on a problem of k mahler diophantine approximation and cantor sets let k denote the middle third cantor set and mathcal a 3 n n 0 1 2 ldots given a real positive function let w mathcal a psi denote the set of real numbers x in the unit interval for which there exist infinitely many p q in mathbb z times mathcal a such that x p q q the analogue of the hausdorff measure version of the duffin schaeffer conjecture is established for w mathcal a psi cap k one of the consequences of this is that there exist very well approximable numbers other than liouville numbers in k an assertion attributed to k mahler explicit examples of irrational numbers satisfying mahler s assertion are also given

decision based order statistic filters the authors propose two adaptive order statistic filters which generate outputs based on the results of hypothesis tests to reduce edge shifting of the median filter a modified median filter mmf is proposed first a hypothesis test is applied to detect the edge location then whenever an edge is detected the mmf generates the output by modifying the median output based on a comparison of median value to the current input value the mmf can retain more fine details than the median filter with the same window size as well as reduce the output mean square error which is caused mainly by edge shifting to suppress nonimpulsive noise as well as impulsive noise an l type structure is combined with the mmf the proposed filter generates the output by using asymmetric alpha trimming on the mmf output in addition to noise suppression it can enhance the gradient of blurred edges provided that the size of the window is greater than the transition width of the blurred edges experimental results for one and two dimensional signals are presented

a general weighted median filter structure admitting negative weights weighted median smoothers which were introduced by edgemore in the context of least absolute regression over 100 years ago have received considerable attention in signal processing during the past two decades although weighted median smoothers offer advantages over traditional linear finite impulse response fir filters it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems in fact weighted median smoothers are analogous to normalized fir linear filters constrained to have only positive weights it is also shown that much like the mean is generalized to the rich class of linear fir filters the median can be generalized to a richer class of filters admitting positive and negative weights the generalization follows naturally and is surprisingly simple in order to analyze and design this class of filters a new threshold decomposition theory admitting real valued input signals is developed the new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real valued filter coefficients the new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures

the duffin schaeffer conjecture with extra divergence given a nonnegative function psi mathbb n to mathbb r let w denote the set of real numbers x such that nx a 0 a consequence of our main result is that w is of full lebesgue measure if there exists an epsilon 0 such that textstyle sum n in mathbb n left frac psi n n right 1 epsilon varphi n infty the duffin schaeffer conjecture is the corresponding statement with epsilon 0 and represents a fundamental unsolved problem in metric number theory another consequence is that w is of full hausdorff dimension if the above sum with epsilon 0 diverges i e the dimension analogue of the duffin schaeffer conjecture is true

affine order statistic filters medianization of linear fir filters this paper introduces a novel data adaptive filtering framework affine order statistic filters affine order statistic filters operate effectively on a wide range of signal statistics are sensitive to the dispersion of the observed data and are therefore particularly useful in the processing of nonstationary signals these properties result from the introduction of a tunable affinity function that measures the affinity or closeness of observation samples in their natural order to their corresponding order statistics the obtained affinity measures are utilized to control the influence of individual samples in the filtering process depending on the spread of the affinity function which is controlled by a single parameter spl gamma affine order statistic filters operate effectively in various environments ranging from gaussian to impulsive the class of affine order statistic filters subsumes the family of weighted order statistic wos affine filters and the class of fir affine filters we focus on a representative of the wos affine filter class the median affine filter whose behavior can be tuned from that of a linear fir filter to that of a robust median filter by narrowing the affinity function to a process referred to as medianization the superior performance of affine order statistic filters is demonstrated in two applications

diophantine analysis in beta dynamical systems and hausdorff dimensions abstract let x n n 1 0 1 be a sequence of real numbers and let n 0 1 be a positive function using the mass transference principle established by beresnevich and velani 1 we prove that for any x 0 1 the hausdorff dimension of the set 1 t n x x n n for infinitely many n n satisfies a so called 0 1 law according to lim sup n log n n or not where t is the transformation

weighted median filters with sigma delta modulation encoding digital decimation filters play a fundamental role in oversampled sigma delta a d decoders in this paper we first show that weighted median wm filtering of a demodulated sequence at the nyquist rate can be implemented concurrently in the a d decoder through a simple modification of the binary time series outputted by the a d modulator the sequence obtained after the sigma delta modulation sdm decoder is shown to be equivalent to wm filtering the multilevel sequence at the nyquist rate second we show that wm filters can be used for sdm decimation filters and that these filters are readily implemented in the sdm binary domain a very promising characteristic of sdm converters equipped with wm decimating filters is that sharp discontinuities edges can be preserved and acquired thus the bandlimited constraint imposed on the input signals can be relaxed making sdm more attractive to a d conversion of signals containing sharp transitions the proposed signal processing algorithms in essence combine a d sigma delta converters and wm filters into a single programmable system

