the g good neighbor conditional diagnosability of star graphs under the pmc and mm model diagnosability plays an important role in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify this paper shows that the g good neighbor conditional diagnosabilities of star graph sn under the pmc and mm model are ng g 1 1 for n4 and 0gn2 our study shows the g good neighbor conditional diagnosability of star graph is several times larger than the classical diagnosability of star graph

metrics on semistable and numerically effective higgs bundles we provide notions of numerical effectiveness and numerical flatness for higgs vector bundles on compact kahler manifolds in terms of fibre metrics we prove several properties of bundles satisfying such conditions and in particular we show that numerically flat higgs bundles have vanishing chern classes and that they admit filtrations whose quotients are stable flat higgs bundles we compare these definitions with those previously given in the case of projective varieties finally we study the relations between numerically effectiveness and semistability establishing semistability criteria for higgs bundles on projective manifolds of any dimension

the c neumann problem as a completely integrable system on an adjoint orbit it is shown by purely lie algebraic methods that the c neumann problem the motion of a material point on a sphere under the influence of a quadratic potential is a completely integrable system of euler poisson equations on a minimal dimensional orbit of a semidirect product of lie algebras 1 the c neumann problem the motion of a point on the sphere s 1 under the influence of a quadratic potential u x ax x x e r a diag al an is a completely integrable hamiltonian system for n 3 this has been shown by c neumann in 1859 12 and for arbitrary n by k uhlenbeck 16 r devaney 3 j moser 10 11 m adler and p van moerbeke 2 in this paper we show how this problem fits naturally in the framework of euler poisson equations 41 5 14 17 proving that the c neumann problem is a hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of lie algebras thus its complete integrability will follow entirely from lie algebraic considerations the equations of motion are i ax1 axx i 1 n 1 1 where the lagrange multiplier a ax x 11x112 is chosen such that x e sn i during the motion set x y and get the equivalent system to 1 1 xi yi yi a1xi ax x 11y112 xi llxll 1 x y 0 1 2 the following crucial remark that motivated the present investigation is due to k uhlenbeck and can be verified without any difficulties lemma 1 1 put x xixj p yixj xiyj system 1 2 is equivalent to x p x p x a lixl 1 x y o 1 3 remark that if one replaces x and a by x id n and a tr a id n respectively where id is the n x n identity matrix equations 1 3 remain unchanged from now on we shall assume that in 1 3 this change has been made so that x p a e sl n the next section gives a lie algebraic interpretation to these equations received by the editors march 12 1980 1980 mathematics subject classification primary 58f07 70h05 secondary 53c15 17b99

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

on the g good neighbor connectivity of graphs abstract connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network g in 1996 fabrega and fiol proposed the g good neighbor connectivity of g in this paper we show that 1 g g n 2 g 2 for 0 g g n 3 2 and graphs with g g 1 2 and trees with g t n n t for 4 t n 2 2 are characterized respectively in the end we get the three extremal results for the g good neighbor connectivity

quadrics on complex riemannian spaces of constant curvature separation of variables and the gaudin magnet integrable systems that are connected with orthogonal separation of variables in complex riemannian spaces of constant curvature are considered herein an isomorphism with the hyperbolic gaudin magnet previously pointed out by one of the authors extends to coordinates of this type the complete classification of these separable coordinate systems is provided by means of the corresponding l matrices for the gaudin magnet the limiting procedures or e calculus which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems

remarks on kowalevski s top we present a review of some results on kowalevski s top kt in classical and quantum mechanics the following items are considered i the generalized kt gkt on the lie algebras so 4 e 3 and so 3 1 ii kowalevski s gyrostat on these algebras iii action of the gkt iv quantum counterparts of the kt v semiclassical quantization of the gkt vi generalization of the kt by chaplygin and goryachev at l 0 and its lax representation unsolved questions are also discussed

lax representation and poisson geometry of the kowalevski top we discuss the poisson structure underlying the two field kowalevski gyrostat and the kowalevski top we start from their lax structure and construct a suitable pencil of poisson brackets which endows these systems with the structure of bi hamiltonian completely integrable systems we study the casimir functions of such pencils and show how it is possible to frame the kowalevski systems within the so called gel fand zakharevich bi hamiltonian setting for integrable systems

a new integrable system on s2 with the second integral quartic in the momenta we construct a new integrable system on the sphere s2 with an additional integral of fourth order in the momenta using standard machinery of the reflection equation theory at the special values of parameters this system coincides with the kowalevski goryachev chaplygin system

einstein hermitian connections on polystable principal bundles over a compact kahler manifold given a compact kahler manifold m and a connected reductive algebraic group g over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i a prinlcipal g bundle over m admits an einstein hermitian connection if and only if the principal bundle is polystable if m is a projective manifold a higgs g bundle over m admits an einstein hermitian connection if and only if the higgs bundle is polystable

