orthogonalization coefficients for the elliott draayer states of the two parametric irreps of su n so n explicit algebraic polynomial expressions for the direct and inverse orthogonalization coefficients of the projected elliott draayer basis states of su n so n are constructed in the case of two parametric irreducible representation irreps they cover the general su 3 so 3 case as well as the two parametric covariant irreps of su 4 restricted to the su 2 su 2 subgroup of spin and isospin the obtained orthonormal bases are equivalent to those constructed by means of the ordered gram schmidt process the orthogonalization coefficients are expressed up to explicitly given elementary factors in terms of the numerator and denominator polynomials related with those that appeared in the orthogonalization coefficients of the paracanonical coupling coefficients isofactors of su 3 and with a functions of louck biedenharn and lohe

equilibrium in continuous time financial markets endogenously dynamically complete markets we prove existence of equilibrium in a continuous time securities market in which the securities are potentially dynamically complete the number of securities is at least one more than the number of independent sources of uncertainty we prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model our result is universal rather than generic dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a brownian filtration and from a mild exogenous nondegeneracy condition on the terminal security dividends the nondegeneracy condition which requires that finding one point at which a determinant of a jacobian matrix of dividends is nonzero is very easy to check we find that the equilibrium prices consumptions and trading strategies are well behaved functions of the stochastic process describing the evolution of information we prove that equilibria of discrete approximations converge to equilibria of the continuous time economy

minimum euclidean distance for combinations of short rate 1 2 convolutional codes and cpfsk modulation continuous phase frequency shift keying cpfsk is a constant amplitude modulation method with good spectral sidelobe properties good error probability properties can be obtained with coherent maximum likelihood detection in this paper we study the euclidean distance properties of signals formed by a conventional rate 1 2 convolutional encoder followed by a binary or 4 level cpfsk modulator the minimum euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length the optimum detector is discussed for rational modulation index values the best obtainable codes are found for the case of short rate 1 2 codes with binary or 4 level cpfsk modulation lists of the best codes are given among the results are that the noncatastrophic rate 1 2 convolutional codes with optimum free hamming distance do not in general give the best euclidean distance with cpfsk

short turbo codes over high order fields two classes of turbo codes constructed on high order finite fields are introduced the codes are derived from a particular protograph sub ensemble of the 2 3 regular low density parity check ldpc code ensemble the first construction results in a parallel concatenation of two non binary time variant accumulators the second construction consists of the serial concatenation of a non binary time variant differentiator with a non binary time variant accumulator and provides a highly structured flexible encoding scheme for 2 4 ldpc codes a cycle graph representation is also provided the proposed codes can be decoded efficiently either as ldpc codes via belief propagation decoding on their bipartite graphs or as turbo codes via the forward backward algorithm applied to the component code trellises by means of the fast fourier transform the proposed codes provide remarkable coding gains more than 1 db at a codeword error rate 10 4 over binary ldpc and turbo codes in the moderate short block length regime

on the projected bases for sp 4 u 2 and the orthogonalization problem the mutual expansion and overlaps of the projected smirnov tolstoy and szpikowski berej bases for the irreducible representations irreps of sp 4 restricted to u 2 are considered the equivalence relation connecting the overlaps of both st and sb bases after the definite substitution of parameters up to an elementary factor is presented the overlaps of the szpikowski berej basis states are rearranged to double sums restricted by the parameters characterizing the multiplicities of the repeating irreps of subgroup and thus representable in polynomial forms the regge type symmetry of these new expressions allows the proof of the earlier conjectured symmetry of the orthogonalization coefficients oc for the family of the biorthogonal bases of sp 4 u 2 su 4 su 2 su 2 and su n so n for two parametric irreps the structure of the orthogonalization coefficients for the szpikowski berej basis is also considered the symmetry of the b c abde functions as the conjectured numerator polynomials

coded continuous phase modulation using ring convolutional codes rate 1 2 convolutional codes over the ring of integers modulo m are combined with m ary continuous phase modulation cpm schemes whose modulation indices are of the form h 1 m an m ary cpm scheme with h 1 m can be modeled by a continuous phase encoder cpe followed by a memoryless modulator mm where the cpe is linear over the ring of integers modulo m the fact that the convolutional code and the cpe are over the same algebra allows the state of the cpe to be fed back and used by the convolutional encoder a modified euclidean distance function that substantially simplifies the search for good codes has been derived and used to find new codes numerical results show that this approach consistently improves the performance as compared to coded schemes using binary convolutional codes with the same decoding complexity

