a note on specialization l preorder of l topological spaces l fuzzifying topological spaces and l fuzzy topological spaces it is shown that for any spatial frame l i e l is a complete lattice generated by the set of all prime elements both the specialization l preorder of l topological spaces introduced by lai and zhang fuzzy preorder and fuzzy topology fuzzy sets and systems 157 2006 1865 1885 and that of l fuzzifying topological spaces introduced by fang and qiu fuzzy orders and fuzzifying topologies internat j approx reasoning 48 2008 98 109 are special cases of that of l fuzzy topological spaces introduced by fang i fuzzy alexandrov topologies and specialization orders fuzzy sets and systems 158 2007 2359 2374

finite length scaling for polar codes consider a binary input memoryless output symmetric channel w such a channel has a capacity call it i w and for any r i w and strictly positive constant pe we know that we can construct a coding scheme that allows transmission at rate r with an error probability not exceeding p e assume now that we let the rate r tend to i w and we ask how we have to scale the blocklength n in order to keep the error probability fixed to p e we refer to this as the finite length scaling behavior this question was addressed by strassen as well as polyanskiy poor and verdu and the result is that n must grow at least as the square of the reciprocal of i w r polar codes are optimal in the sense that they achieve capacity in this paper we are asking to what degree they are also optimal in terms of their finite length behavior since the exact scaling behavior depends on the choice of the channel our objective is to provide scaling laws that hold universally for all binary input memoryless output symmetric channels our approach is based on analyzing the dynamics of the un polarized channels more precisely we provide bounds on the exponent of the number of subchannels whose bhattacharyya constant falls in a fixed interval a b mathematically this can be stated as bounding the sequence 1 n logpr z n a b n n where z n is the bhattacharyya process we then use these bounds to derive tradeoffs between the rate and the block length the main results of this paper can be summarized as follows consider the sum of bhattacharyya parameters of subchannels chosen by the polar coding scheme to transmit information if we require this sum to be smaller than a given value p e 0 then the required block length n scales in terms of the rate r i w as n i w r where is a positive constant that depends on p e and i w we show that 3 579 is a valid choice and we conjecture that indeed the value of can be improved to 3 627 the parameter for the binary erasure channel also we show that with the same requirement on the sum of bhattacharyya parameters the blocklength scales in terms of the rate like n i w r where is a constant that depends on p e and i w and 6

fundamental study complete and directed complete ω categories let w be a commutative unital quantale complete and directed complete w categories are the core objects in quantitative domain theory this paper based on the theory of f completeness for enriched categories presents a systematic investigation of completeness and directed completeness of w categories

the l ordered semigroups based on l partial orders abstract fuzzy frame valued ordered semigroups are investigated with the orders being l partial orders in this framework l left resp l right ideals l bi ideals and l quasi ideals of an l ordered semigroup are discussed the idea of which is applied to the characterizations of regular and duo l ordered semigroups in addition the relationship between an l ordered semigroup and an l topological semigroup is studied as a consequence an isomorphism between the category of l ordered semigroups and the category of l topological semigroups with the topologies being l t 0 and alexandrov is obtained

fuzzy radicals and prime fuzzy ideals of ordered semigroups let s be an ordered semigroup a fuzzy subset of s is by definition an arbitrary mapping f s 0 1 where 0 1 is the usual interval of real numbers motivated by studying prime fuzzy ideals in rings semigroups and ordered semigroups and as a continuation of kehayopulu and tsingelis s works in ordered semigroups in terms of fuzzy subsets in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup s and give a characterization of prime fuzzy ideals of an ordered semigroup s we also introduce the concepts of weakly prime fuzzy ideals completely prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup s and establish the relationship between the five classes of ideals furthermore we characterize weakly prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of s by their level ideals finally we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup s can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it as an application of the results of this paper the corresponding results of semigroups without order are also obtained

