seismic data restoration via data driven tight frame restoration interpolation of missing traces plays a crucial role in the seismic data processing pipeline efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as fourier wavelet curvelet and shearlet transforms most existing methods are based on transforms with a fixed basis we considered an adaptive sparse transform for restoration of data with complex structures in particular we evaluated a data driven tight frame based sparse regularization method for seismic data restoration the main idea of the data driven tight frame tf is to adaptively learn a set of framelet filters from the currently interpolated data under which the data can be more sparsely represented hence the sparsity promoting l1 norm spl1 minimization methods can produce better restoration results by using the learned filters a split inexact uzawa algorithm which can be viewed as a generalization of the alternating direction of multiplier method admm was applied to solve the presented spl1 model numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid our experiments showed that our proposed method obtains the state of the art restoration results in comparison with the traditional fourier based projection onto convex sets the tight frame based method and the recent shearlet regularization admm method

experimental study of the fluctuation theorem in a nonequilibrium steady state the fluctuation theorem ft quantifies the probability of second law violations in small systems over short time scales while this theorem has been experimentally demonstrated for systems that are perturbed from an initial equilibrium state there are a number of studies suggesting that the theorem applies asymptotically in the long time limit to systems in a nonequilibrium steady state the asymptotic application of the ft to such nonequilibrium steady states has been referred to in the literature as the steady state fluctuation theorem or ssft in this paper we demonstrate experimentally the application of the ft to nonequilibrium steady states using a colloidal particle localized in a translating optical trap furthermore we show for this colloidal system that the ft holds under nonequilibrium steady states for all time and not just in the long time limit as in the ssft

common structure of techniques for choosing smoothing parameters in regression problems on tente de relier deux methodes de choix du parametre de lissage l une basee sur la mesure du risque et l autre sur le degre d ajustement de la regression aux donnees

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

on the role of exponential splines in image interpolation a sobolev reproducing kernel hilbert space approach to image interpolation is introduced the underlying kernels are exponential functions and are related to stochastic autoregressive image modeling the corresponding image interpolants can be implemented effectively using compactly supported exponential b splines a tight l2 upper bound on the interpolation error is then derived suggesting that the proposed exponential functions are optimal in this regard experimental results indicate that the proposed interpolation approach with properly tuned signal dependent weights outperforms currently available polynomial b spline models of comparable order furthermore a unified approach to image interpolation by ideal and nonideal sampling procedures is derived suggesting that the proposed exponential kernels may have a significant role in image modeling as well our conclusion is that the proposed sobolev based approach could be instrumental and a preferred alternative in many interpolation tasks

optimized wavelet denoising for self similar alpha stable processes we investigate the performance of wavelet shrinkage methods for the denoising of symmetric alpha stable text s alpha text s self similar stochastic processes corrupted by additive white gaussian noise awgn where alpha is tied to the sparsity of the process the wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean square approximation error mmse estimator we derive the corresponding formula for the expected value of the averaged estimation error we show that the predicted mmse is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process using the calculus of variations we then optimize this criterion to find the best performing wavelet within the extended family of meyer wavelets which are bandlimited these are compared with the daubechies wavelets which are compactly supported in time we find that the wavelets that are shorter in time in particular the haar basis are better suited to denoise the sparser processes say alpha while the bandlimited ones including the held and shannon wavelets offer the best performance for alpha 1 6 the limit corresponding to the gaussian case fbm with alpha 2

chaotic principle an experimental test abstract the chaotic hypothesis discussed in gallavotti and cohen 1995 is tested experimentally in a simple conduction model besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which as the forcing strength grows the attractor ceases to be an anosov system and becomes an axiom a attractor a first text of new predictions is also attempted

steady states invariant measures and response theory using the method of computer simulation we test the predictions of nonlinear response theory for classical systems subject to dissipative external fields we provide convincing numerical evidence that kawasaki methods agree with both the transient time correlation function predictions and with a direct measurement of the nonlinear response furthermore this numerical agreement is observed over a time scale which is sufficiently long for the response to have relaxed to within ensuremath sim 1 of its nonequilibrium steady state value this is in spite of the fact that in the steady state the n particle distribution function ultimately becomes fractal we discuss the normalization of the kawasaki distribution and derive a lagrangian form of the kawasaki response function and show that it is consistent with predictions that are obtained using a natural invariant measure for nonequilibrium steady states c 1995 the american physical society

