on p adic interpolating function for q euler numbers and its derivatives abstract in this paper we study a two variable p adic q l function l p q s t for dirchlet s character with the property that l p q n t e n n q p t 2 q 2 q p n p p q n e n n q p t for positive integers n and t c p with t p 1 and e n n q x generalized euler polynomials finally we prove that l p q s t is analytic in s and t for s c p with s p p 1 1 p 1 and t c p with t p 1

semilinear reaction diffusion systems of several components abstract the homogeneous dirichlet boundary value problem u it u i j 1 n u j p ij i 1 2 n in a bounded domain r n is considered where p ij 0 1 i j n are constants denote by i the identity matrix and p pij which is assumed to be irreducible we find out that whether or not i p is a so called m matrix plays a fundamental role in the blow up theorems

nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction diffusion system abstract we get simple conditions under which the elliptic system u u p 1 v q 1 v u p 2 v q 2 in r n n 3 with p i q i 1 i 1 2 has no positive radially symmetric solutions then by using this nonexistence result we establish blow up estimates for reaction diffusion system of fujita type u t u u p 1 v q 1 v t v u p 2 v q 2 with the homogeneous dirichlet boundary value condition

dedekind carlitz polynomials as lattice point enumerators in rational polyhedra we study higher dimensional analogs of the dedekind carlitz polynomials c u v a b sum limits k 1 b 1 u left frac ka b right v k 1 where u and v are indeterminates and a and b are positive integers carlitz proved that these polynomials satisfy the reciprocity law v 1 rm c u v a b u 1 rm c v u b a u a 1 v b 1 1 from which one easily deduces many classical reciprocity theorems for the dedekind sum and its generalizations we illustrate that dedekind carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the carlitz reciprocity law and various extensions of it our approach gives rise to new reciprocity theorems and computational complexity results for dedekind carlitz polynomials a characterization of dedekind carlitz polynomials in terms of generating functions of lattice points in triangles and a multivariate generalization of the mordell pommersheim theorem on the appearance of dedekind sums in ehrhart polynomials of 3 dimensional lattice polytopes

properties of discrete delta functions and local convergence of the immersed boundary method many problems involving internal interfaces can be formulated as partial differential equations with singular source terms numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions we study the convergence properties of such discretizations for constant coefficient elliptic problems using the immersed boundary method as an example we show how the order of the differential operator order of the finite difference discretization and properties of the discrete delta function all influence the local convergence behavior in particular we show how a recently introduced property of discrete delta functions the smoothing order is important in the determination of local convergence rates

a polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed we prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d dimensional euclidean space previously such algorithms were known for dimensions d 1 2 3 and 4 only

q dedekind type sums related to q zeta function and basic l series abstract the main purpose of this paper is to define new generating functions by applying the mellin transformation formula to these generating functions we define q analogue of riemann zeta function q analogue hurwitz zeta function q analogue dirichlet l function and two variable q l function in particular by using these generating functions we will construct new generating functions which produce q dedekind type sums and q dedekind type sums attached to dirichlet character we also give the relations between these sums and dedekind sums furthermore by using product which is given in this paper we will give the relation between dedekind sums and q l function as well

on the q extension of euler and genocchi numbers abstract carlitz has introduced an interesting q analogue of frobenius euler numbers in l carlitz q bernoulli numbers and polynomials duke math j 15 1948 987 1000 l carlitz q bernoulli and eulerian numbers trans amer math soc 76 1954 332 350 he has indicated a corresponding stadudt clausen theorem and also some interesting congruence properties of the q euler numbers a recent author s study of more general q euler and genocchi numbers can be found in previous publication t kim l c jang h k pak a note on q euler and genocchi numbers proc japan acad ser a math sci 77 2001 139 141 in this paper we give a new construction of q euler numbers which are different from carlitz s q extension and author s q extension in previous publication see t kim l c jang h k pak a note on q euler and genocchi numbers proc japan acad ser a math sci 77 2001 139 141 by using our q extension of euler numbers we can also consider a new q extension of genocchi numbers and obtain some interesting relations between q extension of euler numbers and q extension of genocchi numbers

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

genocchi polynomials associated with the umbral algebra abstract the aim of this paper is to study on the genocchi polynomials of higher order on p the algebra of polynomials in the single variable x over the field c of characteristic zero and p the vector spaces of all linear functional on p by using the action of a linear functional l on a polynomial p x sheffer sequences and appell sequences we obtain some fundamental properties of the genocchi polynomials furthermore we give relations between the first and second kind stirling numbers euler polynomials of higher order and genocchi polynomials of higher order

necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations in this work by using fixed point theorem and monotone iterative technique the author establishes a necessary and sufficient condition of the existence of positive solutions for a class of nonlinear singular elastic beam differential system under reasonable conditions moreover some uniqueness results of positive solution and the estimate of convergent rate of the iterative sequence of solution are obtained

q hardy berndt type sums associated with q genocchi type zeta and q l functions abstract the aim of this paper is to define new generating functions by applying a derivative operator and the mellin transformation to these generating functions we define q analogue of the genocchi zeta function q analogue hurwitz type genocchi zeta function and q genocchi type l function we define partial zeta function by using this function we construct p adic interpolation functions which interpolate generalized q genocchi numbers at negative integers we also define p adic meromorphic functions on c p furthermore we construct new generating functions of q hardy berndt type sums and q hardy berndt type sums attached to dirichlet character we also give some new relations related to these sums

