effective computation of traces determinants and ζ functions for sturm liouville operators abstract the principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents and resolvent differences fredholm determinants functions and function regularized determinants associated with linear operators in a hilbert space in particular we detail the connection between fredholm and function regularized determinants concrete applications of our formalism to general i e three coefficient regular sturm liouville operators on bounded intervals with various separated and coupled boundary conditions and schrodinger operators on a half line are provided and further illustrated with an array of examples

local l factors of motives and regularized determinants for a smooth projective variety x over a global field k consider the completed l function of h w x it is defined as the euler product over all places v of k of the local l factors l hw x s introduced in se in the function field case the generalized lefschetz trace formula in adic cohomology gives a global cohomological expression for the l function this has been used on the following purposes

tadpole graph in covariant closed string field theory a one loop tadpole graph in covariant closed string field theory must generate all inequivalent tori with one puncture in order to determine the region of integration in moduli space we map the tadpole to a two sheeted sphere and use the analyticity of the modular parameter with respect to the complex propagator parameter it is shown that the naive closed string extension of the witten open string vertex fails to give the correct modular region for this one loop amplitude

the polyakov path integral over bordered surfaces the open string amplitudes we use the feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic bdhp string in a position representation the consistent functional integral representation of the open string propagator is derived and evaluated this result is used as a starting point for two kinds of constructions of the off shell multiloop open string amplitudes the general idea of the presented approach is to consider the off shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space

elliptic operators in the functional quantisation for gauge field theories given a gauge theory with gauge groupg acting on a path spacex g andx being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary it is shown that when the action ofg onx is smooth free and proper the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection x x g and the existence of the faddeev popov determinant arising in the functional quantisation of the gauge theory this holds for certain gauge theories with anomalies like bosonic closed string theory in non critical dimension and also holds for a class of gauge theories which includes yang mills theory

on the gribov ambiguity in the polyakov string the global aspects of the gauge fixing in the polyakov path integral for the bosonic string are considered within the ebin fischer mareden approach to the geometry of spaces of riemannian metrics and conformal structures it is shown that for surfaces of higher genus the existence of local conformal gauges is sufficient to derive the globally defined integral over the teichmuller space the generalized faddeev popov procedure for incomplete gauges is formulated and used to derive the global expression for the polyakov path integral in the cases of torus and sphere the gribov ambiguity in the functional integral over surfaces without boundary can be successfully overcome for arbitrary genus

the spectral determinant of the isotropic quantum harmonic oscillator in arbitrary dimensions we show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity we determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large as a consequence we provide an alternative proof of a conjecture posed by bar and schopka concerning the convergence of the determinant of the dirac operator on s n determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the work of moller math ann 343 35 51 2009

algebraic groups over the field with one element in this paper we provide a first realization of an idea of jacques tits from a 1956 paper which first mentioned that there should be a field of characteristique une which is now called mathbb f 1 the field with one element this idea was that every split reductive group scheme over mathbb z should descend to mathbb f 1 and its group of mathbb f 1 rational points should be its weyl group we connect the notion of a torified scheme to the notion of mathbb f 1 schemes as introduced by connes and consani this yields models of toric varieties schubert varieties and split reductive group schemes as mathbb f 1 schemes we endow the class of mathbb f 1 schemes with two classes of morphisms one leading to a satisfying notion of mathbb f 1 rational points the other leading to the notion of an algebraic group over mathbb f 1 such that every split reductive group is defined as an algebraic group over mathbb f 1 furthermore we show that certain combinatorics that are expected from parabolic subgroups of gl n and grassmann varieties are realized in this theory

the notion of dimension in geometry and algebra this talk reviews some mathematical and physical ideas related to the notion of dimension after a brief historical introduction various modern constructions from fractal geometry noncommutative geometry and theoretical physics are invoked and compared glenn gould disapproved of his own recording of goldberg variations there is a lot of piano playing going on there and i mean that as the most disparaging comment possible nyrb oct 7 2004 p 10