thresholding estimators for linear inverse problems and deconvolutions thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded for signals in a set sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence deconvolutions with kernels having a fourier transform which vanishes at high frequencies are examples of unstable inverse problems where a thresholding in a wavelet basis is a suboptimal estimator a new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals this thresholding estimator is used to restore blurred satellite images

a classification of one dimensional cellular automata using infinite computations this paper proposes an application of the infinite unit axiom and grossone introduced by yaroslav sergeyev see sergeyev 2003 2009 2013 2008 2008 15 19 to classify one dimensional cellular automata whereby each class corresponds to a different and distinct dynamical behavior the forward dynamics of a cellular automaton map are studied via defined classes using these classes along with the infinite unit axiom and grossone the number of configurations that equal those of a given configuration in some finite central window under an automaton map can now be computed hence a classification scheme for one dimensional cellular automata is developed whereby determination in a particular class is dependent on the number of elements in their respective forward iteration classes

numerical infinitesimals in a variable metric method for convex nonsmooth optimization abstract the objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems where the objective function is assumed to be convex and not necessarily differentiable for such family of problems the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems we focus on a family of nonsmooth optimization methods based on a variable metric approach and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large as a consequence of nonsmoothness in particular we consider the case treated in the literature where the metric is defined via a diagonal matrix with positive entries we provide the computational results of our implementation on a set of benchmark test problems from scientific literature

infinitesimals and infinites in the history of mathematics a brief survey abstract we will review the main episodes of the history of the infinite in mathematics stressing the asymmetry between infinitesimals and infinites while the actual infinite after being avoided for so long when it became a mathematical object it entered mathematics to stay infinitesimals have had a troubled history of condemnations and resurrections this has probably to do with the ambiguous basic act of mathematization as revealed by the passage from the egyptians ropes to the length without largeness of euclid s definition of the line the main characters of the narration are aristotle archimedes cavalieri euler cantor in the end we will give some information on the last re emergence of infinitesimals in nonstandard analysis and on some quite recent and intriguing new ideas

coexistence of chaotic and non chaotic states in the two dimensional gauss navier stokes dynamics recently gallavotti proposed an equivalence conjecture in hydrodynamics which states that forced damped fluids can be equally well represented by means of the navier stokes equations ns and by means of time reversible modifications of ns called gauss navier stokes equations gns this equivalence conjecture received numerical support in several recent papers concerning two dimensional fluid mechanics the corresponding results rely on the fact that the ns and gns systems only have one attracting set performing similar two dimensional simulations we find that there are conditions to be met by the gns system for this to be the case in particular increasing the reynolds number while keeping fixed the number of fourier modes leads to the coexistence of different attractors this makes difficult a test of the equivalence conjecture but constitutes a spurious effect due to the insufficient spectral resolution with sufficiently fine spectral resolution the steady states are unique and the equivalence conjecture can be conveniently established

a dynamical systems interpretation of the dissipation function t mixing and their relation to thermodynamic relaxation we review the notions of the dissipation function and t mixing for non invariant measures recently introduced for nonequilibrium molecular dynamics models we provide a dynamical systems interpretation for the dissipation function and related results providing new perspectives into results such as the second law inequality we then consider the problem of relaxation within this framework the convergence of time averages along single phase space trajectories as opposed to the convergence of ensemble averages as a first step in this direction we observe that t mixing implies convergence to a unique asymptotic ensemble independent on the initial ensemble in particular the initial ensemble can be concentrated arbitrarily closely to any point in phase space

on the fluctuation relation for nose hoover boundary thermostated systems we discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures the system is a modified lorentz gas in which the fixed scatterers exchange energy with the gas of particles and the thermostats are modelled by two nose hoover thermostats applied at the boundaries of the system the transient fluctuation relation which holds only for a precise choice of the initial ensemble is verified at all times as expected times longer than the mesoscopic scale needed for local equilibrium to be settled are required if a different initial ensemble is considered this shows how the transient fluctuation relation asymptotically leads to the steady state relation when as explicitly checked in our systems the condition found in d j searles et al j stat phys 128 1337 2007 for the validity of the steady state fluctuation relation is verified for the steady state fluctuations of the phase space contraction rate and of the dissipation function a similar relaxation regime at shorter averaging times is found the quantity satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale the quantity appears to begin a monotonic convergence after such times this is consistent with the fact that and differ by a total time derivative and that the tails of the probability distribution function of are gaussian

