a summation formula over the zeros of a combination of the associated legendre functions with a physical application by using the generalized abel plana formula we derive a summation formula for the series over the zeros of a combination of the associated legendre functions with respect to the degree the summation formula for the series over the zeros of the combination of the bessel functions previously discussed in the literature is obtained as a limiting case as an application we evaluate the wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space for the dirichlet boundary conditions the corresponding mode sum contains the series over the zeros of the combination of the associated legendre functions the application of the summation formula allows us to present the wightman function in the form of the sum of two integrals the first one corresponds to the wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell the boundary induced part in the vacuum expectation value of the field squared is investigated for points away from the boundaries the corresponding renormalization procedure is reduced to that for the boundary free part

analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

small mass expansion of functional determinants on the generalized cone in this paper we compute the small mass expansion for the functional determinant of a scalar laplacian defined on the bounded generalized cone in the framework of zeta function regularization we obtain an expression for the functional determinant valid in any dimension for both dirichlet and robin boundary conditions in terms of the spectral zeta function of the base manifold moreover as a particular case we specify the base to be a d dimensional sphere and present explicit results for d 2 3 4 5

the existence and uniqueness of solutions of yang mills equations with bag boundary conditions the cauchy problem for the yang mills equations in the coulomb gauge is studied on a compact connected and simply connected riemannian manifold with boundary an existence and uniqueness theorem for the evolution equations is proven for fields with cauchy data in an appropriate sobolev space the proof is based the hodge decomposition of the yang mills fields and the theory of non linear semigroups

ditopological texture spaces and fuzzy topology i basic concepts this is the first of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of l topological spaces the authors begin by defining the notion of q sets which together with the p sets considered earlier enable the formulation of a powerful concept of duality this plays an important role in the theory of direlations and difunctions which is described here in detail difunctions are then taken as the morphisms of a category dftex whose objects are texture spaces several important subcategories are defined and the closely related construct ftex defined some properties of the functors between these categories are obtained

phase resetting phase locking and bistability in the periodically driven saline oscillator experiment and model the saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water with a small orifice in the center of the bottom of the inner vessel there is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice we develop a very stable i e stationary version of this saline oscillator we first investigate the response of the oscillator to periodic forcing with a train of stimuli text period t mathrm p of large amplitude each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume for t mathrm p sufficiently close to the intrinsic period of the oscillator t 0 there is 1 1 synchronization or phase locking between the stimulus train and the oscillator as t mathrm p is decreased below t 0 one finds the succession of phase locking rhythms 1 1 2 2 2 1 2 2 and 1 1 as t mathrm p is increased beyond t 0 one encounters successively 1 1 1 2 2 4 2 3 2 4 and 1 2 phase locking rhythms we next investigate the phase resetting response in which injection of a single stimulus transiently changes the period of the oscillation by systematically changing the phase of the cycle at which the stimulus is delivered the old phase we construct the new phase old phase curve the phase transition curve from which we then develop a one dimensional finite difference equation map that predicts the response to periodic stimulation these predicted phase locking rhythms are close to the experimental findings in addition iteration of the map predicts the existence of bistability between two different 1 1 rhythms which was then searched for and found experimentally bistability between 1 1 and 2 2 rhythms is also encountered finally with one exception numerical modeling with a phenomenologically derived rayleigh oscillator reproduces all of the experimental behavior

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

di uniformities and hutton uniformities the authors characterize di uniformities on a texture s s in the sense of ozcag and brown di uniform texture spaces appl gen top 4 1 2003 157 192 in terms of functions on the texturing s this characterization enables quasi uniformities in the sense of hutton uniformities on fuzzy topological spaces j math anal appl 58 1977 559 571 to be regarded as di uniformities on the corresponding hutton texture thereby revealing di uniformities as a generalization of hutton quasi uniformities the effect of imposing a complementation on s s is also considered and several important results established

analytic surgery of the zeta function in this paper we study the asymptotic behavior in the sense of meromorphic functions of the zeta function of a laplace type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface separating the manifold into two manifolds with infinite cylindrical ends we also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary forming a manifold with an infinite cylindrical end such singular deformations fall under the category of analytic surgery developed originally by hassell comm anal geom 6 255 289 1998 hassell et al comm anal geom 3 115 222 1995 and mazzeo and melrose geom funct anal 5 14 75 1995 in the context of eta invariants and determinants

symbol calculus and zeta function regularized determinants in this work we use semigroup integral to evaluate zeta function regularized determinants this is especially powerful for nonpositive operators such as the dirac operator in order to understand fully the quantum effective action one should know not only the potential term but also the leading kinetic term in this purpose we use the weyl type of symbol calculus to evaluate the determinant as a derivative expansion the technique is applied both to a spin 0 bosonic operator and to the dirac operator coupled to a scalar field

isochrones and the dynamics of kicked oscillators abstract the interest in studying the isochrone portrait of relaxation oscillators with a stable limit cycle is that knowledge of this portrait implies knowledge of the phase transition curve ptc and the ptc allows one to obtain in the case of impulsive forces a one dimensional approximation of the poincare stroboscopic map of the system in this paper a simple and effective method for finding the isochrone portrait is shown the effectiveness of our method was tested in the reconstruction of an isochrone of the bonhoffer van der pol model studied by a t winfree using the isochrone portrait and simple geometric arguments important properties of the ptc such as its topological degree and the presence of cubic singularities can be predicted without calculation the method is applied to two models van der pol and brusellas we also compute explicitly the ptcs to show how the geometric analysis works the analysis is valid for a large class of impulsive forces

on the resonance structure in a forced excitable system the dynamics of forced excitable systems are studied analytically and numerically with a view toward understanding the resonance or phase locking structure in a singular limit the system studied reduces to a discontinuous flow on a two torus which in turn gives rise to a set valued circle map it is shown how to define rotation numbers for such systems and derive properties analogous to those known for smooth flows the structure of the phase locking regions for a fitzhugh nagumo system in the singular limit is also analyzed a singular perturbation argument shows that some of the general results persist for the nonsingularly perturbed system and some numerical results on phase locking in the forced fitzhugh nagumo equations illustrate this fact the results explain much of the phase locking behavior seen experimentally and numerically in forced excitable systems including the existence of threshold stimuli for phase locking the results are compared with known results for forced oscillatory systems

global dynamics of a stochastic neuronal oscillator nonlinear oscillators have been used to model neurons that fire periodically in the absence of input these oscillators which are called neuronal oscillators share some common response structures with other biological oscillations such as cardiac cells in this study we analyze the dependence of the global dynamics of an impulse driven stochastic neuronal oscillator on the relaxation rate to the limit cycle the strength of the intrinsic noise and the impulsive input parameters to do this we use a markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve which describes the phase shift due to a single isolated impulse previously we derived the markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane here we construct a markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle in both cases the response of the stochastic neuronal oscillator to time varying impulses is described by a product of markov operators furthermore we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density specifically we analyze the dynamics of the number of spikes per unit time based on the properties of the markov operators each markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions this allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator which we show to be based on the dependence of the oscillator on past activity our analysis shows how the duration of the past neuronal activity depends on the relaxation rate the noise strength and the impulsive input parameters

forcing an entire bifurcation diagram case studies in chemical oscillators abstract we study the finite amplitude periodic forcing of chemical oscillators in particular we examine systems that when autonomous i e for zero forcing amplitude exhibit a single stable oscillation using one of the system parameters as a forcing variable by varying it periodically we show through extensive numerical work how the bifurcation diagram of the autonomous system with respect to this parameter affects the qualitative response of the full forced system as the forcing variable oscillates around its midpoint its instantaneous values may cross points such as hopf bifurcation poiints of the autonomous bifurcation diagram so that the characterization of the system as a simple forced oscillator is no longer valid such a neighboring hopf bifurcation of the unforced system is found to set the scene for the interaction of resonance horns and the loss of tori in the full forced system as the amplitude of the forcing grows our test case presented here is the continuous stirred tank reactor cstr with periodically forced coolant temperature

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

algebraic analysis of fuzzy systems in this paper we have developed an algebraic theory suitable for the analysis of fuzzy systems we have used the notions of semiring and semimodule introduced the notion of semilinear space have given numerous examples of them and defined also the notions of linear dependence and independence we have shown that the composition operation which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference can be interpreted as a homomorphism between special semimodules consequently this operation is in a certain sense a linear operation this property formally explains why fuzzy systems are attractive for applications

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

functional determinants for laplacians on annuli and elliptical regions in this article we consider the functional determinant on an annulus and elliptic regions with a variety of boundary conditions known results for the annulus are rederived and extended using contour integration techniques conformal transformations are then used to relate these results to ellipsoidal annuli providing very explicit answers similarly for the ellipse answers are related to those for a disc but explicit results can only be found for small eccentricity

antisymmetric tensor fields on spheres functional determinants and non local counterterms the hodge de rham laplacian on spheres acting on antisymmetric tensor fields is considered explicit expressions for the spectrum are derived in a quite direct way confirming previous results associated functional determinants and the heat kernel expansion are evaluated using this method new non local counterterms in the quantum effective action are obtained which can be expressed in terms of betti numbers

dirac operator on a disk with global boundary conditions we compute the functional determinant for a dirac operator in the presence of an abelian gauge field on a bidimensional disk under global boundary conditions of the type introduced by atiyah patodi singer we also discuss the connection between our result and the index theorem

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

the spectral zeta function for laplace operators on warped product manifolds of the type i f n in this work we study the spectral zeta function associated with the laplace operator acting on scalar functions defined on a warped product of manifolds of the type i f n where i is an interval of the real line and n is a compact d dimensional riemannian manifold either with or without boundary starting from an integral representation of the spectral zeta function we find its analytic continuation by exploiting the wkb asymptotic expansion of the eigenfunctions of the laplace operator on m for which a detailed analysis is presented we apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

determinants finite difference operators and boundary value problems we relate the determinants of differential and difference operators to the boundary values of solutions of the operators previous proofs of related results have involved considering one parameter families of such operators showing the desired quantities are equal up to a constant and then calculating the constant we take a more direct approach for a fixed operator we prove immediately that the two sides of our formulas are equal by using the following simple observation proposition 1 3 let u su n c write u in block form u left begin array 20 c u 11 u 12 u 21 u 22 end array right where u11and u22are square matrices then det u 11 overline det u 22

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

vector fields on a disk with mixed boundary conditions vector fields on a disk satisfying two types of boundary conditions are studied these boundary conditions are selected by brst invariance in electrodynamics they also appear in the de rham complex the construction of the harmonic expansion is presented the eigenfunctions of the vector laplace operator are expressed in terms of fields satisfying pure dirichlet or robin boundary conditions for the case of four dimensional disk several first coefficients of the heat kernel expansion are computed an error in the analytical expression by branson and gilkey is corrected

heat kernel expansion on noncompact domains and a generalized zeta function regularization procedure heat kernel expansion and zeta function regularization are discussed for laplace type operators with discrete spectrum in noncompact domains since a general theory is lacking the heat kernel expansion is investigated by means of several examples it is pointed out that for a class of exponential analytic interactions generically the noncompactness of the domain gives rise to logarithmic terms in the heat kernel expansion then a meromorphic continuation of the associated zeta function is investigated a simple model is considered for which the analytic continuation of the zeta function is not regular at the origin displaying a pole of higher order for a physically meaningful evaluation of the related functional determinant a generalized zeta function regularization procedure is proposed

heat kernel coefficients for chiral bag boundary conditions we study the asymptotic expansion of the smeared l 2 trace of f e tp 2 where p is an operator of dirac type f is an auxiliary smooth smear ing function which is used to localize the problem and chiral bag boundary conditions are imposed special case calculations functorial methods and the theory of and invariants are used to obtain the boundary part of the heat kernel coefficientsa1 and a2

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

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

spectral action for torsion with and without boundaries we derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the skew symmetric torsion and the chiral bag conditions on the boundary the spectral action splits into bulk and boundary parts in the bulk we clarify certain issues of the previous calculations show that many terms in fact cancel out and demonstrate that this cancellation is a result of the chiral symmetry of spectral action on the boundary we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter of the boundary conditions and show that 0 is a critical point of the action in any dimension and at all orders of the expansion

gauge theories on manifolds with boundary the boundary value problem for laplace type operators acting on smooth sections of a vector bundle over a compact riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied the condition of strong ellipticity of this boundary value problem is formulated the resolvent kernel and the heat kernel in the leading approximation are explicitly constructed as a result the previous work in the literature on heat kernel asymptotics is shown to be a particular case of a more general structure for a bosonic gauge theory on a compact riemannian manifold with smooth boundary the problem of obtaining a gauge field operator of laplace type is studied jointly with local and gauge invariant boundary conditions which should lead to a strongly elliptic boundary value problem the scheme is extended to fermionic gauge theories by means of local and gauge invariant projectors after deriving a general condition for the validity of strong ellipticity for gauge theories it is proved that for euclidean yang mills theory and rarita schwinger fields all the above conditions can be satisfied for euclidean quantum gravity however this property no longer holds i e the corresponding boundary value problem is not strongly elliptic some non standard local formulae for the leading asymptotics of the heat kernel diagonal are also obtained it is shown that due to the lack of strong ellipticity the heat kernel diagonal is non integrable near the boundary

