design and analysis of a fuzzy proportional integral derivative controller abstract this paper describes the design principle tracking performance and stability analysis of a fuzzy proportional integral pi plus a derivative d controller first the fuzzy pi d controller is derived from the conventional continuous time linear pi d controller then the fuzzification control rule base and defuzzification in the design of the fuzzy controller are discussed in detail the resulting controller is a discrete time fuzzy version of the conventional pi d controller which has the same linear structure in the proportional integral and derivative parts but has nonconstant gains the proportional integral and derivative gains are nonlinear functions of the input signals the new fuzzy pi d controller thus preserves the simple linear structure of its conventional counterpart yet enhances the self tuning control capability computer simulation results have demonstrated the advantages of the fuzzy controller particularly when the process to be controlled is nonlinear after a brief stability analysis where a simple and realistic sufficient condition for the bounded input bounded output stability of the overall feedback control system was derived several computer simulation results are shown to compare with the conventional pi d controller computer simulation results have shown the new fuzzy controller indeed has satisfactory tracking performance

a fixed point formula and harish chandra s character formula the main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators for proper actions by connected semisimple lie groups on possibly noncompact manifolds with compact quotients for compact groups and manifolds this reduces to the atiyah segal singer fixed point formula other special cases include an index theorem by connes and moscovici for homogeneous spaces and an earlier index theorem by the second author both in cases where the group acting is connected and semisimple as an application of this fixed point formula we give a new proof of harish chandra s character formula for discrete series representations

geometric quantization and families of inner products abstract we formulate a quantization commutes with reduction principle in the setting where the lie group g the symplectic manifold it acts on and the orbit space of the action may all be noncompact it is assumed that the action is proper and the zero set of a deformation vector field associated to the momentum map and an equivariant family of inner products on the lie algebra g of g is g cocompact the central result establishes an asymptotic version of this quantization commutes with reduction principle using an equivariant family of inner products on g instead of a single one makes it possible to handle both noncompact groups and manifolds by extending tian and zhang s witten deformation approach to the noncompact case

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

on multiplicities of graded sequences of ideals abstract we generalize a result from l ein et al math ag 0202303 proving that for an arbitrary graded sequence of zero dimensional ideals the multiplicity of the sequence is equal to its volume this is done using a deformation to monomial ideals as a consequence of our result we obtain a formula which computes the multiplicity of an ideal i in terms of the multiplicities of the initial monomial ideals of the powers i m we use this to give a new proof of the inequality between multiplicity and the log canonical threshold from de fernex et al j alg geom to appear

new design and stability analysis of fuzzy proportional derivative control systems this paper describes the design principle tracking performance and stability analysis of a fuzzy proportional derivative pd controller first the fuzzy pd controller is derived from the conventional continuous time linear pd controller then the fuzzification control rule base and defuzzification in the design of the fuzzy pd controller are discussed in detail the resulting controller is a discrete time fuzzy version of the conventional pd controller which has the same linear structure in the proportional and the derivative parts but has nonconstant gains both the proportional and derivative gains are nonlinear functions of the input signals the new fuzzy pd controller thus preserves the simple linear structure of the conventional pd controller yet enhances its self tuning control capability computer simulation results have demonstrated this advantage of the fuzzy pd controller particularly when the process to be controlled is nonlinear after a detailed stability analysis where a simple and realistic sufficient condition for the bounded input bounded output stability of the overall feedback control system was derived several computer simulation results are compared with the conventional pd controller although the conventional and fuzzy pd controllers are not exactly comparable the authors compare them in order to have a sense of how well the fuzzy pd controller performs for this reason in the simulations several first order and second order linear systems with or without time delays are first used to test the performance of the fuzzy pd controller for step reference inputs the fuzzy pd control systems show remarkable performance as well as if not better than the conventional pd control systems moreover the fuzzy pd controller is compared to the conventional pd controller for a particular second order linear system showing the advantage of the fuzzy pd controller over the conventional one in the sense that in order to obtain the same control performance the conventional pd controller has to employ an extremely large gain while the fuzzy controller uses a reasonably small gain finally in the case of nonlinear systems the authors provide some examples to show that the fuzzy pd controller can track the set points satisfactorily but the conventional pd controller cannot

model construction rule reduction and robust compensation for generalized form of takagi sugeno fuzzy systems this paper presents a systematic procedure of fuzzy control system design that consists of fuzzy model construction rule reduction and robust compensation for nonlinear systems the model construction part replaces the nonlinear dynamics of a system with a generalized form of takagi sugeno fuzzy systems which is newly developed by us the generalized form has a decomposed structure for each element of a sub i and b sub i matrices in consequent parts the key feature of this structure is that it is suitable for constructing if then rules and reducing the number of if then rules the rule reduction part provides a successive procedure to reduce the number of if then rules furthermore we convert the reduction error between reduced fuzzy models and a system to model uncertainties of reduced fuzzy models the robust compensation part achieves the decay rate controller design guaranteeing robust stability for the model uncertainties finally two examples demonstrate the utility of the systematic procedure developed

multiplicity free hamiltonian actions need not be kahler multiplicity free actions are symplectic manifolds with a very high degree of symmetry delzant 2 showed that all compact multiplicity free torus actions admit compatible kahler structures and are therefore toric varieties in this note we show that delzant s result does not generalize to the non abelian case our examples are constructed by applying u 2 equivariant symplectic surgery to the flag variety u 3 t3 we then show that these actions fail a criterion which tolman 9 shows is necessary for the existence of a compatible kahler structure

extended rees algebras and mixed multiplicities the aim of this paper is to study mixed multiplicities of ideals and use them to calculate multiplicities of certain homogeneous ideals of extended rees algebras and to give a complete characterization of those parameter ideals whose extended rees algebras are cohen macaulay with minimal multiplicity at their maximal homogeneous ideals let r m be a local ring of dimension d let i be an m primary ideal it is well known that the length of the artinian ring r f i r f is a polynomial of degree d in r for all large r the coefficient of nd d in this polynomial called the multiplicity of i denoted by e i is a well understood and useful invariant of suppose j is another m primary ideal then it is natural to consider l r fj s for positive integers r and s it is proved in b that for large values of r and s i r fj is given by a polynomial p r s of total degree d in r and s moreover the terms of total degree d in p r s have the form

norm square localization and the quantization of hamiltonian loop group spaces abstract in an earlier article we introduced a new definition for the quantization of a hamiltonian loop group space m involving the equivariant l 2 index of a dirac type operator d on a non compact finite dimensional submanifold y of m in this article we study a deformation of this operator similar to the work of tian zhang and ma zhang we obtain a formula for the index with infinitely many non trivial contributions indexed by the components of the critical set of the norm square of the moment map this is the main part of a new proof of the q r 0 theorem for hamiltonian loop group spaces

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

quantisation commutes with reduction at discrete series representations of semisimple groups abstract using the analytic assembly map that appears in the baum connes conjecture in noncommutative geometry we generalise the guillemin sternberg conjecture that quantisation commutes with reduction to discrete series representations of semisimple groups g with maximal compact subgroups k acting cocompactly on symplectic manifolds we prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements g se the set of elements of g with compact stabilisers this assumption on the image of the momentum map is equivalent to the assumption that m g k n for a compact hamiltonian k manifold n the proof comes down to a reduction to the compact case this reduction is based on a quantisation commutes with induction principle and involves a notion of induction of hamiltonian group actions this principle in turn is based on a version of the naturality of the assembly map for the inclusion k g

a geometric realisation of tempered representations restricted to maximal compact subgroups let g be a connected linear real reductive lie group with compact centre let k g be maximal compact for a tempered representation pi of g we realise the restriction pi k as the k equivariant index of a dirac operator on a homogeneous space of the form g h for a cartan subgroup h g the result in fact applies to every standard representation such a space can be identified with a coadjoint orbit of g so that we obtain an explicit version of kirillov s orbit method for pi k in a companion paper we use this realisation of pi k to give a geometric expression for the multiplicities of the k types of pi in the spirit of the quantisation commutes with reduction principle this generalises work by paradan for the discrete series to arbitrary tempered representations

asymptotic multiplicities of graded families of ideals and linear series abstract we find simple necessary and sufficient conditions on a local ring r of dimension d for the limit lim n l r r i n n d to exist whenever i n is a graded family of m r primary ideals and give a number of applications we also give simple necessary and sufficient conditions on projective schemes over a field k for asymptotic limits of the growth of all graded linear series of a fixed kodaira iitaka dimension to exist

kirillov s orbit method the case of discrete series representations let be a discrete series representation of a real reductive lie group g and let g be a reductive subgroup of g in this paper we give a geometric expression of the g multiplicities in g when the representation is g admissible

mixed multiplicities of ideals versus mixed volumes of polytopes the main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as samuel s multiplicities in particular we can give a purely algebraic proof of bernstein s theorem which asserts that the number of common zeros of a system of laurent polynomial equations in the torus is bounded above by the mixed volume of their newton polytopes

quantization of presymplectic manifolds and circle actions we prove several versions of quantization commutes with reduction for circle actions on manifolds that are not symplectic instead these manifolds possess a weaker structure such as a spinc structure our theorems work whenever the quantization data and the reduction data are compatible this condition always holds if we start from a presymplectic in particular symplectic manifold

additivity of spin c quantization under cutting a g equivariant spin c structure on a manifold gives rise to a virtual representation of the group g called the spin c quantization of the manifold we present a cutting construction for s 1 equivariant spin c manifolds and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces our proof uses kostant type formulas which express the quantization in terms of local data around the fixed point set of the s 1 action

spin structures and proper group actions abstract we generalise atiyah and hirzebruch s vanishing theorem for actions by compact groups on compact spin manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact spin manifolds as corollaries we obtain some vanishing results for an a type genus

localization of the riemann roch character abstract we present a k theoretic approach to the guillemin sternberg conjecture v guillemin and s sternberg invent math 67 1982 515 538 about the commutativity of geometric quantization and symplectic reduction which was proved by e meinrenken j amer math soc 9 1996 373 389 adv math 134 1998 240 277 and tian zhang y tian and w zhang invent math 132 1998 229 259 besides providing a new proof of this conjecture for the full non abelian group action case our methods lead to a generalization for compact lie group actions on manifolds that are not symplectic these manifolds carry an invariant almost complex structure and an abstract moment map

torus fibrations and localization of index iii equivariant version and its applications this paper is the third of the series concerning the localization of the index of dirac type operators in our previous papers we gave a formulation of index of diractype operators on open manifolds under some geometric setting whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds we introduce two equivariant versions of the localization as an application we give a proof of guillemin sternberg s quantization conjecture in the case of torus action

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

the dirac operator on homogeneous spaces and representations of reductive lie groups ii soit g un groupe de lie reductif reel connexe et k un sousgroupe compact maximal contenant un sousgroupe connexe compact donne l de g soit d v l operateur de dirac tordu soit kerd v l espace des spineurs harmoniques l une representation unitaire de g on etudie comment kerd v l se decompose en representations irreductibles unitaires de g quand l est un sous groupe compact

fuzzy pid controller design performance evaluation and stability analysis abstract this paper presents a design for a new fuzzy logic proportional integral derivative pid controller the main motivation for this design was to control some known nonlinear systems such as robotic manipulators which violate the conventional assumption of the linear pid controller this controller is developed by first describing the discrete time linear pid control law and then progressively deriving the steps necessary to incorporate a fuzzy logic control mechanism into the modifications of the pid structure the final version of this new fuzzy pid controller is a computationally efficient analytic scheme suitable for implementation in a real time closed loop digital control numerous computer simulations are included to demonstrate the effectiveness of the controller for both linear and nonlinear systems finally a brief analysis is presented to prove that the controller has bounded input bounded output bibo stability

three dimensional min max gravity based fuzzy pid inference analysis and tuning this paper proposes a novel algorithm to produce an analytical solution for three input fuzzy pid systems the analysis is based on the most common inference method zadeh mamdani s min max gravity fuzzy reasoning two types of fuzzy systems are investigated both types use unevenly distributed triangular shaped fuzzy input memberships in the first type the output fuzzy sets are considered to be uniformly distributed symmetrical triangular type memberships and in the second type the output fuzzy sets are considered unevenly distributed singleton functions a new input transformation technique is developed to reduce the number of input conditions required in defining the fuzzy output the multi phase complexity that exists in conventional fuzzy reasoning analysis is thus avoided an efficient solution algorithm is outlined to produce the general fuzzy output solution using a minimum number of nonlinear expressions with the proposed method the fuzzy output solution for a three input fuzzy system having uniform output membership functions requires only two nonlinear expressions when the output fuzzy sets are unevenly distributed the general three input based solution requires only four nonlinear expressions the fuzzy controller output is finally decomposed into linear and nonlinear parts the output decomposition is useful in defining tuning gains for fuzzy pid systems

on unique and almost unique factorization of complete ideals ii the theory of complete ideals was begun by zariski in 1938 z 1 and further developed in appendix 5 of zs zariski developed a theory of factorization of complete ideals in regular two dimensional local rings in this paper we develop the theory of factorization of complete ideals in complete normal two dimensional local rings let r m be a complete normal two dimensional local ring and let re r be the semi group of complete m primary ideals of r we show how the divisor class group of r ci r is related to factorization in re r we generalize results obtained in c2 where it is assumed that r m is algebraically closed assume that r is any regular local ring of dimension 2 zariski proved that any complete ideal i of r has unique factorization into products of simple complete ideals the distinct simple complete ideals occuring in the factorization of i are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of i this shows that any proper birational morphism x spec r with x normal can be decomposed into a sequence of blowups with irreducible exceptional divisors now assume that r is a complete normal local domain of dimension 2 it is natural to ask if the condition that m r has unique factorization is equivalent to r being a ufd we show theorem 5 that if re r has unique factorization then r is a ufd we also give an example example 1 to show that the converse does not hold however lipman has shown in l that if r is a ufd and if r m is algebraically closed then re r has unique factorization recall that r is a ufd if and only if ci r 0 g6hner g introduced the notion of a semigroup being semi factorial this definition is reproduced in definition 3 of this paper we prove g6hner s conjecture 2 3 of theorem 4 that re r is semifactorial if and only if ci r is a torsion group as a corollary corollary to theorem 4 we obtain that if rim is algebraically closed of characteristic zero then re r is semi factorial if and only if r has a rational singularity another corollary is that ci r is torsion if and only if r satisfies the condition n of muhly and sakuma ms 1r of theorem 4 r satisfies condition n if any proper birational morphism x spec r factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors

fuzzy control theory a nonlinear case abstract sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs fuzzy control rules type of fuzzy logic used for evaluating the fuzzy control rules and the defuzzification algorithm used for the controller output we analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms we prove theoretically that such a fuzzy controller the smallest possible with two inputs error and rate change of error and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional integral pi controller with proportional gain and integral gain changing with error and rate change of error about a setpoint furthermore this fuzzy controller is precisely equivalent to a conventional linear pi controller if a linear defuzzification algorithm is employed computer simulation showed that the performance of the fuzzy controller was almost the same as that of the pi controller when first order and second order linear processes were used furthermore the fuzzy controller was significantly better when a first order with a time delay model was used more importantly the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled but the pi controller was unstable

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

mixed multiplicities of divisorial filtrations abstract suppose that r is an excellent local domain with maximal ideal m r the theory of multiplicities and mixed multiplicities of m r primary ideals extends to possibly non noetherian filtrations of r by m r primary ideals and many of the classical theorems for m r primary ideals continue to hold for filtrations the celebrated theorems involving inequalities continue to hold for filtrations but the good conclusions that hold in the case of equality for m r primary ideals do not hold for filtrations in this article we consider multiplicities and mixed multiplicities of r by m r primary divisorial filtrations we show that some important theorems on equalities of multiplicities and mixed multiplicities of m r primary ideals that are not true in general for filtrations are true for divisorial filtrations we prove that a theorem of rees showing that if there is an inclusion of m r primary ideals i i with the same multiplicity then i and i have the same integral closure also holds for divisorial filtrations this theorem does not hold for arbitrary filtrations the classical minkowski inequalities for m r primary ideals i 1 and i 2 hold quite generally for filtrations if r has dimension two and there is equality in the minkowski inequalities then teissier and rees and sharp have shown that there are powers i 1 a and i 2 b that have the same integral closure this theorem does not hold for arbitrary filtrations the teissier rees sharp theorem has been extended by katz to m r primary ideals in arbitrary dimension we show that the teissier rees sharp theorem does hold for divisorial filtrations in an excellent local domain of dimension two we also show that the mixed multiplicities of divisorial filtrations are anti positive intersection products on a suitable normal scheme x birationally dominating r when r is an algebraic local domain essentially of finite type over a field

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

torus fibrations and localization of index ii local index for acyclic compatible system we give a framework of localization for the index of a dirac type operator on an open manifold suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets each of which has a structure of the total space of a torus bundle under an acyclic condition we define the index of the dirac type operator by using the witten type deformation and show that the index has several properties such as excision property and a product formula in particular we show that the index is localized on the compact set

equivariant indices of operatorname spin c dirac operators for proper moment maps we define an equivariant index of spinc dirac operators on possibly noncompact manifolds acted on by compact connected lie groups our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle

poincare series of resolutions of surface singularities let x spec r be a resolution of singularities of a normal surface singularity spec r with integral exceptional divisors e 1 e r we consider the poincare series g h n t n where h n r x o x n 1 e 1 n r e r we show that if r m has characteristic zero and pic 0 x is a semi abelian variety then the poincare series g is rational however we give examples to show that this series can be irrational if either of these conditions fails

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

analytical structures for fuzzy pid controllers in this paper analytical structures for fuzzy proportional integral derivative pid controllers are derived by using triangular membership functions for inputs singletons or triangular membership functions for output minimum triangular norm maximum or drastic sum triangular conorm mamdani minimum drastic or larsen product inference nonlinear control rules and center of sum defuzzification it is shown that these analytical structures are not suitable for control purpose in this context it is extremely important to note that the analytical structures reported by carvajal et al are also not valid for control

geometric quantization for proper actions we first introduce an invariant index for g equivariant elliptic differential operators on a locally compact manifold m admitting a proper cocompact action of a locally compact group g it generalizes the kawasaki index for orbifolds to the case of proper cocompact actions our invariant index is used to show that an analog of the guillemin sternberg geometric quantization conjecture holds if m is symplectic with a hamiltonian action of g that is proper and cocompact this essentially solves a conjecture of hochs and landsman