numerical and asymptotic analysis of a localized heat source undergoing periodic motion abstract a highly localized heat source moves with simple periodic motion along a one dimensional medium with reactive diffusive properties it is known that the system will experience a blow up in finite time numerical results suggest that this blow up though unavoidable can be delayed by increasing either the amplitude or the frequency of the motion these numerical results are compared to known analytical results numerical and analytical results are also compared for the related case in which the heat source moves at a constant speed in one direction the asymptotic behavior of the temperature of the material at the location of the heat source near the time of blow up is also analyzed it can be of interest to consider the asymptotic behavior of the solution in the context of the numerical solution in order to gain confidence in the numerical results

functional determinant of the massive laplace operator and the multiplicative anomaly after a brief survey of zeta function regularization issues and of the related multiplicative anomaly illustrated with a couple of basic examples namely the harmonic oscillator and quantum field theory at finite temperature an application of these methods to the computation of functional determinants corresponding to massive laplacians on spheres in arbitrary dimensions is presented explicit formulas are provided for the laplace operator on spheres in dimensions and for vector and tensor laplacians on the unitary sphere s4

functional determinants in quantum field theory functional determinants of differential operators play a prominent role in theoretical and mathematical physics and in particular in quantum field theory they are however difficult to compute in non trivial cases for one dimensional problems a classical result of gel fand and yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues here i report recent progress in extending this approach to higher dimensions i e functional determinants of partial differential operators with applications in quantum field theory

zeta determinants on manifolds with boundary abstract through a general theory for relative spectral invariants we study the determinant of global boundary problems of aps type in particular we compute the determinant ratio for dirac laplacian boundary problems in terms of a scattering fredholm determinant over the boundary

functional determinants in higher derivative lagrangian theories motivated by the considerable success of alternative theories of gravity we consider the toy model of a higher derivative lagrangian theory which reduces in the gaussian approximation to the pais uhlenbeck oscillator studied in a recent paper by hawking and hertog phys rev d 65 103515 2002 its euclidean path integral is studied with a certain detail and a pedagogical derivation of the propagator which makes use of a theorem due to forman is consequently proposed

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

blow up solutions to a system of nonlinear volterra equations a system of nonlinear volterra integral equations with convolution kernels is considered estimates are given for the blow up time when conditions are such that the solution is known to become unbounded in finite time for two examples that arise in combustion problems numerical estimates of blow up time are presented additionally the asymptotic behavior of the blow up solution in the key limit is established for the power law and exponential nonlinearity cases

a condition for finite blow up time for a volterra integral equation abstract the paper is devoted to the study of the equation u x x 0 x s 1 g u s ds 0 x 0 where g is continuous and nondecreasing some estimates of u and the interval of the existence of a nontrivial solution u are given

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

meyer vietoris type formula for determinants of elliptic differential operators abstract for a closed codimension one submanifold of a compact manifold m let m be the manifold with boundary obtained by cutting m along let a be an elliptic differential operator on m and b and c be two complementary boundary conditions on if a b is an elliptic boundary valued problem on m then one defines an elliptic pseudodifferential operator r of neumann type on and prove the following factorization formula for the regularized determinants det a det a b k det r with k a local quantity depending only on the jets of the symbols of a b and c along the particular case when m has dimension 2 a is the laplace beltrami operator and b resp c is the dirichlet resp neumann boundary condition is considered

blowing up behavior for a class of nonlinear vies connected with parabolic pdes we study the blowing up behavior of solutions of a class of nonlinear integral equations of volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities we present some analytic results and in the case of the kernel of abel kind with power nonlinearity and fixed initial data we give a numerical approximation by using one point collocation methods by means of the numerical simulations we give the dependence of the blow up time from the parameters of the equation

stability for second order abstract evolution equations on presente une methode generale qui semble bonne pour etudier les proprietes de stabilite d une grande classe d equations non lineaires et qui n a pas besoin de propriete de compacite

analysis of explosion for nonlinear volterra equations this survey paper presents some analytical tools for determining the existence of a blow up solution for a wide class of nonlinear volterra integral equations the related phenomenon of quenching has also been investigated using similar techniques in both cases lower and upper bound estimates for the critical time as well as asymptotic solution behavior in the key limit are established this paper surveys recent results in the analysis of such problems and proposes problems for future study

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

on functional determinants of matrix differential operators with multiple zero modes we generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by mckane and tarlie to differential operators with multiple zero eigenvalues we derive general formulas for such functional determinants of r times r matrix second order differential operators o with 0 n leqslant 2r linearly independent zero modes we separately discuss the cases of the homogeneous dirichlet boundary conditions when the number of zero modes cannot exceed r and the case of twisted boundary conditions including the periodic and anti periodic ones when the number of zero modes is bounded above by 2r in all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r n or 2r n depending on the boundary conditions solutions of the homogeneous equation o h 0 in the spirit of gel fand yaglom approach in instanton calculations the contribution of the zero modes is taken into account by introducing the so called collective coordinates we show that there is a remarkable cancellation of a factor involving scalar products of zero modes between the jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues this cancellation drastically simplifies instanton calculations when one uses our formulas

conditions for blow up of solutions of some nonlinear volterra integral equations in this paper we give necessary and sufficient conditions for blow up of solutions for a particular class of nonlinear volterra equations we also give some examples

