circle and divisor problems and double series of bessel functions abstract in approximately 1915 ramanujan recorded two identities involving doubly infinite series of bessel functions the identities were brought to the mathematical public for the first time when his lost notebook was published in 1988 and are connected with the classical long standing circle and divisor problems respectively we provide a proof of the first identity for the first time by analytically continuing a new kind of dirichlet series delicate estimates of exponential sums are needed and the new methods we introduce may be of independent interest

weighted divisor sums and bessel function series ii abstract on page 335 in his lost notebook ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of bessel functions in each case there are three possible interpretations for the double series in an earlier paper two of the present authors proved the first identity under one possible interpretation in the present paper the second identity is proved under a similar interpretation with one additional assumption moreover under a second interpretation entirely different proofs of both identities depending on weighted or twisted divisor sums are offered the two identities are intimately connected with the classical circle and divisor problems respectively

sequential regularization methods for nonlinear higher index daes sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints this paper extends an earlier work siam j numer anal 33 1996 pp 1921 1940 to nonlinear problems and to differential algebraic equations daes with an index higher than 2 rather than having one winning method this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances we propose sequential regularization methods for index 2 and index 3 daes both with and without constraint singularities in the case of no constraint singularity we prove convergence results numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods the examples include constrained multibody systems

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

on the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method a miscible displacement of one incompressible fluid by another is modeled by a coupled system of partial differential equations the approximation of this system using a galerkin method that makes use of a modified method of characteristics for the concentration equation combined with a mixed finite element method for the pressure equation has been analyzed in comput math appl mech engrg 47 1984 pp 73 92 in this work we prove optimal error estimates in l 2 for this algorithm under milder restrictions in the time step than those required in the previous analysis in particular in the two dimensional case our estimates are valid when the time and space discretization parameters are of the same order which is a reasonable assumption in practice

generalized lambert series and arithmetic nature of odd zeta values it is pointed out that the generalized lambert series sum nolimits n 1 infty n n 2h e n nx 1 studied by kanemitsu tanigawa and yoshimoto can be found on page 332 of ramanujan s lost notebook in a slightly more general form we extend an important transformation of this series obtained by kanemitsu tanigawa and yoshimoto by removing restrictions on the parameters n and h that they impose from our extension we deduce a beautiful new generalization of ramanujan s famous formula for odd zeta values which for n odd and m 0 gives a relation between 2 m 1 and 2 nm 1 a result complementary to the aforementioned generalization is obtained for any even n and m it generalizes a transformation of wigert and can be regarded as a formula for 2 m 1 1 n applications of these transformations include a generalization of the transformation for the logarithm of dedekind eta function z zudilin and rivoal type results on transcendence of certain values and a transcendence criterion for euler s constant

integral representations for the exceptional univariate lommel functions an elementary integral representation for the lommel function s 1 0 z is given and extended to other exceptional cases

discontinuous upwinding and mixed finite elements for two phase flows in reservoir simulation abstract a method for the simulation of incompressible two phase flows in reservoir simulations is presented here its distinctive features are the following the global pressure is introduced to formulate the equations discontinuous upwinding techniques are used to approximate the transport terms of the saturation equation while mixed finite elements are used to approximate the pressure equation and the diffusion term of the saturation equation around wells macroelements which are divided into sectors in which the flow is simulated by a one dimensional model are introduced numerical results are shown

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

mixed methods with dynamic finite element spaces for miscible displacement in porous media abstract miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations we approximate the pressure equation which is elliptic and the concentration equation which is parabolic but normally convection dominated by the mixed methods with dynamic finite element spaces i e different number of elements and different basis functions are adopted at different time levels and the approximate concentration is projected onto the next finite element space in weighted l 2 norm for starting a new time step this allows us to make local grid refinements or unrefinements and basis function improvements two fully discrete schemes are presented and analysed error estimates show that these methods have optimal convergent rate in some sense the efficiency and capability of the dynamic finite element method are commented for accurately solving time dependent problems with localized phenomena such as fronts shocks and boundary layers

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

fourier analysis of stabilized q 1 q 1 mixed finite element approximation we use fourier analysis to investigate the instability of an equal order mixed finite element approximation method for elliptic incompressible flow equations the lack of stability can be attributed to the fact that the associated discrete ladyzhenskaya babuska brezzi lbb constant tends to zero as the mesh size is reduced we develop a stabilization approach that is appropriate to the periodic setting and deduce optimal choices of the associated stabilization parameter