note on two theorems in nonequilibrium statistical mechanics an attempt is made to clarify the difference between a theorem derived by evans and searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

a smoothness priors long ar model method for spectral estimation a new smoothness priors long ar model method approach is taken to the short data span spectral estimation problem an autoregressive ar model that is relatively long compared to the data length is considered the smoothness priors are in the form of the integrated squared derivatives of the ar model whitening filter a smoothness tradeoff parameter or bayesian hyperparameter balances the tradeoff between the infidelity of the ar model to the data and the infidelity of the model to the smoothness constraint the critical computation of the likelihood of the hyperparameters of the bayesian model is realized by a constrained least squares computation numerical examples are shown the results of simulation studies using entropy comparison evaluations of the bayesian and minimum aic ar methods of spectral estimation are also shown

nonparametric fixed interval smoothing with vector splines a computationally efficient algorithm for nonparametric smoothing of vector signals with general measurement covariances is presented this algorithm provides an alternative to the optimal smoothing algorithms that hinge on possibly inaccurate parametric state space models automatic procedures that use the measurements to determine how much to smooth are developed and compared this adaptation allows the data to speak for itself without imposing a gauss markov model structure a nonparametric approach to covariance estimation for the case of independently identically distributed i i d measurement errors is presented monte carlo simulations demonstrate the performance of the algorithm

wavelet frame based surface reconstruction from unorganized points applications of wavelet frames to image restoration problems e g image deblurring and inpainting have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images see e g 13 14 8 however wavelet frames have not yet been used for surface reconstruction problems recently in 6 connections between one of the wavelet frame based image restoration model 8 28 48 and variational models e g the rof model 45 were rigorously established such connections not only grant new insights to wavelet frame based image restorations it also case a geometric explanation to wavelet frame based approaches this leads us to a wavelet frame based model as well as a fast algorithm to reconstruct implicit surfaces from unorganized point sets in r 3 we will demonstrate the effectiveness of the proposed model using several commonly used examples

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

bayesian and regularization methods for hyperparameter estimation in image restoration in this paper we propose the application of the hierarchical bayesian paradigm to the image restoration problem we derive expressions for the iterative evaluation of the two hyperparameters applying the evidence and maximum a posteriori map analysis within the hierarchical bayesian paradigm we show analytically that the analysis provided by the evidence approach is more realistic and appropriate than the map approach for the image restoration problem we furthermore study the relationship between the evidence and an iterative approach resulting from the set theoretic regularization approach for estimating the two hyperparameters or their ratio defined as the regularization parameter finally the proposed algorithms are tested experimentally

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

equilibrium fluctuation relations and transport for irreversible deterministic dynamics abstract in a recent paper m colangeli et al j stat mech p04021 2011 it was argued that the fluctuation relation for the phase space contraction rate could suitably be extended to non reversible dissipative systems we strengthen here those arguments providing analytical and numerical evidence based on the properties of a simple irreversible nonequilibrium baker model we also consider the problem of response showing that the transport coefficients are not affected by the irreversibility of the microscopic dynamics in addition we prove that a form of detailed balance hence of equilibrium holds in the space of relevant variables despite the irreversibility of the phase space dynamics this corroborates the idea that the same stochastic description which arises from a projection onto a subspace of relevant coordinates is compatible with quite different underlying deterministic dynamics in other words the details of the microscopic dynamics are largely irrelevant for what concerns properties such as those concerning the fluctuation relations the equilibrium behavior and the response to perturbations