conical casimir pistons with hybrid boundary conditions in this paper we compute the casimir energy and force for massless scalar fields endowed with hybrid boundary conditions in the setting of the bounded generalized cone by using spectral zeta function regularization methods we obtain explicit expressions for the casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston our general formulas are subsequently specialized to the case in which the piston is modeled by a d dimensional sphere in this particular situation explicit results are given for d 2 3 4 5

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

functional determinants on spheres and sectors the scalar functional determinants on sectors of the two dimensional disc and spherical cap are determined for arbitrary angles rational factors of the whole sphere and hemisphere expressions are also given in low dimensions for both the ordinary and the conformal laplacian

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

conformal invariants for determinants of laplacians on riemann surfaces for a riemann surface with smooth boundaries conformal weyl invariant quantities proportional to the determinant of the scalar laplacian operator are constructed both for dirichlet and neumann boundary conditions the determinants are defined by zeta function regularization the other quantities in the invariants are determined from metric properties of the surface as applications explicit representations for the determinants on the flat disk and the flat annulus are derived

heat kernel coefficients for laplace operators on the spherical suspension in this paper we compute the coefficients of the heat kernel asymptotic expansion for laplace operators acting on scalar functions defined on the so called spherical suspension or riemann cap subjected to dirichlet boundary conditions by utilizing a contour integral representation of the spectral zeta function for the laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure thanks to the well known relation between the zeta function and the heat kernel obtainable via mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions the particular case of a d dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly

spectral analysis and zeta determinant on the deformed spheres we consider a class of singular riemannian manifolds the deformed spheres s n k defined as the classical spheres with a one parameter family g k of singular riemannian structures that reduces for k 1 to the classical metric after giving explicit formulas for the eigenvalues and eigenfunctions of the metric laplacian delta s n k we study the associated zeta functions zeta s delta s n k we introduce a general method to deal with some classes of simple and double abstract zeta functions generalizing the ones appearing in zeta s delta s n k an application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions and in particular zeta 0 delta s n k and zeta 0 delta s n k we give explicit formulas for the zeta regularized determinant in the low dimensional cases n 2 3 thus generalizing a result of dowker 25 and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k

some analytic continuations of the barnes zeta function in two and higher dimensions abstract formulas for the analytic continuation of the barnes zeta function and some affine extensions thereof in two and more dimensions are constructed the expressions are used to deal with determinants of multidimensional harmonic oscillators an example is therewith obtained of the multiplicative anomaly or defect associated with the most common definition due to ray and singer d b ray reidemeister torsion and the laplacian on lens spaces adv math 4 1970 109 126 d b ray i m singer r torsion and the laplacian on riemannian manifolds adv math 7 1971 145 201 d b ray i m singer analytic torsion for complex manifolds ann math 98 1973 154 177 of determinant of a pseudodifferential operator admitting a zeta function m kontsevich s vishik geometry of determinants of elliptic operators in functional analysis on the eve of the 21st century vol 1 1995 pp 173 197

ditopological texture spaces and fuzzy topology iii separation axioms this is the third of three papers which develop various fundamental aspects of the theory of ditopological texture spaces and relate this work with the theory of fuzzy topological spaces in this paper the authors define and study basic separation axioms in general ditopological texture spaces relating these to known separation axioms for bitopological and topological spaces and to the point free separation axioms of b hutton and i reilly for fuzzy topological spaces

zeta determinants for regular singular laplace type operators we discuss a specific class of regular singular laplace type operators with matrix coefficients their zeta determinants were studied by kirsten loya and park manuscr math 125 95 2008 on the basis of the contour integral method with general boundary conditions at the singularity and dirichlet boundary conditions at the regular boundary we complete the arguments of kirsten loya and park by explicitly verifying that the contour integral method indeed applies in the regular singular setup further we extend the zeta determinant computations to generalized neumann boundary conditions at the regular boundary and apply our results to compute zeta determinants of laplacians on a bounded generalized cone with relative ideal boundary conditions