structure optimization and realization of ffsk trellis codes the structure of combinations of l k binary trellis encoders and fast frequency shift keying ffsk modulation is studied a novel method for the optimization of these combinations is proposed it is concluded that regardless of the particular realization the best combinations can always be obtained by using a so called matched convolutional code optimum ffsk codes of rates 1 2 and 2 3 for up to 64 and 32 states respectively are reported it is shown that the best matched encoders for ffsk modulation are also matched and best for any ffsk type scheme where the pulse shape is required to satisfy nyquist s third criterion

coding cpfsk for differential demodulation a model is developed for finding codes for differentially demodulated continuous phase frequency shift keying with modulation index h 1 m where m is the size of the symbol alphabet the results of codes searches show that there is very little loss in squared euclidean distance when comparing differential codes with coherent codes

a decomposition approach to cpm it is shown that any continuous phase modulation cpm system can be decomposed into a continuous phase encoder and a memoryless modulator in such a way that the former is a linear modulo some integer p time invariant sequential circuit and the latter is also time invariant this decomposition is exploited to obtain alternative realizations of the continuous phase encoder and hence of cpm and also to obtain alternative forms of the optimum decoding algorithm when p is a prime p so that the encoder is linear over the finite field gf p it is shown that cascading it with an outside convolutional encoder is equivalent to a single convolutional encoder it is pointed out that the cascade of the modulator the waveform channel which it is assumed is characterized by additive white gaussian noise and the demodulator that operates over one symbol interval yield a discrete memoryless channel that can be studied without the distractions introduced by continuous phase encoding

a search for good convolutional codes to be used in the construction of turbo codes recursive systematic convolutional encoders have been shown to play a crucial role in the design of turbo codes we recall some properties of binary convolutional encoders and apply them to a search for good constituent convolutional codes of turbo codes tables of the best recursive systematic convolutional encoders found are presented for various rates together with the average bit error probability performances of some turbo codes using them

exact formula for the minimum squared euclidean distance of cpfsk the exact expression for the minimum squared euclidean distance of continuous phase frequency shift keying cpfsk with modulation index h

explicit canonical tensor operators and orthonormal coupling coefficients of su 3 the canonical unit su 3 tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators with the renormalization factors expressed in terms of the denominator functions of biedenharn gustafson lohe louck and milne the matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of asherova and smirnov the self conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique the corresponding extreme isoscalar factors of the clebsch gordan wigner coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of su 3 with its considerable simplication for the boundary values of parameters the general isofactors are also expanded in the different ways in terms of their boundary values the new classes of the generaliz

construction of linear ring codes for 6 psk the algebraic construction of linear codes over zm for m psk is considered for the case that m is the product of two distinct primes by means of the chinese remainder theorem crt such codes can be constructed from linear codes over the corresponding prime fields it is shown that for given component codes the crt composition yields psk codes that are at least as good and often much better than the codes obtained from the standard coset coding technique a lower bound on the minimum euclidean distance of 6 psk codes is derived that is based on the hamming weight distribution of the component codes

biorthogonal systems for su4 su2 su2 sun son and sp4 u2 and analytical inversion symmetry analytical inversion symmetry of the biorthogonal systems of su4 contains implies su2 su2 sun contains implies son and sp4 contains implies u2 bases for two parametric covariant and mixed tensor irreducible representation is discovered this symmetry relates the dual isoscalar factors or resubducing coefficients it allowed us to invert by means of a special analytical continuation procedure the non orthogonal isofactors of sun contains implies son for couplings p0 0q to 0 and p10 p20 to 0 as well as the resubducing coefficients transformation brackets for expansion of the su3 contains implies so3 elliott basis states and sp4 contains implies u2 smirnov and tolstoy basis states in terms of the corresponding canonical basis states new expressions for bilinear combinations of sun contains implies son special isofactors are obtained expansion of su3 canonical basis states in terms of su3 contains implies so3 elliott states is found isofactors for coupling of two elliott states are given

