lower bounds for randomized k serverand motion planning algorithms in this paper the authors prove lower bounds on the competitive ratio of randomized algorithms for two on line problems the k server problem suggested by manasse mcgeoch and sleator competitive algorithms for on line problems j algorithms 11 1990 pp 208 230 and an on line motion planning problem due to papadimitriou and yannakakis shortest paths without a map lecture notes in comput sci 372 springer verlag new york 1989 pp 610 620 the authors prove against an oblivious adversary an omega log k lower bound on the competitive ratio of any randomized on line k server algorithm in any sufficiently large metric space an omega log log k lower bound on the competitive ratio of any randomized on line k server algorithm in any metric space with at least k 1 points and an omega log log n lower bound on the competitive ratio of any on line motion planning algorithm for a scene with n obstacles previously no superconstant lower bound on the competitive ratio of randomized on line algorithms was known for any of these problems

connectedness at infinity of complete kahler manifolds one of the main purposes of this paper is to prove that on a complete k ahler manifold of dimension m if the holomorphic bisectional curvature is bounded from below by 1 and the minimum spectrum lambda 1 m ge m 2 then it must either be connected at infinity or isometric to bbb r times n with a specialized metric with n being compact generalizations to complete k ahler manifolds satisfying a weighted poincar e inequality are also being considered

competitive exploration of rectilinear polygons exploring a polygon with robots when the robots do not have knowledge of the surroundings can be viewed as an online problem typical for online problems is that decisions must be made based on past events without complete information about the future in our case the robots do not have complete information about the environment competitive analysis can be used to measure the performance of methods solving online problems the competitive ratio of such a method is the ratio between the method s performance and the performance of the best method having full knowledge of the future we prove constant competitive strategies and lower bounds for exploring a simple rectilinear polygon in the l1 metric

tree exploration with advice we study the amount of knowledge about the network that is required in order to efficiently solve a task concerning this network the impact of available information on the efficiency of solving network problems such as communication or exploration has been investigated before but assumptions concerned availability of particular items of information about the network such as the size the diameter or a map of the network in contrast our approach is quantitative we investigate the minimum number of bits of information bits of advice that has to be given to an algorithm in order to perform a task with given efficiency we illustrate this quantitative approach to available knowledge by the task of tree exploration a mobile entity robot has to traverse all edges of an unknown tree using as few edge traversals as possible the quality of an exploration algorithm a is measured by its competitive ratio i e by comparing its cost number of edge traversals to the length of the shortest path containing all edges of the tree depth first search has competitive ratio 2 and in the absence of any information about the tree no algorithm can beat this value we determine the minimum number of bits of advice that has to be given to an exploration algorithm in order to achieve competitive ratio strictly smaller than 2 our main result establishes an exact threshold number of bits of advice that turns out to be roughly loglogd where d is the diameter of the tree more precisely for any constant c we construct an exploration algorithm with competitive ratio smaller than 2 using at most loglogd c bits of advice and we show that every algorithm using loglogd g d bits of advice for any function g unbounded from above has competitive ratio at least 2

boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

on largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces we apply topological methods to study the smallest non zero number lambda 1 in the spectrum of the laplacian on finite area hyperbolic surfaces for closed hyperbolic surfaces of genus two we show that the set s in mathcal m 2 lambda 1 s frac 1 4 is unbounded and disconnects the moduli space mathcal m 2 using this for genus g ge 3 we show the existence of eigenbranches that start as lambda 1 and eventually becomes frac 1 4

an alternative proof for the equivalence of searcher and 2 searcher it was conjectured in 1992 that the 2 searcher who has two flashlights has the same searching capability as the searcher who has an omni directional light source park et al proved this conjecture affirmatively in 2001 but their proof is rather complicated and not easy to understand in this paper we present an alternative proof which we believe is conceptually more transparent and easier to understand for this purpose we introduce a tool called the 2 link visibility diagram that represents 2 link visibility which has other applications

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

sharp poincare hardy and poincare rellich inequalities on the hyperbolic space abstract we study hardy type inequalities associated to the quadratic form of the shifted laplacian h n n 1 2 4 on the hyperbolic space h n n 1 2 4 being as it is well known the bottom of the l 2 spectrum of h n we find the optimal constant in a resulting poincare hardy inequality which includes a further remainder term which makes it sharp also locally the resulting operator is in fact critical in the sense of 17 a related improved hardy inequality on more general manifolds under suitable curvature assumption and allowing for the curvature to be possibly unbounded below is also shown it involves an explicit curvature dependent and typically unbounded potential and is again optimal in a suitable sense furthermore with a different approach we prove rellich type inequalities associated with the shifted laplacian which are again sharp in suitable senses