wavelet bi frames with uniform symmetry for curve multiresolution processing this paper is about the construction of univariate wavelet bi frames with each framelet being symmetric as bivariate filter banks are used for surface multiresolution processing it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices for open surfaces special multiresolution algorithms are designed to process boundary vertices when the multiresolution algorithms derived from univariate wavelet bi frames are used as the boundary algorithms it is desired that not only the scaling functions but also all framelets be symmetric in addition the algorithms for curve surface multiresolution processing should be given by templates so that they can be easily implemented in this paper first by appropriately associating the lowpass and highpass outputs to the nodes of z we show that both biorthogonal wavelet multiresolution algorithms and bi frame multiresolution algorithms can be represented by templates then using the idea of the lifting scheme we provide frame algorithms given by several iterative steps with each step represented by a symmetric template finally with the given templates of algorithms we obtain the corresponding filter banks and construct bi frames based on their smoothness and vanishing moments two types of symmetric bi frames are studied in this paper in order to provide a clearer picture on the template based procedure for bi frame construction in this paper we also consider the template based construction of biorthogonal wavelets the approach of the template based bi frame construction introduced in this paper can be extended easily to the construction of bivariate bi frames with high symmetry for surface multiresolution processing

alternating direction method for balanced image restoration this paper presents an efficient algorithm for solving a balanced regularization problem in the frame based image restoration the balanced regularization is usually formulated as a minimization problem involving an l2 data fidelity term an l1 regularizer on sparsity of frame coefficients and a penalty on distance of sparse frame coefficients to the range of the frame operator in image restoration the balanced regularization approach bridges the synthesis based and analysis based approaches and balances the fidelity sparsity and smoothness of the solution our proposed algorithm for solving the balanced optimal problem is based on a variable splitting strategy and the classical alternating direction method this paper shows that the proposed algorithm is fast and efficient in solving the standard image restoration with balanced regularization more precisely a regularized version of the hessian matrix of the l2 data fidelity term is involved and by exploiting the related fast tight parseval frame and the special structures of the observation matrices the regularized hessian matrix can perform quite efficiently for the frame based standard image restoration applications such as circular deconvolution in image deblurring and missing samples in image inpainting numerical simulations illustrate the efficiency of our proposed algorithm in the frame based image restoration with balanced regularization

simultaneous iterative image restoration and evaluation of the regularization parameter a nonlinear regularized iterative image restoration algorithm is proposed according to which only the noise variance is assumed to be known in advance the algorithm results from a set theoretic regularization approach where a bound of the stabilizing functional and therefore the regularization parameter are updated at each iteration step sufficient conditions for the convergence of the algorithm are derived and experimental results are shown

fluctuation relation beyond linear response theory the fluctuation relation fr is an asymptotic result onthe distribution of certain observables averaged over timeintervals as and it is a generalization of thefluctuation dissipation theorem to far from equilibrium systemsin a steady state which reduces to the usual green kubo gk relation in the limit of small external non conservative forces fr is a theorem for smooth uniformly hyperbolic systems and it isassumed to be true in all dissipative chaotic enough systemsin a steady state in this paper we develop a theory of finitetime corrections to fr needed to compare the asymptoticprediction of fr with numerical observations which necessarilyinvolve fluctuations of observables averaged over finite timeintervals we perform a numerical test of fr in two cases inwhich non gaussian fluctuations are observable while gk does notapply and we get a non trivial verification of fr that is independent of and different from linear response theory our results are compatible with the theory of finite timecorrections to fr while fr would be observably violated well within the precision of our experiments if such correctionswere neglected

identification and restoration of noisy blurred images using the expectation maximization algorithm a maximum likelihood approach to the blur identification problem is presented the expectation maximization algorithm is proposed to optimize the nonlinear likelihood function in an efficient way in order to improve the performance of the identification algorithm low order parametric image and blur models are incorporated into the identification method the resulting iterative technique simultaneously identifies and restores noisy blurred images

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

a unified approach to superresolution and multichannel blind deconvolution this paper presents a new approach to the blind deconvolution and superresolution problem of multiple degraded low resolution frames of the original scene we do not assume any prior information about the shape of degradation blurs the proposed approach consists of building a regularized energy function and minimizing it with respect to the original image and blurs where regularization is carried out in both the image and blur domains the image regularization based on variational principles maintains stable performance under severe noise corruption the blur regularization guarantees consistency of the solution by exploiting differences among the acquired low resolution images several experiments on synthetic and real data illustrate the robustness and utilization of the proposed technique in real applications

