on a solvable system of difference equations of kth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

a linear domain decomposition method for partially saturated flow in porous media abstract the richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated unsaturated flow in porous media we assume that the medium occupies a bounded lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface this leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at after an euler implicit discretisation of the resulting nonlinear subproblems a linear iterative l type domain decomposition scheme is proposed the convergence of the scheme is proved rigorously in the last part we present numerical results that are in line with the theoretical finding in particular the convergence of the scheme under mild restrictions on the time step size we further compare the scheme to other approaches not making use of a domain decomposition namely we compare to a newton and a picard scheme we show that the proposed scheme is more stable than the newton scheme while remaining comparable in computational time even if no parallelisation is being adopted after presenting a parametric study that can be used to optimise the proposed scheme we briefly discuss the effect of parallelisation and give an example of a four domain implementation

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

the soft feedback equalizer for turbo equalization of highly dispersive channels the complexity of a turbo equalizer based on the bahl cocke jelinek raviv bcjr algorithm is manageable only for mildly dispersive channels having a small amount of memory to enable turbo equalization of highly dispersive channels we propose the soft feedback equalizer sfe the sfe combines linear equalization and soft intersymbol interference cancellation its coefficients are chosen to minimize the mean squared error mse between the equalizer output and the transmitted sequence under a gaussian approximation to the a priori information and the sfe output the resulting complexity grows only linearly with the number of coefficients as opposed to the quadratic complexity of previously reported minimum mse structures we will see that an sfe based turbo equalizer consistently outperforms another structure of similar complexity and can outperform a bcjr based scheme when complexity is taken into account

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

degenerate two phase incompressible flow ii regularity stability and stabilization abstract in this paper we analyze a coupled system of highly degenerate elliptic parabolic partial differential equations for two phase incompressible flow in porous media this system involves a saturation and a global pressure or a total flow velocity first we show that the saturation is holder continuous both in space and time and the total velocity is holder continuous in space uniformly in time applying this regularity result we then establish the stability of the saturation and pressure with respect to initial and boundary data from which uniqueness of the solution to the system follows finally we establish a stabilization result on the asymptotic behavior of the saturation and pressure we prove that the solution to the present system converges in appropriate norms to the solution of a stationary system as time goes to infinity an example is given to show typical regularity of the saturation

a nonlinear time compactness result and applications to discretization of degenerate parabolic elliptic pdes abstract we propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems it is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of simon 49 our approach allows to handle various numerical discretizations both in the space variables and in the time variable in particular we can cope quite easily with variable time steps and with multistep time differentiation methods like e g the backward differentiation formula of order 2 bdf2 scheme we illustrate our approach by proving the convergence of a two point flux finite volume in space and bdf2 in time approximation of the porous medium equation

the richards equation with hysteresis and degenerate capillary pressure abstract we study the richards equation with a dynamic capillary pressure including hysteresis we provide existence and approximation results for degenerate capillary pressure curves p c treating two cases in the first case the permeability function k can be degenerate but the initial saturation does not take the critical values in the second case the permeability function k is strictly positive but the capillary pressure function can be multi valued in both cases the degenerate behavior of p c leads to the physically desired uniform bounds for the saturation variable our approach exploits maximum principles and relies on the corresponding uniform bounds for pressure and saturation a new compactness result for the saturation variable allows to take limits in nonlinear terms the solution concept uses tools of convex analysis

on the positive solutions of the difference equation xn 1 xn 1 1 x nxn 1 the positive solutions of the difference equation x n 1 x n 1 1 x nx n 1 n 0 1 2 are investigated where x 1 and x 0 are the positive real numbers

degenerate two phase incompressible flow i existence uniqueness and regularity of a weak solution abstract this is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data then we prove that this weak solution is unique finally we establish regularity on the weak solution which is needed in the uniqueness proof in particular the holder continuity of the saturation in space and time and the lipschitz continuity of the pressure in space are obtained

optimization of ldpc codes for modulation and detection in layered multi antenna architectures in this paper we first perform the density evolution of ldpc codes in layered multi antenna architectures using bit interleaved coded modulation and detection based on successive decoding and interference cancellation the noise thresholds of ldpc codes are shown to be achieved within 1 2 db of the capacity for these architectures we then present a rate allocation method which optimizes the performance of a layered architecture and find good degree distributions for ldpc codes with the rates allocated by our method simulation results show that the performance for an optimized layered architecture is better by 4 db than that for the layered architecture using ldpc codes with the same rate in all layers

