relative difference families with variable block sizes and their related oocs seven infinite classes of relative difference families with variable block sizes are presented explicitly in particular a balanced gv g k 1 df with g k k k2 k 2 is explicitly given for i k 3 4 5 and every v coprime to 6 ii k 3 4 6 3 5 6 or 3 4 5 6 and every v coprime to 30 as far as the authors are aware these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable weight optical orthogonal codes with more than two weights it is observed however that there are infinitely many values of v for which an optimal v w 1 q ooc exists whatever the set of weights w and the weight distribution sequence q are

optical orthogonal codes with unequal auto and cross correlation constraints an optical orthogonal code ooc is a collection of binary sequences with good auto and cross correlation properties they were defined by salehi and others as a means of obtaining code division multiple access on optical networks up to now all work on oocs have assumed that the constraint placed on the autocorrelation and that placed on the cross correlation are the same we consider codes for which the two constraints are not equal specifically we develop bounds on the size of such oocs and demonstrate constriction techniques for building them the results demonstrate that a significant increase in the code size is possible by letting the autocorrelation constraint exceed the cross correlation constraint these results suggest that for a given performance requirement the optimal ooc may be one with unequal constraints this paper also views oocs with unequal auto and cross correlation constraints as constant weight unequal error protection uep codes with two levels of protection the bounds derived are interpreted from this viewpoint

on a time dependent grade two fluid model in two dimensions we prove the existence of a weak solution of a time dependent grade two fluid model in a plane lipschitz domain and uniqueness of the solution in a convex polygon the method of proof is constructive and can be adapted to the numerical analysis of finite element schemes for solving this problem numerically

exact solutions for electro osmotic flow of viscoelastic fluids in rectangular micro channels transient electro osmotic flow of viscoelastic fluids in rectangular micro channels is investigated the general twofold series solution for the velocity distribution of electro osmotic flow of viscoelastic fluids with generalized fractional oldroyd b constitutive model is obtained by using finite fourier and laplace transforms under three limiting cases the generalized oldroyd b model simplifies to newtonian model fractional maxwell model and generalized second grade model where all the explicit exact solutions for velocity distribution are found through the discrete laplace transform of the sequential fractional derivatives these exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in biomems and lab on a chip devices and thus could benefit the design of these devices

optimal variable weight optical orthogonal codes via difference packings variable weight optical orthogonal code ooc was introduced by yang for multimedia optical cdma systems with multiple quality of service qos requirements in this paper the upper bound on the size of variable weight oocs is improved a cyclic t v w q packing is introduced to construct a variable weight ooc an upper bound for the number of blocks of t v w q packings is obtained and an equivalence between optimal cyclic packing and optimal variable weight optical orthogonal code is established recursive constructions for optimal 2 cp w 1 q v s are also presented by using skew starters and these constructions infinite classes of optimal v w 1 1 2 1 2 oocs are obtained for w 3 4 and 4 5

optical orthogonal codes new bounds and an optimal construction a technique for constructing optimal oocs optical orthogonal codes is presented it provides the only known family of optimal with respect to family size oocs having lambda 2 the parameters n omega lambda are respectively p sup 2m 1 p sup m 1 2 where p is any prime and the family size is p sup m 2 three distinct upper bounds on the size of an ooc are presented that for many values of the parameter set n omega lambda improve upon the tightest previously known bound

compact finite difference scheme and rbf meshless approach for solving 2d rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives abstract this article is devoted to the comparison between a high order difference scheme and radial basis functions rbfs meshless approach for the numerical solution of two dimensional fractional rayleigh stokes problem the fractional derivative of problem is described in the riemann liouville sense in the high order difference scheme we discretize the spaces derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme also in the rbf meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the kansa s approach to approximate the spatial derivatives we prove the stability and convergence for high order difference scheme using fourier analysis and for time discrete scheme in rbf method via energy approach we compare the results of compact finite difference method and rbf meshless approach in terms of accuracy and cpu time also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method

constructions for optimal spl upsi 4 1 optical orthogonal codes four direct constructions three of which are by way of skew starters are given in this correspondence for optimal spl upsi 4 1 optical orthogonal codes ooc s these improve the known existence results concerning optimal spl upsi 4 1 oocs in particular it is shown that an optimal spl upsi 4 1 ooc exists for all positive integers spl upsi spl equiv 6 mod 12 or spl upsi spl equiv 24 mod 48 it is also shown that an optimal 12 spl upsi 4 1 ooc exists for any positive integer spl upsi whose prime factors are all congruent to 1 modulo 4