precoding based network alignment using transform approach for acyclic networks with delay the algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known based on this formulation for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links the output symbols at the sink nodes at any given time instant is a f p m linear combination of the input symbols across different generations where f p m denotes the field over which the network operates p is prime and m is a positive integer we use finite field discrete fourier transform to convert the output symbols at the sink nodes at any given time instant into a f p m linear combination of the input symbols generated during the same generation without making use of memory at the intermediate nodes we call this as transforming the acyclic network with delay into n instantaneous networks n is sufficiently large we show that under certain conditions there exists a network code satisfying sink demands in the usual nontransform approach if and only if there exists a network code satisfying sink demands in the transform approach when the zero interference conditions are not satisfied we propose three precoding based network alignment pbna schemes for three source three destination multiple unicast network with delays 3 s 3 d mun d termed as pbna using transform approach and time invariant local encoding coefficients lecs pbna using time varying lecs and pbna using transform approach and block time varying lecs we derive sets of necessary and sufficient conditions under which throughputs close to n 1 2n 1 n 2n 1 and n 2n 1 are achieved for the three source destination pairs in a 3 s 3 d mun d employing pbna using transform approach and time invariant lecs and pbna using transform approach and block time varying lecs where n is a positive integer for pbna using time varying lecs we obtain a sufficient condition under which a throughput demand of n 1 n n 2 n and n 3 n can be met for the three source destination pairs in a 3 s 3 d mun d where n 1 n 2 and n 3 are positive integers less than or equal to the positive integer n this condition is also necessary when n 1 n 3 n 1 n 2 n where n 1 n 2 n 3

the polyakov path integral over bordered surfaces ii the closed string off shell amplitudes following the general scheme of the convariant path integral quantization of gauge systems two alternative formulations of the first quantized closed bosonic string in a position representation are presented in both approaches the covariant path integral representations of the propagator and of the higher order off shell amplitudes are constructed for a wide class of gauges the explicit formulae for off shell amplitudes are obtained this paper is the continuation of our previous work where the corresponding problems in the open string case were considered 20

spatio temporal coding for wireless communication multipath signal propagation has long been viewed as an impairment to reliable communication in wireless channels this paper shows that the presence of multipath greatly improves achievable data rate if the appropriate communication structure is employed a compact model is developed for the multiple input multiple output mimo dispersive spatially selective wireless communication channel the multivariate information capacity is analyzed for high signal to noise ratio snr conditions the mimo channel can exhibit a capacity slope in bits per decibel of power increase that is proportional to the minimum of the number multipath components the number of input antennas or the number of output antennas this desirable result is contrasted with the lower capacity slope of the well studied case with multiple antennas at only one side of the radio link a spatio temporal vector coding stvc communication structure is suggested as a means for achieving mimo channel capacity the complexity of stvc motivates a more practical reduced complexity discrete matrix multitone dmmt space frequency coding approach both of these structures are shown to be asymptotically optimum an adaptive lattice trellis coding technique is suggested as a method for coding across the space and frequency dimensions that exist in the dmmt channel experimental examples that support the theoretical results are presented

sums of certain series of the riemann zeta function abstract the object of the present paper is to investigate systematically several interesting families of summation formulas involving infinite series of the riemann zeta function many of the various results which are unified and generalized here in a remarkably simple manner have received considerable attention in recent years we also present a brief account of a number of analogous results associated with the hurwitz s generalized zeta function

error bounds and exponential improvement for the asymptotic expansion of the barnes g function in this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the barnes g function using these representations we obtain e

the faddeev popov procedure and application to bosonic strings an infinite dimensional point of view a generalisation of the finite dimensional presentation of the faddeev popov perocedure is derived in an infinite dimensional framework for gauge theories with finite dimensional moduli space using heat kernel regularised determinants it is shown that the infinite dimensional faddeev popov determinant is up to a finite dimensional determinant determined by a choice of a slice canonically determined by the geometrical data defining the gauge theory namely a fibre bundlep p g with structure groupg and the invariance group of a metric structure given on the total spacep the case of closed bosonic string theory is discussed