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

on joint source and channel coding using trellis coded cpm analytical bounds on the channel distortion joint source and channel coding jscc using trellis coded quantization tcq in conjunction with trellis coded continuous phase modulation cpm is studied the channel is assumed to be the additive white gaussian noise awgn channel analytical bounds on the channel distortion for the investigated systems with maximum likelihood sequence detection mlsd are developed the bounds are based on the transfer function technique which was modified and generalized to include continuous amplitude discrete time signals for a memoryless uniform source the constructed bounds for the investigated systems are shown to be asymptotically tight for increasing channel signal to noise ratio snr values for a memoryless nonuniform source the constructed bounds are not as tight as the one for the uniform source however it still can be used as an indication to how the system performs it is concluded that the minimum euclidean distance of the system alone is not enough to evaluate the performance of the considered systems the number of error events having minimum euclidean distance and the total distortion caused by those error events also affect the asymptotic performance this work provides an analysis tool for the investigated systems the analysis method is very general it may be applied to any trellis based jscc schemes

equilibrium pricing in incomplete markets under translation invariant preferences we propose a general discrete time framework for deriving equilibrium prices of financial securities it allows for heterogeneous agents unspanned random endowments and convex trading constraints we give a dual characterization of equilibria and provide general results on their existence and uniqueness in the special case where all agents have preferences of the same type and in equilibrium all random endowments are replicable by trading in the financial market we show that a one fund theorem holds and give an explicit expression for the equilibrium pricing kernel

on biorthogonal and orthonormal clebsch gordan coefficients of su 3 analytical and algebraic approaches minimal biorthogonal systems of the clebsch gordan wigner coefficients of su 3 u 2 are discussed as well as the dual coupled bases the closed system of analytical expressions for the dual isofactors reduced wigner coefficients and the overlaps of coupled states is obtained with the help of analytical inversion symmetry the regge type symmetry of the overlaps and the boundary orthonormal isofactors orthogonalization coefficients is discovered the polynomial structure of the alternative complete algebraic systems of the orthonormal su 3 isofactors characterized by the null spaces symmetries and additional selection rules and obtained by means of the hecht or gram schmidt process is considered the realizations of the external missing label operators of the third and the fourth orders in the minimal coupled bases which lead to preferable algorithms to evaluate the orthonormal su 3 coupling coefficients satisfying different symmetry properties are presented with the help of the 6j coef

on the denominator function for canonical su 3 tensor operators ii explicit polynomial form the canonical resolution of the multiplicity problem for tensor operators in su 3 is equivalent to the map the denominator mapping from the set of all su 3 unit tensor operators to su 3 invariant functions the denominator functions the denominator function vanishes precisely on that characteristic null space that specifies each operator uniquely since for su 3 the characteristic null spaces are known to be simply ordered each denominator function can be expressed up to explicitly known multiplicative factors as a ratio of two successive polynomials in the set gtq t 0 1 q 1 q 0 1 by obtaining explicitly the set of all polynomials gtq this paper completes the construction of all su 3 denominator functions

space time analyticity of weak solutions to linear parabolic systems with variable coefficients analytic smoothing properties of a general strongly parabolic linear cauchy problem of second order in rn 0 t with analytic coefficients in space and time variables are investigated they are expressed in terms of holomorphic continuation of global weak l2 type solutions to the system given 0 0 are constants depending upon t the proof uses the extension of a solution to an l2 type solution in a domain in cn c such that this extension satisfies the cauchy riemann equations the holomorphic extension is thus obtained in a hardy space h2 applications include market completion by european options in finance

the capital asset pricing model and arbitrage pricing theory a unification we present a model of a financial market in which naive diversification based simply on portfolio size and obtained as a consequence of the law of large numbers is distinguished from efficient diversification based on mean variance analysis this distinction yields a valuation formula involving only the essential risk embodied in an asset s return where the overall risk can be decomposed into a systematic and an unsystematic part as in the arbitrage pricing theory and the systematic component further decomposed into an essential and an inessential part as in the capital asset pricing model the two theories are thus unified and their individual asset pricing formulas shown to be equivalent to the pervasive economic principle of no arbitrage the factors in the model are endogenously chosen by a procedure analogous to the karhunen loeve expansion of continuous time stochastic processes it has an optimality property justifying the use of a relatively small number of them to describe the underlying correlational structures our idealized limit model is based on a continuum of assets indexed by a hyperfinite loeb measure space and it is asymptotically implementable in a setting with a large but finite number of assets because the difficulties in the formulation of the law of large numbers with a standard continuum of random variables are well known the model uncovers some basic phenomena not amenable to classical methods and whose approximate counterparts are not already or even readily apparent in the asymptotic setting

explicit orthogonalization of some biorthogonal bases for su n so n and sp 4 u 2 mutually related explicit algebraic polynomial expressions of the orthogonalization coefficients are proposed for the biorthogonal polynomial bargmann moshinsky stretched antistretched quasistretched and their dual bases of the two parametric mixed tensor and covariant irreducible representations irreps of su n restricted to so n as well as for the projected smirnov tolstoy and dual bases of the five dimensional quasispin the orthogonalization coefficients of the essentially simplified gram schmidt process are expressed up to explicitly given elementary factors in terms of the numerator and denominator polynomials represented as compositions of the generalized hypergeometric coefficients 3f2 and 1f0 and equivalent in the diagonal denominator and boundary cases to the a cabde functions of biedenharn and louck the distribution of zeros and the symmetry properties of the introduced polynomials are crucial for the conjectured solutions

