jordan all derivable points in the algebra of all upper triangular matrices abstract let tm n be the algebra of all n n upper triangular matrices we say that l tm n is a jordan derivable mapping at g if st ts s t s t t s t s for any s t tm n with st g an element g tm n is called a jordan all derivable point of tm n if every jordan derivable linear mapping at g is a derivation in this paper we show that every element in tm n is a jordan all derivable point

sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

on liouville systems at critical parameters part 1 one bubble abstract in this paper we consider bubbling solutions to the general liouville system 0 1 g u i k j 1 n a i j j k h j e u j k h j e u j k 1 0 in m i 1 n n 2 where m g is a riemann surface and a a i j n n is a constant non negative matrix and j k j as k among other things we prove the following sharp estimates 1 the location of the blowup point 2 the convergence rate of j k j j 1 n these results are of fundamental importance for constructing bubbling solutions it is interesting to compare the difference between the general liouville system and the su 3 toda system on estimates 1 and 2

the evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non conductors the theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock to solve these equations we use perturbation methods with a perturbation parameter e characterising the initial shock amplitude it is shown that for decaying shock waves if the accompanying second order discontinuity is of order one the straightforward perturbation procedure yields uniformly valid solutions but if the accompanying second order discontinuity is of order e the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel we also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofhunter keller s weakly nonlinear geometrical optics method the two approaches give exactly the same results within their common range of validity the explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity and on what length scale these effects are most pronounced

a linear approximation for the regular reflection of a weak shock at a wedge abstract the problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math lvii 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle between incident shock and wedge is small we can see the weak regular reflection as the disturbance of normal reflection as in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques a delicate a priori estimate and compact arguments we show that the solution of the linearized problem is smooth in the interior and lipschitz continuous up to the degenerate boundary

dispersive nonlinear geometric optics we construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena the principal themes are the following 1 the natural framework for the study of dispersion is wavelength e solutions of systems of partial differential operators in e the natural e characteristic equation and e eikonal equations are not homogeneous this corresponds exactly to the fact that the speeds of propagation which are called group velocities depend on the length of the wave number 2 the basic dynamic equations are expressed in terms of the operator e t as a result growth or decay tends to occur at the catastrophic rate ect e the analysis is limited to conservative or nearly conservative models 3 if a phase x e satisfies the natural e eikonal equation the natural harmonic phases n x e generally do not one needs to impose a coherence hypothesis for the harmonics 4 in typical examples the set of harmonics

diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

solutions containing a large parameter of a quasi linear hyperbolic system of equations and their nonlinear geometric optics approximation it is well known that a quasi linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data 1 a 0 x x r 1 0 x r n 0 here r 1 is a characteristic vector and a 0 x is a smooth scalar function of compact support under the additional requirements that n 2 or 3 and that a 0 x have the vanishing mean with respect to it is shown that a genuine solution exists in a time interval independent of and that the formal solution is asymptotic to the genuine solution as

on maps preserving operators of local spectral radius zero abstract let l x be the algebra of all bounded linear operators on a complex banach space x we describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x we also describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x furthermore we characterize maps not necessarily linear nor surjective on l x which satisfy r t s x 0 if and only if r t s x 0 for every x x and t s l x

surjective linear maps preserving local spectra let b x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize linear surjective and continuous maps on b x preserving different local spectral quantities at a nonzero fixed vector

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

multiplicative mappings at some points on matrix algebras let mn be the algebra of all n n matrices and let mn mn be a linear mapping we say that is a multiplicative mapping at g if st s t for any s t mn with st g fix g mn we say that g is an all multiplicative point if every multiplicative linear bijection at g with in in is a multiplicative mapping in mn where in is the unit matrix in mn we mainly show in this paper the following two results 1 if g mn with detg 0 then g is an all multiplicative point in mn 2 if is an multiplicative mapping at in then there exists an invertible matrix p mn such that either s psp 1 for any s mn or t pttrp 1 for any t mn

low mach number flows in time dependent domains we perform a multiple time scale single space scale analysis of a compressible fluid in a time dependent domain when the time variations of the boundary are small with respect to the acoustic velocity we introduce an average operator with respect to the fast time the averaged leading order variables satisfy modified incompressible equations which are coupled to linear acoustic equations with respect to the fast time we discuss possible initial boundary data for the asymptotic equations inherited from the initial boundary data for the compressible equations

biseparating maps between operator algebras we prove that a biseparating map between spaces b e and some other banach algebras is automatically continuous and a multiple of an algebra isomorphism

maps on matrices compressing the local spectrum in the spectrum abstract let m n c be the algebra of all complex n n matrices let x 0 be a nonzero vector in c n we describe maps on m n c not linear or surjective satisfying t s x 0 t s for all t s m n c we also obtain a similar description by supposing that is surjective and t s x 0 t s for all t s m n c

operators preserving the strong birkhoff james orthogonality on b h abstract we characterize the class of surjective conjugate linear mappings b h b h that preserve the strong birkhoff james orthogonality in both directions we also give characterizations of rank one operators as well as coisometries in terms of the strong birkhoff james orthogonality

a topological degree counting for some liouville systems of mean field type let a a ij n n be an invertible matrix and a 1 a ij n n be the inverse of a in this paper we consider the generalized liouville system n 0 1 g u i n a ij j h j e u j h j e u j 1 0 in m j 1 where 0 h j c 1 m and j and prove that under the assumptions of h 1 and h 2 see introduction the leray schauder degree of 0 1 is equal to m 1 m n n if 1 n satisfies 8 n n n i 1 i 1 i j n a ij i j 8 n 1 n n i 1 i equation 0 1 is a natural generalization of the classic liouville equation and is the euler lagrangian equation of the nonlinear function u 1 2 m 1 i j n a ij g u i g u j n n i 1 m i u i n n i 1 i log m h i e u i the liouville system 0 1 has arisen in many different research areas in mathematics and physics our counting formulas are the first result in degree theory for liouville systems

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

maps on the positive definite cone of a c algebra preserving certain quasi entropies abstract we describe the structure of those bijective maps on the cone of all positive invertible elements of a c algebra with a normalized faithful trace which preserve certain kinds of quasi entropy it is shown that essentially any such map is equal to a jordan isomorphism of the underlying algebra multiplied by a central positive invertible element

a localization technique for linear preserver problems abstract we introduce a new general technique for solving linear preserver problems the idea is to localize a given linear preserver at each non zero vector in such a way we get vector valued linear maps on the space of matrices which inherit certain properties from if we can prove that such induced maps have a standard form then itself has either a standard form or a very special degenerate form we apply this technique to characterize linear preservers of full rank using this technique we further reprove two classical results describing the general form of linear preservers of rank one and linear preservers of the unitary or orthogonal group

the validity of nonlinear geometric optics for weak solutions of conservation laws the method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems the tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth since shock waves typically form in such solutions these assumptions are rarely satisfied in practice nevertheless in a variety of applied contexts these methods give qualitatively reliable answers for discontinuous weak solutions here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable the methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations stability estimates for intermediate asymptotic times and the rapid decay in variation of weak solutions for large asymptotic times

linear maps preserving the minimum and surjectivity moduli of hilbert space operators abstract let b h be the algebra of all bounded linear operators on a complex infinite dimensional hilbert space h for every t b h let m t and q t denote the minimum modulus and surjectivity modulus of t respectively let b h b h be a surjective linear map in this paper we prove that the following assertions are equivalent i m t m t for all t b h ii q t q t for all t b h iii there exist two unitary operators u v b h such that t u t v for all t b h this generalizes the result of mbekhta 7 theorem 3 1 to the non unital case

a spectral characterization of operators having rank k abstract let a b x the algebra of all bounded linear operators on a complex banach space x and let k be a positive integer it is proved that a has rank k if and only if i k 1 m 1 t c m a t for every t b x and some nonzero scalars c 1 c k 1 and ii k m 1 t c m a t for some t b x and some distinct scalars c 1 c k

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

spectral preservers and approximate spectral preservers on operator algebras abstract let a and b be unital primitive banach algebras with minimal idempotents we prove that every surjective spectral isometry from a onto b is of the form where c with 1 and is either an isomorphism or an anti isomorphism from a onto b as an application we show that for all banach spaces x and y the spectral nearisometries and the approximate spectrum preserving maps from l x onto l y are perturbations of actual spectral isometries and spectrum preserving maps respectively

additive maps on standard operator algebras preserving invertibilities or zero divisors let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be an additive surjection from a onto b we prove that if is unital and preserves any one of left right invertibility left right zero divisors left right topological divisors of zero quasi affinity injectivity surjectivity range density lower boundedness and left right maximal ideals in both directions then it has one of the following forms isomorphism conjugate isomorphism anti isomorphism and conjugate anti isomorphism we also show that preserves zero products in both directions if and only if is either an isomorphism or a conjugate isomorphism