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

a local fluctuation theorem abstract a mechanism for the validity of a local version of the fluctuation theorem uniform in the system size is discussed for a reversible chain of weakly coupled anosov systems

lexicographic multi objective linear programming using grossone methodology theory and algorithm abstract numerous problems arising in engineering applications can have several objectives to be satisfied an important class of problems of this kind is lexicographic multi objective problems where the first objective is incomparably more important than the second one which in its turn is incomparably more important than the third one etc in this paper lexicographic multi objective linear programming lmolp problems are considered to tackle them traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single objective function the latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single objective one and the search of correct weights can be very time consuming in this work a new approach for solving lmolp problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed it is shown that a smart application of infinitesimal weights allows one to construct a single objective problem avoiding the necessity to determine finite weights the equivalence between the original multi objective problem and the new single objective one is proved a simplex based algorithm working with finite and infinitesimal numbers is proposed implemented and discussed results of some numerical experiments are provided

multiplicative noise and blur removal by framelet decomposition and l_ 1 based l curve method this paper proposes a framelet based convex optimization model for multiplicative noise and blur removal problem the main idea is to employ framelet expansion to represent the original image and use the variable decomposition to solve the problem because of the nature of multiplicative noise we decompose the observed data into the original image variable and the noise variable to obtain the resulting model the original image variable is represented by framelet and it is determined by using l 1 norm in the selection and shrinkage of framelet coefficients the noise variable is measured by using the mean and the variance of the underlying probability distribution this framelet setting can be applied to analysis synthesis and balanced approaches and the resulting optimization models are convex such that they can be solved very efficiently by the alternating direction of a multiplier method an another contribution of this paper is to propose to select the regularization parameter by using the l 1 based l curve method for these framelet based models numerical examples are presented to illustrate the effectiveness of these models and show that the performance of the proposed method is better than that by the existing methods

on typicality in nonequilibrium steady states from the statistical mechanical viewpoint relaxation of macroscopic systems and response theory rest on a notion of typicality according to which the behavior of single macroscopic objects is given by appropriate ensembles ensemble averages of observable quantities represent the measurements performed on single objects because almost all objects share the same fate in the case of non dissipative dynamics and relaxation toward equilibrium states almost all is referred to invariant probability distributions that are absolutely continuous with respect to the lebesgue measure in other words the collection of initial micro states single systems that do not follow the ensemble is supposed to constitute a set of vanishing phase space volume this approach is problematic in the case of dissipative dynamics and relaxation to nonequilibrium steady states because the relevant invariant distributions attribute probability 1 to sets of zero volume while evolution commonly begins in equilibrium states i e in sets of full phase space volume we consider the relaxation of classical thermostatted particle systems to nonequilibrium steady states we show that the dynamical condition known as omega t mixing is necessary and sufficient for relaxation of ensemble averages to steady state values moreover we find that the condition known as weak t mixing applied to smooth observables is sufficient for ensemble relaxation to be independent of the initial ensemble lastly we show that weak t mixing provides a notion of typicality for dissipative dynamics that is based on the non invariant lebesgue measure and that we call physical ergodicity

numerosities of labelled sets a new way of counting abstract the notions of labelled set and numerosity are introduced to generalize the counting process of finite sets the resulting numbers called numerosities are then used to develop nonstandard analysis the existence of a numerosity function is equivalent to the existence of a selective ultrafilter hence it is independent of the axioms of zfc

an adaptive fixed point proximity algorithm for solving total variation denoising models abstract we study an adaptive fixed point proximity algorithm to solve the total variation denoising model the objective function is a sum of two convex functions and one of them is composed by an affine transformation which is usually a regularization term by decoupling and splitting the problem is changed into two subproblems due to the nonsmooth and nondifferentiability of the subproblem we solve its proximity minimization problem instead of the original one to overcome the staircase effect during the process of denoising an adaptive criterion on proximity parameters is put forward at last we apply the improved algorithm to solve the isotropic total variation denoising model the numerical results are given to illustrate the efficiency of the algorithm

nonlinear programming and grossone quadratic programing and the role of constraint qualifications abstract a novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by sergeyev this novel numeral system is based on the use of a new infinite unit of measure the number grossone indicated by the numeral the number of elements of the set in of natural numbers based on the use of de cosmis and de leone 2012 have then proposed a new exact differentiable penalty function for constrained optimization problems in this paper these results are specialized to the important case of quadratic problems with linear constraints moreover the crucial role of constraint qualification conditions well know in constraint minimization literature is also discussed with reference to the new proposed penalty function

synchronisation of canonical measures for hyperbolic attractors under suitable conditions it is shown how to change the velocity of ac2 axioma attractor so that the sinai ruelle bowen measure coincides with the measure of maximal entropy these measures are obtained as limits of certain closed orbital measures