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

interpolation function of the h q extension of twisted euler numbers in h ozden y simsek i n cangul generating functions of the h q extension of euler polynomials and numbers acta math hungarica in press doi 10 1007 510474 008 7139 1 by using p adic q invariant integral on z p in the fermionic sense ozden et al constructed generating functions of the h q extension of euler polynomials and numbers they defined h q euler zeta functions and h q euler l functions they also raised the following problem find ap adic twisted interpolation function of the generalized twisted h q euler numbers e n g w h q the aim of this paper is to give a partial answer to this problem therefore we constructed twisted h q partial zeta function and twisted p adic h q euler l functions l e p x q h s g 2 m 1 m p 1 g m 1 m x mq h mm s which interpolate h q extension of euler numbers at negative integers l e p x q h n g e n x g n h q p n g n p e n x p g n h q p by using this interpolation function and twisted h q partial zeta function we proved distribution relations of the h q extension of generalized euler polynomials consequently we find a partial answer to the above question

new changhee q euler numbers and polynomials associated with p adic q integrals using non archimedean q integrals on z p defined in t kim on a q analogue of the p adic log gamma functions and related integrals j number theory 76 1999 320 329 t kim q volkenborn integration russ j math phys 9 2002 288 299 we define new changhee q euler polynomials and numbers which are different from those of kim t kim p adic q integrals associated with the changhee barnes q bernoulli polynomials integral transforms spec funct 15 2004 415 420 and carlitz l carlitz q bernoulli and eulerian numbers trans amer math soc 76 1954 332 350 we define generating functions of multiple q euler numbers and polynomials furthermore we construct a multivariate hurwitz type zeta function which interpolates the multivariate q euler numbers or polynomials at negative integers

an adaptive version of the immersed boundary method a computational setting for the immersed boundary method employing an adaptive mesh refinement is presented enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested progressively finer rectangular grid patches which dynamically follow the immersed boundary motion the set of equations describing the interaction between a non stationary viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method specially designed for locally refined meshes to an implicit formulation of the immersed boundary method the main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the immersed boundary method on locally refined meshes this approach is tested for a particular two dimensional model problem for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary and on the associated uniform mesh built with the resolution of the finest level

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

a gaussian like immersed boundary kernel with three continuous derivatives and improved translational invariance the immersed boundary ib method is a general mathematical framework for studying problems involving fluid structure interactions in which an elastic structure is immersed in a viscous incompressible fluid in the ib formulation the fluid described by eulerian variables is coupled with the immersed structure described by lagrangian variables via the use of the dirac delta function from a numerical standpoint the lagrangian force spreading and the eulerian velocity interpolation are carried out by a regularized compactly supported discrete delta function which is assumed to be a tensor product of a single variable immersed boundary kernel ib kernels are derived from a set of postulates designed to achieve approximate grid translational invariance interpolation accuracy and computational efficiency in this note we present a new 6 point immersed boundary kernel that is c 3 and yields a substantially improved translational invariance compared to other common ib kernels

positive solutions for singular boundary value problems of a coupled system of differential equations by using fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions for singular boundary value problems of a coupled system of nonlinear ordinary differential equations

positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right abstract the purpose of this paper is to establish several local existence theorems concerned with n positive solutions for a fourth order two point boundary value problem where n is an arbitrary positive integer and the nonlinear term may be singular in mechanics the problem describes deflection of an elastic beam rigidly fastened on the left and simply supported on the right

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

necessary and sufficient condition for the existence of positive solutions of a coupled system for reaction diffusion equations abstract in this article by using fixed point theorem and monotone iterative technique the author presents necessary and sufficient conditions of the existence of positive solutions for a class of nonlinear singular reaction diffusion system of differential equations under reasonable conditions moreover some uniqueness results of positive solution and estimate of convergent rate of iterative sequence of solution are obtained

an immersed boundary method with divergence free velocity interpolation and force spreading abstract the immersed boundary ib method is a mathematical framework for constructing robust numerical methods to study fluid structure interaction in problems involving an elastic structure immersed in a viscous fluid the ib formulation uses an eulerian representation of the fluid and a lagrangian representation of the structure the lagrangian and eulerian frames are coupled by integral transforms with delta function kernels the discretized ib equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure and to spread structural forces to the fluid it is well known that the conventional ib method can suffer from poor volume conservation since the interpolated lagrangian velocity field is not generally divergence free and so this can cause spurious volume changes in practice the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries the aim of this paper is to greatly reduce the volume error of the ib method by introducing velocity interpolation and force spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least c 1 and satisfies a continuous divergence free condition and that the force spreading operator is the adjoint of the velocity interpolation operator we confirm through numerical experiments in two and three spatial dimensions that this new ib method is able to achieve substantial improvement in volume conservation compared to other existing ib methods at the expense of a modest increase in the computational cost further the new method provides smoother lagrangian forces tractions than traditional ib methods the method presented here is restricted to periodic computational domains its generalization to non periodic domains is important future work

positive solutions for singular m point boundary value problems with positive parameter in this paper some new existence results for positive solutions for a class of singular m point boundary value problems with parameter be obtained