subsets of some families of real functions and their algebrability abstract we consider the following families of real valued functions darboux d quasi continuous q and wi tkowski functions s the aim of this paper is to study the level of algebrability for various intersections of families d q and s and their complements

independent bernstein sets and algebraic constructions abstract we present the method of constructing algebras and linear spaces of 2 c generators using independent bernstein sets as an application we obtain large algebras of special functions i strongly everywhere surjective functions which are not perfectly everywhere surjective ii nowhere continuous darboux functions iii nowhere continuous compact to compact functions iv functions which are continuous precisely on a fixed closed proper subset of r most conclusions obtained in this paper are improvements of some already known results

nonseparable spaceability and strong algebrability of sets of continuous singular functions abstract let cbv denote the banach algebra of all continuous real valued functions of bounded variation defined in 0 1 we show that the set of strongly singular functions in cbv is nonseparably spaceable we also prove that certain families of singular functions constitute strongly c algebrable sets the argument is based on a criterion of strong c algebrability using compositions with exponential like functions

lineability spaceability and additivity cardinals for darboux like functions we introduce the concept of maximal lineability cardinal number ml m of a subset m of a topological vector space and study its relation to the cardinal numbers known as additivity a m homogeneous lineability hl m and lineability l m of m in particular we will describe in terms of l the lineability and spaceability of the families of the following darboux like functions on r n n 1 extendable jones and almost continuous functions

stability of a functional equation deriving from cubic and quadratic functions abstract in this paper we establish the general solution of the functional equation 6 f x y 6 f x y 4 f 3 y 3 f x 2 y 3 f x 2 y 9 f 2 y and investigate the hyers ulam rassias stability of this equation

strong c algebrability of strong sierpinski zygmund smooth nowhere analytic and other sets of functions abstract we present a useful technique of proving strong c algebrability as an outcome we obtain the strong c algebrability of the following sets of functions strong sierpinski zygmund nowhere holder bruckner garg nowhere monotone differentiable a certain baire class smooth and nowhere analytic functions

characterization on practical photon counting receiver in optical scattering communication we characterize the practical photon counting receiver in optical scattering communication with a finite sampling rate and electrical noise finite small pulse width incurs dead time effect that may lead to sub poisson distribution on the recorded pulses we analyze the first order and second order moments on the number of recorded pulses with a finite sampling rate at the receiver side under two cases where the sampling period is shorter than or equal to the pulse width as well as longer than the pulse width moreover we adopt the maximum likelihood ml detection in order to simplify the analysis we adopt binomial distribution approximation on the number of recorded pulses in each slot a tractable holding time and decision threshold selection rule is provided to maximize the minimal kullback leibler kl distance between the two distributions the performances of proposed sub poisson distribution and the binomial approximation are verified by the experimental results the equivalent photon arrival rate and pulse holding time predicted by the sub poisson model and the associated proposed binomial distribution are well validated by the simulation results where the finite sampling rate and electrical noise are considered the proposed holding time and decision threshold selection rule performs close to the optimal ml receiver

on the capacity of the discrete time poisson channel the large inputs asymptotic capacity of a peak power and average power limited discrete time poisson channel is derived using a new firm nonasymptotic lower bound and an asymptotic upper bound the upper bound is based on the dual expression for channel capacity and the notion of capacity achieving input distributions that escape to infinity the lower bound is based on a lower bound on the entropy of a conditionally poisson random variable in terms of the differential entropy of its conditional mean

the bit error performance and information transfer rate of spad array optical receivers in this paper the photon counting characteristics the information rate and the bit error performance of single photon avalanche diode spad arrays are investigated it is shown that for sufficiently large arrays the photocount distribution is well approximated by a gaussian distribution with dead time dependent mean and variance because of dead time the spad array channel is subject to counting losses part of which are due to inter slot interference isi distortions consequently this channel has memory the information rate of this channel is assessed two auxiliary discrete memoryless channels dmcs are proposed which provide upper and lower bounds on the spad array information rate it is shown that in sufficiently large arrays isi is negligible and the bounds are tight under such conditions the spad array channel is precisely modelled as a memoryless channel a discrete time gaussian channel with input dependent mean and variance is adopted and the properties of the capacity achieving input distributions are studied using a numerical algorithm the information rate and the capacity achieving input distributions subject to peak and average power constraints are obtained furthermore the bit error performance of a spad based system with on off keying ook is evaluated for various array sizes dead times and background count levels

algebrability of some families of darboux like functions abstract we prove that the class of all almost continuous in the sense of stallings functions possessing the perfect road property and the class of all almost continuous functions without the perfect road property contain a 2 c generated free algebras we prove also that the family of all extendable functions contains a 2 c dimensional linear space

large free linear algebras of real and complex functions let x be a set of cardinality such that we prove that the linear algebra rx or cx contains a free linear algebra with 2 generators using this we prove several algebrability results for spaces cc and rr in particular we show that the set of all perfectly everywhere surjective functions f c c is strongly 2c algebrable we also show that the set of all functions f r r whose sets of continuity points equals some fixed g set g is strongly 2c algebrable if and only if r g is c dense in itself

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

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

non lipschitz functions with bounded gradient and related problems let e be a topological vector space and let us consider a property p we say that the subset m of e formed by the vectors in e which satisfy p is lineable for certain cardinal finite or infinite if m 0 contains an infinite dimensional linear space of dimension in this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties 1 being continuous on 0 1 a e differentiable with a e bounded derivative and not lipschitz 2 differentiable in r2 r and not enjoying the mean value theorem 3 real valued differentiable on an open connected and non convex set having bounded gradient non lipschitz and therefore not verifying the mean value theorem

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

capacity achieving distributions for the discrete time poisson channel part i general properties and numerical techniques despite being an accepted model for a wide variety of optical channels few general results on optimal signalling for discrete time poisson dtp channels are known among the most significant is that under simultaneous peak and average constraints the capacity achieving distributions are discrete with a finite number of mass points in this paper several fundamental properties of capacity achieving distributions for dtp channels are established in particular we demonstrate that all capacity achieving distributions of the dtp channel have zero as a mass point in the case of only a peak constraint it is further shown that the optimal distribution always has a mass point at the maximum amplitude finally under solely an average power constraint it is shown that a finite number of mass points are insufficient to achieve the capacity in addition to these analytical results a numerical algorithm based on deterministic annealing is presented which can efficiently compute both the channel capacity and the associated optimal input distribution under peak and average power constraints numerical lower bounds based on the envelope of information rates induced by the maxentropic distributions are also shown to be extremely close to the capacity especially in the low power regime

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

on the hyers ulam rassias stability of a quadratic functional equation abstract in this paper we investigate the hyers ulam rassias stability problems of a new quadratic functional equation f x y z f x f y f z f x y f y z f z x furthermore the stability results will be applied to the study of an interesting asymptotic property of the quadratic functions

capacity achieving distributions for the discrete time poisson channel part ii binary inputs discrete time poisson dtp channels exist in many scenarios including space laser communication systems which operate over long distances and which can be corrupted by reflected and scattered light through simulation binary input distributions have been observed to be optimal in many cases however little analytical work exists on conditions for optimality or the form of optimal signalling in this second part the general properties of part i are extended to the case of dtp channels where binary inputs are optimal necessary and sufficient conditions on the optimality of binary i e two mass point distributions are presented by leveraging the general properties of dtp capacity achieving distributions closed form expressions of the capacity achieving distributions are derived in several important special cases including zero dark current and for high dark current numerical results are presented to elucidate the developed analytical work