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

multichannel restoration of single channel images using a wavelet based subband decomposition we present a new matrix vector formulation of a wavelet based subband decomposition this formulation allows for the decomposition of both the convolution operator and the signal in the subband domain with this approach any single channel linear space invariant filtering problem can be cast into a multichannel framework we apply this decomposition to the linear space invariant image restoration problem and propose a family of multichannel linear minimum mean square error lmmse restoration filters these filters explicitly incorporate both within and between subband channel relations of the decomposed image since only within channel stationarity is assumed in the image model this approach presents a new method for modeling the nonstationarity of images experimental results are presented which test the proposed multichannel lmmse filters these experiments show that if accurate estimates of the subband statistics are available the proposed multichannel filters provide major improvements over the traditional single channel filters

extended heat fluctuation theorems for a system with deterministic and stochastic forces heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components a brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity using a langevin equation we find the exact fourier transform of the distribution of these fluctuations for all tau by a saddle point method we obtain analytical results for the inverse fourier transform which for not too small tau agree very well with numerical results from a sampling method as well as from the fast fourier transform algorithm due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid the conventional heat fluctuation theorem is for infinite and for finite tau replaced by an extended fluctuation theorem that differs noticeably and measurably from it in particular for large fluctuations the ratio of the probability for absorption of heat by the particle from the fluid to the probability to supply heat by the particle to the fluid is much larger here than in the conventional fluctuation theorem

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

a general framework for frequency domain multi channel signal processing the authors provide a general framework for performing processing of stationary multichannel mc signals that is linear shift invariant within channel and shift varying across channels emphasis is given to the restoration of degraded signals it is shown that by utilizing the special structure of semiblock circulant and block diagonal matrices mc signal processing can be easily carried out in the frequency domain the generalization of many frequency domain single channel sc signal processing techniques to the mc case is presented it is shown that in mc signal processing each frequency component of a signal and system is presented respectively by a small vector and a matrix of size equal to the number of channels while in sc signal processing each frequency component in both cases is a scalar

the steady state fluctuation relation for the dissipation function we give a proof of transient fluctuation relations for the entropy production dissipation function in nonequilibrium systems which is valid for most time reversible dynamics we then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax producing a unique nonequilibrium steady state although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown if such a steady state exists the generation of the steady state fluctuation relation from the transient relation is shown to be very general it is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation which is needed also for e g the existence of transport coefficients because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities the steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor whose convergence is problematic for systems close to equilibrium we examine some model dynamics that have been considered previously and show how they are described in the context of this work

posterior mean super resolution with a causal gaussian markov random field prior we propose a bayesian image super resolution sr method with a causal gaussian markov random field mrf prior sr is a technique to estimate a spatially high resolution image from given multiple low resolution images an mrf model with the line process supplies a preferable prior for natural images with edges we improve the existing image transformation model the compound mrf model and its hyperparameter prior model we also derive the optimal estimator not the joint maximum a posteriori map or the marginalized maximum likelihood ml but the posterior mean pm from the objective function of the l2 norm based mean square error peak signal to noise ratio point estimates such as map and ml are generally not stable in ill posed high dimensional problems because of overfitting whereas pm is a stable estimator because all the parameters in the model are evaluated as distributions the estimator is numerically determined by using the variational bayesian method the variational bayesian method is a widely used method that approximately determines a complicated posterior distribution but it is generally hard to use because it needs the conjugate prior we solve this problem with simple taylor approximations experimental results have shown that the proposed method is more accurate or comparable to existing methods

fluctuation theorem for stochastic systems the fluctuation theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics it has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems in the present paper we show that the fluctuation theorem is also valid for a class of stochastic nonequilibrium systems the theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics numerical tests verify the theoretical result

wavelet deblurring algorithms for spatially varying blur from high resolution image reconstruction abstract high resolution image reconstruction refers to reconstructing a higher resolution image from multiple low resolution samples of a true image in chan et al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 we considered the case where there are no displacement errors in the low resolution samples i e the samples are aligned properly and hence the blurring operator is spatially invariant in this paper we consider the case where there are displacement errors in the low resolution samples the resulting blurring operator is spatially varying and is formed by sampling and summing different spatially invariant blurring operators we represent each of these spatially invariant blurring operators by a tensor product of a lowpass filter which associates the corresponding blurring operator with a multiresolution analysis of l 2 r 2 using these filters and their duals we derive an iterative algorithm to solve the problem based on the algorithmic framework of chanet al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 our algorithm requires a nontrivial modification to the algorithms in chan et al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 which apply only to spatially invariant blurring operators our numerical examples show that our algorithm gives higher peak signal to noise ratios and lower relative errors than those from the tikhonov least squares approach

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

fluctuation theorems for entropy production in open systems we derive a fluctuation theorem to describe entropy fluctuations in steady states of systems with density gradients due to open boundaries the fluctuations are related to the growth rate of the phase space density instead of the phase space contraction rate explicit derivations are presented for a multibaker map but the arguments are rather general and should hold for a much wider class of dynamical systems a comparison with recent results for stochastic systems is also given

application of the gallavotti cohen fluctuation relation to thermostated steady states near equilibrium the fluctuation relation of the gallavotti cohen fluctuation theorem gcft concerns fluctuations in the phase space compression rate of dissipative reversible dynamical systems it has been proven for anosov systems but it is expected to apply more generally this raises the question of which non anosov systems satisfy the fluctuation relation we analyze time dependent fluctuations in the phase space compression rate of a class of n particle systems that are at equilibrium or in near equilibrium steady states this class does not include anosov systems or isoenergetic systems however it includes most steady state systems considered in molecular dynamics simulations of realistic systems we argue that the fluctuations of the phase space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the gcft although the discrepancies become somewhat smaller as the systems move further from equilibrium in contrast similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

metamathematical investigations on the theory of grossone we propose an axiomatization of sergeyev s theory of grossone trying to comply with his methodological principles we find that a simplified form of his divisibility axiom is sufficient we use for easier readability a second order language and a predicative second order logic our theory is not finitely axiomatizable and is a conservative extension of peano s arithmetic

lyapunov spectra and nonequilibrium ensembles equivalence in 2d fluid mechanics we perform numerical experiments to study the lyapunov spectra of dynamical systems associated with the navier stokes ns equation in two spatial dimensions truncated over the fourier basis recently new equations called gns equations have been introduced and conjectured to be equivalent to the ns equations at large reynolds numbers the lyapunov spectra of the ns and of the corresponding gns systems overlap adding evidence in favor of the conjectured equivalence already studied and partially extended in previous papers we make use of the lyapunov spectra to study a fluctuation relation which had been proposed to extend the fluctuation theorem to strongly dissipative systems preliminary results towards the formulation of a local version of the fluctuation formula are also presented

chaotic hypothesis fluctuation theorem and singularities the chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them however its application to physics problems requires attention and can lead to apparent inconsistencies in particular there are several cases that have been considered in the literature in which singularities are built in the models for instance when among the forces there are lennard jones potentials which are infinite in the origin and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded the situation is well understood in certain special cases in which the system is subject to gaussian noise here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived the main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider ie deterministic systems with thermostat forces acting according to gauss principle for the constraint of constant total kinetic energy isokinetic gaussian thermostats close and far from equilibrium near equilibrium it even predicts a fluctuation relation which in deterministic cases with more general thermostat forces ie not necessarily of gaussian isokinetic nature extends recent relations obtained in situations in which the thermostatting forces satisfy gauss principle this relation agrees where expected with the fluctuation theorem for perfectly chaotic systems the results are compared with some recent works in the literature

l minimization for wavelet frame based image restoration the theory of tight wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems the success of wavelet frame based models including balanced approach 18 7 and analysis based approach 11 31 50 is due to their capability of sparsely approximating piecewise smooth functions like images motivated by the balanced approach and analysis based approach we shall propose a wavelet frame based l0 minimization model where the l0 norm of the frame coefficients is penalized we adapt the penalty decomposition pd method of 40 to solve the proposed optimization problem some convergence analysis of the adapted pd method will also be provided numerical results showed that the proposed model solved by the pd method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images