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

osgood type condition for the volterra integral equations with bounded and nonincreasing kernels abstract this paper provides the necessary and sufficient osgood type condition for the existence of blow up solutions of volterra equation with kernels being nonincreasing and bounded functions examples of such equations related to models of anomalous diffusion as well as some integral estimates of blow up time are also presented

almost periodic solutions for a semilinear quasi autonomous hyperbolic problem on demontre des conditions suffisantes pour la stabilite locale pour garantir la presque periodicite d une solution globale

blow up behavior of collocation solutions to hammerstein type volterra integral equations we analyze the blow up behavior of one parameter collocation solutions for hammerstein type volterra integral equations vies whose solutions may blow up in finite time to approximate such solutions and the corresponding blow up time we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow up behavior is the same as the one for the exact solution based on the local convergence of the collocation methods for vies we present the convergence analysis for the numerical blow up time numerical experiments illustrate the analysis

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

p determinant regularization method for elliptic boundary problems an expression for thep determinant of the quotient of two differential elliptic operators with boundary conditions is given in terms of the boundary values of their solutions applications to physical examples are considered

regularization of functional determinants using boundary perturbations the formalism which has been developed to give general expressions for the determinants of differential operators is extended to the physically interesting situation where these operators have a zero mode which has been extracted in the approach adopted here this mode is removed by a novel regularization procedure which allows remarkably simple expressions for these determinants to be derived

a new upper estimation for the blow up time of solutions of a volterra integral equation and its application to the modeling of the formation of shear bands abstract the estimation of the blow up time of a class of volterra integral equations of convolution type is studied a new upper estimate of the blow up time expressed in terms of the convergence of a certain integral is found moreover it turns out that the aforementioned upper estimate can be applied to an even more general class of volterra integral equations like the one describing the formation of shear bands in steel

dimensional influence on blow up in a superdiffusive medium thermal blow up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one two and three dimensions an analysis of a nonlinear model of this problem reveals that the occurrence of a blow up depends upon the spatial dimension of the superdiffusive medium in one dimension a blow up always occurs in two and three dimensions a blow up can be avoided if the superdiffusive properties are sufficiently enhanced the asymptotic growth of the temperature near blow up is determined for energy sources whose output increases in either an algebraic or an exponential manner the results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

thermal blow up in a subdiffusive medium the problem of thermal blow up in a subdiffusive medium is examined within the framework of a fractional heat equation with a nonlinear source term this model establishes that a thermal blow up always occurs when a finite strip of subdiffusive material is exposed to the effects of a localized high energy source this behavior is distinctly different from the classical diffusion case in which a blow up can be avoided by locating the site of the energy source sufficiently close to one of the cold ends of the strip the asymptotic growth of the solution near blow up is determined for a nonlinear source whose output increases with temperature in either an algebraic or exponential manner the blow up growth rate is found to depend upon the anomalous diffusion parameter that defines the subdiffusive medium this suggests that such media might be characterized by their response to a reaction diffusion process

simplified vacuum energy expressions for radial backgrounds and domain walls we extend our previous results of simplified expressions for functional determinants for radial schrodinger operators to the computation of vacuum energy or mass corrections for static but spatially radial backgrounds and for domain wall configurations our method is based on the zeta function approach to the gel fand yaglom theorem suitably extended to higher dimensional systems on separable manifolds we find new expressions that are easy to implement numerically for both zero and non zero temperatures

determinants of dirac operators with local boundary conditions we study functional determinants for dirac operators on manifolds with boundary we give for local boundary conditions an explicit formula relating these determinants to the corresponding green s functions we finally apply this result to the case of a bidimensional disk under baglike conditions

on the gluing problem for the spectral invariants of dirac operators abstract in this paper we solve the gluing problem for the determinant of a dirac laplacian to do so we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems the analysis of the resolvent of the dirac operator and the introduction of an auxiliary model problem moreover as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis

recent results on blow up and quenching for nonlinear volterra equations in this survey paper the author examines nonlinear volterra integral equations of the second kind with solutions that blow up or quench the focus is on analytical results although a few words about numerical solutions for such equations are provided the integral equations arise in the mathematical modeling of thermal processes within a reactive diffusive medium the scope of this review is on the published literature between 1997 and 2005 serving as an update to a previous review by the same author