convergence rate location and z2 condition for fully bubbling solutions to su n 1 toda systems abstract it is well known that the study of su n 1 toda systems is important not only to chern simons models in physics but also to the understanding of holomorphic curves harmonic sequences or harmonic maps from riemann surfaces to c p n one major goal in the study of su n 1 toda system on riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions in this article we use a unified approach to study fully bubbling solutions to general su n 1 toda systems and we prove three major sharp estimates important for constructing bubbling solutions the closeness of blowup solutions to entire solutions the location of blowup points and a z 2 condition

on a regularization of the compressible euler equations for an isothermal gas abstract we consider a leray type regularization of the compressible euler equations for an isothermal gas the regularized system depends on a small parameter 0 using riemann invariants we prove the existence of smooth solutions for the regularized system for every 0 the regularization mechanism is a non linear bending of characteristics that prevents their finite time crossing we prove that in the 0 limit the regularized solutions converge strongly however based on our analysis and numerical simulations the limit is not the unique entropy solution of the euler equations the numerical method used to support this claim is derived from the riemann invariants for the regularized system this method is guaranteed to preserve the monotonicity of characteristics

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

derivable mappings at unit operator on nest algebras abstract let a be an operator subalgebra with the unit operator i in b h we say that a linear mapping from a into itself is a derivable mapping at i if st s t s t for any s t a with st i in this paper we show the following main result every strongly operator topology continuous derivable mapping at i on a nest algebra alg n is an inner derivation

a subcell resolution method for viscous systems of conservation laws a subcell resolution scheme for viscous systems of conservation laws is presented the scalar model is extended to systems via weakly nonlinear geometrical optics approximations for parabolic perturbations of hyperbolic conservation laws and the roe field by field decomposition

the influence of differencing and cfl number on implicit time dependent non linear calculations abstract this paper is concerned with the stability and precision of transient implicit finite difference algorithms the test problem is the burgers equation the influence of space discretisation and initial values is studied linear analysis provides incomplete stability conditions for both continuous and weak solutions and we derive a simple criterion of augmented stability taking into account the non linearity of the problem the real stability limits appear to be a combination of the linear stability analysis criterion and of this augmented stability criterion

additive idempotence preservers abstract let a be a local matrix algebra over a field of characteristic different from 2 b an arbitrary algebra and x y banach spaces additive mappings a b which preserve idempotents are characterized the result is then used in classifying additive surjections b x b y which preserve idempotents and do not annihilate finite rank operators

a priori estimates for d4 and f4 toda systems abstract in this article we establish the a priori bounds of the toda systems with arbitrarily many singular sources associated to the simple lie algebras of type d 4 and f 4 extending the previous works of lin et al 27 20 24 for the a n b n c n and g 2 type lie algebras the problem of obtaining a priori estimates can be reduced to locating the local mass of blowup solutions the key step is to calculate the local mass near each blowup points efficiently a precise estimate on the total mass of entire solutions of toda systems with many singular sources is an indispensable step in our argument

all derivable points in matrix algebras abstract let m n be the algebra of all n n matrices we say that an element g m n is an all derivable point in m n if every derivable linear mapping at g i e st s t s t for any s t m n with st g is a derivation we mainly show in this paper that a matrix g is an all derivable point in m n if and only if g 0

characterizations of isomorphisms and derivations of some algebras abstract let be a zero product preserving bijective bounded linear map from a unital algebra a onto a unital algebra b such that 1 k we show that if a is a csl algebra on a hilbert space or a j lattice algebra on a banach space then there exists an isomorphism from a onto b such that k for a nest algebra a in a factor von neumann algebra we characterize the linear maps on a such that x y x y 0 for all x y a with x y 0

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

a note on spectrum preserving maps abstract let a and b be unital semisimple banach algebras if m 2 a b is a bijective spectrum preserving linear map then is a jordan homomorphism

2 local automorphisms of operator algebras abstract a not necessarily continuous linear or multiplicative function from an algebra a into itself is called a 2 local automorphism if agrees with an automorphism of a at each pair of points in a in this paper we study when a 2 local automorphism of a c algebra or a standard operator algebra on a locally convex space is an automorphism