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

the duffin schaeffer conjecture with extra divergence abstract the duffin schaeffer conjecture is a fundamental unsolved problem in metric number theory it asserts that for every non negative function n r for almost all reals x there are infinitely many coprime solutions a n to the inequality n x a n provided that the series n 1 n n n is divergent in the present paper we prove that the conjecture is true under the extra divergence assumption that divergence of the series still holds when n is replaced by n log n e for some e 0 this improves a result of beresnevich harman haynes and velani and solves a problem posed by haynes pollington and velani

kurzweil type metrical diophantine properties in the field of formal laurent series abstract in this article we consider diophantine properties of the orbit of irrational rotations over the field of formal laurent series f q x 1 more precisely for a given f f q x 1 and a sequence r n we investigate the size of the set g f q x 1 min deg q n q f g r n for infinitely many n n in the sense of the haar measure and the hausdorff dimension

weighted iterative truncated mean filter the iterative truncated arithmetic mean itm filter was proposed recently it offers a way to estimate the sample median by simple arithmetic computing instead of the time consuming data sorting in this paper a rich class of filters named weighted itm witm filters are proposed by iteratively truncating the extreme samples the output of the witm filter converges to the weighted median proper stopping criterion makes the witm filters own merits of both the weighted mean and median filters and hence outperforms the both in some applications three structures are designed to enable the witm filters being low band and high pass filters properties of these filters are presented and analyzed experimental evaluations are carried out on both synthesis and real data to verify some properties of the witm filters

fir median hybrid filters with predictive fir substructures a class of finite impulse response fir median hybrid fmh filters that contain linear fir substructures to estimate the current signal value using forward and backward prediction is introduced the output of the overall filter is the median of the predicted values and the actual signal value in the middle of the filter window predictors maximizing the signal to noise ratio on signal sections described by an lth order polynominal are derived the ramp enhancement filters are shown to attenuate the noise on a ramp signal better than the standard median sm filters the new predictive fmh filters are shown to have root signals which do not exist for the sm filters e g triangular waves by combining the level and the ramp enhancement fmh filters a filter is obtained which attenuates noise on constant and ramp signals the noise attenuation on ramp signals is better than with the sm filter and the predictive fmh filter has novel and meaningful root structures the number of arithmetic operations needed to implement the predictive fmh filter grows linearly with the length of the filter

a localized jarnik besicovitch theorem abstract fundamental questions in diophantine approximation are related to the hausdorff dimension of sets of the form x r x where 1 and x is the diophantine approximation exponent of an irrational number x we go beyond the classical results by computing the hausdorff dimension of the sets x r x f x where f is a continuous function our theorem applies to the study of the approximation exponents by various approximation families it also applies to functions f which are continuous outside a set of prescribed hausdorff dimension

a dynamical dimension transference principle for dynamical diophantine approximation diophantine approximation in dynamical systems concerns the diophantine properties of the orbits in classic diophantine approximation the powerful mass transference principle established by beresnevich and velani provides a general principle to the dimension for a limsup set in this paper we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system more precisely let x t be a topological dynamical system where x is a compact metric space and t x rightarrow x is an expanding continuous transformation given y o in x we consider the following limsup set mathcal w t f driven by the dynamical system x t begin aligned big x in x x in b z e s n f log t z text for some z in t n y o text with infinitely many n in mathbb n big end aligned where log t is a function reflecting the local conformality of the transformation t f is a non negative continuous function over x and s n f log t z denotes the ergodic sum f log t z cdots f log t t n 1 z by proposing a dynamical ubiquity property assumed on the system x t we obtain that the dimensions of x and mathcal w t f are both related to the bowen manning mccluskey formulae namely the solution to the pressure functions begin aligned texttt p t log t 0 text and texttt p t log t f 0 text respectively end aligned we call this phenomenon a dynamical dimension transference principle because of its partial analogy with the mass transference principle this general principle unifies and extends some known results which were considered only separatedly before these include the b adic expansions expanding rational maps over julia sets inhomogeneous diophantine approximation on the triadic cantor set and finite conformal iterated function systems

definable and contractible contracts this paper analyzes bayesian normal form games in which players write contracts that condition their actions on the contracts of other players these contracts are required to be representable in a formal language this is accomplished by constructing contracts which are definable functions of the godel code of every other player s contract we provide a complete characterization of the set of allocations supportable as pure strategy bayesian equilibria of this contracting game when information is complete this characterization provides a folk theorem in general the set of supportable allocations is smaller than the set supportable by a centralized mechanism designer

when the sieve works we are interested in classifying those sets of primes p such that when we sieve out the integers up to x by the primes in pc we are left with roughly the expected number of unsieved integers in particular we obtain the first general results for sieving an interval of length x with primes including some in x x using methods motivated by additive combinatorics

mechanism games with multiple principals and three or more agents we consider a class of mechanism games in which there are multiple principals and three or more agents for a mechanism game in this class a sort of folk theorem holds there is a threshold value for each of the principals such that an allocation is achieved at a pure strategy sequential equilibrium of the game if and only if i it is incentive compatible and ii it attains an expected utility for each principal that is greater than or equal to the threshold value for the principal

bohr sets and multiplicative diophantine approximation in two dimensions gallagher s theorem is a strengthening of the littlewood conjecture that holds for almost all pairs of real numbers we prove an inhomogeneous fiber version of gallagher s theorem sharpening and making unconditional a result recently obtained conditionally by beresnevich haynes and velani the idea is to find large generalized arithmetic progressions within inhomogeneous bohr sets extending a construction given by tao this precise structure enables us to verify the hypotheses of the duffin schaeffer theorem for the problem at hand via the geometry of numbers