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

more on a rational second order difference equation abstract here we give natural explanations for some recent relations for the solutions of the following difference equation z n 2 z n 1 3 z n 2 2 z n 1 2 n n 0 and extend them for the case of complex initial values we also present some other properties related to solutions of the equation as well as some properties of the solutions of an associate difference equation of first order

on a max difference equation abstract in this paper we study the existence of generalized invariants and the periodicity of the positive solutions of max equations x n 1 max a n n i n k 1 x i b n n i n k x i n 0 1 where an bn are sequences of positive numbers x k x k 1 x0 0 and k 2 3

on a system of difference equations of odd order solvable in closed form it is shown that the next system of difference equationsx n c ny n 2 k 1 a n b ny n 2 k 1 i 1 k 1y n 2 i 1 x n 2 i y email protected nx n 2 k 1 a email protected nx n 2 k 1 i 1 k 1x n 2 i 1 y n 2 email protected n 0 where a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 2k 1 are real numbers is solved in closed form and the behavior of its well defined solutions when all the coefficients in the system are constant is described

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua this paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow as those involving rainfall infiltration and seepage faces besides the penalty technique we also consider the novel regularization of these conditions by means of the more effective augmented lagrangian method the performance of the so obtained finite element method is carefully investigated in terms of accuracy and ill conditioning effects including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact in this way we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization the proposed method and the estimated coefficient ranges are validated in further numerical examples involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam these applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed newton raphson method hence the present approach should be preferred to the commonly used iterative switching between the imposed flow and the imposed pressure conditions which often leads to spurious oscillations and convergence failures

order of convergence estimates for an euler implicit mixed finite element discretization of richards equation we analyze a discretization method for a class of degenerate parabolic problems that includes the richards equation this analysis applies to the pressure based formulation and considers both variably and fully saturated regimes to overcome the difficulties posed by the lack in regularity we first apply the kirchhoff transformation and then integrate the resulting equation in time we state a conformal and a mixed variational formulation and prove their equivalence this will be the underlying idea of our technique to get error estimates a regularization approach is combined with the euler implicit scheme to achieve the time discretization again equivalence between the two formulations is demonstrated for the semidiscrete case the lowest order raviart thomas mixed finite elements are employed for the discretization in space error estimates are obtained showing that the scheme is convergent

two phase flow equations with outflow boundary conditions in the hydrophobic hydrophilic case we introduce an approximation procedure and provide existence results for two phase flow equations in porous media the medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations our main interest is the outflow boundary condition which models an interface with open space the approximate system introduces standard boundary conditions and can be used in numerical schemes it allows the derivation of maximum principles this is the basis for the derivation of the limiting system in the form of a variational inequality

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

eventual periodicity of some systems of max type difference equations the periodicity of the next system of max type difference equations xn 1 max1 i1 m1f1i1xn ki1 1 1 1 xn ki1 2 1 2 xn ki1 l 1 l n xn t1s 1 xn 2 max1 i2 m2f2i2xn ki2 1 2 1 xn ki2 2 2 2 xn ki2 l 2 l n xn t2s 2 xn l max1 il mlflilxn kil 1 l 1 xn kil 2 l 2 xn kil l l l n xn tls l n n0n n0 where s l mj tj kij h j n j h 1 l j h 1 l 1 2 l 1 2 l is a permutation of 1 2 l 1 2 l and fjij 0 l n0 0 fjij 0 l n0 0 j 1 l j 1 l ij 1 mj ij 1 mj is studied it is shown that under some conditions posed on functions fjijfjij all positive solutions of the system are eventually periodic with period ts for some t nt n

