the poisson multiple access channel the poisson multiple access channel mac models many to one optical communication through an optical fiber or in free space for this model we compute the capacity region for the two user case as a function of the allowed peak power focusing on the maximum throughput we generalize our results to the case where the users are subjected to an additional average power constraint and to the many users case we show that contrary to the gaussian mac in the poisson mac the maximum throughput is bounded in the number of users we quantify the loss that is incurred when time division multiple access tdma is employed and show that while in the two user case and in the absence of dark current the penalty is rather mild the penalty can be quite severe in the many users case in the presence of large dark current we introduce a generalized tdma technique that mitigates this loss to a large extent

rings and modules which are stable under automorphisms of their injective hulls it is proved among other results that a prime right nonsingular ring in particular a simple ring r is right self injective if rr is invariant under automorphisms of its injective hull this answers two questions raised by singh and srivastava and clark and huynh an example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption also shown is that automorphism invariant modules are precisely pseudo injective modules answering a recent question of lee and zhou furthermore rings whose cyclic modules are automorphism invariant are investigated

a rodriguez formula and integration measure for the quantum deformed legendre functions quantum deformed analogs of the legendre functions are constructed from shift operator representations of the quantum deformed orbital angular momentum operators a rodriguez formula is found as well as the measure used to orthonormalize the functions

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

solutions of the q deformed schrodinger equation for special potentials solutions of the q deformed schrodinger equation are presented for the following potentials shifted oscillator isotropic oscillator rosen morse ii eckart ii and poschl teller i and ii potentials various properties of solutions to such equations are discussed including the limit case q 1 that corresponds to the non deformed schrodinger equation

modules which are coinvariant under automorphisms of their projective covers abstract in this paper we study modules coinvariant under automorphisms of their projective covers we first provide an alternative and in fact a more succinct and conceptual proof for the result that a module m is invariant under automorphisms of its injective envelope if and only if given any submodule n of m any monomorphism f n m can be extended to an endomorphism of m and then as a dual of it we show that over a right perfect ring a module m is coinvariant under automorphisms of its projective cover if and only if for every submodule n of m any epimorphism m m n can be lifted to an endomorphism of m

second order q difference equations solvable by factorization method by solving an infinite nonlinear system of q difference equations one constructs a chain of q difference operators the eigenproblems for the chain are solved and some applications including the one related to q hahn orthogonal polynomials are discussed it is shown that in the limit q 1 the present method corresponds to the one developed by infeld and hull

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

optimum waveform signal sets with amplitude and energy constraints it is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another this problem is considered in the space of square integrable functions for signals which have finite duration and are constrained in the ranges of their values as well as in energy corresponding to each of the following cases we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set and construct sets of signals that attain both bounds simultaneously begin list item em case a energy constraint only the average energy of the waveforms in the signal set is at most sigma where 0 leq sigma item em case b energy and peak amplitude constraints the average energy of the waveforms in the signal set is leq sigma 0 leq sigma and the absolute value of each waveform is at most 1 item em case c energy and value constraints the average energy of the waveforms in the signal set is at most b 2 sigma a 2 1 sigma and each waveform takes values in the set a b where 0 leq a and 0 leq sigma leq 1 end list cases a and b are applicable to signal design for communication in channels with additive noise say gaussian and case c is applicable to signal design for optical channels where the signal represents the intensity of a photon stream the general character of the results is that the minimum distance behaves like gamma sigma in cases a and b and like gamma sigma 1 sigma in case c with gamma a suitable constant

time quantization and q deformations we extend to quantum mechanics the technique of stochastic subordination by means of which one can express any semi martingale as a time changed brownian motion as examples we considered two versions of the q deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules

directed information causal estimation and communication in continuous time a notion of directed information between two continuous time processes is proposed a key component in the definition is taking an infimum over all possible partitions of the time interval which plays a role no less significant than the supremum over space partitions inherent in the definition of mutual information properties and operational interpretations in estimation and communication are then established for the proposed notion of directed information for the continuous time additive white gaussian noise channel it is shown that duncan s classical relationship between causal estimation error and mutual information continues to hold in the presence of feedback upon replacing mutual information by directed information a parallel result is established for the poisson channel the utility of this relationship is demonstrated in computing the directed information rate between the input and output processes of a continuous time poisson channel with feedback where the channel input process is constrained to be constant between events at the channel output finally the capacity of a wide class of continuous time channels with feedback is established via directed information characterizing the fundamental limit on reliable communication