trilinear resonant interactions of semilinear hyperbolic waves an important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions or approximate solutions one of the difficulty is known to be the nonlinear interaction of oscillations this phenomenon has been studied for special classes of solutions such as oscillatory functions with phases and amplitudes this is the domain of geometric optics in some cases general families of bounded solutions have also been considered the present paper enters the later category but is highly inspired by the results of nonlinear geometric optics geometric optics provides a precise description of the asymptotic structure of solutions to both linear e g wh and nonlinear j hk mr hmr jmr 1 jmr 2 s hyperbolic pdes whose initial data u 0 x u0 x eu1 x 0 x e have rapid oscillations with prescribed phases 0 and a single prescribed scale 1e such solutions are important in both theory and applications l kk in the lowest approximation oscillations having different spatial scales interact only when m equals zero since otherwise the amplitudes of higher frequency oscillations become negligible relative to those of lower frequencies taking m 0 requires l estimates systems that are well posed in l occur mostly in one spatial dimension for such systems arbitrary uniformly bounded sequences of initial data may be considered but comparatively little is known about the asymptotic structure of the resulting solutions since the oscillations in such solutions may be extremely complex a natural problem is to determine their non oscillatory part which may be identified with the weak limit of the sequence of solutions the highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general of course some information about those oscillations will generally be needed in order to determine the weak limit the asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity for entropy solutions of a genuinely nonlinear conservation law t 1 or of a pair of such equations in one spatial dimension dp oscillations of amplitude o 1 do not persist for positive times so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data although oscillations do persist in linear systems the weak limit of a sequence of solutions to such

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

justification of multidimensional single phase semilinear geometric optics for semilinear strictly hyperbolic systems lu f x u we construct and justify high frequency nonlinear asymptotic expansions of the form u e x e j u j x x e lu e f x u e 0 the study of the principal term of such expansions is called nonlinear geometric optics in the applied literature we show 1 formal expansions with periodic profiles u j can be computed to all orders ii the equations for the profiles from i are solvable and iii there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion

characterizations of automorphisms of operator algebras on banach spaces abstract let x be a complex banach space of dimension greater than one and denote by b x the algebra of all the bounded linear operators on x it is shown that if b x b x is a multiplicative map not assumed linear and if is sufficiently close to a linear automorphism of b x in some uniform sense then it is actually an automorphism

invertibility preserving linear maps on j subspace lattice algebras it is shown that every unital surjective linear map between j subspace lattice algebras which preserves invertibility in both directions is a jordan isomorphism this result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras

spherical isometries of finite dimensional c algebras abstract in this paper it is shown that every surjective isometry between the unit spheres of two finite dimensional c algebras extends to a real linear jordan isomorphism followed by multiplication operator by a fixed unitary element this gives an affirmative answer to tingley s problem between two finite dimensional c algebras moreover we show that if two finite dimensional c algebras have isometric unit spheres then they are isomorphic

on jordan all derivable points of b h abstract let h be a hilbert space and b h the algebra of all bounded linear operators on h in this note we show that 0 is a generalized jordan all derivable point of b h if h is infinite dimensional for any hilbert space h we also show that i is a jordan all derivable point of b h

unital and multiplicatively spectrum preserving surjections between semi simple commutative banach algebras are linear and multiplicative abstract let t be a surjective map from a unital semi simple commutative banach algebra a onto a unital commutative banach algebra b suppose that t preserves the unit element and the spectrum f g of the product of any two elements f and g in a coincides with the spectrum t f t g then b is semi simple and t is an isomorphism the condition that t is surjective is essential an example of a non linear and non multiplicative unital map from a commutative c algebra into itself such that t f t g f g holds for every f g are given we also show an example of a surjective unital map from a commutative c algebra onto itself which is neither linear nor multiplicative such that t f t g f g holds for every f g

characterizations of derivations and jordan derivations on banach algebras abstract let a be an algebra and m be an a bimodule let x be in a and a m be a linear map which satisfies ab a b a b for all a b a with ab x it is shown that is a jordan derivation if is continuous and x is left or right invertible also it is shown that is a derivation if x is idempotent such that for m m the condition x a i x m 0 implies i x m 0 and the condition mx a i x 0 implies mx 0

n waves in elasticity we study the large time behavior of solutions of the one dimensional equations of elasticity typically a nonconvex system of conservation laws we show that solutions with compact initial support converge in l1 to a superposition of n waves at algebraic rate likewise we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order our analysis uses approximate scalar laws together with l1 stability of scalar solutions and decay in total variation of the wave speed

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

on linear maps preserving generalized invertibility and related properties let h be an infinite dimensional complex hilbert space b h be the algebra of all bounded linear operators on h we study surjective linear maps on b h preserving generalized invertibility we also investigate surjective linear maps preserving fredholm respectively semi fredholm operators our results improve those of mbekhta rodman and semrl

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

all derivable points in continuous nest algebras let a be an operator algebra on a hilbert space we say that an element g a is an all derivable point of a for the strong operator topology if every strong operator topology continuous derivable linear mapping at g i e st s t s t for any s t algn with st g is a derivation let n be a continuous nest on a complex and separable hilbert space h we show in this paper that every orthogonal projection operator p m 0 m n is an all derivable point of algn for the strong operator topology

approximately spectrum preserving maps let x and y be superreflexive complex banach spaces and let b x and b y be the banach algebras of all bounded linear operators on x and y respectively if a bijective linear map b x b y almost preserves the spectra then it is almost multiplicative or anti multiplicative furthermore in the case where x y is a separable complex hilbert space such a map is a small perturbation of an automorphism or an anti automorphism

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

preservers of local spectra of matrix sums abstract let m n c be the set of n n complex matrices and for a nonzero vector e c n and t m n c let t e denote the local spectrum of t at e we characterize surjective maps on m n c which satisfy t s e t s e for all matrices t and s we arrive at the same conclusion by supposing that t s e t s e for all matrices t and s without the surjectivity assumption on continuous maps from m n c onto itself that preserve the local spectral radius of the sum of matrices at a nonzero fixed vector are also characterized

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

linear rank and corank preserving maps on b h and an application to semigroup isomorphisms of operator ideals abstract we characterize linear rank k nonincreasing rank k preserving and corank k preserving maps on b h the algebra of all bounded linear operators on the hilbert space h this unifies and extends finite dimensional results and results on linear rank 1 non increasing and rank 1 preserving maps in the infinite dimensional case we conclude with an application to semigroup isomorphisms of operator ideals

linear maps on mn c preserving the local spectrum let x0 cn be a nonzero vector we prove that if a linear map mn c mn c preserves the local spectrum at x0 i e t x0 t x0 for all t mn c then there exists an invertible matrix a such that a x0 x0 and t ata 1 for every t

additive maps derivable at some points on j subspace lattice algebras abstract let l be a j subspace lattice on a real or complex banach space dim x with x 2 and alg l be the associated j subspace lattice algebra let alg l alg l be an additive map it is shown that if is derivable at zero point i e ab a b a b whenever ab 0 then a a a a where is an additive derivation and is a scalar if is generalized derivable at zero point i e ab a b a b a i b whenever ab 0 then is a generalized derivation it is also shown that if x is complex then every linear map derivable at unit operator on alg l is a derivation

on a spectral characterization of rank one matrices abstract let a be an n n complex matrix with a 2 0 it is proved that a has rank one if and only if t a t ca t for every rank one matrix t and some fixed scalar c 0 or 1 the condition a 2 0 cannot be dropped in case n 4

on positive linear maps preserving invertibility theorem a positive linear map 4 between two c algebras is a jordan homomorphism if d preserves invertibility and the range of is a c algebra a counterexample is given for the case that the range of q is not assumed to be a c algebra this answers a question raised by b russo proc amer math sot 17

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

quantum renyi relative entropies on density spaces of c algebras their symmetries and their essential difference abstract we extend the definitions of different types of quantum renyi relative entropy from the finite dimensional setting of density matrices to density spaces of c algebras we show that those quantities which trivially coincide in the classical commutative case are essentially different on non commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces besides we determine the symmetry groups of density spaces corresponding to each of those quantum renyi relative entropies and find that they are identical similar results concerning the umegaki and the belavkin staszewksi relative entropies are also presented

invertibility preservers on central simple algebras abstract we solve kaplansky s problem concerning the structure of linear preservers of invertibility in the special case of maps on central simple algebras

characterizing isomorphisms in terms of completely preserving invertibility or spectrum let a and b be standard operator algebras on complex banach spaces x and y respectively in this paper we characterize the surjective maps completely preserving the invertibility in both directions and the surjective maps completely preserving the spectrum from a to b we show that a surjective map from a to b is a ring isomorphism if and only if it is unital and completely preserves the invertibility of operators in both directions is an isomorphism if and only if it completely preserves the spectrum of operators

all derivable points of operator algebras abstract let a be an operator subalgebra in b h where h is a hilbert space we say that an element z a is an all derivable point of a for the norm topology strongly operator topology etc if every norm topology strongly operator topology etc continuous derivable linear mapping at z i e st s t s t for any s t a with st z is a derivation in this paper we show that every invertible operator in the nest algebra alg n is an all derivable point of the nest algebra for the strongly operator topology we also prove that every nonzero element of the algebra of all 2 2 upper triangular matrixes is an all derivable point of the algebra

bures isometries between density spaces of c algebras abstract we describe the precise structure of all surjective a priori nonlinear or better say nonaffine bures isometries between density spaces of c algebras equipped with faithful traces it turns out that those maps are closely related to linear jordan isomorphisms between the underlying algebras beside density spaces we consider the problem also on the positive definite and positive semidefinite cones of c algebras