resolution of singularities of threefolds in positive characteristic ii abstract together with vincent cossart olivier piltant resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings j algebra 320 3 2008 1051 1082 this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields

reduced storage vq via secondary quantization this paper introduces methods for reducing the table storage required for encoding and decoding with unstructured vector quantization uvq or tree structured vector quantization tsvq specifically a low storage secondary quantizer is used to compress the code vectors and test vectors of the primary quantizer the relative advantages of uniform and nonuniform secondary quantization are investigated a linde buzo gray lbg like algorithm that optimizes the primary uvq codebook for a given secondary codebook and another that jointly optimizes both primary and secondary codebooks are presented in comparison to conventional methods it is found that significant storage reduction is possible typically a factor of two to three with little loss of signal to noise ratio snr moreover when reducing dimension is considered as another method of reducing storage it is found that the best strategy is a combination of both the method of secondary quantization is also applied to tsvq to reduce the table storage required for both encoding and decoding it is shown that by exploiting the correlation among the test vectors in the tree both encoder and decoder storage can be significantly reduced with little loss of snr by a factor of about four or two relative to the conventional method of storing test vectors or test hyperplanes

variable length constrained storage tree structured vector quantization constrained storage vector quantization csvq introduced by chan and gersho 1990 1991 allows for the stagewise design of balanced tree structured residual vector quantization codebooks with low encoding and storage complexities on the other hand it has been established by makhoul et al 1985 riskin et al 1991 and by mahesh et al see ieee trans inform theory vol 41 p 917 30 1995 that variable length tree structured vector quantizer vltsvq yields better coding performance than a balanced tree structured vector quantizer and may even outperform a full search vector quantizer due to the nonuniform distribution of rate among the subsets of its input space the variable length constrained storage tree structured vector quantization vlcs tsvq algorithm presented in this paper utilizes the codebook sharing by multiple vector sources concept as in csvq to greedily grow an unbalanced tree structured residual vector quantizer with constrained storage it is demonstrated by simulations on test sets from various synthetic one dimensional 1 d sources and real world images that the performance of vlcs tsvq whose codebook storage complexity varies linearly with rate can come very close to the performance of greedy growth vltsvq of riskin et al and mahesh et al the dramatically reduced size of the overall codebook allows the transmission of the code vector probabilities as side information for source adaptive entropy coding

alphabet constrained vector quantization alphabet constrained rate distortion theory is extended to coding of sources with memory two different cases are considered when only the size of the codebook is constrained and when the codevector values are also held fixed for both cases nth order constrained alphabet rate distortion functions are defined and a convergent algorithm for their evaluation is presented specific simulations using ar 1 sources show that performance near the rate distortion bound is possible using a reproduction alphabet consisting of a small number of codevectors it is also shown that the additional constraint of holding the codevector values fixed does not degrade performance of the coder in relation to the size only constrained case this observation motivates the development of a fixed codebook vector quantizer called the alphabet and entropy constrained vector quantizer the performance of which is comparable to the entropy constrained vector quantizer a number of examples using an ar 1 and a speech source are presented to corroborate the theory

computer simulation of geometric stable distributions we present a new method for computer simulation of strictly geometric stable random variables the method is based on their representation as a product of two independent random variables with explicit distribution functions coupled with the inversion method we also extend the method to the multivariate case by deriving new representation and simulation algorithm for multivariate linnik distribution

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

analytic and asymptotic properties of linnik s probability densities ii abstract in 1953 linnik introduced the probability density p x defined in terms of its characteristic function t 1 1 t 0 recently this density has received several applications in this paper the expansions of p x into convergent series in terms involving log x x k x k k 0 1 2 are obtained and the asymptotic behaviour of p x at 0 and is investigated with respect to these expansions and to the asymptotic behaviour at 0 the cases i 1 is an integer ii 1 is a non integer rational number and iii is an irrational number are quite distinct the first part of the paper dealt with preliminaries asymptotic behavior at and case i the second part deals with cases ii and iii

analytic and asymptotic properties of generalized linnik probability densities is a characteristic function of a symmetric probability density p since a 4 then the family of symmetric linnik densities p a g 0 2 had several a w x probabilistic applications see e g 2 7 analytic and asymptotic properw x ties of the densities p were studied in 8 a w x in 1984 klebanov maniya and melamed 9 introduced the concept of geometric strict stability and proved that the family of geometrically strictly stable densities coincides with the family of densities with characteristic functions

places of algebraic function fields in arbitrary characteristic we consider the zariski space of all places of an algebraic function field f k of arbitrary characteristic and investigate its structure by means of its patch topology we show that certain sets of places with nice properties e g prime divisors places of maximal rank zero dimensional discrete places lie dense in this topology further we give several equivalent characterizations of fields that are large in the sense of f pop s annals paper embedding problems over large fields we also study the question whether a field k is existentially closed in an extension field l if l admits a k rational place in the appendix we prove the fact that the zariski space with the zariski topology is quasi compact and that it is a spectral space

on the rate of convergence to rosenblatt type distribution the main result of the article is the rate of convergence to the rosenblatt type distributions in non central limit theorems specifications of the main theorem are discussed for several scenarios in particular special attention is paid to the cauchy generalized linnik s and local global distinguisher random processes and fields direct analytical methods are used to investigate the rate of convergence in the uniform metric

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

universal source coding with codebook transmission a universal source coding system with vector quantizer codebook transmissions is studied using high resolution quantization theory conditions are derived for the optimal tradeoff between quantizer resolution and the information rate used to transmit codebooks a formula that tightly bounds the mean squared error of the universal coding system as a function of the time between codebook transmissions is experimentally verified and found to be tight and a new and simpler derivation is given other research in the literature has proposed vector quantizing the transmitted codebooks one conclusion we prove here is that under some reasonable conditions uniform scalar quantization of the transmitted codebooks performs as well as vector quantizing them experimental results are given that support the analytic derivations

source coding bounds using quantizer reproduction levels corresp constraining the reproduction alphabet to be of small size in encoding continuous amplitude memoryless sources has been shown to give very small degradation from the ideal performance of the rate distortion bound the optimum fixed size reproduction alphabet and its individual letter probabilities are required in order to encode the source with performance approaching that of theory these can be found through a somewhat lengthy but convergent algorithm given reasonably chosen fixed sets of reproduction letters and or their probabilities we define new rate distortion functions which are coding bounds under these alphabet constraints we calculate these functions for the gaussian and laplacian sources and the squared error distortion measure and find that performance near the rate distortion bound is achievable using a reproduction alphabet consisting of a small number of optimum quantizer levels

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

anomalous subdiffusion with multispecies linear reaction dynamics we have introduced a set of coupled fractional reaction diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics the model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics the effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms one consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species the general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail we have carried out monte carlo simulations corresponding to these special cases and we find excellent agreement with theory

the tree of quadratic transforms of a regular local ring of dimension two abstract let d be a 2 dimensional regular local ring and let q d denote the quadratic tree of 2 dimensional regular local overrings of d we explore the topology of the tree q d and the family r d of rings obtained as intersections of rings in q d if a is a finite intersection of rings in q d then a is noetherian and the structure of a is well understood however other rings in r d need not be noetherian the two main goals of this paper are to examine topological properties of the quadratic tree q d and to examine the structure of rings in the set r d

the algebra and model theory of tame valued fields a henselian valued field k is called a tame field if its algebraic closure k is a tame extension that is the ramification field of the normal extension k k is algebraically closed every algebraically maximal kaplansky field is a tame field but not conversely we develop the algebraic theory of tame fields and then prove ax kochen ershov principles for tame fields this leads to model completeness and completeness results relative to value group and residue field as the maximal immediate extensions of tame fields will in general not be unique the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy ax kochen ershov principles the results of this paper have been applied to gain insight in the zariski space of places of an algebraic function field and in the model theory of large fields

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

the constructible topology on spaces of valuation domains we consider properties and applications of a compact hausdorff topology called the ultrafilter topology defined on an arbitrary spectral space and we observe that this topology coincides with the constructible topology if k is a field and a a subring of k we show that the space zar k a of all valuation domains having k as quotient field and containing a endowed with the zariski topology is a spectral space by giving in this general setting the explicit construction of a ring whose zariski spectrum is homeomorphic to zar k a we extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings as a very particular case we prove that two collections of valuation domains of k with the same ultrafilter closure represent as an intersection the same integrally closed domain

constrained storage quantization of multiple vector sources by codebook sharing a codebook sharing technique called constrained storage vector quantization csvq is introduced this technique offers a convenient and optimal way of trading off performance against storage the technique can be used in conjunction with tree structured vector quantization vq and other structured vq techniques that alleviate the search complexity obstacle the effectiveness of csvq is illustrated for coding transform coefficients of audio signals with multistage vq

fine coarse vector quantization a fast method for searching an unstructured vector quantization vq codebook is introduced and analyzed the method fine coarse vector quantization fcvq operates in two stages a fine structured vq followed by a table lookup coarse unstructured vq its rate distortion arithmetic complexity and storage are investigated using analytical and experimental means optimality condition and an optimizing algorithm are presented the results of experiments with both uniform scalar quantization and tree structured vq tsvq as the first stage are reported comparisons are made with other fast approaches to vector quantization especially tsvq it is found that when rate distortion arithmetic complexity and storage are all taken into account fcvq outperforms tsvq in a number of cases in comparison to full search quantization fcvq has much lower arithmetic complexity at the expense of a slight increase in distortion and a substantial increase in storage the increase in mean squared error over full search decays as a negative power of the available storage

on the cole cole relaxation function and related mittag leffler distribution abstract in the framework of the one dimensional fractal time random walk ftrw relaxation model we rigorously show that the frequency domain response takes in both nonbiased and biased walks the only possible cole cole form the underlying reason for this is the specific form of the relaxation function the survival probability of a relaxing system determined in this model by the mittag leffler distribution we provide also analytical formulas for the propagators of the nonbiased and biased ftrws

multiscale conservation laws driven by levy stable and linnik diffusions asymptotics shock creation preservation and dissolution recent work has shown that the solutions of the fractal conservation laws driven by levy alpha stable diffusions exhibit shocks for bounded odd and convex on the positive half line initial data when the parameter alpha 1 we study the analogous situation for the levy alpha linnik diffusions in which case the local behavior is strikingly different although we are able to establish analytically that the large time behavior of the two types of conservation laws are similar but the main new insights obtained via large scale numerical experiments is that for any 0 alpha le 2 the conservation laws driven by alpha linnik diffusions display shocks that do not dissipate over time while those for alpha stable diffusion 0 alpha le 1 do we formulate rigorous conjectures based on these numerical experiments

local time of levy random walks a path integral approach the local time of a stochastic process quantifies the amount of time that sample trajectories x spend in the vicinity of an arbitrary point x for a generic hamiltonian we employ the phase space path integral representation of random walk transition probabilities in order to quantify the properties of the local time for time independent systems the resolvent of the hamiltonian operator proves to be a central tool for this purpose in particular we focus on the local times of levy random walks levy flights which correspond to fractional diffusion equations

value groups residue fields and bad places of rational function fields we classify all possible extensions of a valuation from a ground field k to a rational function field in one or several variables over k we determine which value groups and residue fields can appear and we show how to construct extensions having these value groups and residue fields in particular we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated in the case of a rational function field k x in one variable we consider the relative algebraic closure of k in the henselization of k x with respect to the given extension and we show that this can be any countably generated separable algebraic extension of k in the tame case we show how to determine this relative algebraic closure finally we apply our methods to power series fields and the p adics

every place admits local uniformization in a finite extension of the function field we prove that every place p of an algebraic function field f k of arbitrary characteristic admits local uniformization in a finite extension f of f we show that f f can be chosen to be galois after a finite purely inseparable extension of the ground field k instead of being galois the extension can also be chosen such that the induced extension fp fp of the residue fields is purely inseparable and the value group of f only gets divided by the residue characteristic if f lies in the completion of an abhyankar place then no extension of f is needed our proofs are based solely on valuation theoretical theorems which are of particular importance in positive characteristic they are also applicable when working over a subring r k and yield similar results if r is regular and of dimension smaller than 3

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

sliding block and random source coding with constrained size reproduction alphabets recent research has shown that nearly optimum ratedistortion function performance can be obtained in encoding memoryless continuous amplitude sources using a constrained size reproduction alphabet in this paper we use a four letter reproduction alphabet in encoding the gaussian and laplacian sources with the squared error distortion measure we introduce a sliding block code generator decoder and perform extensive coding simulations at a rate of one bit source symbol the average distortions in those simulations are much below the optimum quantization distortion and with much smaller search intensity are about the same or lower than distortions reported previously in comparable research the slidingblock code results are also compared to those of parallel random coding simulations with the same reproduction alphabet and rate

vector quantizers and predictive quantizers for gauss markov sources low rate vector quantizers are designed and simulated for highly correlated gauss markov sources and the resulting performance is compared with arnstein s optimized predictive quantizer and with huang and schultheiss optimized transform coder two implementations of vector quantizers are considered full search vector quantizers which are optimal but require large codebook searches and tree searched vector quantizers which are suboptimal but require far less searching the various systems are compared on the basis of performance complexity and generality of design techniques