factorization method for difference equations of hypergeometric type on nonuniform lattices we study the factorization of the hypergeometric type difference equation of nikiforov and uvarov on nonuniform lattices an explicit form of the raising and lowering operators is derived and some relevant examples are given

mutual information relative entropy and estimation in the poisson channel let be a nonnegative random variable and let the conditional distribution of a random variable given be poisson for a parameter we identify a natural loss function such that 1 the derivative of the mutual information between and with respect to is equal to the minimum mean loss in estimating based on regardless of the distribution of 2 when is estimated based on by a mismatched estimator that would have minimized the expected loss had the integral over all values of of the excess mean loss is equal to the relative entropy between and for a continuous time setting where is a nonnegative stochastic process and the conditional law of given is that of a non homogeneous poisson process with intensity function under the same loss function 1 the minimum mean loss in causal filtering when is equal to the expected value of the minimum mean loss in noncausal filtering smoothing achieved with a channel whose parameter is uniformly distributed between 0 and bridging the two quantities is the mutual information between and 2 this relationship between the mean losses in causal and noncausal filtering holds also in the case where the filters employed are mismatched i e optimized assuming a law on which is not the true one bridging the two quantities in this case is the sum of the mutual information and the relative entropy between the true and the mismatched distribution of thus relative entropy quantifies the excess estimation loss due to mismatch in this setting these results are parallel to those recently found for the gaussian channel the i mmse relationship of guo the relative entropy and mismatched estimation relationship of verdu and the relationship between causal and noncasual mismatched estimation of weissman

automorphism invariant non singular rings and modules abstract a ring a is a right automorphism invariant right non singular ring if and only if a s t where s a right self injective regular ring and t is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients over a ring a each direct sum of automorphism invariant non singular right modules is an automorphism invariant module if and only if the factor ring of the ring a with respect to its right goldie radical is a semiprime right goldie ring

capacity and error exponent for the direct detection photon channel ii for pt i see ibid vol 34 no 6 p 1449 61 1980 the discussion of the capacity and error exponent of the direct detection optical channel is continued the channel input in a t second interval is a waveform satisfying certain peak and average power constraints for the optical signals the channel output is a poisson process with an intensity parameter that accounts for the dark current component an upper bound is obtained on the error exponent which coincides with the lower bound thus this channel is one for which the error exponent can be known exactly

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

zero rate reliability of the exponential server timing channel we determine the reliability function of the exponential server timing channel estc in the limit as the data rate approaches zero the limit shows that at low rates the estc is strictly more reliable than the poisson channel without dark current answering a question arikan posed in these transactions the proof employs a distance metric over inputs to timing channels that parallels euclidean and hamming distance for conventional channels a consequence of the proof is that bounded distance decoding with distance measured according to this metric is exponentially optimum for the estc in the low rate regime we also prove the straight line bound for the channel and a bound on the reliability of timing channels with general service distributions in the limit as the data rate approaches zero

an improved achievable region for the discrete memoryless two user multiple access channel with noiseless feedback an achievable region for the two user discrete memoryless multiple access channel dmmac with noiseless feedback is proposed the proposed region includes the cover leung region with the inclusion being for some channels strict this inner bound is demonstrated for the ideal two user poisson multiple access channel with noiseless feedback in which case it is shown to improve on the cover leung rate sum

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

on zero rate error exponents of finite state channels with input dependent states we derive a single letter formula for the zero rate reliability error exponent of a finite state channel whose state variable depends deterministically and recursively on past channel inputs where the code complies with a given channel input constraint special attention is then devoted to the important special case of the gaussian channel with intersymbol interference where more explicit results are obtained

the covariant linear oscillator and generalized realization of the dynamical su 1 1 symmetry algebra an exactly solvable problem for the finite difference schrodinger equation in the relativistic configurational space is considered the appropriate finite difference generalization of the factorization method is developed the theory of new special functions the relativistic hermite polynomials in which the solutions are expressed is constructed

information capacity of the poisson channel the information capacity of the poisson channel with random or time varying noise intensity is obtained for time varying peak and average constraints on the encoder intensity the channel model is specified and some definitions from information theory are given causal feedback is shown not to increase the channel capacity for the case of nonrandom noise intensity for random noise intensity use of causal feedback does increase capacity the extent of that increase is quantified jamming is considered and the optimal jamming signal is given poisson channels with thinning are introduced