zeta function for the laplace operator acting on forms in a ball with gauge boundary conditions the laplace operator acting on antisymmetric tensor fields in a d dimensional euclidean ball is studied gauge invariant local boundary conditions absolute and relative ones in the language of gilkey are considered the eigenfuctions of the operator are found explicitly for all values of d using in a row a number of basic techniques as mellin transforms deformation and shifting of the complex integration contour and pole compensation the zeta function of the operator is obtained from its expression in particular 0 and 0 are evaluated exactly a table is given in the paper for d 3 4 8 the functional determinants and casimir energies are obtained for d 3 4 6

zeta function regularization for a scalar field in a compact domain we express the zeta function associated with the laplacian operator on s1r m in terms of the zeta function associated with the laplacian on m where m is a compact connected riemannian manifold this gives formulae for the partition function of the associated physical model at low and high temperature for any compact domain m furthermore we provide an exact formula for the zeta function at any value of r when m is a d dimensional box or a d dimensional torus this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated with the physical models at finite temperature

some remarks on fuzzy powerset operators fuzzy powerset operators related to morphisms of underlying ground categories of categories of fuzzy topological spaces are considered and some properties of such operators related to isomorphisms fuzzy points and order reversing involutions are verified

bifurcation analysis of a periodically forced relaxation oscillator differential model versus phase resetting map we compare the dynamics of the periodically forced fitzhugh nagumo oscillator in its relaxation regime to that of a one dimensional discrete map of the circle derived from the phase resetting response of this oscillator the phase resetting map the forcing is a periodic train of gaussian shaped pulses with the width of the pulses much shorter than the intrinsic period of the oscillator using numerical continuation techniques we compute bifurcation diagrams for the periodic solutions of the full differential equations with the stimulation period being the bifurcation parameter the period 1 solutions which belong either to isolated loops or to an everywhere unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes merge together to form a single branch at larger stimulation amplitudes as a consequence of the fast slow nature of the oscillator this merging occurs at virtually the same stimulation amplitude for all the period 1 loops again using continuation we show that this stimulation amplitude corresponds in the circle map to a change of topological degree from one to zero we explain the origin of this coincidence and also discuss the translational symmetry properties of the bifurcation diagram

ditopological texture spaces and fuzzy topology ii topological considerations this is the second of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of l topological spaces in the first paper in this series subtitled basic concepts the authors presented the notions of direlation and difunction between textures and introduced the category dftex the construct ftex and several related categories in this paper the category dfditop of ditopological texture spaces and bicontinuous difunctions is defined and the forgetful functor u dfditop dftex shown to be topological several properties are discussed including the existence of products and coproducts an equivalence with the category of classical hutton spaces is presented and the paper ends with a consideration of l valued sets and topologies from the viewpoint of ditopological texture spaces

resonance in excitable systems under step function forcing ii subharmonic solutions and persistence abstract in an earlier paper othmer and watanabe 1994 we studied the existence and stability of harmonic solutions in a fitzhugh nagumo type model under step function forcing in this paper we study subharmonic solutions and show that a number of phenomena arise which do not occur in invertible circle maps in particular we show that the rotation number may be non monotonic or discontinuous in one parameter families and that stable periodic solutions with different rotation numbers coexist on open sets in parameter space the latter result shows how an experimental observation first due to mines 1913 can be understood in the context of a flow we also show that the table results for the singular system persist in the non singular system in a sense made precise later

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

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

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

transporting many valued sets along many valued relations the transport of l sets along an l relation r is considered with l any residuated lattice sometimes with further properties as to be a frame or an mv algebra pairs of powerset operators related to r are considered and their basic properties are studied in a comprehensive discussion including some results already known a few examples and suggestions for application and further development are also given

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

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

period adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator the response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code in this paper we study a neurobiologically motivated relaxation oscillator with appropriately identified fast and slow coordinates that admits an explicit mathematical analysis an application of geometric singular perturbation theory shows the existence of an attracting invariant manifold which is used to construct the fenichel normal form for the system this facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so called extended isochronal map the isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response mode locked quasiperiodic and chaotic the bifurcation structure of the system is seen to be extremely rich and supports period adding bifurcations separated by windows of both chaos and periodicity a bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system

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

spectral asymmetry on the ball and asymptotics of the asymmetry kernel let be the dirac operator on a d 2d dimensional ball with radius r we calculate the spectral asymmetry for d 2 and d 4 when local chiral bag boundary conditions are imposed with these boundary conditions we also analyse the small t asymptotics of the heat trace where p is an operator of dirac type and f is an auxiliary smooth smearing function

phase reduction of weakly perturbed limit cycle oscillations in time delay systems abstract the phase reduction method is applied to a general class of weakly perturbed time delay systems exhibiting periodic oscillations the adjoint equation with an appropriate initial condition for the infinitesimal phase response curve of a time delay system is derived the method is demonstrated numerically for the mackey glass equation as well as for a chaotic rossler system subject to a delayed feedback control dfc we show that the profile of the phase response curve of a periodic orbit stabilized by the dfc algorithm does not depend on the control matrix this property is universal and holds for any dynamical system subject to the dfc

conformal transformations and the effective action in the presence of boundaries the conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary the conformal transformation of the effective action is obtained a novel derivation of the boundary term in the gauss bonnet chern theorem is detailed

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

effective action in spherical domains the effective action on an orbifolded sphere is computed for minimally coupled scalar fields the results are presented in terms of derivatives of barnes functions and it is shown how these may be evaluated numerical values are shown an analytical heat kernel derivation of the cesaro fedorov formula for the number of symmetry planes of a regular solid is also presented

extended real dicompactness and an application to hutton spaces abstract the authors extend the notion of real dicompactness from the class of nearly plain textures to the considerably larger class of almost plain textures and show how this may be used to define a notion of realcompactness for hutton spaces

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

finite temperature properties of the dirac operator under local boundary conditions we study the finite temperature free energy and fermion number for dirac fields in a one dimensional spatial segment under two different members of the family of local boundary conditions defining a self adjoint euclidean dirac operator in two dimensions for one of such boundary conditions compatible with the presence of a spectral asymmetry we discuss in detail the contribution of this part of the spectrum to the zeta regularized determinant of the dirac operator and thus to the finite temperature properties of the theory

synthetic analysis of periodically stimulated excitable and oscillatory membrane models many excitable neuronal membranes become oscillatory when stimulated by large enough dc currents in this paper we investigate how the transition from excitable to oscillatory regimes affects the response of the membrane to periodic pulse trains to this end we examine how the dynamics of periodically stimulated fitzhugh nagumo neuron model changes as the system switches from excitability to oscillation we show that despite the important change in the asymptotic dynamics of the unperturbed model p q phase locking i e the model membrane discharges q times in p interstimulus intervals and q input output intervals repeat periodically regions in the stimulus period stimulus amplitude parameter plane arnold tongues change continuously when the model changes from excitable to oscillatory we provide further evidence for the continuous change of the arnold tongues by using an analytically tractable one dimensional map that approximates the poincar e map of the forced system we argue that the smooth change in the arnold tongues results from the fact that despite the qualitative difference between the asymptotic dynamics of unforced excitable and oscillatory regimes other aspects of the dynamics such as the wave form of individual action potentials are similar in the two regimes

periodic forcing of a limit cycle oscillator fixed points arnold tongues and the global organization of bifurcations the effects of periodic pulsatile stimuli on a nonlinear limit cycle oscillation are analyzed for various relaxation rates to the limit cycle oscillation in the infinite relaxation limit the effects of periodic stimuli are analyzed by consideration of the bifurcations of circle maps in the case of a finite relaxation rate it is necessary to analyze two dimensional maps of the disk however the simple structure of the limit cycle allows us to carry out detailed theoretical and numerical analyses of the dynamics using the brouwer fixed point theorem we show that for any finite nonzero frequency and amplitude of the stimulus there will be a period 1 fixed point associated with a period 1 phase locking we use analytical and numerical methods to determine the stability boundary of the period 1 fixed point as a function of the stimulation frequency amplitude and relaxation rate to the limit cycle these results combined with numerical studies give insight into the changes in the global organization of the phase locking zones as a function of the relaxation rate to the limit cycle

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

symmetric tensor eigenspectrum of the laplacian on n spheres the eigenvalues and degeneracies of the covariant laplacian acting on symmetric tensors of rank m 2 defined on n spheres with n 3 are given