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

analytic perturbation of the taylor spectrum let tf z tm z z g c ck be analytic families of bounded operators in a complex banach space x such that for each z g the operators ti z and tj z i j l n commute main result if k z denotes the taylor spectrum of the tuple tf z tm z then the set valued function k g 2c is analytic analyticity of such set valued functions is defined here by a simultaneous local maximum property of fc tuples of complex polynomials on the graph of k introduction background and results the following result due to the author is the starting point of this paper theorem ift z z g g c c is an analytic family of bounded operators on a banach space then the set valued function k z sp t z is analytic in the following sense the set 1 u z w zgg w k z

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

semilinear subdiffusion with memory in the one dimensional case abstract for 0 1 we consider the semilinear integro differential equation on the one dimensional domain a b in the unknown u u x t d t u l 1 u 0 t k t s l 2 u s d s f x t u g x t where d t is the caputo fractional derivative and l 1 and l 2 are uniform elliptic operators with time dependent smooth coefficients under certain structural conditions on the nonlinearity f the global existence and uniqueness of classical solutions to the related initial boundary value problems are established via the so called continuation argument approach the key point is looking for suitable a priori estimates of the solution in the fractional holder spaces

combinatorial constructions of optimal optical orthogonal codes with weight 4 a v k spl lambda optical orthogonal code c is a family of 0 1 sequences of length v and weight k satisfying the following correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda for any x x sub 0 x sub 1 x sub v 1 spl isin c and any integer i spl ne 0 mod v 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda for any x x sub 0 x sub 1 x sub v 1 spl isin c y y sub 0 y sub 1 y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v a v k spl lambda optical orthogonal code ooc with spl lfloor 1 k spl lfloor v 1 k 2 spl lfloor v 2 k 2 spl lfloor spl middot spl middot spl middot spl lfloor v spl lambda k spl lambda spl rfloor m spl rfloor spl rfloor spl rfloor codewords is said to be optimal oocs are essential for success of fiber optic code division multiple access cdma communication systems the use of an optimal ooc enables the largest possible number of asynchronous users to transmit information efficiently and reliably in this paper various combinatorial constructions for optimal v 4 1 oocs such as those via skew starters and weil s theorem on character sums are given for v spl equiv 0 mod 12 these improve the known existence results on optimal oocs in particular it is shown that an optimal v 4 1 ooc exists for any positive integer v spl equiv 0 mod 24

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

numerical analysis of the rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives australian research council dp0559807 national natural science foundation of china 10271098 nsf of china tianyuan fund for mathematics 10726061

bounds and constructions of optimal n 4 2 1 optical orthogonal codes in this paper a tight upper bound on the maximum possible code size of n 4 2 1 oocs and some direct and recursive constructions of optimal n 4 2 1 oocs attaining the upper bound are give

an improved tau method for the multi dimensional fractional rayleigh stokes problem for a heated generalized second grade fluid abstract we develop efficient algorithms based on the legendre tau approximation for one and two dimensional fractional rayleigh stokes problems for a generalized second grade fluid the time fractional derivative is described in the riemann liouville sense discussions on the l 2 convergence of the proposed method are presented numerical results for one and two dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis and to illustrate the efficiency of the proposed algorithms

optical orthogonal codes design analysis and applications an optical orthogonal code ooc is a family of 0 1 sequences with good auto and cross correlation properties i e the autocorrelation of each sequence exhibits the thumbtack shape and the cross correlation between any two sequences remains low throughout the use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably the thumbtack shaped autocorrelation facilitates the detection of the desired signal and low profile cross correlation reduces interference from unwanted signals theoretical upper and lower bounds on the maximum possible size of oocs are derived methods for the design and analysis of oocs using tools from projective geometry the greedy algorithm iterative constructions algebraic coding theory block design and various other combinational disciplines are discussed

new families of optimal variable weight optical orthogonal codes with high weights the optical orthogonal codes oocs have been widely used as spreading codes in communication systems employing the unipolar transmission they are classified into constant weight oocs cw oocs and variable weight oocs vw oocs according to the number of distinct hamming weights which their codewords have in this paper we present a new generic construction of vw oocs of length q 1 n from a cw ooc of length n where q is a prime power and gcd q 1 n 1 as a result three new families of optimal vw oocs with a maximum correlation value 1 are obtained in particular these families can have the codewords of high weights while most of the previously known optimal vw oocs have only codewords of weight less than 8

the uniqueness of the complete norm topology in complete normed nonassociative algebras abstract a theorem on uniqueness of the complete norm topology for complete normed nonassociative algebras is proved this theorem contains the well known one by johnson for associative banach algebras and the recent analogous result by aupetit for banach jordan algebras

numerical algorithm for the space time fractional fokker planck system with two internal states the fractional fokker planck system with multiple internal states is derived in xu and deng math model nat phenom mathbf 13 10 2018 where the space derivative is laplace operator if the jump length distribution of the particles is power law instead of gaussian the space derivative should be replaced with fractional laplacian this paper focuses on solving the two state fokker planck system with fractional laplacian we first provide a priori estimate for this system under different regularity assumptions on the initial data then we use l 1 scheme to discretize the time fractional derivatives and finite element method to approximate the fractional laplacian operators furthermore we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions finally the effectiveness of the designed scheme is verified by one and two dimensional numerical experiments

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

optical orthogonal codes their bounds and new optimal constructions a v k spl lambda sub a spl lambda sub c optical orthogonal code ooc c is a family of 0 1 sequences of length v and weight k satisfying the following two correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda sub a for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c and any integer i not equivalent 0 mod v and 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda sub b for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c y y sub 0 y sub 1 spl middot spl middot spl middot y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v the study of optical orthogonal codes is motivated by an application in optical code division multiple access communication systems in this paper upper bounds on the size of an optical orthogonal code are discussed several new constructions for optimal optical orthogonal codes with weight k spl ges 4 and correlation constraints spl lambda sub a spl lambda sub c 1 are described by means of optimal cyclic packings many new infinite series of such optimal optical orthogonal codes are thus produced

on linear maps preserving generalized invertibility and related properties let h be an infinite dimensional complex hilbert space b h be the algebra of all bounded linear operators on h we study surjective linear maps on b h preserving generalized invertibility we also investigate surjective linear maps preserving fredholm respectively semi fredholm operators our results improve those of mbekhta rodman and semrl

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

a new recursive construction for optical orthogonal codes we present a new recursive construction for n spl omega spl lambda sub a spl lambda sub c optical orthogonal codes for the case of spl lambda sub a spl lambda sub c spl lambda this recursive construction enlarges the original family with spl lambda unchanged and produces a new family of asymptotically optimal codes if the original family is asymptotically optimal we call a code asymptotically optimal following the definition of o moreno et al see ibid vol 41 p 448 55 1995 if as n the length of code goes to infinity the ratio of the number of codewords to the corresponding johnson bound approaches unity

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

a new class of optimal optical orthogonal codes with weight five a v k 1 optical orthogonal code ooc or briefly a v k 1 ooc c is a family of 0 1 sequences of length v and weight k satisfying the following two properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les 1 for any x x sub 0 x sub 1 x sub v 1 spl isin c and any integer i spl ne 0 mod v 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les 1 for any x x sub 0 x sub 1 x sub v 1 spl isin c y y sub 0 y sub 1 y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are reduced modulo v a v k 1 ooc is optimal if it contains spl lfloor v 1 k k 1 spl rfloor codewords in this note we establish that there exists an optimal 3 sup s 5v 5 1 ooc for any nonnegative integer s whenever visa product of primes congruent to 1 modulo 4 this improves the known existence results concerning optimal oocs

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

the uniqueness of the complete norm topology in banach algebras and banach jordan algebras abstract in this paper we give a very simple subharmonic proof of an extension of the famous theorem of b e johnson on the equivalence of complete norms in semi simple banach algebras this proof avoids irreducible representations so that it can be adapted to the situation of banach jordan algebras in order to give a similar result