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

fir median hybrid filters this paper introduces a new generalized class of median type filters which we call fir median hybrid fmh filters the input signal x n is filtered with m linear phase fir filters and the output of the fmh filter is the median of the outputs of the fir filters the statistical properties of the fmh filters are analyzed for input signals with gaussian double exponential and uniform density functions it is shown that fmh filters have essentially the same statistical properties and same type of root signals as the standard median sm filters fmh filters preserve corner points and ramps and attenuate impulsive type noise components effectively an interesting subclass of the fmh filters requiring only one scaling multiplier two additions and three compare swap operations irrespective of the filter length is introduced in the examples speed improvement by a factor of 450 is obtained over an otherwise approximately equivalent sm filter

a multifractal mass transference principle for gibbs measures with applications to dynamical diophantine approximation let mu be a gibbs measure of the doubling map t of the circle for a mu generic point x and a given sequence r n subset of r consider the intervals t n x r n mod 1 t n x r n mod 1 in analogy to the classical dvoretzky covering of the circle we study the covering properties of this sequence of intervals this study is closely related to the local entropy function of the gibbs measure and to hitting times for moving targets a mass transference principle is obtained for gibbs measures that are multifractal such a principle was proved by beresnevich and velani ann math 164 2006 971 992 for monofractal measures in the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence in particular we can describe the occurrence of atypical relatively long words our results have a direct and deep number theoretical interpretation via inhomogeneous dyadic diophantine approximation by numbers belonging to a given dyadic diophantine class less

a mass transference principle and the duffin schaeffer conjecture for hausdorff measures a hausdorff measure version of the duffin schaeffer conjecture in metric number theory is introduced and discussed the general conjecture is established modulo the original conjecture the key result is a mass transference principle which allows us to transfer lebesgue measure theoretic statements for lim sup subsets of r k to hausdorff measure theoretic statements in view of this the lebesgue theory of lim sup sets is shown to underpin the general hausdorff theory this is rather surprising since the latter theory is viewed to be a subtle refinement of the former

analysis of the properties of median and weighted median filters using threshold logic and stack filter representation the deterministic properties of weighted median wm filters are analyzed threshold decomposition and the stacking property together establish a unique relationship between integer and binary domain filtering the authors present a method to find the weighted median filter which is equivalent to a stack filter defined by a positive boolean function because the cascade of wm filters can always be expressed as a single stack filter this allows expression of the cascade of wm filters as a single wm filter a direct application is the computation of the output distribution of a cascade of wm filters the same method is used to find a nonrecursive expansion of a recursive wm filter as applications of theoretical results several interesting deterministic and statistical properties of wm filters are derived

edge preserving filtering by combining nonlinear mean and median filters a very general class of nonlinear filters called mapping order statistics filters mosfs is introduced a subclass of the mosfs called l sub p mean median filters l sub p mmfs is discussed the l sub p mmf shows better properties in preserving edges and attenuating positive and negative impulse noise simultaneously than do the nonlinear mean filters in the analysis of the deterministic properties of the l sub p mmf the concept of a monotonic trend region is introduced a necessary condition for a signal to be a root signal is given this necessary condition is also valid for the known finite impulse response median hybrid filter

on root structures of median and median type filters in this correspondence we analyze the root structures of the standard median sm filters the recursive median rm filters and the fir median hybrid fmh filters it is shown that these filters have many oscillatory infinitely long root signals when a section of an oscillatory root is present in a signal the filter s noise attenuation of the filter is not as good as predicted by statistical measures we also suggest methods to prevent the possible harmful effect caused by the existence of these roots

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

explicit salem sets in r2 abstract we construct explicit i e non random examples of salem sets in r 2 of dimension s for every 0 s 2 in particular we give the first explicit examples of salem sets in r 2 of dimension 0 s 1 this extends a theorem of kaufman

the duffin schaeffer conjecture with extra divergence ii in 2012 the authors set out a programme to prove the duffin schaeffer conjecture for measures arbitrarily close to lebesgue measure in this paper we take a new step in this direction given a non negative function psi mathbb n rightarrow mathbb r let w psi denote the set of real numbers x such that nx a 0 our main result is that w psi is of full lebesgue measure if there exists a c 0 such that begin aligned sum n ge 16 frac varphi n psi n n exp c log log n log log log n infty end aligned