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

lattice of partially ordered fuzzy subalgebras abstract fuzzy subalgebras are defined to be mappings from an algebra to a poset which can be a lattice and also the interval 0 1 so that every level subset is an ordinary subalgebra every fuzzy subalgebra has a canonical representation a mapping from the algebra to the poset of levels necessary and sufficient conditions under which a fuzzy subalgebra can be explicitly described by a suitable formula in lattice theoretic terms are given classes of algebras for which the poset of fuzzy subalgebras is a lattice are characterized

l fuzzy topological groups abstract in this paper the theory is applied of the q neighborhood system of a fuzzy point which was introduced in 3 and the fuzzy topological groups and their subgroups normal subgroups and quotients are discussed then some results introduced in 1 and 7 are improved

regular articlelowen spaces an l fuzzy topological space x is called a lowen space if has a basis consisting of leveled characteristic functions that is to say the elements in of the form au a l u x are a basis for in the case l 0 1 x is a lowen space if and only if x is a fuzzy neighborhood space in the sense of lowen it is proved in this paper that the category of lowen spaces is a simultaneously bireflective and bicoreflective subcategory of the category of l fuzzy topological spaces and it is isomorphic to the category of l fuzzifying topological spaces several characterizations of these spaces are given

kernel systems on l ordered sets abstract this paper presents fuzzy versions of kernel operators and kernel systems on l ordered sets and establishes a one to one correspondence between them

regular ordered semigroups in terms of fuzzy subsets given a set s a fuzzy subset of s or a fuzzy set in s is by definition an arbitrary mapping f s 0 1 where 0 1 is the usual interval of real numbers if the set s bears some structure one may distinguish some fuzzy subsets of s in terms of that additional structure this important concept of a fuzzy set was first introduced by zadeh fuzzy groups have been first considered by rosenfeld fuzzy semigroups by kuroki a theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed some results on ordered groupoids semigroups have been already given by the same authors in n kehayopulu m tsingelis fuzzy sets in ordered groupoids semigroup forum 65 2002 128 132 n kehayopulu m tsingelis the embedding of an ordered groupoid into a poe groupoid in terms of fuzzy sets inform sci 152 2003 231 236 n kehayopulu m tsingelis fuzzy bi ideals in ordered semigroups inform sci 171 2004 13 28 where s has been endowed with the structure of an ordered semigroup and defined fuzzy analogous for several notions that have been proved to be useful in the theory of ordered semigroups the characterization of regular rings in terms of right and left ideals is well known the characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left right ideals and quasi ideals is well known as well the characterization of regular le semigroups that is lattice ordered semigroups having a greatest element in terms of right ideal elements and left ideal elements or right left and quasi ideal elements is also known in the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals then we characterize the regular ordered semigroups in terms of fuzzy right fuzzy left ideals and fuzzy quasi ideals the paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of fuzzy ordered semigroups some of our results are true for ordered groupoids in general

wide angle parabolic equation solutions to two range dependent benchmark problems numerical solutions are presented to two benchmark problems involving acoustic propagation in range dependent media the two problems deal with 1 upslope propagation in a wedge shaped channel with a penetrable bottom and 2 propagation in a plane parallel waveguide with a range varying sound speed profile the solutions are based on a pair of wide angle variable density parabolic equations one of which is solved using a finite difference algorithm while the other is solved using a split step algorithm

a stone type duality for st0 stratified alexandrov l topological spaces abstract we show that for every complete lattice a both the set of completely prime elements and the set of completely coprime elements are one to one corresponding to ch a the set of complete lattice homomorphisms from a to the two element lattice 2 a complete lattice is called completely generated a cg lattice for short if it is generated by the set of completely prime elements or equivalently by the set of completely coprime elements then we restudy the duality between the category of t 0 alexandrov topological spaces and the category of cg lattices by means of ch a with these preparations for a frame l as the truth value table we introduce s t 0 separation axiom for stratified alexandrov l topological spaces and finally establish a duality between the category of s t 0 stratified alexandrov l topological spaces and the category of completely generated complete l ordered sets we also investigate some properties of the s t 0 axiom for example it is hereditary by closed subspaces and productive

quantitative domains via fuzzy sets part i continuity of fuzzy directed complete posets this paper deals with quantitative domain theory via fuzzy sets it examines the continuity of fuzzy directed complete posets dcpos for short based on complete residuated lattices first we show that a fuzzy partial order in the sense of fan and zhang and an l order in the sense of belohlavek are equivalent to each other then we redefine the concepts of fuzzy directed subsets and continuous fuzzy dcpos we also define and study fuzzy galois connections on fuzzy posets we investigate some properties of continuous fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy double downward arrow operator has a right adjoint we define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy way below relation is the smallest approximating fuzzy auxiliary relation