on the difference equation xn xn kb cxn 1 xn k abstract we describe the behavior of well defined solutions of the difference equation x n x n k b cx n 1 x n k n n 0 where k n parameters b c and the initial values x k x 1 are real numbers these results among others improve and extend our results in s stevic more on a rational recurrence relation appl math e notes 4 2004 80 85

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

on difference equations with powers as solutions and their connection with invariant curves abstract we look for nonlinear systems of difference equations which have power functions as solutions and we find a connection to the theory of invariant curves and to self reversed polynomials in particular we construct algebraic solutions of a special equation of invariant curves

study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form in this paper we study the boundedness the persistence and the asymptotic behavior of the positive solutions of the following systems of two difference equations of exponential form xn 1 a be yn c yn 1 y n 1 d e xn f xn 1

on a solvable system of difference equations of fourth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j n n 0 j 1 k and initial values x i j i j 1 k are real numbers and where s n 1 k is defined by s km j j 1 j 1 k 1 s km k 1 m n 0 can be solved in closed form in an elegant way some applications of obtained formulas for the case when the sequences a n j b n j and c n j are k periodic are also given

dynamics of a rational difference equation abstract in this note we investigate the solution of the difference equation x n 1 x n 1 a x n 1 x n n 0 1 2 where x 1 x 0 r and a 0 moreover we discuss the stability properties and semi cycle behavior of this solution

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

on the system of difference equations x n x n 1 y n 2 a y n 2 b y n 1 y n y n 1 x n 2 c x n 2 d x n 1 we show that the following system of difference equations x n x n 1 y n 2 a y n 2 b y n 1 y n y n 1 x n 2 c x n 2 d x n 1 n n 0 where parameters a b c d as well as initial values x 2 x 1 y 2 y 1 are real numbers is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions we also give natural explanations and considerably extend some recent results in the literature

on a difference equation with maximum abstract this paper studies the asymptotic behavior of positive solutions to the difference equation x n max 1 x n 1 a x n 2 n 0 1 where 0 0 we prove that every positive solution of this equation approaches x 1 or is eventually periodic with period 4 according as a 1 or a 1

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

on a symmetric system of max type difference equations we study behavior of positive solutions of the following max type system of difference equationsx n 1 maxc y n px n 1 p y n 1 maxc x n py n 1 p email protected n 0 where p email protected 0 extending some results in the literature among other results we prove that if p email protected 0 1 then every positive solution of the system converges to 1 1

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

novel reduced state bcjr algorithms bcjr algorithm is an exact and efficient algorithm to compute the marginal posterior distributions of state variables and pairs of consecutive state variables of a trellis structure due to its overwhelming complexity reduced complexity variations such as the m bcjr algorithm have been developed in this paper we propose improvements upon the conventional m bcjr algorithm based on modified active state selection criteria we propose selecting the active states based on estimates of the fixed lag smoothed distributions of the state variables we also present gaussian approximation techniques for the low complexity estimation of these fixed lag smoothed distributions the improved performance over the m bcjr algorithm is shown via computer simulations

on the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

on the dynamics of the difference equation xn 1 1bnxn xn 1 in this note we study the following non autonomous equation x n 1 1b nx n x n 1 n 0 1 where b n 0 is a period 2 sequence and the initial values x 1 x 0 0 we show that the unique prime period 2 solution of this equation is globally asymptotically stable which answer open problem 5 proposed by camouzis and ladas 6

on the dynamics of two exponential type systems of difference equations in this paper we study the asymptotic behavior of the positive solutions of the systems of the two difference equations i x n 1 a by n 1e x n y n 1 c dx n 1e y n ii x n 1 a by n 1e y n y n 1 c dx n 1e x n where the constants a b c d are positive real numbers and the initial values x 1 x 0 y 1 y 0 are also positive real numbers

scheduling for an adaptive number of iterations in turbo equalizers combined with ldpc decoders a new scheduling to reduce the complexity of a turbo equalizer te combined with a low density parity check ldpc decoder is proposed simple metrics and criteria for the proposed scheduling are derived from its convergence behavior the te with the proposed scheduling monitors the status of its iterative process and determines whether additional equalization is needed or not based on the decision it adjusts performing equalization and reduces the execution number of equalization consequently this leads to lowering the overall complexity of the te numerical results show that the proposed scheduling has lower complexity than conventional schedulings while they have almost the same performance

on a system of difference equations which can be solved in closed form we present an elegant method for solving the system of difference equationsx n 1 1x email protected n y n email protected n 1y n email protected n 0 where parameters a and b and the initial values x 0 and y 0 are positive numbers and by using obtained formulae we show that every solution of the system converges to a two periodic solution

on the application of factor graphs and the sum product algorithm to isi channels in this paper based on the application of the sum product sp algorithm to factor graphs fgs representing the joint a posteriori probability app of the transmitted symbols we propose new iterative soft input soft output siso detection schemes for intersymbol interference isi channels we have verified by computer simulations that the sp algorithm converges to a good approximation of the exact marginal apps of the transmitted symbols if the fg has girth at least 6 for isi channels whose corresponding fg has girth 4 the application of a stretching technique allows us to obtain an equivalent girth 6 graph for sparse isi channels the proposed algorithms have advantages in terms of complexity over optimal detection schemes based on the bahl cocke jelinek raviv bcjr algorithm they also allow a parallel implementation of the receiver and the possibility of a more efficient complexity reduction the application to joint detection and decoding of low density parity check ldpc codes is also considered and results are shown for some partial response magnetic channels also in these cases we show that the proposed algorithms have a limited performance loss with respect to that can be obtained when the optimal serial bcjr algorithm is used for detection therefore for their parallel implementation they represent a favorable alternative to the modified parallel bcjr algorithm proposed in the literature for the application to magnetic channels

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

dynamics of a kth order rational difference equation in this article we investigate the dynamics of the difference equation x n 1 x n ka x n kx n where x k x 0 0 and a 0 k is any positive integer in some particular cases we allow for zero initial conditions

on a solvable rational system of difference equations abstract we show that the system of difference equations x n 1 x n y n k y n k 1 a n b n x n y n k y n 1 y n x n k x n k 1 c n d n y n x n k n n 0 where k n the sequences a n b n c n d n n n 0 and initial values x i y i i 0 k are real numbers can be solved in closed form by using obtained formulae we investigate asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant

eventually constant solutions of a rational difference equation we describe all the solutions of a rational difference equation from putnam s mathematical competition which are eventually equal to its positive equilibrium email protected 1 as a consequence we give a new elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one moreover we show that almost all solutions of the equation are not eventually equal to one

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

error estimates for a mixed finite element discretization of some degenerate parabolic equations we consider a numerical scheme for a class of degenerate parabolic equations including both slow and fast diffusion cases a particular example in this sense is the richards equation modeling the flow in porous media the numerical scheme is based on the mixed finite element method mfem in space and is of one step implicit in time the lowest order raviart thomas elements are used here we extend the results in radu et al siam j numer anal 42 1452 1478 2004 schneid et al numer math 98 353 370 2004 to a more general framework by allowing for both types of degeneracies we derive error estimates in terms of the discretization parameters and show the convergence of the scheme the features of the mfem especially of the lowest order raviart thomas elements are now fully exploited in the proof of the convergence the paper is concluded by numerical examples

on a class of third order nonlinear difference equations this paper studies the boundedness character of the positive solutions of the difference equation x n 1 a x n px n 1 qx n 2 r n n 0 where the parameters a p q and r are positive numbers

quasi cyclic ldpc codes for fast encoding in this correspondence we present a special class of quasi cyclic low density parity check qc ldpc codes called block type ldpc b ldpc codes which have an efficient encoding algorithm due to the simple structure of their parity check matrices since the parity check matrix of a qc ldpc code consists of circulant permutation matrices or the zero matrix the required memory for storing it can be significantly reduced as compared with randomly constructed ldpc codes we show that the girth of a qc ldpc code is upper bounded by a certain number which is determined by the positions of circulant permutation matrices the b ldpc codes are constructed as irregular qc ldpc codes with parity check matrices of an almost lower triangular form so that they have an efficient encoding algorithm good noise threshold and low error floor their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices

difference equations and nonexistence of solutions of an integral equation abstract by using difference equations we give some easily applicable sufficient conditions for the non existence of nonnegative solutions satisfying some additional conditions of the integral equation f x x a x t b x t f t p d t where p 1 and x is a measurable space with probability measure

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

binary intersymbol interference channels gallager codes density evolution and code performance bounds we study the limits of performance of gallager codes low density parity check ldpc codes over binary linear intersymbol interference isi channels with additive white gaussian noise awgn using the graph representations of the channel the code and the sum product message passing detector decoder we prove two error concentration theorems our proofs expand on previous work by handling complications introduced by the channel memory we circumvent these problems by considering not just linear gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols we compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify by simulation that the thresholds represent accurate predictions of the performance of the iterative sum product algorithm for finite but large block lengths we also demonstrate that for high rates the thresholds are very close to the theoretical limit of performance for gallager codes over isi channels if c denotes the capacity of a binary isi channel and if c sub i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed i i d binary random variables c sub i i d spl les c we prove that the maximum information rate achievable by the sum product decoder of a gallager coset code is upper bounded by c sub i i d the last topic investigated is the performance limit of the decoder if the trellis portion of the sum product algorithm is executed only once this demonstrates the potential for trading off the computational requirements and the performance of the decoder

on a third order system of difference equations abstract we show that the system of difference equations x n 1 a 1 x n 2 b 1 y n z n 1 x n 2 c 1 y n 1 a 2 y n 2 b 2 z n x n 1 y n 2 c 2 z n 1 a 3 z n 2 b 3 x n y n 1 z n 2 c 3 n n 0 where the parameters a i b i c i i 1 2 3 and initial values x j y j z j j 0 1 2 are real numbers can be solved developing further the results in the literature

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

on the nonautonomous difference equation xn 1 an xn 1pxnq abstract in this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation x n 1 a n x n 1 p x n q n 0 1 where an is a positive bounded sequence p q 0 and x 1 x0 are positive numbers

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied

regularization of outflow problems in unsaturated porous media with dry regions we study a porous medium with saturated unsaturated and dry regions described by richards equation for the saturation s and the pressure p due to a degenerate permeability coefficient k x s and a degenerate capillary pressure function pc x s the equations may be of elliptic parabolic or of ode type we construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system an example shows that the convergence fails in general our approach provides an existence result for the outflow problem in the case of x dependent coefficients and a method for a numerical approximation

on boundedness of the difference equation xn 1 pn xn 3s 1xn s 1 with period k coefficients in this note we study the following non autonomous equation xn 1 pn xn 3s 1xn s 1 n 0 1 where pn 0 is a period k sequence and s 1 2 with gcd 3s k 1 and the initial conditions x 3s 1 x0 0 we show that every positive solution of this equation is bounded

strong solutions to the richards equation in the unsaturated zone abstract we study the unsaturated case of the richards equation in three space dimensions with dirichlet boundary data we first establish an a priori l estimate with its help by means of a fixed point argument we prove global in time existence of a unique weak solution in sobolev spaces finally we are able to improve the regularity of this weak solution in order to gain a strong one