the g good neighbor diagnosability of n k star graphs many large scale multiprocessor or multi computer systems take interconnection networks as underlying topologies fault diagnosis is especially important to identify fault tolerability of such systems the g good neighbor conditional diagnosability such that every fault free node has at least g fault free neighbors is a novel measure of diagnosability in this paper we show that the g good neighbor diagnosability of the n k star graph s n k under the pmc model 2 k n 1 and 1 g n k and the comparison model 2 k n 1 and 2 g n k is n g k 1 1 respectively in addition we derive that 1 good neighbor diagnosability of s n k under the comparison model is n k 2 for 3 k n 1 and n 4 as a supplement we also derive that the g good neighbor diagnosability of the n 1 star graph s n 1 1 g n 2 1 and n 4 under the pmc model and the comparison model is n 2 1 respectively we explore fault diagnosability of multiprocessor systems based on combinatorial network theory we establish the g good neighbor diagnosability of the n k star graph s n k under the pmc model we establish the g good neighbor diagnosability of the n k star graph s n k under the comparison model

lie point symmetries and first integrals the kowalevski top we show how the lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations the method of reduction increase of order developed by nucci j math phys 37 1772 1775 1996 is essential noether s theorem is neither necessary nor considered the most striking example we present is the relationship between lie group analysis and the famous first integral of the kowalevski top

a special case of neumann s system and the kowalewski chaplygin goryachev top the authors present l operators both for the special case of neumann s system and the kowalewski chaplygin goryachev top these are quantum systems integrable by the quantum inverse scattering r matrix method l operators being 2 2 matrices satisfy an algebra generated by an r matrix of the xxx type the close connection between the two models is demonstrated they carry out a non obvious separation of variables and also give a dynamical group scheme for the eigenstate problem of neumann s system this separation differs from those in euler angles and allows them to find eigenenergies in an effective way

the g good neighbor and g extra diagnosability of networks abstract diagnosability of a multiprocessor system is an important research topic the system and interconnection network has a underlying topology which usually presented by a graph g v e in 2012 a measurement for fault tolerance of the graph was proposed by peng et al this measurement is called the g good neighbor diagnosability that restrains every fault free node to contain at least g fault free neighbors in 2016 zhang et al proposed a new measurement for fault diagnosis of the graph namely the g extra diagnosability which restrains that every fault free component has at least g 1 fault free nodes a fault set f v is called a g good neighbor faulty set if the degree d v g for every vertex v in g f a g good neighbor cut of g is a g good neighbor faulty set f such that g f is disconnected the minimum cardinality of g good neighbor cuts is said to be the g good neighbor connectivity of g a fault set f v is called a g extra faulty set if every component of g f has at least g 1 vertices a g extra cut of g is a g extra faulty set f such that g f is disconnected the minimum cardinality of g extra cuts is said to be the g extra connectivity of g the g good neighbor extra diagnosability and g good neighbor extra connectivity of many well known graphs have been widely investigated in this paper we show the relationship between the g good neighbor extra diagnosability and g good neighbor extra connectivity of graphs

the 1 2 good neighbor conditional diagnosabilities of regular graphs abstract fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems in 2012 peng et al proposed a new measure for the fault diagnosis of systems namely g good neighbor conditional diagnosability which requires that any fault free vertex has at least g fault free neighbors in the system the g good neighbor conditional diagnosabilities of a graph g under the pmc model and the mm model are denoted by t g pmc g and t g mm g respectively in this paper we first determine that t g pmc g t g mm g if g 2 second we establish a general result on the 1 2 good neighbor conditional diagnosabilities of some regular graphs as applications the 1 2 good neighbor conditional diagnosabilities of bc graphs folded hypercubes and four classes of cayley graphs namely unicyclic transposition graphs wheel transposition graphs complete transposition graphs and tree transposition graphs are determined under the pmc model and the mm model in addition we determine the r2 connectivities of bc graphs and folded hypercubes

new generalizations of all the known integrable problems in rigid body dynamics in the present paper we consider a quite general problem of motion of a rigid body under the action of an axially symmetric combination of potential and gyroscopic forces special versions of this problem are the classical problems of motion of a heavy body gyrostat or a body in a liquid another example is the problem of motion of a heavy magnetized and electrically charged body under the combined action of gravitational magnetic electric and lorentz forces we introduce a method based on the invariance of the equations of motion under certain transformations this method enables a systematic construction of several integrable problems generalizing all the known general for arbitrary initial conditions and conditional on a single level of the cyclic integral integrable problems as well as particular solutions five general integrable cases are constructed that generalize known ones by the inclusion of one to four additional free parameters fifteen conditional integrable cases generalizing all the known general and conditional cases and containing an additional arbitrary function are introduced the latter type of generalization can be applied to any one of a large collection of particular solutions known in many versions of the problem considered the method also enables the construction of the explicit solutions of the equations of motion in the generalized cases from those of the original cases detailed physical interpretation is given for some cases

projective manifolds whose tangent bundles are numerically effective in 1979 mori mo proved the so called hartshorne frankel conjecture every projective n dimensional manifold with ample tangent bundle is isomorphic to the complex projective space p a differential geometric analogon assuming the existence of a k ihler metric on x with positive holomorphic bisectional curvature is independently due to siu yau sy thus it seems natural to classify projective manifolds x whose tangent bundle tx satisfy a degenerate condition of ampleness numerical effectivity abbreviated by nef this means that the tautological quotient line bundle d 1 on f tx is numerically effective i e c 0

the g good neighbor conditional diagnosability of hypercube under pmc model abstract processor fault diagnosis plays an important role in multiprocessor systems for reliable computing and the diagnosability of many well known networks has been explored for example hypercubes crossed cubes mobius cubes and twisted cubes of dimension n all have diagnosability n the conditional diagnosability of n dimensional hypercube q n is proved to be 4 n 2 1 under the pmc model in this paper we study the g good neighbor conditional diagnosability of q n under the pmc model and show that it is 2 g n g 2 g 1 for 0 g n 3 the g good neighbor conditional diagnosability of q n is several times larger than the classical diagnosability

lax representation for the systems of s kowalevskaya type we describe a number of dynamical systems that are generalizations of the s kowalevskaya system and admit the lax representation

the g good neighbor diagnosability of locally twisted cubes abstract diagnosability of a multiprocessor system is one important measure of the reliability of interconnection networks in 2012 peng et al proposed the g good neighbor diagnosability that restrains every fault free node containing at least g fault free neighbors the locally twisted cube l t q n is applied widely in this paper we give that the g good neighbor diagnosability of l t q n is 2 g n g 1 1 under the pmc model and the mm model for n 3 and 0 g n 3

simultaneous separation for the kowalevski and goryachev chaplygin gyrostats in the special case of zero square integral the kowalevski gyrostat and goryachev chaplygin gyrostat share a simple separation of variables originated from the 4 4 lax matrix

semistable and numerically effective principal higgs bundles we study miyaoka type semistability criteria for principal higgs g bundles e on complex projective manifolds of any dimension we prove that e has the property of being semistable after pullback to any projective curve if and only if certain line bundles obtained from some characters of the parabolic subgroups of g are numerically effective one also proves that these conditions are met for semistable principal higgs bundles whose adjoint bundle has vanishing second chern class in a second part of the paper we introduce notions of numerical effectiveness and numerical flatness for principal higgs bundles discussing their main properties for non higgs principal bundles we show that a numerically flat principal bundle admits a reduction to a levi factor which has a flat hermitian yang mills connection and as a consequence that the cohomology ring of a numerically flat principal bundle with coefficients in r is trivial to our knowledge this notion of numerical effectiveness is new even in the case of non higgs principal bundles

poisson maps and integrable deformations of the kowalevski top we construct a poisson map between manifolds with linear poisson brackets corresponding to the lie algebras e 3 and so 4 using this map we establish a connection between the deformed kowalevski top on e 3 proposed by sokolov and the kowalevski top on so 4 the connection between these systems leads to the separation of variables for the deformed system on e 3 and yields the natural 5 5 lax pair for the kowalevski top on so 4

on a trajectory isomorphism of the kowalevski gyrostat and the clebsch problem for the kowalevski gyrostat a change of variables similar to that for the kowalevski top is done we establish one to one correspondence between solutions of the kowalevski gyrostat and the clebsch system and demonstrate that kowalevski variables for the gyrostat practically coincide with elliptic coordinates on a sphere for the clebsch case

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

kowalewski s top on the lie algebras o 4 e 3 and o 3 1 the authors extend the lax pair for kowalewski s top kt obtained by haine and horozov 1987 to the lie algebras o 4 and o 3 1 the results are expressed in terms of a solution of the neumann system they derive formulae for the action variables by analogy with the e 3 kt case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables

g good neighbor conditional diagnosability measures for 3 ary n cube networks the diagnosability of a parallel system is defined as the maximum number of faulty processors or nodes that the system can guarantee to identify in this study we investigate the g good neighbor conditional diagnosability which indicates that every fault free node in a system contains at least g fault free neighbors compared with the conventional diagnosability g good neighbor conditional diagnosability improves accuracy in measuring the reliability of interconnection networks in heterogeneous environments we apply the pmc and mm models to study the g good neighbor conditional diagnosability of 3 ary n cube networks which represent a family of popular parallel systems such as ibm s blue gene and cray t3d the findings made in this study facilitate accurate reliability measurements in modern parallel systems powered by 3 ary n cube networks specifically our results show that the g good neighbor conditional diagnosability of 3 ary n cube is g 2 2 n g 1 1 and g 1 2 4 n 2 g 1 1 when the g value is even and odd respectively the r g connectivity of 3 ary n cube networks was determined for n 3 and 0 g n 1 applying the pmc model and the mm model we investigated and measured the g good neighbor conditional diagnosability of 3 ary n cube networks for n 4 and 0 g n 1