performed

fixed points of asymptotic contractions abstract let r r be a contractive gauge function in the sense that is continuous s 0 and if f m m satisfies d f x f y d x y for all x y in a complete metric space m d then f always has a unique fixed point it is proved that if t m m satisfies d t n x t n y n d x y x y m where each n is continuous and n uniformly on the range of d then t has a unique fixed point and moreover all of the picard iterates of t converge to this fixed point

pointwise lipschitzian mappings in uniformly convex and uniformly smooth banach spaces abstract let x be a uniformly convex and uniformly smooth banach space let t c c be an asymptotic pointwise nonexpansive mapping where c is a bounded closed and convex subset of x in this paper we investigate conditions sufficient for the weak convergence of the generalized mann and ishikawa processes to a fixed point of t

explicit solutions of grassmannian σ models classical solutions of the two dimensional grassmannian models of respectively euclidean and minkowskian type are compared with each other some explicit solutions of both types are constructed and some of their graphs are presented

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

normal structure and orbital fixed point conditions abstract in some sense all of the results discussed in this paper are an outgrowth of the concept of normal structure introduced by brodskii and milman in 1948 and following related idea introduced by belluce and kirk in 1969 a mapping t of a metric space k into itself is said to have diminishing orbital diameters if given any x k it is the case that lim n d i a m o t n x d i a m o x whenever d i a m o x 0 where o x x t x t 2 x among other things belluce and kirk observed that the assumption of diminishing orbital diameters on a mapping t k k is sufficient to assure that t has a fixed point if t is nonexpansive and if k is a weakly compact convex subset of a banach space it was later shown by kirk that the convexity assumption on k can be dropped the purpose of this discussion is to illustrate how the above ideas are related to a number of similar ones that have been introduced more recently among these is the following concept introduced recently by amini harandi et al a mapping t k k of a subset k of a banach space is said to be nonexpansive wrt orbits if for all x y k t x t y r x o y where r x o y sup x u u o y it is shown that if k is a weakly compact subset of a banach space and if t k k is nonexpansive wrt orbits then the following condition always assures the existence of a fixed point for t for each x k with t x x inf m n lim sup n t m x t n x d i a m o x some open questions are also discussed

differential invariants for quasi linear and semi linear wave type equations abstract in this paper we consider the class of wave type equations u tt f x t u u xx g x t u u x u t and two special cases of it we derive the equivalence transformations for these equations and using these transformations we obtain differential invariants and invariant equations we employ these invariants or and invariant equations to classify all equations of this general class that can be mapped into a simple linear hyperbolic or into an elliptic equation additional applications are presented

optimal system invariant solutions and evolution of weak discontinuity for isentropic drift flux model abstract inequivalent classes of similarity solutions for isentropic drift flux two phase flow model are obtained by using one dimensional optimal classifications the optimality of this classification is shown in a constructive manner i e by constructing various invariant functions including conditional and numerical invariants further with the help of invariant solution we discuss the evolution of weak discontinuity which displays the nature of the general solution

on some differential invariants for a family of diffusion equations the equivalence transformation algebra le and some of its differential invariants for the class of equations ut h u ux x f x u ux h 0 are obtained using these invariants we characterize subclasses which can be mapped by means of an equivalence transformation into the well studied family of equations vt vkvx x

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

application of weak equivalence transformations to a group analysis of a drift diffusion model a group analysis of a class of drift diffusion systems is performed in account of the presence of arbitrary constitutive functions we look for lie symmetries starting from the weak equivalence transformations applications to the transport of charges in semiconductors are presented and a special class of solutions is given for particular doping profiles

symmetry analysis and exact invariant solutions for a class of energy transport models of semiconductors the symmetry classification of a class of energy transport models for semiconductors is performed reduced systems and examples of exact invariant solutions are shown

one local retracts and banach operator pairs in metric spaces abstract in this paper we first introduce the concept of nr map and then use this concept to establish the existence of common fixed points for banach operator pairs in the context of uniformly convex geodesic metric spaces new proofs of main results theorems 2 1 and 3 5 of chen and li j chen z li banach operator pair and common fixed points for nonexpansive maps nonlinear anal 74 2011 3086 3090 are presented further we prove de marr s theorem for the family of symmetric banach operator pairs in metric spaces and r trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity

on a countable dimensional subalgebra of the equivalence algebra for equations vtt f x vx vxx g x vx the analysis of the equations vtt f x vx vxx g x vx initiated by ibragimov et al j math phys 32 2988 1991 is continued the preliminary group classification of these equations with respect to a countable dimensional subalgebra of their equivalence algebra is presented this subalgebra is an extension of the 10 dimensional subalgebra considered by ibragimov et al the result of the work is a table of nonequivalent equations possessing an additional symmetry

lie symmetries of nonlinear multidimensional reaction diffusion systems ii we present a complete description of the classical lie symmetries of a coupled system of partial differential equations comprising a pair of semilinear reaction diffusion equations with constant diffusivities and arbitrary nonlinearities in the reaction terms in any number of spatial dimensions part i cherniha r m and king j r 2000 j phys a math gen 33 267 82 j phys a math gen 33 7839 41 addressed the case of unequal diffusivities here we complete the analysis by treating the case of equal diffusivities in which the symmetry structure is richer still such models arise in the description of numerous physical chemical and biological systems and we also indicate the possible application in such contexts of some of the specific cases arising from the group classification specifically a variety of lie s ansatze and exact solutions of the so called reaction diffusion systems of a type that arises in mathematical biology are constructed

demiclosed principle and δ convergence theorems for total asymptotically nonexpansive mappings in cat 0 spaces abstract the purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove the demiclosed principle for this kind of mappings in cat 0 spaces as a consequence we obtain a convergence theorem of the krasnoselskii mann type iteration for total asymptotically nonexpansive mappings in this setting our results extend and improve the corresponding recent results announced by many authors

fixed point iteration processes for asymptotic pointwise nonexpansive mappings in banach spaces abstract let x be a uniformly convex banach space with the opial property let t c c be an asymptotic pointwise nonexpansive mapping where c is bounded closed and convex subset of x in this paper we prove that the generalized mann and ishikawa processes converge weakly to a fixed point of t in addition we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex banach spaces both processes converge strongly to a fixed point

a group classification of the general second order coupled diffusion system we classify second order systems of coupled heat diffusion equations the underlying system contains several unknown functions the forms of which are specified via the method of group classification in particular the canonical forms of the low dimensional lie algebras and the lie algebras of higher dimension provide a means to specify the functional forms of these arbitrary functions

group classification of unsteady boundary layer equations of a class of non newtonian fluids two dimensional unsteady boundary layer equations of a general model of non newtonian fluids were investigated in this study in this model the shear stress is taken as an arbitrary function of the velocity gradient group classification of the equations with respect to shear stress is done using two different approaches 1 classical theory 2 equivalence transformations both approaches yield identical results it is found that the principle lie algebra extends only for cases of newtonian and power law flows

group classification of a generalization of the heath equation abstract the complete group classification of a generalization of the heath model is carried out by connecting it to the heat equation with nonlinear source examples of invariant solutions are given under the terminal and the barrier option condition

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

equivalence groups for balance equations abstract we consider a system of balance equations and by using ovsiannikov s method construct the most general equivalence algebra moreover some special cases are presented

classifying evolution equations a lie point symmetry classification of evolution equations in 1 1 time space dimensions was presented a combination of the standard lie algorithm for point symmetry and the equivalence group of th

a mapping of a generalisation of a thin film equation on a moving substrate abstract lie group analysis is used to determine an invertible transformation between a generalised thin film equation u t u n m u x m x 0 and a generalised thin film equation on a moving substrate u t u n m u x m v t u x 0 where v t is the substrate velocity consequently we obtain new solutions for the motion of a hard contact lens on a thin film of tears where 1 3 n m 3

group classification of three dimensional boundary layer equations of a class of non newtonian fluids group classification of three dimensional boundary layer equations of a class of non newtonian fluids in which thee stress is an arbitrary function of rates of strain is made group classification of the equations with respect to shear stress is done by using equivalence transformations special cases such as arbitrary stress newtonian fluids and power law fluids are treated

second order differential invariants of a family of diffusion equations an equivalence transformation algebra for a class of nonlinear diffusion equations is found after having obtained the second order differential invariants with respect to we get some results which allow us to linearize a subclass of the equations considered