piecemeal graph exploration by a mobile robot we study how a mobile robot can learn an unknown environment in a piecemeal manner the robot s goal is to learn a complete map of its environment while satisfying the constraint that it must return every so often to its starting position for refueling say the environment is modeled as an arbitrary undirected graph which is initially unknown to the robot we assume that the robot can distinguish vertices and edges that it has already explored we present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph g v e in which the robot explores every vertex and edge in the graph by traversing at most o e v1 o 1 edges this nearly linear algorithm improves on the best previous algorithm in which the robot traverses at most o e v2 edges we also give an application of piecemeal learning to the problem of searching a graph for a treasure

on the local langlands conejcture mathrm gl n the cyclic case let f be a locally compact nonarchimedean field and p its residue characteristic let f be a separable algebraic closure of f wf the weil group of f over f e a character of fx of order n and e the corresponding character of wf then there exists a canonical map from the set of equivalence classes a of irreducible degree n representations of wf such that e a a to the set of equivalence classes 7r of admissible irreducible supercuspidal representations of gl n f verifying odet x 7t this map is shown to be bijective and to preserve e factors for pairs

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

poisson equation on complete manifolds abstract we develop heat kernel and green s function estimates for manifolds with positive bottom spectrum the results are then used to establish existence and sharp estimates of the solution to the poisson equation on such manifolds with ricci curvature bounded below as an application we show that the curvature of a steady gradient ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above

the polygon exploration problem we present an on line strategy that enables a mobile robot with vision to explore an unknown simple polygon we prove that the resulting tour is less than 26 5 times as long as the shortest watchman tour that could be computed off line our analysis is doubly founded on a novel geometric structure called angle hull let d be a connected region inside a simple polygon p we define the angle hull of d cal ah d to be the set of all points in p that can see two points of d at a right angle we show that the perimeter of cal ah d cannot exceed in length the perimeter of d by more than a factor of 2 this upper bound is tight

optimality and competitiveness of exploring polygons by mobile robots a mobile robot represented by a point moving along a polygonal line in the plane has to explore an unknown polygon and return to the starting point the robot has a sensing area which can be a circle or a square centered at the robot this area shifts while the robot moves inside the polygon and at each point of its trajectory the robot sees explores all points for which the segment between the robot and the point is contained in the polygon and in the sensing area we focus on two tasks exploring the entire polygon and exploring only its boundary we consider several scenarios both shapes of the sensing area and the manhattan and the euclidean metrics we focus on two quality benchmarks for exploration performance optimality the length of the trajectory of the robot is equal to that of the optimal robot knowing the polygon and competitiveness the length of the trajectory of the robot is at most a constant multiple of that of the optimal robot knowing the polygon most of our results concern rectilinear polygons we show that optimal exploration is possible in only one scenario that of exploring the boundary by a robot with square sensing area starting at the boundary and using the manhattan metric for this case we give an optimal exploration algorithm and in all other scenarios we prove impossibility of optimal exploration for competitiveness the situation is more optimistic we show a competitive exploration algorithm for rectilinear polygons whenever the sensing area is a square for both tasks regardless of the metric and of the starting point finally we show a competitive exploration algorithm for arbitrary convex polygons for both shapes of the sensing area regardless of the metric and of the starting point

complete manifolds with positive spectrum ii in this paper we continued our investigation of complete manifolds whose spectrum of the laplacian has an optimal positive lower bound in particular we proved a splitting type theorem for n dimensional manifolds that have a finite volume end this can be viewed as a study of the equality case of a theorem of cheng

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

optimal graph exploration without good maps a robot has to visit all nodes and traverse all edges of an unknown undirected connected graph using as few edge traversals as possible the quality of an exploration algorithm a is measured by comparing its cost number of edge traversals to that of the optimal algorithm having full knowledge of the graph the ratio between these costs maximized over all starting nodes in the graph and over all graphs in a given class u is called the overhead of algorithm a for the class u of graphs we consider three scenarios providing the robot with varying amount of information the robot may either know nothing about the explored graph or have an unlabeled isomorphic copy of it an unanchored map or have such a copy with a marked starting node an anchored map for all of the above scenarios we construct natural exploration algorithms that have smallest or in one case close to smallest overhead while for the class of all graphs depth first search turns out to be an optimal algorithm for all scenarios the situation for trees is much different we show that under the scenario without any knowledge dfs is still optimal for trees but this is not the case if a map is available under the scenario with an unanchored map we show that optimal overhead is at least 3 but strictly below 2 and thus dfs is not optimal under the scenario with an anchored map we construct an optimal algorithm for trees and show that its overhead is 3 2 we also consider exploration of the class of lines simple paths in this case depth first search remains optimal for the scenario without any knowledge with overhead 2 under the scenario with an unanchored map we construct an optimal algorithm and show that its overhead is 3 finally under the scenario with an anchored map we construct an optimal algorithm and show that its overhead is 7 5 an important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

vc dimension of exterior visibility in this paper we study the vapnik chervonenkis vc dimension of set systems arising in 2d polygonal and 3d polyhedral configurations where a subset consists of all points visible from one camera in the past it has been shown that the vc dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes here we consider the case of exterior visibility where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary we present results for the cases of cameras lying on a circle containing a polygon vc dimension 2 or lying outside the convex hull of a polygon vc dimension 5 the main result of this paper concerns the 3d case we prove that the vc dimension is unbounded if the cameras lie on a sphere containing the polyhedron hence the term exterior visibility

