existence of solutions and multiple solutions for a class of weighted p r laplacian system abstract this paper investigates the existence of solutions for a class of weighted p r laplacian ordinary system boundary value problems the sufficient conditions for the existence of solutions have been given via leray schauder s degree and the existence of multiple solutions has been discussed via critical point theory as an application we discussed the existence of the radial solutions for the p x laplacian partial differential system boundary value problems

analysis of the block error probability of concatenated polar code ensembles in this paper we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check crc codes separated by an interleaver in the short and moderate block length regimes the analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding the analysis is carried out by introducing the concatenated polar code cpc ensembles whose distance properties can be analyzed for sufficiently short block lengths by means of the uniform interleaver approach at moderate block lengths we resort to the monte carlo simulations results show that if the inner polar code possesses a low minimum distance and the outer crc code has a sufficiently large amount of redundancy then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting cpc

advances in the formulation of the rotation free basic shell triangle a family of rotation free three node triangular shell elements is presented the simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle an enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes this allows to compute an assumed linear membrane strain field which improves the in plane behaviour of the original element a simple and economic version of the element using a single integration point is presented the efficiency of the different rotation free shell triangles is demonstrated in many examples of application including linear and non linear analysis of shells under static and dynamic loads the inflation and de inflation of membranes and a sheet stamping problem

fully nonlinear fourth order equations with functional boundary conditions abstract the aim of this paper consists in to give sufficient conditions to ensure the existence and location of the solutions of a nonlinear fully fourth order equation with functional boundary conditions the arguments make use of the upper and lower solutions method a laplacian operator and a fixed point theorem an application of the beam theory to a nonlinear continuous model of the human spine allows to estimate its deformation under some loading forces

existence of solutions for weighted p r laplacian system boundary value problems this paper investigates the existence of solutions for weighted p r laplacian ordinary system boundary value problems via leray schauder degree the sufficient conditions for the existence of solutions be given as an application we obtain the existence of the radial solutions for the p x laplacian partial differential system dirichlet problems

a vertex based finite volume method applied to non linear material problems in computational solid mechanics a vertex based finite volume fv method is presented for the computational solution of quasi static solid mechanics problems involving material non linearity and infinitesimal strains the problems are analysed numerically with fully unstructured meshes that consist of a variety of two and threedimensional element types a detailed comparison between the vertex based fv and the standard galerkin fe methods is provided with regard to discretization solution accuracy and computational efficiency for some problem classes a direct equivalence of the two methods is demonstrated both theoretically and numerically however for other problems some interesting advantages and disadvantages of the fv formulation over the galerkin fe method are highlighted

on the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions abstract we prove the existence of extremal solutions for the third order discontinuous functional nonlinear problem u t f t u u t u t for a e t a b u a a a r l 1 u u u a u b u a 0 l 2 u u a u b u b 0 by using a fixed point theorem after having established some existence results for some auxiliary second order nonlinear problems we observe that together with the discontinuities allowed on the spacial variable u with adequate modifications of technical type analogous results can be obtained when the equation has a second member not necessarily continuous in u

existence of solutions of boundary value problem for 3rd order nonlinear system by appropriate bounding function pair and modified functions using the theory of differential inequality and the theory of topological degree this paper presents the existence criterions of solutions for the system of 3rd order differential equations under nonlinear boundary conditions and extends many known results

list decoding of polar codes we describe a successive cancellation list decoder for polar codes which is a generalization of the classic successive cancellation decoder of arikan in the proposed list decoder l decoding paths are considered concurrently at each decoding stage where l is an integer parameter at the end of the decoding process the most likely among the l paths is selected as the single codeword at the decoder output simulations show that the resulting performance is very close to that of maximum likelihood decoding even for moderate values of l alternatively if a genie is allowed to pick the transmitted codeword from the list the results are comparable with the performance of current state of the art ldpc codes we show that such a genie can be easily implemented using simple crc precoding the specific list decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit and then uses a pruning procedure to discard all but the l most likely paths however straightforward implementation of this algorithm requires omega l n 2 time which is in stark contrast with the o n log n complexity of the original successive cancellation decoder in this paper we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem we devise an efficient numerically stable implementation of the proposed list decoder that takes only o l n log n time and o l n space

bounding functions methods for fully nonlinear boundary value problems abstract using the bounding functions method and the theory of topological degree this paper presents the existence criterion of solution for third order bvp with nonlinear boundary conditions and extends the existing results