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

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

boundary conditions for the parabolic equation in a range dependent duct the parabolic wave equation in a range dependent duct with a rigid bottom is considered the range dependence will introduce amplitude errors in addition to the usual phase error of the local modes it is shown that the amplitude error increases by local mode angle and slope of the bottom and is inversely proportional to the depth imposing the vanishing of the normal derivative of the pressure at the bottom which is the physically correct boundary condition might also lead to ill conditioning of the parabolic model various ways to replace the boundary condition u n 0 are discussed and their accuracy is assessed both numerically and theoretically

channel polarization through the lens of blackwell measures each memoryless binary input channel bic can be uniquely described by its blackwell measure which is a probability distribution on the unit interval 0 1 with mean 1 2 conversely any such probability distribution defines a bic the evolution of the blackwell measure under arikan s polar transform is derived for general bics and is analogous to density evolution as cited in the literature the present analysis emphasizes functional equations consequently the evolution of a variety of channel functionals is characterized including the symmetric capacity bhattacharyya parameter moments of information density hellinger affinity gallager s reliability function the hirschfeld gebelein renyi maximal correlation and the bayesian information gain the evolution of measure is specialized for symmetric bics according to their decomposition into binary symmetric sub channels bscs which simplifies iterative computations and the construction of polar codes it is verified that as a consequence of the blackwell sherman stein theorem all channel functionals text i f that can be expressed as an expectation of a convex function f with respect to the blackwell measure of a channel polarize in each iteration due to the polar transformation on the class of symmetric bics moreover for f either convex or non convex a necessary and sufficient condition is established to determine whether the random process associated with each text i f is a martingale submartingale or supermartingale represented via functional inequalities in terms of f this condition is numerically verifiable for all text i f and can generate analytical proofs to exhibit one such proof it is shown that the random process associated with the squared maximal correlation parameter is a supermartingale and converges almost surely on the unit interval 0 1

discrete non local boundary conditions for split step pade approximations of the one way helmholtz equation this paper deals with the efficient numerical solution of the two dimensional one way helmholtz equation posed on an unbounded domain in this case one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of the so called discrete transparent boundary conditions for state of the art parabolic equation methods namely a split step discretization of the high order parabolic approximation and the split step pade algorithm of collins finally several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach

continuity in quantitative domains based on the notion of an l fuzzy partially ordered set constructed in an l fuzzy approach to quantitative domain i generalized ordered set valued in frame and adjunction theory fuzzy systems and mathematics 14 2000 6 7 and by introducing the concepts of an l fuzzy directed set and the join of an l fuzzy set w r t the l fuzzy partial order l fuzzy domains are defined and the generalized scott topology on an l fuzzy domain is built this approach is similar to flagg s logic approach to quantitative domain theory a logical approach to quantitative domain theory elsevier amsterdam 1996 submitted for publication in addition the concepts of stratified approximation and a basis for an l fuzzy domain are proposed and a notion of a continuous l fuzzy domain is developed it is proved that if l is a completely distributive lattice in which 1 is irreducible and the well below relation is multiplicative then the stratified interpolation property holds in a continuous l fuzzy domain x e and ax 0 a 1 x x is a base for the generalized scott topology on x e

join completions of l ordered sets in this paper we build and characterize the join completions of an l ordered set which generalizes the join completions of a crisp poset the universal property of join completions is studied a categorical characterization of the dedekind macneille completion for an l ordered set is presented

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

fuzzy complete lattices in this paper based on l fuzzy posets previously introduced by the third author l fuzzy complete lattices are defined which are generalizations of usual complete lattices and coincide with wagner s complete and cocomplete w categories enriched over the frame l and are consequently a special kind of complete w lattices defined by lai and zhang however tarski fixed point theorem for the l fuzzy complete lattices is proved in a different way from that by lai and zhang furthermore some fuzzy powerset operators are suggested they are not only generalizations of ordinary powerset operators but also generalizations of l valued zadeh powerset operators and their properties are discussed

generalized partial orders for polar code bit channels we study partial orders pos for the synthesized bit channels of polar codes first we give an alternative proof of an existing po for bit channels with the same hamming weight and use the underlying idea to extend the bit channel ordering to some additional cases in particular the bit channel ordering for a given code block length is used to generate additional bit channel ordering relationships for larger block lengths generalizing previously known pos next we consider pos especially for the binary erasure channel bec we identify a symmetry property of the bhattacharyya parameters of complementary bit channel pairs on the bec and provide a condition for the alignment of polarized sets of bit channels for the bec and general binary input memoryless symmetric bms channels numerical examples and further properties about the pos for the bit channels with different hamming weights are provided to illustrate the new pos the bit channels with universal ordering positions which are independent of the channel erasure probability are verified for all of the code block lengths finally we show the threshold behavior of the bhattacharyya parameters of some bit channels by approximating the threshold values the corresponding value for a bit channel can be used to determine whether it is good or bad when the underlying channel is known

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

polar codes speed of polarization and polynomial gap to capacity we prove that for all binary input symmetric memoryless channels polar codes enable reliable communication at rates within an additive gap e 0 to the shannon capacity with a block length construction complexity and decoding complexity all bounded by a polynomial in 1 e polar coding gives the first known explicit construction with rigorous proofs of all these properties previous constructions were not known to achieve capacity with less than exp 1 e decoding complexity except for erasure channels we establish the capacity achieving property of polar codes via a direct analysis of the underlying martingale of conditional entropies without relying on the martingale convergence theorem this step gives rough polarization noise levels e for the good channels which can then be adequately amplified by tracking the decay of the channel bhattacharyya parameters our effective bounds imply that polar codes can have block length and encoding decoding complexity bounded by a polynomial in 1 e the generator matrix of such polar codes can be constructed in polynomial time by algorithmically computing an adequate approximation of the polarization process

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

quantitative domains via fuzzy sets part ii fuzzy scott topology on fuzzy directed complete posets this paper pursues an investigation on quantitative domains via fuzzy sets initiated by the first author this time we define an l topology called the fuzzy scott topology on fuzzy dcpos and investigate its properties scott convergence of stratified l filters is also defined and studied we show that a fuzzy dcpo x e is continuous if and only if for any stratified l filter on x scott convergence coincides with the convergence with respect to the fuzzy scott topology at last we show that the category of fuzzy dcpos with fuzzy scott continuous maps is cartesian closed

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

exact discrete nonlocal boundary conditions for high order pade parabolic equations the nonlocal boundary conditions nlbcs for high order finite difference parabolic equations pes are obtained by z transformation of the discrete pe in a homogeneous medium the considered nlbcs include the free space radiation condition possibly with a density jump at the nlbc interface the nlbc at an arbitrary impedance interface and the nlbcs for sources and the starting field beyond the nlbc interface the derivation is presented for the multiterm pade pe model owwe one way wave equation but the developed technique is applicable to a broad class of finite difference pes the obtained nlbcs are exact for the given finite difference scheme they are not limited by the order of pade approximation or by the pe steps in range and depth the nlbc convolution coefficients are calculated by the numerical inverse z transformation the accuracy and performance of the algorithm are analyzed for several benchmark problems the solution is robust for the range steps over 25 wavelengths and or the approximations up to the tenth order the nlbcs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution precomputation of the nlbc coefficients may be required in time demanding applications the results are compared with earlier proposed nlbcs for high order pes

q fuzzy subsets on ordered semigroups in this paper we introduce the concept of a q fuzzy subset of an ordered semigroup where a quantale q replaces the unit interval we define a binary operation on the fuzzy power set that makes it a quantale in order to discuss the relation between the referential set and the fuzzy power set the notion of an ordered q fuzzy point of an ordered semigroup is introduced in terms of ordered q fuzzy points we prove that any ordered semigroup can be embedded into a quantale and also build a functor between the category of ordered semigroups and the category of quantales

unconditionally stable discretization schemes of non reflecting boundary conditions for the one dimensional schrodinger equation this paper addresses the problem of the construction of stable approximation schemes for the one dimensional linear schrodinger equation set in an unbounded domain after a study of the initial boundary value problem in a bounded domain with a transparent boundary condition some unconditionally stable discretization schemes are developed for this kind of problem the main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator the proposed semi discretization of this operator yields with a very different point of view the one proposed by yevick friese and schmidt j comput phys 168 2001 433 two possible choices of transparent boundary conditions based on the dirichlet neumann dn and neumann dirichlet nd operators are presented to preserve the stability of the fully discrete scheme conform galerkin finite element methods are employed for the spatial discretization finally some numerical tests are performed to study the respective accuracy of the different schemes

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

transparent boundary conditions for iterative high order parabolic equations recently a new approach to the construction of high order parabolic approximations for the helmholtz equation was developed these approximations have the form of the system of iterative parabolic equations where the solution of the n th equation is used as an input term for the n 1 th equation in this study the transparent boundary conditions for such systems of coupled parabolic equations are derived the existence and uniqueness of the solution of the initial boundary value problem for the system of iterative parabolic equations with the derived boundary conditions are proved the well posedness of this problem is also established and an unconditionally stable finite difference scheme for its solution is proposed