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

multilevel coded cpfsk systems for awgn and fading channels in this paper we introduce multilevel coded continuous phase frequency shift keying cpfsk systems for both additive white gaussian noise and fading channels based on multilevel coding and multistage decoding techniques these schemes are designed under the constraint m spl ges 2p for m ary cpfsk with modulation index j p in order to maintain the phase continuity property after multilevel coding we use some specific set partitioning rules we construct examples to show that the proposed systems outperform the corresponding conventional one level schemes for the fading case we consider an ideal fading channel where adjacent fading symbols are assumed to be independent and a correlated fading channel where bit interleavers are needed at each coding level the computer simulation results show that the proposed systems have bit error performance and decoder complexity advantages over the corresponding reference codes taken from the literature it is also shown that there is only a slight degradation on the bit error performance for correlated fading compared to the ideal fading case

explicit orthonormal clebsch gordan coefficients of su 3 the construction of the explicit algebraic polynomial expressions for the nonmultiplicity free orthonormal clebsch gordan wigner coefficients of su 3 u 2 is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the hecht or gram schmidt process the direct and inverse orthogonalization coefficients the first of them being equivalent to the boundary orthonormal isofactors are expressed up to explicitly given multiplicative factors in terms of the numerator and denominator polynomials related with the auxiliary a function of louck biedenharn and lohe that appears as a fragment of the denominator g functions of canonical su 3 tensor operators

overlaps of the biorthogonal 3 coupling coefficients and related basic hypergeometric and other q factorial series the intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group su 3 with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated some expansion coefficients such as new prime overlap functions equivalent to the bilinear combinations of the different boundary isofactors are expressed in terms of the balanced saalschutzian basic or classical hypergeometric series these new overlap functions their analytical inversion some known triangular expansion matrices transition matrices related to definite racah coefficients the well poised or series and compositions of q factorial series resembling the well poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non orthonormal systems of and su 3 isofactors which may be orthonormalized to the paracanonical or with a definite caution canonical version

a new physical basis for the irreducible representations of the orthogonal group so 5 in the quasi spin formalism new physical state vectors for the irs of the group so 5 are constructed in an analytical form the state vectors are labelled by the quantum numbers of the isospin its third component by the number of particles and by a fourth properly defined quantum number the previously constructed bases have been compared with the new construction from the same point of view two examples illustrate the new construction

rate 1 2 component codes for nonbinary turbo codes the iterative decoding structure and component maximum a posteriori decoders used for decoding binary concatenated codes can be extended to the nonbinary domain this paper considers turbo codes over nonbinary rings specifically ternary quaternary penternary hexernary and octernary codes the best rate 1 2 component codes are determined using a practical search algorithm the performance of the resulting rate 1 3 turbo codes on an additive white gaussian noise channel using q ary phase shift keying modulation is given

nonstandard construction of the stochastic integral and applications to stochastic differential equations ii h j keisler has recently used a nonstandard theory of lto integration due to r m anderson to construct solutions of ito integral equations by solving an associated internal difference equation in this paper we use the same general approach to find solutions y t of semimartingale integral equations of the form y t c h t c f s y w dz s where z is a given semimartingale h is a right continuous process and f s w is continuous on the space of right continuous functions with left limits with the topology of uniform convergence on compacts in addition we generalize keisler s continuity theorem and give necessary and sufficient conditions for an internal martingale to be s continuous introduction this paper is a continuation of hoover and perkins 5 notation and definitions presented in 5 are used without further introduction and references to that paper are made by simply citing the number of the theorem or lemma due to the consecutive numbering of the two papers no ambiguity will arise finally the reader is referred to 0 of 5 for a brief description of the contents of this work 8 continuous local martingales we assume throughout this section that all processes are r or r valued since the main results theorems 8 1 8 5 and 8 6 then follow immediately for higher dimensions we will show theorem 8 1 that a continuous local martingale has a q1 local martingale lifting for any internal filtration t recall that in general theorem 5 6 asserted only that such a lifting exists for some internal filtration for this reason unless otherwise stated the internal filtration under consideration will be it i t e t which was used in the original definition of t note that if x t x q m is an s continuous process i e x w is s continuous a s then st x has continuous paths a s and x is a uniform lifting of st x in the sense of keisler 6 the following lifting theorem was first proved in a more specialized setting in panetta 9 the proof given here is different received by the editors january 22 1982 the results of this paper were presented at the second victoria symposium on nonstandard analysis victoria canada june 16 20 1980 a ms mos subject cllassifications 1970 primary 60h20 02h25 secondary 60g45

orthogonalization of the projected states and isofactors the explicit direct and inverse orthogonalization coefficients ocs are derived for the projected basis states of the two parametric covariant irreducible representations of including elliott s n 3 case and draayer with the alternative choices of the gram schmidt processes began respectively from the lowest vergados or highest absolute value of the intrinsic parameter k the different direct and inverse ocs the latter equivalent to the definite boundary isofactors or the boundary resubducing coefficients are found to be connected by the definite analytical continuation contraction procedures with mutually related ocs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of the orthogonalization coefficients considered are presented as definite algebraic polynomial structures under the square root with the linear factors and numerator denominator polynomials in terms of the partition dependent functions of biedenharn and louck

serially concatenated continuous phase modulation with convolutional codes over rings in this paper we investigate serially concatenated continuous phase modulation sccpm with convolutional codes cc over rings the transmitted signals are disturbed by additive white gaussian noise the properties for systems with both infinite and finite block lengths are investigated for an infinite length system we check the convergence threshold using the extrinsic information transfer chart for a finite length system we use union bound techniques to estimate the error floors in the union bound analysis we consider both the order and the position of nonzero permuted symbols a simple method for determining a cpm error event through the sum of the input symbol sequence is shown thus we can determine if the output symbol sequence of an error event in the ring cc can form an error event in cpm two properties concerning the interleaver gain ig are investigated a recursive search algorithm for the maximal ig is shown compared with previous sccpm with a binary cc the proposed system shows an improvement concerning the convergence threshold or error floors

trellis coded continuous phase frequency shift keying with ring convolutional codes trellis coding techniques are applied to continuous phase frequency shift keying cpfsk a new coding scheme based on convolutional codes on the ring of integers modulo p is shown to be a natural way to apply trellis coding to cpfsk previous work has decomposed cpfsk into two parts a linear encoder with memory called the continuous phase encoder cpe and a memoryless modulator mm where the cpe often has a code structure defined over the ring of integers modulo p the combination of a modulo p convolutional channel encoder ce and the cpe is a linear modulo p encoder design examples are given for rate 1 2 coded quaternary cpfsk with modulation indexes 1 2 and 1 4 and rate 2 3 coded octal cpfsk with modulation index 1 8 combinations are optimized in the normalized minimum euclidean distance sense for a given total number of states in the overall mlse receiver numerical results show that this new coding scheme consistently obtains better performance than previous schemes

coupling coefficients of sun son for on scalar microscopic theory of collective states boson realisations of all on scalar states which span a subspace of the hilbert space for the collective degrees of freedom of the nucleus and are labelled by irreducible representations of the chain of subgroups u3 contains implies u2 contains implies u1 of the complementary group sp 6 r are given two general expressions for isoscalar factors reduced wigner coefficients of sun contains implies son needed for constructing on scalar basis functions depending on the microscopic collective variables in the oscillator basis are derived these isoscalar factors couple three symmetric representations of sun un p10 p20 p30 to the representation h1h2h3 of sun subduced to the son on scalar representation and more generally to the son irreducible representation fff which appears in the case of the closed shells

design of coded cpfsk modulation systems for bandwidth and energy efficiency consideration is given to the problems related to the design of m ary continuous phase frequency shift keying cpfsk systems with modulation index h j m combined with eternal rate r binary convolution encoders the following questions are raised and answered 1 how should different encoder modulator systems be compared and how can comparable systems be recognized from the system parameters i e m h and r 2 what are the limits on the information rate per unit bandwidth versus signal to noise ratio when reliable transmission is required 3 how does one choose the system parameters m h and r when the overall system has to achieve a specified performance and 4 how does one design the external rate r binary convolutional encoder to put in front of the m ary cpfsk modulation system with h j m a simple approximation for the bandwidth of a cpfsk signal is given and shown to be sufficiently accurate for system design purposes the design of the external convolutional encoder is carried out in a novel way that leads to fewer states in the combined encoder modulator system and thus yields improved performance for a given demodulation decoding complexity compared to previous approaches for the design of coded cpfsk systems