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

competitive online approximation of the optimal search ratio how efficiently can we search an unknown environment for a goal in an unknown position how much would it help if the environment were known we answer these questions for simple polygons and for undirected graphs by providing online search strategies that are as good as the best offline search algorithms up to a constant factor for other settings we prove that no such online algorithms exist we introduce a natural measure which gives reasonable results and is more realistic than pure pessimistic competitive analysis

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

a sharp estimate for the bottom of the spectrum of the laplacian on kahler manifolds on a complete noncompact kahler manifold we prove that the bottom of the spectrum for the laplacian is bounded from above by m 2 if the ricci curvature is bounded from below by 2 m 1 then we show that if this upper bound is achieved then the manifold has at most two ends these results improve previous results on this subject proved by p li and j wang in l w3 and l w under assumptions on the bisectional curvature

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

how to learn an unknown environment i the rectilinear case we consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it we seek algorithms that achieve a bounded ratio of the worst case distance traversed in order to see all visible points of the environment thus creating a map divided by the optimum distance needed to verify the map if we had it in the beginning the situation is complicated by the fact that the latter off line problem the problem of optimally verifying a map is np hard although we show that there is no such competitive algorithm for general obstacle courses we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it we restrict ourselves to the rectilinear case where each side of the obstacles and the room is parallel to one of the coordinates and the robot must also move either parallel or perpendicular to the sides in a subsequent paper we will discuss the extension to polygons of general shapes we also discuss the off line problem for simple rectilinear polygons and find an optimal solution in the l 1 metric in polynomial time in the case where the entry and the exit are different points

tight bounds for undirected graph exploration with pebbles and multiple agents we study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents the vertices of the graph are unlabeled and cannot be distinguished by the agents but the edges incident to a vertex have locally distinct labels the graph is explored when all vertices have been visited by at least one agent in this setting it is known that for a single agent without pebbles log n bits of memory are necessary and sufficient to explore any graph with at most n vertices we are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph we give tight results for both questions showing that for a single agent with constant memory log log n pebbles are necessary and sufficient for exploration we further prove that using collaborating agents instead of pebbles does not help as log log n agents with constant memory each are necessary and sufficient for exploration for the upper bounds we devise an algorithm for a single agent with constant memory that explores any n vertex graph using o log log n pebbles even when n is not known a priori the algorithm terminates after polynomial time and returns to the starting vertex we further show that the algorithm can be realized with additional constant memory agents rather than pebbles implying that o log log n agents with constant memory can explore any n vertex graph for the lower bound we show that the number of agents needed for exploring any graph with at most n vertices is already log log n when we allow each agent to have at most o log n 1 e bits of memory for any e 0 our argument also implies that a single agent with sublogarithmic memory needs log log n pebbles to explore any n vertex graph

broadcast in the rendezvous model in many large distributed or mobile networks broadcast algorithms are used to update information stored at the nodes in this paper we propose a new model of communication based on rendezvous and analyze a multi hop distributed algorithm to broadcast a message in a synchronous setting in the rendezvous model two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously thus nodes u and v obtain a rendezvous at a meeting point if m is the number of meeting points the network can be modeled by a graph of n vertices and m edges at each round every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities rendezvous enable an exchange of information between the two entities we get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast we show that for any graph the expected number of rounds is between ln n and o n2 for these two bounds we prove that there exist some graphs for which the expected number of rounds is either o ln n or n2 for specific topologies additional bounds are given

optimal hardy weight for second order elliptic operator an answer to a problem of agmon abstract for a general subcritical second order elliptic operator p in a domain r n or noncompact manifold we construct hardy weight w which is optimal in the following sense the operator p w is subcritical in for all 1 null critical in for 1 and supercritical near any neighborhood of infinity in for any 1 moreover if p is symmetric and w 0 then the spectrum and the essential spectrum of w 1 p are equal to 1 and the corresponding agmon metric is complete our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators the constructed hardy weight is given by an explicit simple formula involving two distinct positive solutions of the equation p u 0 the existence of which depends on the subcriticality of p in

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr