strong convergence for the modified mann s iteration of λ strict pseudocontraction abstract in this paper for an strict pseudocontraction t we prove strong convergence of the modified mann s iteration defined by x n 1 n u n x n 1 n n n tx n 1 n x n where n n and n in 0 1 satisfy i 0 n k 2 with lim inf n n k 2 n 0 ii lim n n 0 and n 1 n iii lim sup n n 1 our results unify and improve some existing results

strong convergence of modified mann iterations abstract the mann iterations for nonexpansive mappings have only weak convergence even in a hilbert space we propose two modifications of the mann iterations in a uniformly smooth banach space one for nonexpansive mappings and the other for the resolvent of accretive operators the two modified mann iterations are proved to have strong convergence

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

partial rigidity of degenerate cr embeddings into spheres abstract in this paper we study degenerate cr embeddings f of a strictly pseudoconvex hypersurface m c n 1 into a sphere s in a higher dimensional complex space c n 1 the degeneracy of the mapping f will be characterized in terms of the ranks of the cr second fundamental form and its covariant derivatives in 2004 the author together with x huang and d zaitsev established a rigidity result for cr embeddings f into spheres in low codimensions a key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere partial rigidity in the 2004 paper it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is n here n is the cr dimension of m then f m is contained in a complex plane of dimension n d 1 the converse of this statement is also true as is easy to see when the total rank d exceeds n it is no longer true in general that f m is contained in a complex plane of dimension n d 1 as can be seen by examples in this paper we carry out a systematic study of degenerate cr mappings into spheres we show that when the ranks of the second fundamental form and its covariant derivatives exceed the cr dimension n then partial rigidity may still persist but there is a defect k that arises from the ranks exceeding n such that f m is only contained in a complex plane of dimension n d k 1 moreover this defect occurs in general as is illustrated by examples

faster and simpler algorithms for multicommodity flow and other fractional packing problems this paper considers the problem of designing fast approximate combinatorial algorithms for multicommodity flows and other fractional packing problems we present new faster and much simpler algorithms for these problems

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

proper holomorphic mappings from the two ball to the three ball we prove that a proper mapping of the two ball in cn into the three ball which is c2 on the closed two ball is equivalent to one of four normalized polynomial mappings this improves the known result of faran the proof is basic using taylor expansions

convergence theorems of iterative algorithms for continuous pseudocontractive mappings in a real reflexive and strictly convex banach space e with a uniformly g teaux differentiable norm we introduced the viscosity iterative process xt given by h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 to continuous pseudocontractive self mapping and proved that the xt strongly converges x f t as t 0 and x is a unique solution in f t to the following variational inequality f x x j z x 0for all z f t we also proposed the modified implicit iteration process xn for continuous pseudocontractive self mapping and show that xn strongly converges x f t the results presented extended and improved the corresponding results of h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291

new strong convergence theorems for nonexpansive nonself mappings without boundary conditions the aim of this work is to establish the strong convergence of explicit viscosity like methods for a nonexpansive nonself mapping without boundary conditions defined in a banach space this gets rid of the restriction and dependence on the implicit anchor like continuous path x t in the existing literature

strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration in this paper we study the strong convergence of viscosity iteration and modified viscosity iteration process for strongly continuous semigroup of uniformly lipschitzian asymptotically pseudocontractive mappings

rigidity of cr maps between shilov boundaries of bounded symmetric domains our goal is to establish what seems to be the first rigidity result for cr embeddings between shilov boundaries of bounded symmetric domains of higher rank the result states that any such cr embedding is the standard linear embedding up to cr automorphisms our basic assumption extends precisely the well known optimal bound for the rank one case there are no other restrictions on the ranks in particular the difficult case when the target rank is larger than the source rank is also allowed

strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings abstract let x be a real reflexive and strictly convex banach space with a uniformly g teaux differentiable norm first purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self mapping t and prove that this iterative process converges strongly to x f t x x x t x where x is the unique solution in f t to the following variational inequality f x x j v x 0 for all v f t second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self mapping t and prove that these iterative schemes strongly converge to the same point x f t basically we show that if the perturbation mapping is nonexpansive then the convergence property of the iterative process holds in this respect the results presented here extend improve and unify some very recent theorems in the literature see l c zeng j c yao implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings nonlinear anal 64 2006 2507 2515 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 y s song r d chen convergence theorems of iterative algorithms for continuous pseudocontractive mappings nonlinear anal 2006

iterative convergence to cesaro means for continuous pseudocontractive mappings abstract let f be a fixed strongly pseudocontractive mapping and t be a continuous pseudocontractive mapping with f t 0 the sequence z m is iteratively defined as follows z m t m f z m 1 t m 1 m 1 j 0 m t j z m m 0 where t m 0 1 satisfies the condition lim m t m 0 we prove that z m converges strongly to the unique solution p to some variational inequality in f t our results develop and complement the corresponding ones by matsushita daishi kuroiwa j math anal appl 294 2004 206 214 and shioji takahashi arch math 72 1999 354 359 and moore nnoli j math anal appl 260 2001 269 278 and su li appl math comput 181 2006 332 341 and song chen appl math comput 186 2007 1120 1128 and wangkeeree appl math comput 201 2008 239 249

initial monomial invariants of holomorphic maps we study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals we start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics giving a readily computable invariant in this important case for example the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball next we associate to each subspace x subset mathcal o u a generic initial monomial subspace which is invariant under biholomorphic transformations and multiplication by nonzero functions the generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

volume preserving mappings between hermitian symmetric spaces of compact type abstract in this paper we establish the rigidity result for local holomorphic volume preserving maps from an irreducible hermitian symmetric space of compact type into its cartesian products

approximating the stochastic knapsack problem the benefit of adaptivity we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size our goal is to compute a solution policy that maximizes the expected value of items successfully placed in the knapsack where the final overflowing item contributes no value we consider both nonadaptive policies that designate a priori a fixed sequence of items to insert and adaptive policies that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far an important facet of our work lies in characterizing the benefit of adaptivity for this purpose we advocate the use of a measure called the adaptivity gap the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy we bound the adaptivity

the equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics we consider the class of levi nondegenerate hypersurfaces m in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i that admit a local cr transversal embedding near a point p e m into a standard nondegenerate hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i with codimension k n n small compared to the cr dimension n of m we show that for hypersurfaces in this class there is a normal form which is closely related to the embedding such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric we also show that if the signature of m and that of the standard hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i are the same then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

rigidity of proper holomorphic mappings between certain unbounded non hyperbolic domains abstract the fock bargmann hartogs domain d n m 0 in c n m is defined by the inequality w 2 e z 2 where z w c n c m which is an unbounded non hyperbolic domain in c n m recently yamamori gave an explicit formula for the bergman kernel of the fock bargmann hartogs domains in terms of the polylogarithm functions and kim ninh yamamori determined the automorphism group of the domain d n m in this article we obtain rigidity results on proper holomorphic mappings between two equidimensional fock bargmann hartogs domains our rigidity result implies that any proper holomorphic self mapping on the fock bargmann hartogs domain d n m with m 2 must be an automorphism

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

on iterations methods for zeros of accretive operators in banach spaces in this paper three iterations are designed to approach zeros of set valued accretive operators in banach spaces the first one is the continuous picard type iteration involving the resolvent the second one is the approximate picard type iteration involving the resolvent and the third one is the halpern type iteration involving the resolvent some strong convergence theorems for three iterations are proved

an explicit iteration for zeros of accretive operators abstract in this paper for lipschitz accretive operator a an iteration scheme is defined as follows x n 1 1 n x n n u n ax n its strong convergence is established for finding some zero of a whenever n n 0 1 satisfying conditions lim n n 0 n 1 n lim n n 0 furthermore some applications for equilibrium problems are given also in particular the iteration coefficient is simpler and more general

rigidity of proper holomorphic self mappings of the pentablock abstract the pentablock is a hartogs domain in c 3 over the symmetrized bidisc in c 2 the domain is a bounded inhomogeneous pseudoconvex domain which does not have a c 1 boundary recently agler lykova young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock and kosinski fully described the group of holomorphic automorphisms of the pentablock the aim of the present study is to prove that any proper holomorphic self mapping of the pentablock must be an automorphism

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

on the adaptivity gap of stochastic orienteering the input to the stochastic orienteering problem gupta et al in soda pp 1522 1538 2012 consists of a budget b and metric v d where each vertex v in v v v has a job with a deterministic reward and a random processing time drawn from a known distribution the processing times are independent across vertices the goal is to obtain a non anticipatory policy originating from a given root vertex to run jobs at different vertices that maximizes expected reward subject to the total distance traveled plus processing times being at most b an adaptive policy can choose the next vertex to visit based on observed random instantiations whereas a non adaptive policy is just given by a fixed ordering of vertices the adaptivity gap is the worst case ratio of the optimal adaptive and non adaptive rewards we prove an varomega left log log b 1 2 right loglogb 1 2 lower bound on the adaptivity gap of stochastic orienteering this provides a negative answer to the o 1 adaptivity gap conjectured by gupta et al 2012 and comes close to the o log log b o loglogb upper bound this result holds even on a line metric we also show an o log log b o loglogb upper bound on the adaptivity gap for the correlated stochastic orienteering problem where the reward of each job is random and possibly correlated to its processing time using this we obtain an improved quasi polynomial time min log n log b cdot tilde o log 2 log b min logn logb o log2logb approximation algorithm for correlated stochastic orienteering

d angelo conjecture in the third gap interval we show that the d angelo conjecture holds in the third gap interval more precisely we prove that the degree of any rational proper holomorphic map from mathbb b n to mathbb b 4n 6 with n ge 7 is not more than 3

equivalent theorems of the convergence between proximal type algorithms abstract the main aim of this work is to give the equivalence of the convergence between proximal type algorithms our convergence analysis covers rockafellar type proximal point algorithm with a weak contraction for finding zeros of accretive operators as well as browder and halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in banach spaces we also provide simple proofs of some results of wong sahu and yao n c wong d r sahu j c yao solving variational inequalities involving nonexpansive type mappings nonlinear anal 69 12 2008 4732 4753

dimension and rigidity of quasi fuchsian representations let fo c so n 1 n 2 be a cocompact lattice and p fo f be an injective representation into a convex cocompact discrete isometric subgroup of a noncompact rank 1 symmetric space the hausdorff dimension 6 f of the limit set of f p fo satisfies 6 f 6 fo n 1 we prove that equality holds if and only if p is a fuchsian representation i e f preserves a totally geodesic copy of hri in hr this generalizes the result of 2 and settles a question raised by tukia 43 p 428 actually we prove a more general result in the context of variable negative curvature strikingly there are no quasi fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry that is for a cocompact lattice fo c su n 1 n 2 and an injective representation p fo su m 1 n m 2n 1 with f p fo convex cocompact we prove that one always has 6 f 6 fo and moreover f must stabilize a totally geodesic copy of h in h this can be viewed as a global generalization of goldman and millson s local rigidity theorem see 20 another global generalization was obtained by k corlette 5 various other related rigidity results are also obtained

rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains we prove that any proper holomorphic mapping from d p p 1 i to d p p i p geq 3 is necessarily a totally geodesic isometric embedding with respect to their bergman metrics and therefore is the standard linear embedding up to their automorphisms the proof of the main result in this paper will be achieved by a differential geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their bergman metrics

strong convergence of iteration algorithms for a countable family of nonexpansive mappings abstract for a countable family t n n 1 of nonexpansive mappings a strong convergence of halpern type iteration is shown in order to find a common fixed point of t n n 1 in a reflexive banach space when n 0 1 satisfies the conditions c1 lim n n 0 and c2 n 1 n and several examples satisfying the condition b is given also we also research the strong convergence of the proximal point algorithm finally we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings

rigidity and holomorphic segre transversality for holomorphic segre maps let mathcal h n and mathcal h mathcal n denote the complexifications of heisenberg hypersurfaces in mathbf c n and mathbf c n respectively we show that non degenerate holomorphic segre mappings from mathcal h n into mathcal h n with n leq2n 2 possess a partial rigidity property as an application we prove that the holomorphic segre non transversality for a holomorphic segre map from mathcal h n into mathcal h n with n leq2n 2 propagates along segre varieties we also give an example showing that this propagation property of holomorphic segre transversality fails when n 2n 2

convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings abstract in this paper a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real banach space using an implicit iteration scheme recently introduced by xu and ori h k xu r g ori an implicit iteration process for nonexpansive mappings numer fuct anal optim 22 2001 767 773 in condition n 0 1 and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of browder petryshyn type is obtained the results presented extend and improve the corresponding results of m o osilike m o osilike implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps j math anal appl 294 2004 73 81

symmetries in cr complexity theory abstract we introduce the hermitian invariant group f of a proper rational map f between the unit ball in complex euclidean space and a generalized ball in a space of typically higher dimension we use properties of the groups to define the crucial new concepts of essential map and the source rank of a map we prove that every finite subgroup of the source automorphism group is the hermitian invariant group of some rational proper map between balls we prove that f is non compact if and only if f is a totally geodesic embedding we show that f contains an n torus if and only if f is equivalent to a monomial map we show that f contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers we also establish a monotonicity result the group after intersecting with the unitary group does not decrease when a tensor product operation is applied to a polynomial proper map we give a necessary condition for f when the target is a generalized ball to contain automorphisms that move the origin

on reich s strong convergence theorems for resolvents of accretive operators let e be a real banach space with norm i then an operator a c e x e with domain d a and range r a is said to be accretive if xi xz 0 an accretive operator a c e x e is m accretive if r i ra e for all r 0 where z is the identity if a is accretive we can define for each positive r a singlevalued mapping j r z ra d a by j z ra it is called the resolvent of a in 6 reich proved the following theorem let e be a uniformly smooth banach space and let a c e x e be m accretive if 0 e r a then for each x in e the strong lim z x exists and belongs to a io he remarked also that the assumption that a is m accretive can be replaced with the assumption that cl a the closure of d a is convex and that a satisfies the range condition r z ra 1 cl d a for all r 0 in this paper we first prove a theorem that generalizes simultaneously the above results furthermore we generalize another reich strong convergence theorem asserting that the strong lim z x exists in uniformly convex and uniformly smooth banach spaces though our proofs are similar to those in 6 they are slightly simple on account of using banach limits

on the c regularity of cr mappings of positive codimension abstract the present paper tackles the c regularity problem for cr maps h m m between c smooth cr submanifolds m m embedded in complex spaces of possibly different dimensions for real hypersurfaces m c n 1 and m c n 1 with n n 1 and m strongly pseudoconvex we prove that every cr transversal map of class c n n 1 that is nowhere c on some non empty open subset of m must send this open subset to the set of d angelo infinite points of m as a corollary we obtain that every cr transversal map h m m of class c n n 1 must be c smooth on a dense open subset of m when m is of d angelo finite type another consequence establishes the following boundary regularity result for proper holomorphic maps in positive codimension given c n 1 and c n 1 pseudoconvex domains with smooth boundaries and both of d angelo finite type n n 1 any proper holomorphic map h that extends c n n 1 smoothly up to must be c smooth on a dense open subset of more generally for cr submanifolds m and m of higher codimensions our main result describes the impact of the existence of a nowhere smooth cr map h m m on the cr geometry of m allowing to extend the previously mentioned results in the hypersurface case to any codimension as well as deriving a number of regularity results for cr maps with d angelo infinite type targets

local holomorphic isometries of a modified projective space into a standard projective space rational conformal factors we consider local modifications omega n f omega d of the fubini study metric with associated 1 1 form omega n on an open subset omega subset mathbb c mathbb p n induced by a local holomorphic mapping f omega rightarrow mathbb p d our main result is that there are gaps in potential dimensions m such that the modification can be obtained as h omega m for some local holomorphic mapping h omega rightarrow mathbb c mathbb p m we also consider the case of rational conformal factors

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

rational proper holomorphic maps from b n into b 2 n in this paper we will investigate rational proper holomorphic maps from b n into b2 n when n 4

rigidity of proper holomorphic mappings between equidimensional hua domains hua domain named after chinese mathematician loo keng hua is defined as a domain in mathbb c n fibered over an irreducible bounded symmetric domain omega subset mathbb c d with the fiber over z in omega being a n d dimensional generalized complex ellipsoid sigma z in general a hua domain is a nonhomogeneous bounded pseudoconvex domain without smooth boundary the purpose of this paper is twofold firstly we obtain what seems to be the first rigidity results on proper holomorphic mappings between two equidimensional hua domains secondly we determine the explicit form of the biholomorphisms between two equidimensional hua domains as a special conclusion of this paper we completely describe the group of holomorphic automorphisms of the hua domain

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

approximation algorithms for orienteering and discounted reward tsp in this paper we give the first constant factor approximation algorithm for the rooted orienteering problem as well as a new problem that we call the discounted reward traveling salesman problem tsp motivated by robot navigation in both problems we are given a graph with lengths on edges and rewards on nodes and a start node s in the orienteering problem the goal is to find a path starting at s that maximizes the reward collected subject to a hard limit on the total length of the path in the discounted reward tsp instead of a length limit we are given a discount factor gamma and the goal is to maximize the total discounted reward collected where the reward for a node reached at time t is discounted by gamma t this problem is motivated by an approximation to a planning problem in the markov decision process mdp framework under the commonly employed infinite horizon discounted reward optimality criterion the approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one time events and nonrepeatable rewards such as for package deliveries we also consider tree and multiple path variants of these problems and provide approximations for those as well although the unrooted orienteering problem where there is no fixed start node s has been known to be approximable using algorithms for related problems such as k tsp in which the amount of reward to be collected is fixed and the total length is approximately minimized ours is the first to approximate the rooted question solving an open problem in e m arkin j s b mitchell and g narasimhan proceedings of the 14 th acm symposium on computational geometry 1998 pp 307 316 and b awerbuch y azar a blum and s vempala siam j comput 28 1998 pp 254 262 we complement our approximation result for orienteering by showing that the problem is apx hard

homotopy equivalence for proper holomorphic mappings abstract we introduce several homotopy equivalence relations for proper holomorphic mappings between balls we provide examples showing that the degree of a rational proper mapping between balls in positive codimension is not a homotopy invariant in domain dimension at least 2 we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite by contrast when the target dimension is at least twice the domain dimension it is well known that there are uncountably many spherical equivalence classes we generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps we introduce whitney sequences a precise analogue in higher dimensions of the notion of finite blaschke product in one dimension we show that terms in a whitney sequence are homotopic to monomial mappings and we establish an additional result about the target dimensions of such homotopies

super rigidity for cr embeddings of real hypersurfaces into hyperquadrics we consider holomorphic mappings sending a given levi nondegenerate pseudoconcave hypersurface m in into a nondegenerate hyperquadric of the same signature in and show that if m is sufficiently close to a hyperquadric in a certain sense then any two such mappings differ only by an automorphism of the hyperquadric

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

uniqueness of certain polynomials constant on a line abstract we study a question with connections to linear algebra real algebraic geometry combinatorics and complex analysis let p x y be a polynomial of degree d with n positive coefficients and no negative coefficients such that p 1 when x y 1 a sharp estimate d 2 n 3 is known in this paper we study the p for which equality holds we prove some new results about the form of these sharp polynomials using these new results and using two independent computational methods we give a complete classification of these polynomials up to d 17 the question is motivated by the problem of classification of cr maps between spheres in different dimensions

iterative construction for common fixed points of finitely many nonexpansive mappings abstract many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a banach space in this paper under the hypotheses that e is a reflexive and strictly convex banach spaces with a uniformly g eaux differentiable norm the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self mappings defined on a closed convex subset k of e