on the spectrum of the laplace operator of metric graphs attached at a vertex spectral determinant approach we consider a metric graph made of two graphs and attached at one point we derive a formula relating the spectral determinant of the laplace operator in terms of the spectral determinants of the two subgraphs the result is generalized to describe the attachment of n graphs the formulae are also valid for the spectral determinant of the schrodinger operator

existence and asymptotic behavior of multi dimensional quantum hydrodynamic model for semiconductors this paper is devoted to the study of the existence and the time asymptotic of multi dimensional quantum hydrodynamic equations for the electron particle density the current density and the electrostatic potential in spatial periodic domain the equations are formally analogous to classical hydrodynamics but differ in the momentum equation which is forced by an additional nonlinear dispersion term due to the quantum bohm potential and are used in the modelling of quantum effects on semiconductor devices we prove the local in time existence of the solutions in the case of the general nonconvex pressure density relation and large and regular initial data furthermore we propose a subsonic type stability condition related to one of the classical hydrodynamical equations when this condition is satisfied the local in time solutions exist globally in time and converge time exponentially toward the corresponding steady state since for this problem classical methods like for instance the friedrichs theory for symmetric hyperbolic systems cannot be used we investigate via an iterative procedure an extended system which incorporates the one under investigation as a special case in particular the dispersive terms appear in the form of a fourth order wave type equation

fermionic determinants for chiral bag like two dimensional systems this paper evaluates the determinant of a dirac like operator for a system of fermions confined to a region of two dimensional euclidean space and subject to bag like boundary conditions in the framework of the function regularization method use is made of seeley s developments for the resolvent of such operators a relation is derived between the green s function and the finite part of the diagonal element of the kernel of the power z of the operator for z 1 which amounts to having information about the whole seeley s series

functional determinants from wronski green functions a general technique is developed for calculating functional determinants of second order differential operators with dirichlet periodic and antiperiodic boundary conditions without the knowledge of spectral properties as an example we give explicit formulas for a harmonic oscillator with an arbitrary time dependent frequency where our result is a generalization of the gel fand yaglom famous formula for dirichlet boundary conditions our technique is based on the wronski s construction of green functions which does not require spectral knowledge our final formula expresses the ratios of functional determinants in terms of an ordinary 2 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with second order differential operators for ratios of determinants encountered in semiclassical fluctuations around a classical solution the result can further be expressed in terms of the classical solution special properties of operators with a zero mode are exhibited

moduli spaces of meromorphic functions and determinant of laplacian the hurwitz space is the moduli space of pairs x f where x is a compact riemann surface and f is a meromorphic function on x we study the laplace operator delta df 2 of the flat singular riemannian manifold x df 2 we define a regularized determinant for delta df 2 and study it as a functional on the hurwitz space we prove that this functional is related to a system of pde which admits explicit integration this leads to an explicit expression for the determinant of the laplace operator in terms of the basic objects on the underlying riemann surface the prime form theta functions the canonical meromorphic bidifferential and the divisor of the meromorphic differential df the proof has several parts that can be of independent interest as an important intermediate result we prove a decomposition formula of the type of burghelea friedlander kappeler for the determinant of the laplace operator on flat surfaces with conical singularities and euclidean or conical ends we introduce and study the s matrix s lambda of a surface with conical singularities as a function of the spectral parameter lambda and relate its behavior at lambda 0 with the schiffer projective connection on the riemann surface x we also prove variational formulas for eigenvalues of the laplace operator of a compact surface with conical singularities when the latter move

positive solutions to a nonlinear abel type integral equation on the whole line abstract we consider the existence of positive solutions to the nonlinear integral equation where g is a continuous nondecreasing function such that g 0 0 we show that the equation always has nontrivial solutions and we give a necessary and sufficient condition for the existence of solutions u such that u x we also provide a condition which ensures that all the nontrivial solutions experience the blow up behaviour

decay and stability for nonlinear hyperbolic equations this paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation the la salle invariance principle has been used to obtain the stability results the first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence in the second part under some restrictions sufficient conditions for precompactness of the orbits and decay of solutions are given an existence and uniqueness theorem is proved in the appendix some examples are given

functional determinants for radial operators we derive simple new expressions in various dimensions for the functional determinant of a radially separable partial differential operator thereby generalizing the one dimensional result of gel fand and yaglom to higher dimensions we use the zeta function formalism and the results agree with what one would obtain using the angular momentum cutoff method based on radial wkb the final expression is numerically equal to an alternative expression derived in a feynman diagrammatic approach but is considerably simpler

numerical simulation of blowup in nonlocal reaction diffusion equations using a moving mesh method in this paper we implement the moving mesh pde method for simulating the blowup in reaction diffusion equations with temporal and spacial nonlinear nonlocal terms by a time dependent transformation the physical equation is written into a lagrangian form with respect to the computational variables the time dependent transformation function satisfies a parabolic partial differential equation usually called moving mesh pde mmpde the transformed physical equation and mmpde are solved alternately by central finite difference method combined with a backward time stepping scheme the integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach the monitor function in mmpdes plays a key role in the performance of the moving mesh pde method the dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle a variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests

zeta function on surfaces of revolution in this paper we applied the contour integral method for the zeta function associated with a differential operator to the laplacian on a surface of revolution using the wkb expansion we calculated the residues and values of the zeta function at several important points the results agree with those obtained from the heat kernel expansion we also obtained a closed form formula for the determinant of the laplacian on such a surface

blow up collocation solutions of nonlinear homogeneous volterra integral equations in this paper collocation methods are used for detecting blow up solutions of nonlinear homogeneous volterra hammerstein integral equations to do this we introduce the concept of blow up collocation solution and analyze numerically some blow up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied finally we discuss the relationships between necessary conditions for blow up of collocation solutions and exact solutions

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

multiplicity results for a p laplacian discrete problems of kirchhoff type this paper is concerned with the existence and multiplicity solutions for a discrete boundary value problem of p kirchhoff type our technical approach is based on variational methods

ζ determinants of laplacians with neumann and dirichlet boundary conditions in this paper we derive a formula for the ratio of the determinants of the laplacian with neumann and dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the determinant of the dirichlet to neumann map

zeta function of self adjoint operators on surfaces of revolution in this article we analyze the zeta function for the laplace operator on a surface of revolution a variety of boundary conditions separated and unseparated are considered formulas for several residues and values of the zeta function as well as for the determinant of the laplacian are obtained the analysis is based upon contour integration techniques in combination with a wkb analysis of solutions of related initial value problems

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

nonlinear volterra integral equations and the schroder functional equation we show an interesting connection between a special class of volterra integral equations and the famous schroder equation the basic results provide criteria for the existence of nontrivial as well as blow up solutions of the volterra equation expressed in terms of the convergence of some integrals examples related to volterra equations with power and exponential nonlinearities are presented

blow up time for solutions to some nonlinear volterra integral equations the problem of the estimating of a blow up time for solutions of volterra nonlinear integral equation with convolution kernel is studied new estimates lower and upper are found and moreover the procedure for the improvement of the lower estimate is presented main results are illustrated by examples the new estimates are also compared with some earlier ones related to a shear band model

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

functional determinants for ordinary differential operators the authors study the analytic properties of regularized functional determinants for ordinary differential operators to this end a new regularization scheme is introduced relations between different approaches are analysed and the connection between the presence of divergent traces and regularization differences is discussed

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

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

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

reformulating the schrodinger equation as a shabat zakharov system we reformulate the second order schrodinger equation as a set of two coupled first order differential equations a so called shabat zakharov system sometimes called a zakharov shabat system there is considerable flexibility in this approach and we emphasize the utility of introducing an auxiliary condition or gauge condition that is used to cut down the degrees of freedom using this formalism we derive the explicit but formal general solution to the schrodinger equation the general solution depends on three arbitrarily chosen functions and a path ordered exponential matrix if one considers path ordering to be an elementary process then this represents complete quadrature albeit formal of the second order linear ordinary differential equation

blow up solutions in one dimensional diffusion models abstract a sufficient and necessary condition for the existence of blow up solutions of a certain class of nonlinear volterra integral equations with kernels arising from various diffusion models and with nonlinearities satisfying the condition g 0 0 is given in the form of the so called generalized osgood condition such nonlinearities were not permitted in the original articles of olmstead and roberts where these diffusion models were introduced

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

critical metrics for the determinant of the laplacian in odd dimensions let m be a closed compact n dimensional manifold with n odd we calculate the first and second variations of the zeta regularized determinants det a and det l as the metric on m varies where a denotes the laplacian on functions and l denotes the conformal laplacian we see that the behavior of these functionals depends on the dimension indeed every critical metric for 1 n 1 2det a or 1 n 1 21 det l has finite index consequently there are no local maxima if n 4rm 1 and no local minima if n 4m 3 we show that the standard 3 sphere is a local maximum for det a while the standard 4m 3 sphere with m 1 2 is a saddle point by contrast for all odd n the standard n sphere is a local extremal for det l an important tool in our work is the canonical trace on odd class operators in odd dimensions this trace is related to the determinant by the formula det q tr log q and we prove some basic results on how to calculate this trace

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

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

functional determinants for general sturm liouville problems simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest we extend the range of situations for which these hold to cover systems of self adjoint operators of the sturm liouville type with arbitrary linear boundary conditions the results hold whether or not the operators have negative eigenvalues the physically important case of functional determinants of operators with a zero mode but where that mode has been extracted is studied in detail for the same range of situations as when no zero mode exists the method of proof uses the properties of generalized zeta functions the general form of the final results is the same for the entire range of problems considered