parameter estimation in bayesian high resolution image reconstruction with multisensors we consider the estimation of the unknown parameters for the problem of reconstructing a high resolution image from multiple undersampled shifted degraded frames with subpixel displacement errors we derive mathematical expressions for the iterative calculation of the maximum likelihood estimate of the unknown parameters given the low resolution observed images these iterative procedures require the manipulation of block semi circulant bsc matrices that is block matrices with circulant blocks we show how these bsc matrices can be easily manipulated in order to calculate the unknown parameters finally the proposed method is tested on real and synthetic images

identification of image and blur parameters for the restoration of noncausal blurs an optimal statistical parameter estimation technique is presented for the identification of unknown image and blur model parameters the development leads to an autoregressive moving average arma model identification problem where the image model coefficients define the ar part and the blur parameters define the ma part conditional maximum likelihood estimates of the unknown parameters are derived both in the absence and in the presence of observation noise the proposed algorithms constitute a generalization of previous work on blur identification in that they are able to locate the zero loci of the blurred image spectrum on the entire z 1 z 2 plane simulation results as well as photographically blurred images processed with the proposed algorithms are shown as examples

reversibility coarse graining and the chaoticity principle we describe a way of interpreting the chaotic principle of gc1 more extensively than it was meant in the original works mathematically the analysis is based on the dynamical notions of axiom a and axiom b and on the notion of axiom c that we introduce arguing that it is suggested by the results of an experiment bgg on chaotic motions physically we interpret a breakdown of the anosov property of a time reversible attractor replaced as a control parameter changes by an axiom a property as a spontaneous breakdown of the time reversal symmetry the relation between time reversal and the symmetry that remains after the breakdown is analogous to the breakdown of t invariance while tcp still holds

regularization preconditioners for frame based image deblurring with reduced boundary artifacts thresholding iterative methods were recently successfully applied to image deblurring problems in this paper we investigate the modified linearized bregman algorithm mlba used in image deblurring problems with a proper treatment of the boundary artifacts we consider two standard approaches the imposition of boundary conditions and the use of the rectangular blurring matrix the fast convergence of the mlba depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block bccb matrix diagonalized by the discrete fourier transform we show that the standard approach based on the bccb preconditioner may provide low quality restored images and we propose different preconditioning strategies that improve the quality of the restoration and save some computational cost at the same time motivated by a recent nonstationary preconditioned iteration we propose a new algorithm that combines such a method with the mlba we p

optimal estimation of direction in regression models with large number of parameters we consider the problem of estimating the optimal direction in regression by maximizing the probability that the scalar product between the vector of unknown parameters and the chosen direction is positive the estimator maximizing this probability is simple in form and is especially useful for situations where the number of parameters is much larger than the number of observations we provide examples which show that this estimator is superior to state of the art methods such as the lasso for estimating the optimal direction

dynamical ensembles equivalence in fluid mechanics abstract an attempt to put together various theoretical mathematical or experimental results recently developed in apparently unrelated subjects namely ruelle s 1980 approach to turbulence the body of nose hoover type of molecular dynamics experiments evans and morriss 1990 evans et al 1993 bonetto et al 1996 mathematical results on lyapunov exponents the pairing rule dressler 1988 and dettman and morriss 1995 and experimental results on them sarman et al 1992 bohr et al 1996 theoretical as well as mathematical results on fluctuations frisch and parisi 1984 multifractality and gallavoti and cohen 1995 chaotic hypothesis the key idea that we try to clarify is that of dynamical ensembles as a generalization of the classical equilibrium ensembles arguing that they should be identified with the srb distributions and that they share several properties with the classical ensembles most of the results invoked here did not deal directly with the navier stokes equations and yet they seem to have a lot to do with them as we shall argue here the discussion will focus on the navier stokes and dissipative euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis the ideas developed to a great extent from the efforts put in interpreting the experimental results in bonetto et al 1996

the use of grossone in mathematical programming and operations research the concepts of infinity and infinitesimal in mathematics date back to anciens greek and have always attracted great attention very recently a new methodology has been proposed by sergeyev 10 for performing calculations with infinite and infinitesimal quantities by introducing an infinite unit of measure expressed by the numeral grossone an important characteristic of this novel approach is its attention to numerical aspects in this paper we will present some possible applications and use of in operations research and mathematical programming in particular we will show how the use of can be beneficial in anti cycling procedure for the well known simplex method for solving linear programming problems and in defining exact differentiable penalty functions in nonlinear programming