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

the polyakov path integral over bordered surfaces iii the brst extended closed string off shell amplitudes the geometrical approach to the functional integral over faddeev popov ghost fields is developed and applied to construct the brst extension of the off shell closed string amplitudes in the constant curvature gauge in this gauge the overlap path integral for off shell amplitudes is evaluated it leads to the nonlocal sewing procedure generating all off shell amplitudes from the cubic interaction vertex the general scheme of the reconstruction of a covariant closed string field theory from the off shell amplitudes is discussed within the path integral framework

higher selberg zeta functions in the paper kw2 we introduced a new type of selberg zeta function for establishing a certain identity among the non trivial zeroes of the selberg zeta function and of the riemann zeta function we shall call this zeta function a higher selberg zeta function the purpose of this paper is to study the analytic properties of the higher selberg zeta function z s especially to obtain the functional equation we also describe the gamma factor of z s in terms of the triple sine function explicitly and further determine the complete higher selberg zeta function with having a discussion of a certain generalized zeta regularization

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

dependent coordinates in path integral measure factorization the transformation of the path integral measure under the reduction procedure in dynamical systems with a symmetry is considered investigation is carried out in the case of the wiener type path integrals that are used for description of the diffusion on a smooth compact riemannian manifold with the given free isometric action of the compact semisimple unimodular lie group the transformation of the path integral which factorizes the path integral measure is based on the application of the optimal nonlinear filtering equation from the stochastic theory the integral relation between the kernels of the original and reduced semigroups is obtained

the determinant of one dimensional polyharmonic operators of arbitrary order abstract we obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator p n 1 n x 2 n on 0 t with dirichlet boundary conditions and n a positive integer and show that it satisfies the asymptotics log det p n n 2 log n 7 3 2 2 3 2 log t 4 n 2 o n for large n this is a consequence of sharp upper and lower bounds for log det p n valid for all n and which coincide in the terms up to order n these results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

the integration of g invariant functions and the geometry of the faddeev popov procedure various versions of the fubini theorem on the principal fibre bundle are derived the formal generalizations of these theorems are used as a basic tool for the investigation of the geometrical setting of the faddeev popov procedure in the cases of pure yang mills theories spinless particle and polyakov s bosonic string

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

on functional determinants of laplacians in polygons and simplicial complexes the functional determinant of an elliptic operator with positive discrete spectrum may be defined ase z 0 wherez s the zeta function is the sum sum limits n lambda n delta analytically continued ins in this paperz 0 is calculated for the laplace operator with dirichlet boundary conditions inside polygons with the topology of a disc in the euclidean plane our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries our expression can be viewed as the energy for a system of static point particles corresponding to the corners of the polygon with self energy and pair interaction energy we have completely explicit closed expressions for triangles and regular polygons with an arbitrary number of sides among these there are five special cases three triangles the square and the circled where thez 0 are known by other means one special case fixes an integration constant and the other provide four independent analytical checks on our calculation

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

the perturbative approach to path integrals a succinct mathematical treatment we study finite dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory this involves a proper understanding of how wick s theorem allows one to evaluate integrals perturbatively i e as a series expansion in a formal parameter irrespective of convergence properties we establish invariance properties of such a wick expansion under coordinate changes and the action of a lie group of symmetries and we use this to study essential features of path integral manipulations including coordinate changes ward identities schwinger dyson equations faddeev popov gauge fixing and eliminating fields by their equation of motion we also discuss the asymptotic nature of the wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively

the geometrical interpretation of the faddeev popov determinant in polyakov s theory of random surfaces in the context of polyakov s theory the geometrical approach in which the functional measure is defined as a formal volume form is developed the special version of the fubini theorem on manifolds is derived which provides the geometrical interpretation of the faddeev popov method the conformal gauge is studied within this framework as a result the explicit form of functional measure in effective liouville quantum field theory is obtained

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and