on barnes multiple zeta and gamma functions abstract we show how various known results concerning the barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions the pertinent functions involve laplace transforms and their asymptotics is obtained by exploiting this we also demonstrate how barnes multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations both of these new approaches to the barnes functions give rise to novel integral representations

generalization of selberg s frac 3 16 theorem and affine sieve an analogue of the well known frac 3 16 lower bound for the first eigenvalue of the laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non elementary subgroup l of sl 2 z the proof in the case that the hausdorff of the limit set of l is bigger than frac 1 2 is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting in the case that delta is less than frac 1 2 we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function these spectral gaps are then applied to sieving problems on orbits of such groups

wave decay on convex co compact hyperbolic manifolds for convex co compact hyperbolic quotients x gamma backslash mathbb h n 1 we analyze the long time asymptotic of the solution of the wave equation u t with smooth compactly supported initial data f f 0 f 1 we show that if the hausdorff dimension of the limit set is less than n 2 then u t c delta f e delta frac n 2 t gamma delta n 2 1 e delta frac n 2 t r t where c delta f in c infty x and r t mathcal o t infty we explain in terms of conformal theory of the conformal infinity of x the special cases delta in n 2 mathbb n where the leading asymptotic term vanishes in a second part we show for all epsilon 0 the existence of an infinite number of resonances and thus zeros of selberg zeta function in the strip n delta epsilon rm re lambda delta as a byproduct we obtain a lower bound on the remainder r t for generic initial data f

translation representations for automorphic solutions of the wave equation in non euclidean spaces i this paper deals with the spectral theory of the laplace beltrami operator acting on automorphic functions in n dimensional hyperbolic space hn we study discrete subgroups which have a fundamental polyhedron f with a finite number of sides and infinite volume concerning these we have shown previously that the spectrum of contains at most a finite number of point eigenvalues in 1 2 n 1 2 0 and none less than 1 2 n 1 2 here we prove that the spectrum of is absolutely continuous and of infinite multiplicity in 1 2 n 1 2 our approach uses the non euclidean wave equation introduced by faddeev and pavlov energy ef is defined as ut ut u lu where the bracket is the l2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric energy is conserved under the group of operator u t relating initial data to data at time t we construct two isometric representations of the space of automorphic data by l2 r n which transmute the action of u t into translation these representations are given explicitly in terms of integrals of the data over horospheres in part ii we shall show the completeness of these representations utt lu 0 l 1 2 n 1 2

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

state feedback impulse elimination of linear time varying singular systems abstract linear time varying singular systems e t x a t x b t u are treated a necessary and sufficient condition for the existence of linear state feedback to eliminate impulsive behavior is derived the condition obtained is a natural extension of existing time invariant theory

derivative and proportional state feedback for linear descriptor systems with variable coefficients abstract we study linear descriptor systems with rectangular variable coefficient matrices using local and global equivalence transformations we introduce normal and condensed forms and get sets of characteristic quantities these quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and or proportional state feedback or not regularizable by feedback means for us that there exists a feedback which makes the closed loop system uniquely solvable for every consistent initial vector

shintani barnes zeta and gamma functions abstract we show that shintani s work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations we re prove many of shintani s formulas and prove several new ones among the latter is a generalization to the shintani barnes gamma functions of raabe s 1843 formula 0 1 log x dx log 2 and a further generalization to the shintani zeta functions these explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions we also relate barnes triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

rarefied elliptic hypergeometric functions abstract two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved they generalize evaluations of the type i and ii elliptic beta integrals attached to the root system c n in a special n 1 case the simplest p 0 limit is shown to lead to a new class of q hypergeometric identities symmetries of a rarefied elliptic analogue of the euler gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed some extensions of the latter function to c n and a n root systems and corresponding symmetry transformations are considered an application of the rarefied type ii c n elliptic hypergeometric function to some eigenvalue problems is briefly discussed

elliptic hypergeometry of supersymmetric dualities we give a full list of known mathcal n 1 supersymmetric quantum field theories related by the seiberg electric magnetic duality conjectures for su n sp 2n and g 2 gauge groups many of the presented dualities are new not considered earlier in the literature for all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals this gives a systematic extension of the related romelsberger and dolan osborn results equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals about half of them were proven earlier and another half represents new challenging conjectures in particular we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author

the zeta functions of ruelle and selberg for hyperbolic manifolds with cusps in this paper we study the ruelle zeta function and the selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps in particular we show that they have meromorphic extensions to mathbb c and satisfy functional equations we also derive the order of the singularity of the ruelle zeta function at the origin to prove these results we completely analyze the weighted unipotent orbital integrals on the geometric side of the selberg trace formula when test functions are defined for the fundamental representations