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase 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

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

asymptotic pointwise contractions a simple and elementary without using the ultrapower technique proof for the existence of a fixed point of an asymptotic pointwise contraction is provided the existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex banach space is also investigated

a symmetry classification for a class of 2 1 nonlinear wave equation abstract in this paper a symmetry classification of a 2 1 nonlinear wave equation u t t f u u x x u y y 0 where f u is a smooth function on u using lie group method is given the basic infinitesimal method for calculating symmetry groups is presented and used to determine the general symmetry group of this 2 1 nonlinear wave equation

group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations

the accommodation of traveling waves of fisher s type to the dynamics of the leading tail traveling wave solutions for equations of fisher s type are given by a connection from a saddle point to a node and their velocity for large times is known to be determined by the spatial decay rate of the initial conditions far ahead of the wave to understand this phenomenon in simple terms initial conditions are considered that correspond to a traveling wave but with an exponential decay rate far ahead of the wave that varies slowly in the region referred to as the leading tail the wave that results is supersonic in the sense that the speed of the core of the traveling wave is greater than the velocity of the characteristics or group velocity in the leading tail consequently the speed of the core adjusts to accommodate the slowly varying decay rate in the leading tail a generalization is also considered that includes fisher s equation for which traveling waves with a stable exponential tail corresponding to a node are supersonic

a group analysis approach for a nonlinear differential system arising in diffusion phenomena we consider a class of second order partial differential equations which arises in diffusion phenomena and following a new approach we look for a lie invariance classification via equivalence transformations a class of exact invariant solutions containing an arbitrary function is obtained

a mathematical model for spatially varying extracellular matrix alignment orientation of extracellular matrix fibers in the skin is a keyingredient of tissue appearance and function and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin in this paper the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them the model extends previous work in which spatial variation was excluded numerical simulations of the model are presented which show spatial variations in alignment over long transients but with spatiallyuniform behavior in the long term this is investigated further via asymptotic analysis using the angular diffusion coefficient as a small parameter this method enables calculation of the form of the steadystate orientation peaks observed numerically by considering behavior at large times the rate of approach to these peaks is shown to be exponential extension of this analysis to the spatially varying model confirms that long time behavior will be spatiallyuniform except in one special and biologicallyunrealistic case the authors conclude that the spatiallyvary ing alignment patterns observed in skin are in fact slow transients and biological implications of the modeling are discussed

lie group classification of the nonlinear generalized kompaneets equations with two functional parameters we study the nonlinear generalized kompaneets equations gkes with two functional parameters by using the lie ovsiannikov algorithm we carried out the group classification of the equations it is shown that the kernel algebra of the full groups of the gkes is the one dimensional lie algebra we obtained nine non equivalent up to transformations from the equivalence group nonlinear gkes that allow wider invariance algebras than the kernel one we found a number of exact solutions of those gkes which have the maximal symmetry properties

mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces abstract let m d be a complete 2 uniformly convex metric space let c be a nonempty bounded closed and convex subset of m and let t c c be an asymptotic pointwise nonexpansive mapping in this paper we prove that the modified mann iteration process defined by x n 1 t n t n x n 1 t n x n converges in a weaker sense to a fixed point of t

spreading of a thin film with suction or blowing including surface tension effects in this paper we derive a fourth order nonlinear partial differential equation modelling the effects of suction blowing on the evolution of the free surface of a thin viscous film on a porous base gravity and surface tension effects are included the suction blowing is modelled as an arbitrary function v n t r the lie group classification method is used to determine a first order quasi linear partial differential equation satisfied by v n t r this equation is coupled with the model equation and solved numerically new results are obtained which show how the free surface and the apparent contact angle vary with suction blowing

complete group classification of a class of nonlinear wave equations preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by ibragimov torrisi and valenti j math phys 32 2988 2995 1991 10 1063 1 529042 in this paper the partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one dimensional lie symmetry extensions related to a fixed finite dimensional subalgebra of the infinite dimensional equivalence algebra of the class under consideration in the present paper we implement the complete group classification of the same class up to both usual and general point equivalence using the algebraic method of group classification this includes the complete preliminary group classification of the class and finding those lie symmetry extensions which are not associated with subalgebras of the equivalence algebra the complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite dimensi