a note on a class of problems for a higher order fully nonlinear equation under one sided nagumo type condition abstract the purpose of this work is to establish existence and location results for the higher order fully nonlinear differential equation u n t f t u t u t u n 1 t n 2 with the boundary conditions u i a a i for i 0 n 3 u n 1 a b u n 1 b c or u i a a i for i 0 n 3 c 1 u n 2 a c 2 u n 1 a b c 3 u n 2 b c 4 u n 1 b c with a i b c r for i 0 n 3 and c 1 c 2 c 3 c 4 real positive constants it is assumed that f a b r n 1 r is a continuous function satisfying one sided nagumo type conditions which allows an asymmetric unbounded behaviour on the nonlinearity the arguments are based on the leray schauder topological degree and lower and upper solutions method

the method of lower and upper solutions for third order periodic boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem l n u t t u t u i 0 u i 2 i r i 0 n 1 with ln an nth order linear operator and a caratheodory function we obtain sufficient conditions to guarantee the validity of the monotone method for this problem for this we study the sign of the green function of the operator t 1n with tnu lnu mu m 0 furthermore existence results are obtained we show that the results obtained are optimal given a lower solution and an upper solution we obtain the existence of solution between and when either or this general study is applied to the third order problem u t pu t qu t t u t u i 0 u i 2 i r i 0 1 2 with p q r for this problem we obtain the best value on the constant m to guarantee that the green function of the operator t 13 is positive

solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions abstract we prove an existence and location result for the third order functional nonlinear boundary value problem u t f t u u t u t for t a b 0 l 0 u u u t 0 0 l 1 u u u a u a 0 l 2 u u u b u b with t 0 a b given f i c i r 2 r is a l 1 caratheodory function and l 0 l 1 l 2 are continuous functions depending functionally on u and u the arguments make use of an a priori estimate on u lower and upper solutions method and degree theory applications to a multipoint problem and to a beam equation will be presented

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

construction of polar codes for arbitrary discrete memoryless channels it is known that polar codes can be efficiently constructed for binary input channels at the same time existing algorithms for general input alphabets are less practical because of high complexity we address the construction problem for the general case and analyze an algorithm that is based on successive reduction of the output alphabet size of the subchannels in each recursion step for this procedure we estimate the approximation error as o mu 1 q 1 where q is the input alphabet size and mu is the quantization parameter i e the maximum size of the subchannel output alphabet allowed by the algorithm the complexity of the code construction scales as o n mu 2 log mu where n is the length of the code we also show that if the polarizing operation relies on modulo q addition it is possible to merge subsets of output symbols without any loss in subchannel capacity performing this procedure before each approximation step results in a further speed up of the code construction and the resulting codes have smaller gap to capacity we also show that a similar acceleration can be attained for polar codes over finite field alphabets experimentation shows that the suggested construction algorithms can be used to construct long polar codes for alphabets of size q 16 and more with acceptable loss of the code rate for a variety of polarizing transforms

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

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

relaxed polar codes polar codes are the latest breakthrough in coding theory as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary input discrete memoryless channels polar encoding and successive cancellation decoding have the complexities of n log n for code length n although the complexity bound of n log n is asymptotically favorable we report in this work methods to further reduce the encoding and decoding complexities of polar coding the crux is to relax the polarization of certain bit channels without performance degradation we consider schemes for relaxing the polarization of both very good and very bad bit channels in the process of channel polarization relaxed polar codes are proved to preserve the capacity achieving property of polar codes analytical bounds on the asymptotic and finite length complexity reduction attainable by relaxed polarization are derived for binary erasure channels we show that the computation complexity can be reduced by a factor of six while preserving the rate and error performance we also show that relaxed polar codes can be decoded with significantly reduced latency for additive white gaussian noise channels with medium code lengths we show that relaxed polar codes can have lower error probabilities than conventional polar codes while having reduced encoding and decoding computation complexities

a finite volume format for structural mechanics summary a general finite volume method fvm for the analysis of structural problems is presented it is shown that the fvm can be considered to be a particular case of finite elements with a non galerkin weighting for structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals both displacement and mixed fv formulations are presented for static and dynamic problems the finite volume method fvm evolved in the early seventies via finite difference approxima tions on non orthogonal grids the popularity of the fvm has been extensive in the field of computational fluid dynamics cfd and heat transfer on the contrary in the field of computational solid mechanics csm the use of the fvm has never achieved such acceptance an early attempt to use fv concepts in csm is due to wilkins6 as an alternative approximation to derivatives in a cell in this he defines the average gradient of a function u in a volume l2 as using the well known divergence theorem such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early hydrocodes of the lawrence livermore laboratory the reasons for the unpopularity of the fvm in structural mechanics is understandable as finite volumes are well known to be less accurate than galerkin based finite elements for self adjoint elliptic problems a comparison between fvm and fem has been recently pre sented by zienkiewicz and oiiate here the authors show that fvm and fem share concepts such as mesh discretization and interpolation giving precisely the same discretized systems of equations for some particular cases this coincidence also shows clearly for 2 d and 3 d structural problems as detailed in this work here however surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of sides in the mesh this leads to an overall solution cost very similar to that of fe computations note that for a refined mesh of three node triangles the number of sides is 1 5 times that of elements this fact suggests that computational speed is not a prior one of the keys to the possible success of fvm in structural problems however the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which in the authors opinion may be worth exploring in detail indeed it

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

performance limits and practical decoding of interleaved reed solomon polar concatenated codes a scheme for concatenating the recently invented polar codes with non binary mds codes as reed solomon codes is considered by concatenating binary polar codes with interleaved reed solomon codes we prove that the proposed concatenation scheme captures the capacity achieving property of polar codes while having a significantly better error decay rate we show that for any e 0 and total frame length n the parameters of the scheme can be set such that the frame error probability is less than 2 n 1 e while the scheme is still capacity achieving this improves upon 2 n 0 5 e the frame error probability of arikan s polar codes the proposed concatenated polar codes and arikan s polar codes are also compared for transmission over channels with erasure bursts we provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder on the other hand it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity achieving properties of polar codes are preserved we also propose decoding algorithms for concatenated polar codes which significantly improve the error rate performance at finite block lengths while preserving the low decoding complexity

randomized polar subcodes with optimized error coefficient a method for construction of polar subcodes with reduced error coefficient is presented the proposed approach relies on explicit enumeration of low weight non zero codewords in a polar code and construction of dynamic freezing constraints which define a subcode not containing most of these codewords the obtained codes provide a large performance gain in the high snr region compared to non optimized polar subcodes and polar codes with crc

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

scaling exponent of list decoders with applications to polar codes motivated by the significant performance gains which polar codes experience under successive cancellation list decoding their scaling exponent is studied as a function of the list size in particular the error probability is fixed and the tradeoff between the block length and back off from capacity is analyzed a lower bound is provided on the error probability under rm map decoding with list size l for any binary input memoryless output symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large then it is shown that under rm map decoding although the introduction of a list can significantly improve the involved constants the scaling exponent itself i e the speed at which capacity is approached stays unaffected for any finite list size in particular this result applies to polar codes since their minimum distance tends to infinity as the block length increases a similar result is proved for genie aided successive cancellation decoding when transmission takes place over the binary erasure channel namely the scaling exponent remains constant for any fixed number of helps from the genie note that since genie aided successive cancellation decoding might be strictly worse than successive cancellation list decoding the problem of establishing the scaling exponent of the latter remains open

shortened polar codes an optimization algorithm for finding a shortening pattern and a set of frozen symbols for polar codes is proposed the structure of polar codes is exploited to eliminate many equivalent shortening patterns thus reducing the search space a reduced complexity suboptimal algorithm is proposed for finding shortening patterns for long polar codes shortened codes obtained with the proposed method are shown to outperform low density parity check ldpc codes

weight distributions for successive cancellation decoding of polar codes in this paper we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes though we do not compute the distance spectrum of polar codes the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels also we establish the minimal distance between two cosets associated with two paths that differ in two positions this can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding

on distance properties of convolutional polar codes a lower bound on the minimum distance of convolutional polar codes is provided the bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix moreover a construction of convolutional polar subcodes is proposed which provides improved performance under successive cancellation list decoding for sufficiently large list size the decoding complexity of convolutional polar subcodes appears to be lower compared with arikan polar subcodes with the same performance the error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of arikan polar subcodes with the same list size

an edge based finite volume scheme for saturated unsaturated groundwater flow abstract the application of a computationally efficient finite volume method for modelling groundwater flow in saturated unsaturated soils is presented the flow equation is solved by applying a vertex centred edge based finite volume scheme and the discretisation of the flow domain is achieved using structured and unstructured grids an acceleration algorithm is employed which utilises an explicit non linear loop for the primary variable update the scheme is employed to solve a two dimensional groundwater flow problem and the results obtained are compared against published solutions it is shown that the edge based finite volume scheme captures the saturation front more efficiently than the published finite element solution comput geosci 27 2 2001 179 moreover an improvement in the solution is shown by the application of a compact stencil scheme to calculate second derivative terms as compared to the standard edge based finite volume scheme

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

finite volumes and finite elements two good friends in this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic convective diffusion and fluid flow problems the paper shows that both procedures share a number of features like mesh discretization and approximation moreover it is shown that in many cases both techniques are completely equivalent

unified scaling of polar codes error exponent scaling exponent moderate deviations and error floors consider the transmission of a polar code of block length n and rate r over a binary memoryless symmetric channel w and let p mathrm e be the block error probability under successive cancellation decoding in this paper we develop new bounds that characterize the relationship of the parameters r n p mathrm e and the quality of the channel w quantified by its capacity i w and its bhattacharyya parameter z w in previous work two main regimes were studied in the error exponent regime the channel w and the rate r are fixed and it was proved that the error probability p mathrm e scales roughly as 2 sqrt n in the scaling exponent approach the channel w and the error probability p mathrm e are fixed and it was proved that the gap to capacity i w r scales as n 1 mu here mu is called scaling exponent and this scaling exponent depends on the channel w a heuristic computation for the binary erasure channel bec gives mu 3 627 and it was shown that for any channel w 3 579 le mu le 5 702 our contributions are as follows first we provide the tighter upper bound mu le 4 714 valid for any w with the same technique we obtain the upper bound mu le 3 639 for the case of the bec this upper bound approaches very closely the heuristically derived value for the scaling exponent of the erasure channel second we develop a trade off between the gap to capacity i w r and the error probability p mathrm e as the functions of the block length n in other words we neither fix the gap to capacity error exponent regime nor the error probability scaling exponent regime but we do consider a moderate deviations regime in which we study how fast both quantities as the functions of the block length n simultaneously go to 0 third we prove that polar codes are not affected by error floors to do so we fix a polar code of block length n and rate r then we vary the channel w and study the impact of this variation on the error probability we show that the error probability p mathrm e scales as the bhattacharyya parameter z w raised to a power that scales roughly like sqrt n this agrees with the scaling in the error exponent regime

how to construct polar codes a method for efficiently constructing polar codes is presented and analyzed although polar codes are explicitly defined straightforward construction is intractable since the resulting polar bit channels have an output alphabet that grows exponentially with the code length thus the core problem that needs to be solved is that of faithfully approximating a bit channel with an intractably large alphabet by another channel having a manageable alphabet size we devise two approximation methods which sandwich the original bit channel between a degraded and an upgraded version thereof both approximations can be efficiently computed and turn out to be extremely close in practice we also provide theoretical analysis of our construction algorithms proving that for any fixed e 0 and all sufficiently large code lengths n polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n

existence result for a third order ode with nonlinear boundary conditions in presence of a sign type nagumo control abstract in this work we provide an existence and location result for the third order nonlinear differential equation u t f t u t u t u t where f a b r 3 r is a continuous function and two types of boundary conditions u a a u b u b 0 u a b or u a a u a u a 0 u b c with r 2 r continuous functions monotonous in the second variable and a b c r we assume that f satisfies a sign type nagumo condition which allows an asymmetric unbounded behaviour on the nonlinearity the arguments used concern leray schauder degree theory and lower and upper solutions technique

first and second order discontinuous functional differential equations with impulses at fixed moments abstract we give sufficient conditions for the existence of extremal solutions to discontinuous and functional differential equations with impulses our main results are new even for ordinary differential equations without impulses

dynamic scflip decoding of polar codes this paper proposes a generalization of the recently introduced successive cancellation flip scflip decoding of polar codes characterized by a number of extra decoding attempts where one or several positions are flipped from the standard sc decoding to make such an approach effective we first introduce the concept of higher order bit flips and propose a new metric to determine the bit flips that are more likely to correct the trajectory of the sc decoding we then propose a generalized scflip decoding algorithm referred to as dynamic scflip d scflip which dynamically builds a list of candidate bit flips while guaranteeing that the next attempt has the highest probability of success among the remaining ones simulation results show that d scflip is an effective alternative to sc list decoding of polar codes by providing very good error correcting performance with an average computation complexity close to the one of the sc decoder

efficient design and decoding of polar codes polar codes are shown to be instances of both generalized concatenated codes and multilevel codes it is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance in some cases this also results in complexity reduction it is shown that gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

performance bounds of concatenated polar coding schemes a concatenated coding scheme over binary memoryless symmetric bms channels using a polarization transformation followed by outer sub codes is analyzed achievable error exponents and upper bounds on the error rate are derived the first bound is obtained using outer codes which are typical random linear codes as a byproduct of this bound it determines the rates of the outer codes a lower bound on the error exponent that holds for all bms channels with a given capacity is then derived improved bounds and approximations for finite blocklength codes using channel dispersions normal approximation as well as converse and approximate converse results are also obtained the bounds are compared with actual simulation results from the literature for the cases considered when transmitting over the binary input additive white gaussian noise channel there was only a small gap between the channel dispersion based approximation and the actual error rate of concatenated bch polar codes

existence of solution for discontinuous third order boundary value problems abstract in this paper we obtain existence results for the problem u q u f t u with boundary conditions u a a u b b u a c and u a u b u a u b u a c we assume f a caratheodory function q l r 0 such that 1 q l loc r 0 and suppose the existence of lower and upper solutions the existence of solution for the first considered conditions is obtained as a consequence of the fixed points theorems we obtain the solution of the second problem as a limit of solutions of the first case for the first problem the monotone method is developed

a third order boundary value problem with one sided nagumo condition abstract in this paper we present an existence and location result for the third order separated boundary value problem u t f t u t u t u t u a a u a 0 u b 0 where f a b r 3 r is a continuous function and a r one sided nagumo condition lower and upper solutions a priori estimates and leray schauder degree play an important role in the arguments

soft decision decoding of reed muller codes as generalized multiple concatenated codes constructs reed muller codes by generalized multiple concatenation of binary block codes of length 2 as a consequence of this construction a new decoding procedure is derived that uses soft decision information the algorithm is designed for low decoding complexity and is applicable to all reed muller codes it gives better decoding performance than soft decision bounded distance decoding its decoding complexity is much lower than that of maximum likelihood trellis decoding of reed muller codes especially for long codes

existence and uniqueness of solutions for third order nonlinear boundary value problems in this paper we present some general results of existence and uniqueness of solutions of nonlinear two point boundary value problems for third order nonlinear differential equations by using the shooting method as applications we give certain concrete sufficient conditions for the existence and uniqueness

discretisation procedures for multi physics phenomena procedures are described for solving the equations governing a multi physics process finite volume techniques are used to discretise using the same unstructured mesh the equations of fluid flow heat transfer with solidification and solid deformation these discretised equations are then solved in an integrated manner the computational mechanics environment physica which facilitates the building of multi physics models is described comparisons between model predictions and experimental data are presented for the casting of metal components

existence of multiple solutions for weighted p r laplacian equation dirichlet problems abstract this paper deals with the existence of multiple solutions for weighted p r laplacian equation dirichlet problems several sub super solution type results have been given via leray schauder s degree method in particular we give a counterexample which means that 2 in paper j henderson h b thompson existence of multiple solutions for second order boundary value problems j differential equations 166 2000 443 454 is not an open subset of c 0 1

construction of polar codes with sublinear complexity consider the problem of constructing a polar code of block length n for a given transmission channel w previous approaches require one to compute the reliability of the n synthetic channels and then use only those that are sufficiently reliable however we know from two independent works by schurch and by bardet et al that the synthetic channels are partially ordered with respect to degradation hence it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem we show that if we take advantage of the partial order we can construct a polar code by computing the reliability of roughly a fraction 1 log 3 2 n of the synthetic channels in particular we prove that n log 3 2 n is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor log log n this set of roughly n log 3 2 n synthetic channels is universal in the sense that it allows one to construct polar codes for any w and it can be identified by solving a maximum matching problem on a bipartite graph our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set as such this method is general and it can be used to further improve the complexity of the construction problem in case a refined partial order on the synthetic channels of polar codes is discovered

extremal solutions for third order nonlinear problems with upper and lower solutions in reversed order abstract this paper deals with the existence of extremal solutions for the third order nonlinear boundary value problem u t f t u t t a b u a a u a b u b c in the presence of a pair of lower and upper solutions in reversed order here r r is an increasing homeomorphism f a b r r is a carathedory function and a b c r the proof follows from monotone iterative techniques which are based on suitable anti maximum principles for adequate operators to deduce such results we study some related problems coupled with boundary value conditions of the form p 0 u a q 0 u a a p 1 u b q 1 u b b u a c and u a a u a b u b c

extremality and comparison results for discontinuous third order functional initial boundary value problems extremality and comparison results are derived for third order functional initial boundary value problems differential equations of problems may involve discontinuous nonlinearities

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

a weak weak formulation for large displacements beam statics a finite volumes approximation this paper addresses the problem of the numerical solution of beam statics undergoing large displacements a kinematic analysis outlines the beam geometrical model through the definition of its lagrangian co ordinate and strain parameters a definition of the stress parameters a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain the equations governing the beam equilibrium are introduced and their weak form is derived these equations are then proved to be equivalent to the primal and mixed form of principle of virtual work the numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form the weak weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain some numerical examples prove the capability of present formulation in handling actual problems

fourth order problems with fully nonlinear boundary conditions differential inequality method bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth order differential equations under nonlinear boundary conditions and many existing results are complemented

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation m r grossinho a b 1 st a tersian c 2 a departamento de matem atica iseg universidade t ecnica de lisboa rua do quelhas 6 1200 lisboa portugal b cmaf universidade de lisboa av prof gama pinto 2 1699 lisboa codex portugal c center of applied mathematics and informatics university of rousse 8 studentska str 7017 rousse bulgaria

polar codes bounds on bhattacharyya parameters and their applications this paper proposes a novel lower bound on the bhattacharyya parameter of synthesized channels in polar codes the proposed bound is a result of a newly defined merge operation that reduces the cardinality of output alphabets of a symmetric binary input discrete memoryless channel by repeated applications of the proposed merge operation we establish a universal bound on the bhattacharyya parameters of synthesized channels that holds for any symmetric binary input discrete memoryless transmission channel we then discuss some applications of the new bound for example using the derived universal lower bound along with the existing upper bound we find a partial order among the bhattacharyya parameters of those synthesized channels that cannot be ordered through existing results this ordering can help efficient design of polar codes in addition some extremal behaviors of the binary symmetric channel are also proved

a stabilized finite point method for analysis of fluid mechanics problems in this paper a meshless procedure termed the finite point method for solving convection diffusion and fluid flow type problems is presented the method is based on the use of a weighted least square interpolation procedure together with point collocation for evaluating the approximation integrals special emphasis is given to the stabilization of the convective terms and the neumann boundary condition which has been found to be essential to obtain accurate results some examples of application to diffusive and convective transport and compressible flow problems using quadratic fp interpolations are presented

a novel pressure velocity formulation and solution method for fluid structure interaction problems in the standard approach for simulating fluid structure interaction problems the solution of the set of equations for solids provides the three displacement components while the solution of equations for fluids provides the three velocity components and pressure in the present paper a novel reformulation of the elastodynamic equations for hookean solids is proposed so that they contain the same unknowns as the navier stokes equations namely velocities and pressure a separate equation for pressure correction is derived from the constitutive equation of the solid material the system of equations for both media is discretised using the same method finite volume on collocated grids and the same iterative technique simple algorithm is employed for the pressure velocity coupling with this approach the continuity of the velocity field at the interface is automatically satisfied a special pressure correction procedure that enforces the compatibility of stresses at the interface is also developed the new method is employed for the prediction of pressure wave propagation in an elastic tube computations were carried out with different meshes and time steps and compared with available analytic solutions as well as with numerical results obtained using the flugge equations that describe the deformation of thin shells for all cases examined the method showed very good performance

uniqueness comparison and existence results for third order initial boundary value problems abstract we derive maximum principles for a nonlinear third order differential operator and prove uniqueness comparison and existence results for different types of third order initial boundary value problems including existence results for functional problems

solvability of some third order boundary value problems with asymmetric unbounded nonlinearities abstract in this paper we present existence and location results for the third order separated boundary value problems u t f t u t u t u t with the boundary conditions u a a u a b u b c or u a a c 1 u a c 2 u a b c 3 u b c 4 u b c with c 1 c 2 c 3 c 4 r and a b c r we assume f a b r 3 r is a continuous function satisfying one sided nagumo type condition which allows an asymmetric unbounded behaviour the arguments used concern leray schauder degree and lower and upper solution techniques

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when

convolutional polar kernels a family of polarizing kernels is presented together with polynomial complexity algorithm for computing scaling exponent the proposed convolutional polar kernels are based on convolutional polar codes also known as b mera codes for these kernels a polynomial complexity algorithm is proposed to find weight spectrum of unrecoverable erasure patterns needed for computing scaling exponent as a result we obtain scaling exponent and polarization rate for convolutional polar kernels of size up to 1024