approximations to stochastic programs with complete recourse a two stage stochastic convex program with relatively complete recourse in l infty space is replaced by a mathematical programming problem in a finite dimensional space using discrete convergence of mappings conditions are presented that guarantee the convergence of the sequence of solutions of approximate problems to the solution of the initial problem

analytic properties of zeta functions and subgroup growth it has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group g although the story began much before the recent activity grew out of a paper by dan segal in 36 the story so far has been well documented in lubotzky s subsequent survey paper in 30 in 24 the second author of this article segal and smith introduced the zeta function of a group as a tool for understanding this growth of subgroups let an g be the number of subgroups of index n in the finitely generated group g and sn g a1 g an g be the number of subgroups of index n or less the zeta function is defined as the dirichlet series with coefficients an g and has a natural interpretation as a noncommutative generalization of the dedekind zeta function of a number field

uniformity and functional equations for local zeta functions of mathfrak k split algebraic groups we study the local zeta functions of an algebraic group mathcal g defined over mathfrak k together with a faithful mathfrak k rational representation rho for a finite extension mathfrak k of bbb q these are given by integrals over mathfrak p adic points of cal g determined by rho for a prime mathfrak p of mathfrak k we prove that the local zeta functions are almost uniform for all mathfrak k split groups whose unipotent radical satisfies a certain lifting property this property is automatically satisfied if mathcal g is reductive we provide a further criterion in terms of invariants of mathcal g and rho which guarantees that the local zeta functions satisfy functional equations for almost all primes of mathfrak k we obtain these results by using a mathfrak p adic bruhat decomposition of iwahori and matsumoto to express the zeta function as a weighted sum over the weyl group w associated to mathcal g of generating functions over lattice points of a polyhedral cone the functional equation reflects an interplay between symmetries of the weyl group and reciprocities of the combinatorial object we construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations our work generalizes results of igusa and du sautoy and lubotzky and has implications for zeta functions of finitely generated torsion free nilpotent groups

conservative linear systems unitary colligations and lax phillips scattering multidimensional generalizations connections between conservative linear systems lax phillips scattering and operator model theory are well known a common thread in all the theories is a contractive analytic operator valued function on the unit disc t z having a representation of the form t z d z c i za 1 b known as the transfer or frequency response function in the system theory community the scattering function in the mathematical physics community and the characteristic operator function in the operator theory community here we consider analogues of this circle of ideas in the more general setting of multidimensional systems multi evolution scattering systems multivariable function theoretic operator theory three particular extensions are discussed from the point of view of system theory these involve 1 a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex euclidean space 2 a non commutative multidimensional linear system with evoluti

inner functions on the bidisk and associated hilbert spaces abstract matrix valued inner functions on the bidisk have a number of natural subspaces of the hardy space on the torus associated to them we study their relationship to agler decompositions regularity up to the boundary and restriction maps into one variable spaces we give a complete description of the important spaces associated to matrix rational inner functions the dimension of some of these spaces can be computed in a straightforward way and this ends up having an application to the study of three variable rational inner functions examples are included to highlight the differences between the scalar and matrix cases

functional equations and uniformity for local zeta functions of nilpotent groups we investigate in this paper the zeta function inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i associated to a nilpotent group introduced in gss this zeta function counts the subgroups h whose profinite completion is isomorphic to the profinite completion inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i by representing inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i as an integral with respect to the haar measure on the algebraic automorphism group g of the lie algebra associated to and by generalizing some recent work of igusa i we give under some assumptions on an explicit finite form for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i in terms of the combinatorial data of the root system of g and information about the weights of various representations of g as a corollary of this finite form we are able to prove 1 a certain uniformity in p confirming a question raised in gss and 2 a functional equation that the local factors satisfy inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i this functional equation is perhaps the most important result of the paper as it is a new feature of the theory of zeta functions of groups

functional models of γn contractions and characterization of γn isometries abstract we show that a weighted bergman space on the polydisc splits up into the orthogonal direct sum of subspaces each corresponding to an irreducible representation of the symmetric group the multiplication by the n tuple of elementary symmetric functions is a n contraction on each of these subspaces a necessary condition for a commuting n tuple to be a n contraction is obtained in terms of pencil of operators we provide a model theory for n isometries a beurling lax halmos type representation for the invariant subspaces of n isometries is given

scenario tree generation for multiperiod financial optimization by optimal discretization multiperiod financial optimization is usually based on a stochastic model for the possible market situations there is a rich literature about modeling and estimation of continuous state financial processes but little attention has been paid how to approximate such a process by a discrete state scenario process and how to measure the pertaining approximation error in this paper we show how a scenario tree may be constructed in an optimal manner on the basis of a simulation model of the underlying financial process by using a stochastic approximation technique consistency relations for the tree may also be taken into account

enumerating graded ideals in graded rings associated to free nilpotent lie rings we compute the zeta functions enumerating graded ideals in the graded lie rings associated with the free d generator lie rings mathfrak f c d of nilpotency class c for all c leqslant 2 and for c d in 3 3 3 2 4 2 we apply our computations to obtain information about mathfrak p adic reduced and topological zeta functions in particular pertaining to their degrees and some special values

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

igusa type functions associated to finite formed spaces and their functional equations we study symmetries enjoyed by the polynomials enumerating non degenerate flags in finite vector spaces equipped with a non degenerate alternating bilinear hermitian or quadratic form to this end we introduce igusa type rational functions encoding these polynomials and prove that they satisfy certain functional equations some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on coxeter groups of type a while our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements we formulate a precise conjecture how in this situation too the polynomials may be described in terms of parabolic length functions

zeta functions of three dimensional p adic lie algebras we give an explicit formula for the subalgebra zeta function of a general three dimensional lie algebra over the p adic integers mathbb z p to this end we associate to such a lie algebra a ternary quadratic form over mathbb z p the formula for the zeta function is given in terms of igusa s local zeta function associated to this form

unitary colligations reproducing kernel hilbert spaces and nevanlinna pick interpolation in several variables abstract recently j agler studied the class s d of scalar valued analytic functions of d complex variables f for which f t 1 t d has norm at most 1 for any collection of d commuting contractions t 1 t d on a hilbert space h among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d variable linear system as well as a nevanlinna pick interpolation theorem for this class of functions in this note we examine the system theory aspects and uniqueness of the transfer function representation and give a simpler proof of the nevanlinna pick interpolation theorem for the class s d and obtain a d variable version of the toeplitz corona theorem by using ideas of arov and grossman introduced for 1 variable problems as a bonus we obtain a collection of linear fractional maps which parametrize the set of all s d solutions of an interpolation problem

scenario reduction in stochastic programming an approach using probability metrics given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

stability of multistage stochastic programs quantitative stability of linear multistage stochastic programs is studied it is shown that the infima of such programs behave locally lipschitz continuous with respect to the sum of an l r distance and of a distance measure for the filtrations of the original and approximate stochastic input processes various issues of the result are discussed and an illustrative example is given consequences for the reduction of scenario trees are also discussed

epi convergent discretizations of multistage stochastic programs in many dynamic stochastic optimization problems in practice the uncertain factors are best modeled as random variables with an infinite support this results in infinite dimensional optimization problems that can rarely be solved directly therefore the random variables stochastic processes are often approximated by finitely supported ones scenario trees which result in finite dimensional optimization problems that are more likely to be solvable by available optimization tools this paper presents conditions under which such finite dimensional optimization problems can be shown to epi converge to the original infinite dimensional problem epi convergence implies the convergence of optimal values and solutions as the discretizations are made finer our convergence result applies to a general class of convex problems where neither linearity nor complete recourse are assumed

classes of tuples of commuting contractions satisfying the multivariable von neumann inequality we obtain a decomposition for multivariable schur class functions on the unit polydisk which to a certain extent is analogous to agler s decomposition for functions from the schur agler class as a consequence we show that d tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d variable von neumann inequality for an arbitrary operator valued bounded analytic function on the polydisk also this decomposition yields a necessary condition for solvability of the finite data nevanlinna pick interpolation problem in the schur class on the unit polydisk 2008 elsevier inc all rights reserved

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

the scenario generation algorithm for multistage stochastic linear programming a multistage stochastic linear program mslp is a model of sequential stochastic optimization where the objective and constraints are linear when any of the random variables used in the mslp are continuous the problem is infinite dimensional to numerically tackle such a problem we usually replace it with a finite dimensional approximation even when all the random variables have finite support the problem is often computationally intractable and must be approximated by a problem of smaller dimension one of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different from the problem stated with the original tree the scenario generation sg algorithm proposed in this paper is a finite element method that addresses this problem for the class of mslp with random right hand sides

scenario tree modeling for multistage stochastic programs an important issue for solving multistage stochastic programs consists in the approximate representation of the multivariate stochastic input process in the form of a scenario tree in this paper we develop stability theory based heuristics for generating scenario trees out of an initial set of scenarios they are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps conditions are established implying closeness of optimal values of the original process and its tree approximation respectively by relying on a recent stability result in heitsch romisch and strugarek siam j optim 17 511 525 2006 for multistage stochastic programs numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management

finitely generated groups p adic analytic groups and poincare series this is equivalent to the coefficients apn g satisfing a linear recurrence relation with constant coefficients for sufficiently large n these poincare series were first studied in the case when g is a finitely generated torsion free nilpotent group in a paper by grunewald segal and smith gssm there the authors established that for such a group g p s is a rational function in p8s in that setting the functions g p s are the local factors associated with the dirichlet series

epi convergent discretizations of multistage stochastic programs via integration quadratures this paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi converge to the original problem as the discretizations are made finer epi convergence implies in particular that the cluster points of the first stage solutions of the discretized problems are optimal first stage solutions of the original problem the discretization procedures apply to a general class of nonlinear stochastic programs where the uncertain factors are driven by time series models using existing routines for numerical integration allows for an easy and efficient implementation of the procedures

functional equations for zeta functions of groups and rings we introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings the specific form of these formulae enables us to deduce local functional equations more precisely we prove local functional equations for the subring zeta functions associated to rings the subgroup conjugacy and representation zeta functions of finitely generated torsion free nilpotent or t groups and the normal zeta functions of t groups of class 2 in particular we solve the two problems posed in 9 section 5 we deduce our theorems from a blueprint result on certain p adic integrals which generalises work of denef and others on igusa s local zeta function the malcev correspondence and a kirillov type theory developed by howe are used to linearise the problems of counting subgroups and representations in t groups respectively

barycentric scenario trees in convex multistage stochastic programming this work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support based on earlier results sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation convergence of the approximate solutions is proven under the stated assumptions preliminary computational results are outlined

scenario reduction in stochastic programming given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

combining sampling based and scenario based nested benders decomposition methods application to stochastic dual dynamic programming nested benders decomposition is a widely used and accepted solution methodology for multi stage stochastic linear programming problems motivated by large scale applications in the context of hydro thermal scheduling in 1991 pereira and pinto introduced a sampling based variant of the benders decomposition method known as stochastic dual dynamic programming sddp in this paper we embed the sddp algorithm into the scenario tree framework essentially combining the nested benders decomposition method on trees with the sampling procedure of sddp this allows for the incorporation of different types of uncertainties in multi stage stochastic optimization while still maintaining an efficient solution algorithm we provide an illustration of the applicability of our method towards a least cost hydro thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the panama power system incorporating both electricity demand and inflow uncertainties

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

on stability of multistage stochastic programs we study the quantitative stability of linear multistage stochastic programs under perturbations of the underlying stochastic processes it is shown that the optimal values behave lipschitz continuously with respect to an l p distance in order to establish continuity of the recourse function with respect to the current state of the stochastic process we assume continuity of the conditional distributions in terms of a fortet mourier metric the main stability result holds for nonanticipative approximations of the underlying process and thus represents a rigorous justification of established discretization techniques

on the scenario tree optimal value error for stochastic programming problems stochastic programming problems generally lead to large scale programs if the number of random outcomes is large or if the problem has many stages a way to tackle them is provided by scenario tree

zeta functions of groups zeros and friendly ghosts we consider zeta functions defined as euler products of w p p s over all primes p where w x y is a polynomial such zeta functions arise naturally in the context of zeta functions of algebraic groups and nilpotent groups the analytic behavior of such euler products is controlled by something we call the ghost polynomial of w x y the ghost is defined from the zeros of w p y as p ranges over all primes when the euler product of the ghost is meromorphic on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i an important property in relation to the analytic behavior of the original zeta function we say that w x y has a friendly ghost we give an explicit expression for the ghosts of classical reductive groups and prove that they are friendly we also consider the ghosts of exceptional reductive groups and zeta functions of nilpotent groups the concept of the ghost implies a new dichotomy for the representations of an algebraic group according to whether their associated zeta function has a friendly ghost or not

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

zeta functions for analytic mappings log principalization of ideals and newton polyhedra in this paper we provide a geometric description of the possi ble poles of the igusa local zeta function z s f associated to an analytic mapping f f1 fl u k n k l and a locally constant function with support in u in terms of a log principalizaton of the k x ideal if f1 fl typically our new method provides a much shorter list of possible poles compared with the previous methods we determine the largest real part of the poles of the igusa zeta function and then as a corollary we ob tain an asymptotic estimation for the number of solutions of an arbitrary sys tem of polynomial congruences in terms of the log canonical threshold of the subscheme given by if we associate to an analytic mapping f f1 fl a newton polyhedron f and a new notion of non degeneracy with respect to f the novelty of this notion resides in the fact that it depends on one newton polyhedron and khovanskii s non degeneracy notion depends on the newton polyhedra of f1 fl by constructing a log principalization we give an explicit list for the possible poles of z s f l 1 in the case in which f is non degenerate with respect to f

a family of class 2 nilpotent groups their automorphisms and pro isomorphic zeta functions the pro isomorphic zeta function zeta wedge gamma s of a finitely generated nilpotent group gamma is a dirichlet generating function that enumerates finite index subgroups whose profinite completion is isomorphic to that of gamma such zeta functions can be expressed as euler products of p adic integrals over the mathbb q p points of an algebraic automorphism group associated to gamma in this way they are closely related to classical zeta functions of algebraic groups over local fields

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

two stage linear decision rules for multi stage stochastic programming multi stage stochastic linear programs mslps are notoriously hard to solve in general linear decision rules ldrs yield an approximation of an mslp by restricting the decisions at each stage to be an affine function of the observed uncertain parameters finding an optimal ldr is a static optimization problem that provides an upper bound on the optimal value of the mslp and under certain assumptions can be formulated as an explicit linear program similarly as proposed by kuhn et al math program 130 1 177 209 2011 a lower bound for an mslp can be obtained by restricting decisions in the dual of the mslp to follow an ldr we propose a new approximation approach for mslps two stage ldrs the idea is to require only the state variables in an mslp to follow an ldr which is sufficient to obtain an approximation of an mslp that is a two stage stochastic linear program 2slp we similarly propose to apply ldr only to a subset of the variables in the dual of the mslp which yields a 2slp approximation of the dual that provides a lower bound on the optimal value of the mslp although solving the corresponding 2slp approximations exactly is intractable in general we investigate how approximate solution approaches that have been developed for solving 2slp can be applied to solve these approximation problems and derive statistical upper and lower bounds on the optimal value of the mslp in addition to potentially yielding better policies and bounds this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static ldr approach a computational study on two example problems demonstrates that using a two stage ldr can yield significantly better primal policies and modestly better dual policies than using policies based on a static ldr

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

canonical agler decompositions and transfer function realizations a seminal result of agler proves that the natural de branges rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift invariant pieces agler s decomposition is non constructive a problem remedied by work of ball sadosky vinnikov which uses scattering systems to produce agler decompositions through concrete hilbert space geometry this method while constructive so far has not revealed the rich structure shown to be present for special classes of functions inner and rational inner functions in this paper we show that most of the important structure present in these special cases extends to general bounded analytic functions we give characterizations of all agler decompositions we prove the existence of coisometric transfer function realizations with natural state spaces and we characterize when schur functions on the bidisk possess analytic extensions past the boundary in terms of associated hilbert spaces

positive extensions fejer riesz factorization and autoregressive filters in two variables in this paper we treat the two variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial this problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes we show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition as a corollary of the main result a necessary and sufficient condition for the existence of a spectral fejerriesz factorization of a strictly positive two variable trigonometric polynomial is given in terms of the fourier coefficients of its reciprocal tools in the proofs include a specific two variable kronecker theorem based on certain elements from algebraic geometry as well as a two variable christoffel darboux like formula the key ingredient is a matrix valued polynomial that appears in a parametrized version of the schur cohn test for stability the results also have consequences in the theory of two variable orthogonal polynomials where a spectral matching result is obtained as well as in the study of inverse formulas for doubly indexed toeplitz matrices finally numerical results are presented for both the autoregressive filter problem and the factorization problem

a stability result for linear markovian stochastic optimization problems in this paper we propose a semi metric for markov processes that allows to bound optimal values of linear markovian stochastic optimization problems similar to existing notions of distance for general stochastic processes our distance is based on transportation metrics as opposed to the extant literature the proposed distance is problem specific i e dependent on the data of the problem whose objective value we want to bound as a result we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature we derive several properties of the proposed semi metric and demonstrate its use in a stylized numerical example

zeta functions of groups and enumeration in bruhat tits buildings we introduce a new method to calculate local normal zeta functions of finitely generated torsion free nilpotent groups it is based on an enumeration of vertices in the bruhat tits building for sln qp it enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations examples include formulae for non uniform normal zeta functions