geodesic ptolemy spaces and fixed points abstract in this paper we study the regularity of geodesic ptolemy spaces and apply our findings to metric fixed point theory it is an open question whether such spaces with a continuous midpoint map are cat 0 spaces we prove that if a certain uniform continuity is imposed on such a midpoint map then these spaces if complete are reflexive that is the intersection of decreasing families of bounded closed and convex subsets is nonempty and that bounded sequences have unique asymptotic centers these properties will then be applied to yield a series of fixed point results specific to cat 0 spaces

nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion abstract we consider a simplified chemotaxis model of tumor angiogenesis described by a keller segel system on the two dimensional infinite cylindrical domain x y r s where s is the circle of perimeter 0 the domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located the system is known to allow planar traveling wave solutions of an invading type in this paper we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if is sufficiently small the same result for the corresponding system in one dimension was obtained by li li wang 2014 17 our result solves the problem remained open in chae choi kang lee 2018 3 at which only linear stability of the planar traveling waves was obtained under certain artificial assumption

traveling wave solutions of a mathematical model for tumor encapsulation the formation of a capsule of dense fibrous extracellular matrix around a solid tumor is a key prognostic indicator in a wide range of cancers however the cellular mechanisms underlying capsule formation remain unclear one hypothesized mechanism is the expansive growth hypothesis which suggests that a capsule may form by the rearrangement of existing extracellular matrix without new matrix production a mathematical model was recently proposed to study the implications of this hypothesis perumpanani sherratt and norbury nonlinearity 10 1997 pp 1599 1614 the model consists of conservation equations for tumor cells and extracellular matrix and exhibits traveling wave solutions in which a pulse of extracellular matrix corresponding to a capsule moves in parallel with the advancing front of the tumor in this paper the author presents a detailed study of traveling wave behavior in the model deriving conditions for the existence of traveling waves and their key properties numerical meth

lie group theory and analytical solutions for the axially accelerating string problem abstract transverse vibrations of a string moving with time dependent velocity v t have been investigated analytical solutions of the problem are found using the systematic approach of lie group theory group classification with respect to the arbitrary velocity function has been performed using a newly developed technique of equivalence transformations from the symmetries of the partial differential equation the method for deriving exact solutions for the arbitrary velocity case is shown special cases of interest such as constant velocity constant acceleration harmonically varying velocity and exponentially decaying velocity are investigated in detail finally for a simply supported strip approximate solutions are presented for the exponentially decaying and harmonically varying cases

symmetries and integrability of a fourth order euler bernoulli beam equation the complete symmetry group classification of the fourth order euler bernoulli ordinary differential equation where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement is obtained we perform the lie and noether symmetry analysis of this problem in the lie analysis the principal lie algebra which is one dimensional extends in four cases viz the linear exponential general power law and a negative fractional power law it is further shown that two cases arise in the noether classification with respect to the standard lagrangian that is the linear case for which the noether algebra dimension is one less than the lie algebra dimension as well as the negative fractional power law in the latter case the noether algebra is three dimensional and is isomorphic to the lie algebra which is sl 2 r this exceptional case although admitting the nonsolvable algebra sl 2 r remarkably allows for a two parameter family of exact solut

group classification of nonlinear wave equations we perform complete group classification of the general class of quasilinear wave equations in two variables this class may be seen as a generalization of the nonlinear d alembert liouville sin sinh gordon and tzitzeica equations we derive a number of new genuinely nonlinear invariant models with high symmetry properties in particular we obtain four classes of nonlinear wave equations that admit five dimensional invariance groups

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

on asymptotic pointwise contractions in metric spaces abstract we discuss the existence of fixed points of asymptotic pointwise mappings in metric spaces this is the nonlinear version of some known results proved in banach spaces we also discuss the case of multivalued mappings

gauge equivalence of sigma models with non compact grassmannian manifolds the gauge equivalence ge of sigma models associated with non compact grassmannian manifolds is investigated with emphasis on the necessary restrictions for the choice of gauge elements in such cases the importance of ge in solving a non linear system with the help of inverse scattering data of its gauge related counterpart is demonstrated the gauge relations between generalised landau lifshitz ll and non linear schrodinger nls type equations and also between non linear sigma models and generalised sine sinh gordon equations for non compact su p q s u u v u s t manifolds are established using h gauge invariance of ll the ge is extended to some higher order specific non linear systems the gauge connection among various ll and nls equations are schematically represented along with the recovery of earlier results important new results some with significant non compact structures are discovered

exact solutions of some nonlinear evolution equations using symbolic computations abstract in this paper we present a solution methodology that utilizes symbolic computations to obtain analytic solutions of some nonlinear evolution equations by balancing the nonlinear and the dispersive effects the solution method is demonstrated by obtaining solutions to burgers equation the nonlinear heat equation the modified kdv equation and the kuramoto sivashinsky equation

on asymptotic pointwise nonexpansive mappings in modular function spaces we prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces

hodograph type transformations abstract we consider second order evolution equations of the form u t f 1 u u x u xx f 2 u u x and third order of the form u t f 1 u u x u xxx f 2 u u x u xx for each of these equations we construct point transformations which relate them with an equation of the same form these mappings have the property that one of the old independent variable depends on the new dependent variable this property leads us to hodograph type transformations in most of the results presented both the hodograph type transformation and the relating equations contain arbitrary functions and or arbitrary parameters certain choices of these functions and parameters make one of the two relating equations linear or integrable or a well known equation for the second order evolution equations we construct hodograph type transformations which connect them with a class of linear equations which are not of the same form

classical nonlinear σ models on grassmann manifolds of compact or noncompact type grassmannian models are reexamined in the light of a new geometrical result namely the cartan immersion of a riemannian symmetric space g h into its isometry group g is not always a diffeomorphism onto the set m introduced by eichenherr and forger nucl phys b 164 528 1980 this is the case in particular for grassmann manifolds of all types their set m is shown to be disconnected and its structure is completely analyzed as a consequence a new constraint must be introduced in the backlund transformation bt method proposed by harnad saint aubin and shnider commun math phys 92 329 1984 it is shown however to always be satisfied for grassmann manifolds of compact type i e g compact but the problem remains open in most other cases on the other hand the bt method is extended to the euclidean regime

lie symmetries and reductions of multi dimensional boundary value problems of the stefan type a new definition of lie invariance for nonlinear multi dimensional boundary value problems bvps is proposed by the generalization of known definitions to much wider classes of bvps the class of 1 3 dimensional nonlinear bvps of the stefan type modeling the process of melting and evaporation of metals is studied in detail using the definition proposed the group classification problem for this class of bvps is solved and some reductions with physical meaning to bvps of lower dimensionality are made examples of how to construct exact solutions of the 1 3 dimensional nonlinear bvp with the correctly specified coefficients are presented

symmetry analysis and exact invariant solutions for the drift diffusion model of semiconductors abstract the symmetry classification of the drift diffusion models for semiconductors is performed reduced systems and examples of exact invariant solutions are shown

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

reduction of balance laws to conservation laws by means of equivalence transformations a class of partial differential equations a conservation law and two balance laws with three independent variables and involving six arbitrary continuously differentiable functions is considered in the framework of equivalence transformations these are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent dependent and additional variables representing values taken by the arbitrary elements projecting the admitted symmetries into the space of independent and dependent variables we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements in particular we are interested in finding an equivalent autonomous system of conservation laws it is shown how the results apply to some physical problems

a group analysis via weak equivalence transformations for a model of tumour encapsulation a symmetry reduction of a pdes system describing the expansive growth of a benign tumour is obtained via a group analysis approach the presence in the model of three arbitrary functions suggests the use of lie symmetries by using the weak equivalence transformations an invariant classification is given which allows us to reduce the initial pdes system to an odes system numerical simulations show a realistic enough description of the physical process

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

on properties of the navier stokes equations the main feature of our analysis consists in the fact that the equation of state is taking into consideration and notice that a consequent application of the lie group based similarity method put us into a position to construct the most general classes of similarity solutions for our problem in addition we could give a group classification with respect to the function f r and of some special classes of solution discussed

preliminary group classification for the nonlinear wave equation utt f x u uxx g x u abstract a classification is given for the nonlinear wave equation u t t f x u u x x g x u admitting an extension by one dimension of the principal lie algebra of the equation under consideration the paper is one of the few applications of an algebraic approach to the problem of group classification the method of preliminary group classification a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained the result of the work is a wide class of equations summarized in table form

homogeneous kahler manifolds paving the way towards new supersymmetric sigma models homogeneous kahler manifolds give rise to a broad class of supersymmetric sigma models containing as a rather special subclass the more familiar supersymmetric sigma models based on hermitian symmetric spaces in this article all homogeneous kahler manifolds with semisimple symmetry groupg are constructed and are classified in terms of dynkin diagrams explicit expressions for the complex structure and the kahler structure are given in terms of the lie algebra g ofg it is shown that for compactg one can always find an einstein kahler structure which is unique up to a constant multiple and for which the kahler potential takes a simple form

equivalence groupoid and group classification of a class of variable coefficient burgers equations abstract we study admissible transformations and lie symmetries for a class of variable coefficient burgers equations we combine the advanced methods of splitting into normalized subclasses and of mappings between classes that are generated by families of point transformations parameterized by arbitrary elements of the original classes a nontrivial differential constraint on the arbitrary elements of the class of variable coefficient burgers equations leads to its partition into two subclasses which are related to normalized classes via families of point transformations parameterized by subclasses arbitrary elements one of the mapped classes is proved to be normalized in the extended generalized sense and its effective extended generalized equivalence group is found using the mappings between classes and the algebraic method of group classification we carry out the group classification of the initial class with respect to its equivalence groupoid

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

lim s theorems for multivalued mappings in cat 0 spaces let x be a complete cat 0 space we prove that if e is a nonempty bounded closed convex subset of x and t e k x a nonexpansive mapping satisfying the weakly inward condition i e there exists p e such that p 1 tx ie x x e 0 1 then t has a fixed point in banach spaces this is a result of lim on asymptotic centers and fixed points of nonexpansive mappings canad j math 32 1980 421 430 the related result for unbounded r trees is given

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

symmetry preserving parameterization schemes methods for the design of physical parameterization schemes that possess certain invariance properties are discussed these methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations the demand that the averaged equation is invariant with respect to a subalgebra of the maximal lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra given no prescribed symmetry group the direct group classification problem is relevant within this framework the algebraic method or direct integration of determining equations for lie symmetries can be applied for cumbersome parameterizations a preliminary group classification can be carried out the methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity

fixed point and convergence theorems for different classes of generalized nonexpansive mappings in cat 0 spaces in this paper we prove the existence of common fixed points in cat 0 spaces for three different classes of generalized nonexpansive mappings including a quasi nonexpansive single valued mapping a pointwise asymptotically nonexpansive mapping and a multivalued mapping satisfying the conditions e and c l for some email protected 0 1 moreover we introduce an iterative process for these mappings and prove convergence and strong convergence theorems for such an iterative process in cat 0 spaces

group invariant solutions and optimal systems for multidimensional hydrodynamics the group properties of the three dimensional 3 d one temperature hydrodynamic equations including nonlinear conduction and a thermal source are presented a subgroup corresponding to axisymmetric geometry is chosen and the details of the construction of the one and two dimensional optimal systems are shown the two dimensional optimal system is used to generate 23 intrinsically different reductions of the 2 d partial differential equations to ordinary differential equations these ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations two example analytic solutions are presented a 2 d axisymmetric flow with a p2 asymmetry and a 3 d spiraling flow

solutions of euclidean σ models on noncompact grassmann manifolds a method for constructing explicit solutions of classical two dimensional euclidean models on grassmann manifolds of noncompact type and their supersymmetric extensions is presented it generalizes the holomorphic method developed by din zakrzewski and sasaki in the compact case by taking properly into account the geometry of indefinite metric spaces the solutions obtained are local in the sense that they may become singular on certain curves in the plane and the signature of the solution may change across such a curve