lagrangian optimization of two description scalar quantizers in this paper we study the problem of optimal design of balanced two description fixed rate scalar quantizer 2dsq under the constraint of convex codecells using a graph based approach to model the problem we show that the minimum expected distortion of the 2dsq is a convex function of the number of codecells in the side quantizers this property allows the problem to be solved by lagrangian minimization for which the optimal lagrangian multiplier exists given a trial multiplier we exploit a monotonicity of the objective function and develop a simple and fast dynamic programming technique to solve the parameterized problem to further improve the algorithm efficiency we propose an rd guided search strategy to find the optimal lagrangian multiplier in our experiments on distributions of interest for signal compression applications the proposed algorithm improves the speed of the fastest algorithm so far by a factor of o k logk where k is the number of codecells in each side quantizer we also assess the impact on the optimality of the convex codecell constraint using a published performance analysis of 2dsq at high rates we show that asymptotically this constraint does not preclude optimality for l 2 distortion measure when channels have a higher than 0 12 loss rate

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

on finite size scaling in the presence of dangerous irrelevant variables a scaling hypothesis on finite size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long range interaction and general geometryld d d a characteristic length which obeys a universal finite size scaling relation is defined the general conjectures are based on exact results for the mean spherical model with inverse power law interaction

first order transitions in spherical models finite size scaling finite size behavior near the first order phase boundary of ferromagnetic spherical models is investigated for block and cylinder shaped systems ind dimensions the bulk thermodynamic singularities are rounded and asymptotically for large size obey appropriate scaling laws both short range interactions and long range couplings decaying like 1 rd with 0 are analyzed the short range results agree precisely with a recently developed scaling theory foro n symmetric systems in the limitn more generally the scaling functions are universal depending only on explicit aspects of the shape and interactions enter only in the spin wave or goldstone mode contributions which appear technically as corrections to scaling an appendix analyzes the truncation error in the approximation by many fold sums of multivariate integrals with integrands diverging like jaj j2 as 0

finite size scaling investigations in the quantum varphi4 model with long range interaction in this paper we study in detail the critical behaviour of the n quantum 4 model with long range interaction decaying with distances r by a power law as r d in the large n limit the zero temperature critical behaviour is discussed its alteration by finite temperature and or finite sizes in the space is studied the scaling behaviours are studied in different regimes depending upon whether the finite temperature or the finite sizes of the system lead a number of results for the correlation length critical amplitudes and the finite size shift for different dimensionalities between the lower d 3 2 critical dimensions are calculated

on anomalous diffusion and the fractional generalized langevin equation for a harmonic oscillator the fractional generalized langevin equation fgle is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two parameter mittag leffler noise the solution of this fgle is discussed by means of the laplace transform methodology and the kernels are presented in terms of the three parameter mittag leffler functions recent results associated with a generalized langevin equation are recovered

uniqueness of locally optimal quantizer for log concave density and convex error weighting function it is desired to encode a random variable x using an n level quantizer q to minimize the expected distortion e rho x q x i where the error weighting function rho is convex strictly increasing and continuously differentiable it is shown that if x has a log concave density then there exists a unique locally optimal quantizer q ast and lloyd s method i may be used to find q ast trushkin had earlier shown this result for the error weighting functions rho t equiv t and rho t euiv t 2

finite size effects in a quantum exactly soluble model for structural phase transition abstract some consequences of finite size scaling are examined within an exactly soluble model of structural phase transition explicit expressions for the susceptibility for classical and quantum limits are derived it is shown that both the cases have the same scaling behaviour

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

successive refinement of information characterization of the achievable rates let r spl middot be the rate distortion function assume that we want to describe a source with distortion no larger than spl delta sub 1 from the rate distortion theory we know that we need to do so at a rate r sub 1 no smaller than r spl delta sub 1 spl lsqb bits symbol spl rsqb if it turns out that a more accurate description at distortion spl delta sub 2 spl delta sub 2

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on the structure of optimal entropy constrained scalar quantizers the nearest neighbor condition implies that when searching for a mean square optimal fixed rate quantizer it is enough to consider the class of regular quantizers i e quantizers having convex cells and codepoints which lie inside the associated cells in contrast quantizer regularity can preclude optimality in entropy constrained quantization this can be seen by exhibiting a simple discrete scalar source for which the mean square optimal entropy constrained scalar quantizer ecsq has disconnected and hence nonconvex cells at certain rates in this work new results concerning the structure and existence of optimal ecsqs are presented one main result shows that for continuous sources and distortion measures of the form d x y spl rho x y where spl rho is a nondecreasing convex function any finite level ecsq can be regularized so that the resulting regular quantizer has the same entropy and equal or less distortion regarding the existence of optimal ecsqs we prove that under rather general conditions there exists an almost regular optimal ecsq for any entropy constraint for the squared error distortion measure and sources with piecewise monotone and continuous densities the existence of a regular optimal ecsq is shown

sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure e f x eta the expected value of an error weighting function f x eta where x is a random variable to be quantized where the probability density function p x describing x is continuous and positive on some finite or infinite interval and zero outside it and where eta is the quantization error the function f x eta is assumed convex and symmetric in eta and zero only for eta 0 it is shown that in the cases of f x eta eta 2 and f x eta eta the simple condition of concavity of in p x is sufficient for uniqueness of a locally optimal quantizer

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

mathematical analysis of carbon nanotube model in this paper we present a recently developed mathematical model for a short double wall carbon nanotube the model is governed by a system of two coupled hyperbolic equations and is reduced to an evolution equation this equation defines a dissipative semi group we show that the semi group generator is an unbounded nonselfadjoint operator with compact resolvent moreover this operator is a relatively compact perturbation of a certain selfadjoint operator

t 0 finite size scaling for a quantum system with long range interaction abstract the general idea of finite size scaling are tested by the example of the pure quantum version of an exactly solvable model of structural phase transition with long range interaction decaying at large distances r as r d where d is the space dimensionality and 0 the finite size shift of the quantum parameter playing the same role as that of the temperature for classical systems is calculated for the general geometry ld d d with 0 0 2

network vector quantization we present an algorithm for designing locally optimal vector quantizers for general networks we discuss the algorithm s implementation and compare the performance of the resulting network vector quantizers to traditional vector quantizers vqs and to rate distortion r d bounds where available while some special cases of network codes e g multiresolution mr and multiple description md codes have been studied in the literature we here present a unifying approach that both includes these existing solutions as special cases and provides solutions to previously unsolved examples

four branch vibrational spectrum of double walled carbon nanotube model asymptotic distribution of the discrete spectrum eigenvalues for a recently developed model of a double walled carbon nanotube is presented in this paper the corresponding initial boundary value problem is reduced to an evolution equation whose dynamics generator is a non self adjoint matrix differential operator with a purely discrete spectrum it is shown in the paper that the entire spectrum asymptotically splits into four spectral branches asymptotic representation is derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity to prove the results a two step procedure involving the construction of the left and right reflection matrices has been used

generalized mittag leffler functions in the theory of finite size scaling for systems with strong anisotropy and or long range interaction the difficulties arising in the investigation of finite size scaling in d dimensional o n systems with strong anisotropy and or long range interaction decaying with the interparticle distance r as r d 0 2 are discussed some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries the analysis is simplified further by expressing this general form in terms of a generalization of the mittag leffler special functions this turned out to be very useful for the extraction of asymptotic finite size behaviours of the thermodynamic functions

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

finite size scaling for the mean spherical model with inverse power law interaction a new analytical technique based on integral transformations with mittag leffler type kernels is used to derive the finite size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential the asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied

test of renormalization predictions for universal finite size scaling functions we calculate universal finite size scaling functions for systems with an n component order parameter and algebraically decaying interactions just as previously found for short range interactions this leads to a singular ensuremath varepsilon expansion where ensuremath varepsilon is the distance to the upper critical dimension subsequently we check the results by numerical simulations of spin models in the same universality class our systems offer the essential advantage that ensuremath varepsilon can be varied continuously allowing an accurate examination of the region where ensuremath varepsilon is small the numerical calculations turn out to be in striking disagreement with the predicted singularity

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

on properties of locally optimal multiple description scalar quantizers with convex cells it is known that the generalized lloyd method is applicable to locally optimal multiple description scalar quantizer mdsq design however it remains unsettled when the resulting mdsq is also globally optimal we partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed rate mdsq of convex cells under trushkin s sufficient conditions for the uniqueness of locally optimal fixed rate single description scalar quantizer this result holds for fixed rate multiresolution scalar quantizer mrsq of convex cells as well thus the well known log concave probability density function pdf condition can be extended to the multiple description and multiresolution cases

quantization as histogram segmentation optimal scalar quantizer design in network systems an algorithm for scalar quantizer design on discrete alphabet sources is proposed the proposed algorithm can be used to design fixed rate and entropy constrained conventional scalar quantizers multiresolution scalar quantizers multiple description scalar quantizers and wyner ziv scalar quantizers the algorithm guarantees globally optimal solutions for conventional fixed rate scalar quantizers and entropy constrained scalar quantizers for the other coding scenarios the algorithm yields the best code among all codes that meet a given convexity constraint in all cases the algorithm run time is polynomial in the size of the source alphabet the algorithm derivation arises from a demonstration of the connection between scalar quantization histogram segmentation and the shortest path problem in a certain directed acyclic graph

on the symmetry of universal finite size scaling functions in anisotropic systems we postulate a symmetry of universal finite size scaling functions under a certain anisotropic scale transformation which connects the properties of a finite two dimensional 2d system at criticality with generalized aspect ratio 1 to a system with 1 the symmetry is formulated within a finite size scaling theory and expressions for several universal amplitude ratios are derived the predictions are confirmed within the exactly solvable weakly anisotropic 2d ising model and are checked within the strongly anisotropic 2d dipolar in plane ising model using monte carlo simulations

stability of a flexible structure with destabilizing boundary conditions the euler bernoulli beam model with non dissipative boundary conditions of feedback control type is investigated components of the two dimensional input vector are shear and moment at the right end and components of the observation vector are time derivatives of displacement and slope at the right end the codiagonal matrix depending on two control parameters relates input and observation the paper contains five results first asymptotic approximation for eigenmodes is derived second the main identity is established it provides a relation between mode shapes of two systems one with non zero control parameters and the other one with zero control parameters third when one control parameter is positive and the other one is zero the main identity yields stability of all eigenmodes though the system is non dissipative fourth the stability of eigenmodes is extended to the case when one control parameter is positive and the other one is sufficiently small finally existence and properties of deadbeat modes are investigated

finite size effects in the approximating hamiltonian method abstract the husimi temperley mean spherical model in which each two particles interact with equal strength is considered this model is shown to be equivalent to a d dimensional model with periodic boundary conditions and interaction potential j r where j r r d as r 0 being a parameter in the limit 0 it is found that the approximating hamiltonian method yields singular finite size scaling functions both in the neighbourhood of the critical point and near a first order phase transition a modification of this method is suggested which allows for all the essential configurations and reproduces the exact finite size scaling near a first order phase transition

sequential scalar quantization of vectors an analysis proposes an efficient vector quantization vq technique called sequential scalar quantization ssq the scalar components of the vector are individually quantized in a sequence with the quantization of each component utilizing conditional information from the quantization of previous components unlike conventional independent scalar quantization isq ssq has the ability to exploit intercomponent correlation at the same time since quantization is performed on scalar rather than vector variables ssq offers a significant computational advantage over conventional vq techniques and is easily amenable to a hardware implementation in order to analyze the performance of ssq the authors appeal to asymptotic quantization theory where the codebook size is assumed to be large closed form expressions are derived for the quantizer mean squared error mse these expressions are used to compare the asymptotic performance of ssq with other vq techniques the authors also demonstrate the use of asymptotic theory in designing ssq for a practical application color image quantization where the codebook size is typically small theoretical and experimental results show that ssq far outperforms isq with respect to mse while offering a considerable reduction in computation over conventional vq at the expense of a moderate increase in mse

universal ratio of magnetization moments in two dimensional ising models the authors calculate the universal critical point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic ising models on the square and on the triangular lattices periodic boundary conditions are used in accordance with the four fold and six fold rotational symmetries of the respective lattices these results which are obtained by means of an analysis of finite size data computed with a transfer matrix technique have an accuracy of the order of one millionth this analysis is also applied to rectangular systems with arbitrary aspect ratios

on the design of optimal noisy channel scalar quantizer with random index assignment the general approach in a noisy channel scalar quantizer design is an iterative descent algorithm which guarantees only a locally optimal solution while sufficient conditions under which the local optimum becomes a global optimum are known in the noiseless channel case such sufficient conditions were not derived for the noisy counterpart moreover efficient globally optimal design techniques for general discrete distributions in the noiseless case exist however they seem not to extend to the noisy scenario when a fixed index assignment is assumed recently the design of noisy channel scalar quantizer with random index assignment ria was proposed using a locally optimal iterative algorithm in this paper we derive sufficient conditions for the uniqueness of a local optimum which thus guarantee the global optimality of the solution these sufficient conditions are satisfied for a log concave probability density function which is additionally symmetric around its mean furthermore we show that assuming an ria the globally optimal design for general discrete sources can also be carried out efficiently

finite size scaling for systems with long range interactions abstract the present review is devoted to the fundamental problems of finite size scaling due to the presence of long range interactions the attention is focused on the precise formulation of critical finite size scaling in the case that the bulk correlation length is identically infinite the hypotheses based on the notion of a finite size scaling length are formulated in a way which treats the cases of short and long range interactions on equal grounds as well as the standard and modified above the upper critical dimensionality finite size scaling the general conjectures are tested on the exactly solvable mean spherical model with power law interaction special attention is paid to the adequate mathematical techniques the formulation of finite size scaling at first order phase transitions is also discussed since the finite size scaling length enters into the definition of the relevant scaled external field variable provided the system has d 0 infinite dimensions the review closes with a summary and short discussion of some open problems

quantizer monotonicities and globally optimal scalar quantizer design new monotonicity properties of optimal scalar quantizers are discussed these monotonicities reveal a globally optimal scalar quantizer structure depending on the probability mass functions and on the number of quantizer levels by incorporating the monotone quantizer structure into a dynamic programming process the time complexities of previous algorithms for designing globally optimal scalar quantizers can be significantly reduced for very general classes of distortion measures

modified finite size scaling for anharmonic crystals with quantum fluctuations abstract the modified finite size scaling for dimensions d d where d is the upper critical dimensionality is verified in the closed vicinity of both the classical and the quantum multicritical points by the example of a quantum model for an anharmonic cyrstal confined to a fully finite block geometry under periodic boundary conditions unified scaling equations corresponding to the two kinds of correlation lengths related with the fixed dimensionless temperature t when the quantum parameter is varied and with fixed when t is varied are obtained

finite size scaling for the correlation function of the spherical model with long range interactions finite size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry ld d d periodic boundary conditions and long range interactions decaying like r d at large distances r with arbitrary real parameters 0 2 d 2 d 0 the analytical technique used is based on integral transformations with kernels of mittag leffler type it makes it possible to easily generalize a number of results available only for short range interactions or in special cases of the ld d d geometry the effective correlation length is identified at arbitrary temperatures from the finite size large distance asymptotic behavior of the pair correlation function it is explicitly shown that the finite size scaling functions are not singular in the e expansion when e 0 neither at d e nor at d 2 e moreover the applicability of the e expansion is tested in the specific finite size case of d e the definition of the scaled fiel

on the finite size scaling in quantum critical phenomena abstract an exactly solvable d dimentional model of a structural phase transition confined to a geometry ld d d are subjected to periodic boundary conditions is considered in the low temperature region it is shown that in this region quantum effects are essential and that borderline dimensionalities between which finite size scaling holds do not coincide with the lower and upper critical dimensionalities for the bulk system a generalized version of the dimensional cross over rule is established i e the finite size scaling behaviour of a d dimensional system finite in d d d directions in the quantum limit is equivalent to the finite size scaling behaviour of the classical d 1 dimensional system also finite in d directions

on two stage sequential coding of correlated sources we study the problem of two stage sequential coding tssc which is an extension of sequential coding of correlated sources let x and y be dependent random variables the network contains two encoders and two decoders 1 a y encoder with input y 2 an x encoder with inputs x and y 3 a y decoder that reconstructs y and 4 an x decoder that reconstructs x the first stage is traditional sequential coding where the y encoder describes y to both the x decoder and y decoder and the x encoder describes x and y to the x decoder at the second stage the y encoder refines the description of y and the x encoder refines the description of x the tssc model is a theoretical abstraction of scalable video coding here y and x represent successive frames of a video sequence and the two stages together give an embedded description that allows the video to be decoded at two distinct rates we give an inner bound on the rate distortion region for this tssc model the tight bound on the rate distortion region is derived when y must be reconstructed losslessly in the usual shannon sense in the second stage we also study the minimum total rate of the tssc model and show that the minimum total rate of one stage sequential coding cannot be achieved at both stages for jointly gaussian sources this theoretical result can shed light on the rate distortion performance behavior of scalable video coding widely noted by practitioners

universal amplitudes in finite size scaling for an anharmonic crystal abstract we consider an exactly solvable d dimensional lattice model of an anharmonic crystal confined to a geometry l d d d and subject to periodic boundary conditions the general idea of finite size scaling on the whole phase diagram is tested and the universal amplitudes of the correlation length near the upper and lower critical dimensions in both the classifical and the quantum multicritical points are computed a detailed analysis is given in terms of the dimensions d and d of the lattice and parameter of the harmonic force decreasing at long distances as 1 r d 0 2

an investigation of finite size scaling for systems with long range interaction the spherical model a method is suggested for the derivation of finite size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr d whered is the space dimensionality and 0 it allows for a unified treatment of short range 2 and long range 2 interaction the asymptotic analysis is illustrated by the mean spherical model of general geometryld d d subject to periodic boundary conditions the fisher privman equation of state is generalized to arbitrary real values ofd 0 d it is shown that thee expansion may be used to study the breakdown of standard finite size scaling at the borderline dimensionalities

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established

design of optimal scalar quantizer for sequential coding of correlated sources this paper addresses the design of a sequential scalar quantizer ssq for finite alphabet correlated sources in the fixed rate fr and entropy constrained ec cases the optimization problem is formulated as the minimization of a weighted sum of distortions and rates the proposed solution is globally optimal for the class of ssqs with convex cells and is based on solving the minimum weight path mwp problem in the ec case respectively a length constrained mwp problem in the fr case in a series of weighted directed acyclic graphs the asymptotic time complexity is o k 1 2 k 2 2 where k 1 and k 2 are the respective sizes of the alphabets of the two sources additionally it is proved that by applying the proposed algorithms to discretizations of correlated sources with continuous joint probability density function the performance approaches that of the optimal ec ssq respectively fr ssq with convex cells for the original sources as the accuracy of the discretization increases extensive experiments performed with correlated gaussian sources validate the effectiveness in practice of the proposed approach in approximating the optimal ssq for the case of continuous alphabet sources