on the lower bound of net driving power in controlled duct flows we examine mathematically the lower bound of the net driving power i e the summation of pumping and actuation powers of a controlled duct flow under a constant flow rate the net power in a duct with arbitrary cross section in the presence of the inertial term blowing suction from the wall and arbitrary body forces can be decomposed into four terms 1 dissipation due to the velocity profile of the stokes flow in other words pumping power for the stokes flow 2 dissipation due to deviation of the mean velocity from the stokes flow profile 3 dissipation due to velocity fluctuations and 4 correlation between the wall pressure of the stokes flow and the time averaged blowing suction velocity of these the first three terms are shown to be non negative while the sign of the fourth term is indefinite the fourth term vanishes in cases where the duct has a constant shape cross section such as cylindrical pipes and plane channels namely in such cases the lower bound of net power is exactly given by the dissipation rate pumping power of the stokes flow at the same flow rate

robust controllability for linear uncertain descriptor systems in this paper under assumptions that the linear nominal descriptor system is regular and controllable some sufficient conditions are proposed to preserve the assumed properties when both structured elemental and unstructured norm bounded parameter uncertainties are added into the nominal descriptor system besides another sufficient conditions are also presented to preserve the assumed properties for a class of linear descriptor systems having structured uncertainties in the structure information matrix as well as having both structured and unstructured parameter uncertainties in the system matrix and the input matrix simultaneously the corresponding results for the dual observability robustness problems are straightforward extensions three numerical examples are given to illustrate the applications of the proposed sufficient conditions and it is shown that the proposed sufficient conditions could be less conservative than the existing ones reported recently in the literature

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

time domain solution to descriptor variable systems the time domain solution to the fast subsystem of the descriptor variable system of the form ex spl dot ax bu with e singular is given by laplace inverse transformation a new and significant feature of the solution is that it contains impulse terms excited by the values of sufficiently smooth input u and its derivatives at the initial time point based on the solution the notions of consistent initial conditions classical solutions and impulse free solutions are discussed perfectly the impulse controllability is discussed also which admits the same interpretation as the controllability at infinity in frequency domain

the laplace operator on a hyperbolic manifold ii eisenstein series and the scattering matrix on considere des varietes m isometriques au quotient de l espace hyperbolique a n dimensions h n par un groupe discret geometriquement fini t sans elements paraboliques et sans elements d ordre fini on demontre que la serie de eisenstein associee admet un prolongement mesomorphe au plan complexe et on etudie l operateur diffusion associe

resonances for the laplacian on products of two rank one riemannian symmetric spaces let x x 1 times x 2 be a direct product of two rank one riemannian symmetric spaces of the noncompact type we show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension the resolvent of the laplacian of x can be lifted to a holomorphic function on a riemann surface which is a branched covering of mathbb c in all other cases the resolvent of the laplacian of x admits a singular meromorphic lift the poles of this function are called the resonances of the laplacian we determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances the ranges of these operators are finite dimensional and explicitly realized as direct sums of finite dimensional irreducible spherical representations of the group of the isometries of x

classical and generalized solutions of time dependent linear differential algebraic equations abstract a coordinate free reduction procedure is developed for linear time dependent differential algebraic equations that transforms their solutions into solutions of smaller systems of ordinary differential equations the procedure applies to classical as well as distribution solutions in the case of analytic coefficients the hypotheses required for the reduction not only are necessary for the validity of the existence and uniqueness results but even allow for the presence of singularities straightforward extensions including undetermined systems and systems with nonanalytic coefficients are also discussed

the laplace operator on hyperbolic three manifolds with cusps of non maximal rank where f is a geometrically finite discrete group of orientation preserving isometries which may contain parabolic elements of non maximal rank we prove the existence of a continuum eigenfunction expansion and show that in the spectral parameter the eigenfun ctions have meromorphic continuations to 112 1 these eigenfunctions are eisen stein series for f the continuous spectrum of laplace operators on hyperbolic manifolds was first analysed by number theorists as a means of studying eisenstein series maass and roelke i m r considered the case of finite volume manifolds in two dimen sions increasingly more general hyperbolic manifolds were studied by faddeev if patterson pal lax and phillips lpi lp4 mandouvalos m1 m3 and others recently agmon l a mazzeo and melrose mm and perry p proved the meromorphic continuation of the eisenstein series for geometrically finite groups with fundamental domains of infinite volume provided the groups have no parabolic elements or all the parabolic elements have maximal rank in this case the boundary at infinity which roughly speaking labels the directions at infinity in which scattering can occur is a smooth compact manifold or in the presence of parabolic elements of maximal rank the union of a smooth compact manifold with a finite number of points in fact the work of mazzeo and melrose and recent work of agmon applies to more general manifolds which have a similar structure near infinity however none of the proofs of meromorphic continuation of eigenfun ctions apply when the group has parabolic elements of non maximal rank we will study the laplace operator on a three dimensional hyperbolic manifold when parabolic elements of non maximal rank are present in this case the research supported by the national science and engineering research council of canada research supported in part by n s f grant dms 8911242 research supported in part by n s f grant dms 8802668

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

controllability and observability of linear time varying singular systems controllability and observability of analytically solvable linear time varying singular systems are considered our analysis includes the possible impulsive zero input response based on the definitions of controllability and observability naturally extended from the ones adopted in the time invariant setting sufficient and necessary conditions for these two properties are derived furthermore according to the specific definitions of these concepts used here the authors show that the dual relationship between controllability and observability does not hold

impulse analysis of linear time varying singular systems impulsive part of solution to initial value problem of linear time varying singular systems is given explicitly impulse observability and impulse controllability for the systems are redefined

upper bounds on the number of resonances for non compact riemann surfaces let x be a riemannian surface of finite geometric type and with hyperbolic ends the resolvent x s 1 s 1 re s 1 of the laplacian on x extends to a meromorphic family of operators on c and its poles are called resonances we prove an optimal polynomial bound for their counting function

scattering poles for asymptotically hyperbolic manifolds for a class of manifolds x that includes quotients of real hyperbolic n 1 dimensional space by a convex co compact discrete group we show that the resonances of the meromorphically continued resolvent kernel for the laplacian on x coincide with multiplicities with the poles of the meromorphically continued scattering operator for x in order to carry out the proof we use shmuel agmon s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations

finite time stability of linear time varying singular systems with impulsive effects in this paper we introduce a new concept of finite time stability of linear time varying singular systems with impulses at fixed times and present new results for the above mentioned class of system in the form of sufficient conditions a sufficient condition is given in terms of matrix inequalities which gives the opportunity of fitting the finite time control problem in the general framework via impulsive control a numerical example is presented to illustrate the efficiency of the proposed result

time dependent linear daes with discontinuous inputs existence and uniqueness results are proved for initial value problems associated with linear time varying differential algebraic equations the right hand sides are chosen in a space of distributions allowing for solutions exhibiting discontinuities as well as impulses this approach also provides a satisfactory answer to the problem of inconsistent initial conditions of crucial importance for physical applications furthermore our theoretical results yield an efficient numerical procedure for the calculation of the jump and impulse of a solution at a point of discontinuity numerical examples are given

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

classical and quantum resonances for hyperbolic surfaces for compact and for convex co compact oriented hyperbolic surfaces we prove an explicit correspondence between classical ruelle resonant states and quantum resonant states except at negative integers where the correspondence involves holomorphic sections of line bundles

a renormalized index theorem for some complete asymptotically regular metrics the gauss bonnet theorem abstract we study the gauss bonnet theorem as a renormalized index theorem for edge metrics these metrics include the poincare einstein metrics of the ads cft correspondence and the asymptotically cylindrical metrics of the atiyah patodi singer index theorem we use renormalization to make sense of the curvature integral and the dimensions of the l 2 cohomology spaces as well as to carry out the heat equation proof of the index theorem for conformally compact metrics even mod x m we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

kinetic boundary conditions in the lattice boltzmann method derivation of the lattice boltzmann method from the continuous kinetic theory x he and l s luo phys rev e 55 r6333 1997 x shan and x he phys rev lett 80 65 1998 is extended in order to obtain boundary conditions for the method for the model of a diffusively reflecting moving solid wall the boundary condition for the discrete set of velocities is derived and the error of the discretization is estimated numerical results are presented which demonstrate convergence to the hydrodynamic limit in particular the knudsen layer in the kramers problem is reproduced correctly for small knudsen numbers

technical communique impulse observability and impulse controllability of linear time varying singular systems impulse controllability and impulse observability of linear time varying singular systems in a standard canonical form are examined we focus on the fast subsystems with constant rank n matrices sufficient and necessary conditions are derived and presented directly in terms of system parameters conditions obtained for these two properties are algebraically dual to each other

scattering phase asymptotics with fractal remainders for a riemannian manifold m g which is isometric to the euclidean space outside of a compact set and whose trapped set has liouville measure zero we prove weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate for axiom a geodesic flows this gives a polynomial improvement over the known remainders we also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis and provide similar asymptotics for hyperbolic quotients using the selberg zeta function

eta invariant and selberg zeta function of odd type over convex co compact hyperbolic manifolds abstract we show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type z o associated to the spinor bundle on an odd dimensional convex co compact hyperbolic manifold h 2 n 1 as a byproduct we do a full analysis of the spectral and scattering theory of the dirac operator on asymptotically hyperbolic manifolds we show that there is a natural eta invariant d associated to the dirac operator d over a convex co compact hyperbolic manifold h 2 n 1 and that exp i d z o 0 thus extending millson s formula to this setting under some assumption on the exponent of convergence of poincare series for the group we also define an eta invariant for the odd signature operator and we show that for schottky 3 dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the zograf factorization formula relating two natural kahler potentials for weil petersson metric on schottky space

resolvent of the laplacian on geometrically finite hyperbolic manifolds for geometrically finite hyperbolic manifolds n 1 we prove the meromorphic extension of the resolvent of laplacian poincare series einsenstein series and scattering operator to the whole complex plane we also deduce the asymptotics of lattice points of in large balls of n 1 in terms of the hausdorff dimension of the limit set of

equidistribution of eisenstein series on convex co compact hyperbolic manifolds the quantum ergodic theorem due to schnirelman 4 colin de verdiere 1 and zelditch 5 says that on any compact riemannian manifold x whose geodesic flow isergodic one can find a full density sequence j of eigenvalues of the laplacian delta x such that the corresponding normalized eigenfunctions j are equidistributed i e for all f l 2 x we have lim j rightarrow infty int limits x f z vert psi j z vert 2 dv z int limits x f z dv z where dv is the normalized volume measure

regularization of linear descriptor systems with variable coefficients we study linear descriptor control systems with rectangular variable coefficient matrices we introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback we show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system plus some solution components that are constrained to be zero

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

holographic formula for q curvature ii we extend the holographic formula for the critical q curvature in graham and juhl 2007 9 to all q curvatures moreover we confirm a conjecture of juhl 2009 11

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

scattering and inverse scattering on ach manifolds we study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds we show the existence of the scattering operator as a meromorphic family of operators in the heisenberg calculus on the boundary which is a contact manifold with a pseudohermitian structure then we define the radiation fields as in the real asymptotically hyperbolic case and reconstruct the scattering operator from those fields as an application we show that the manifold including its topology and the metric are determined up to invariants by the scattering matrix at all energies

the output nulling space projected dynamics and system decomposition for linear time varying singular systems a decomposition of a given linear time varying singular control system is developed by defining the output nulling space with respect to a given output structure relevant subspaces and the dynamics on them are described by computable projection operators obtained from information in the system s derivative array the relevant projectors are generated as solutions of a homogeneous linear matrix differential equation an algorithm for obtaining the system decomposition is outlined in a pointwise manner

determinants and conformal anomalies of gjms operators on spheres the conformal anomalies and functional determinants of the branson gjms operators p2k on the d dimensional sphere are evaluated in explicit terms for any d and k such that k d 2 if d is even the determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly which vanishes for odd d taking the mode system on the sphere as the union of neumann and dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non existence of super critical operators 2k d for even d significant use is made of the barnes zeta function the results are given in terms of ratios of determinants of operators on a d 1 dimensional bulk dual sphere for odd dimensions the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations yielding an integral over a plancherel measure the n d determinant ratio is also found explicitly for even dimensions ehrhart polynomials are encountered

functional determinants generalized btz geometries and selberg zeta function we continue the study of a special entry in the ads cft dictionary namely a holographic formula relating the functional determinant of the scattering operator in an asymptotically locally anti de sitter space to a relative functional determinant of the scalar laplacian in the bulk a heuristic derivation of the formula involves a one loop quantum effect in the bulk and the corresponding sub leading correction at large n on the boundary we presently explore the formula in the background of a higher dimensional version of the euclidean btz black hole obtained as a quotient of hyperbolic space by a discrete subgroup of isometries generated by a loxodromic or hyperbolic element consisting of dilation temperature and torsion angles twist the bulk computation is done using heat kernel techniques and fractional calculus at the boundary we acquire a recursive scheme that allows us to successively include rotation blocks in spacelike planes in the embedding space the determinants are compactly expressed in terms of an associated patterson selberg zeta function and a connection to quasi normal frequencies is discussed

boundary induced nonlinearities at small reynolds numbers we investigate the importance of boundary slip at finite reynolds numbers for mixed boundary conditions nonlinear effects are induced by the non homogeneity of the boundary condition and change the symmetry properties of the flow with an overall mean flow reduction to explain the observed drag modification exact reciprocal relations in the presence of heterogeneous boundary conditions are derived the small reynolds number limit predicts a reduction of the mean flow rate from the creeping flow to be proportional to the second power of the reynolds number to further support our theoretical analysis numerical simulations with the lattice boltzmann method lbm and finite difference method fdm are performed and reveal a pronounced numerical efficiency of lbm with respect to fdm