feedback codes with uniformly bounded codeword lengths and zero error capacities a certain class of variable length codes with feedback whose codeword lengths are uniformly upper bounded is considered for this class of variable length codes it is shown that the zero error capacity region for the single user and multiuser channels with feedback can be extended up to the ordinary average error capacity under some conditions if variable length codes semiblock codes are used in place of fixed length codes this condition is different from that of m v burnashev 1976 for variable length codes with feedback but without any uniform bound on the codeword lengths it is also shown that the capacity region for variable length feedback codes coincides with that for fixed length feedback codes

on the reliability function of the ideal poisson channel with noiseless feedback a coding scheme for the channel under peak power and average power constraints on the input is presented and its asymptotic error exponent is shown to coincide at all rates below capacity with the sphere packing error exponent which for the case at hand is known to be unachievable without feedback for rates below the critical rate an upper bound on the error exponent achievable with feedback is also derived and shown under a capacity reducing average power constraint to coincide with the error exponent achieved by the proposed coding scheme in such a case the coding scheme is asymptotically optimal thus the ideal poisson channel limited by a capacity reducing average power constraint provides a nontrivial example of a channel for which the reliability function is known exactly both with and without feedback it is shown that a slight modification of the coding scheme to one of random transmission time can achieve zero error probability for any rate lower than the ordinary average error channel capacity

feedback does not affect the reliability function of a dmc at rates above capacity corresp asymptotically coincident upper and lower bounds on the exponent of the largest possible probability of correct decoding for block codes of any given rate above capacity have been given by dueck and korner their method is extended to show that the same bounds also hold in the presence of noiseless feedback

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

dual automorphism invariant modules a module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automor phism of m this paper introduces the notion of dual of such modules we call a module m to be a dual automorphism invariant module if whenever k1 and k2 are small submodules of m then any epimorphism m k1 m k2 with small kernel lifts to an endomorphism of m in this paper we give vari ous examples of dual automorphism invariant module and study its properties in particular we study abelian groups and prove that dual automorphism invariant abelian groups must be reduced it is shown that over a right perfect ring r a lifting right r module m is dual automorphism invariant if and only if m is quasi projective all our rings have identity element and modules are right unital a right r module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automorphism of m equivalently m is an automorphism invariant module if for any automorphismofe m m m where e m is the injective hull of m see 6 and 10

factorization of the hypergeometric type difference equation on non uniform lattices dynamical algebra we argue that one can factorize the difference equation of hypergeometric type on non uniform lattices in the general case it is shown that in the most cases of q linear spectrum of the eigenvalues this directly leads to the dynamical symmetry algebra suq 1 1 whose generators are explicitly constructed in terms of the difference operators obtained in the process of factorization thus all models with the q linear spectrum some of them but not all previously considered in a number of publications can be treated in a unified form

slq 2 realizations for kepler and oscillator potentials and q canonical transformations the realizations of the lie algebra corresponding to the dynamical symmetry group so 2 1 of the kepler and oscillator potentials are q deformed the q canonical transformation connecting two realizations is given and a general definition for the q canonical transformation is deduced a q schrodinger equation for a kepler like potential is obtained from the q oscillator schrodinger equation the energy spectrum and the ground state wavefunction are calculated

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

quantum algebra from generalized q hermite polynomials abstract in this paper we discuss new results related to the generalized discrete q hermite ii polynomials h n x q introduced by mezlini et al in 2014 our aim is to give a continuous orthogonality relation a q integral representation and an evaluation at unity of the poisson kernel for these polynomials furthermore we introduce q schrodinger operators and we give the raising and lowering operator algebra corresponding to these polynomials our results generate a new explicit realization of the quantum algebra su q 1 1 using the generators associated with a q deformed generalized para bose oscillator

generally deformed oscillator isospectral oscillator system and hermitian phase operator the generally deformed oscillator gdo and its multiphoton realization as well as the coherent and squeezed vacuum states are studied we discuss in particular the gdo depending on a complex parameter q therefore we call it q gdo together with the finite dimensional cyclic representations as a realistic physical system of a gdo the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator it is pointed out that starting from the q gdo with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically the new creation and annihilation operators of the pegg barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography