markov chains of nonlinear markov processes and an application to a winner takes all model for social conformity we discuss nonlinear markov processes defined on discrete time points and discrete state spaces using markov chains in this context special attention is paid to the distinction between linear and nonlinear markov processes we illustrate that the chapman kolmogorov equation holds for nonlinear markov processes by a winner takes all model for social conformity

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

entropy production and kullback leibler divergence between stationary trajectories of discrete systems the irreversibility of a stationary time series can be quantified using the kullback leibler divergence kld between the probability of observing the series and the probability of observing the time reversed series moreover this kld is a tool to estimate entropy production from stationary trajectories since it gives a lower bound to the entropy production of the physical process generating the series in this paper we introduce analytical and numerical techniques to estimate the kld between time series generated by several stochastic dynamics with a finite number of states we examine the accuracy of our estimators for a specific example a discrete flashing ratchet and investigate how close the kld is to the entropy production depending on the number of degrees of freedom of the system that are sampled in the trajectories

sharp bounds for the cubic parsell vinogradov system in two dimensions abstract we prove a sharp decoupling for a certain two dimensional surface in r 9 as an application we obtain the full range of expected estimates for the cubic parsell vinogradov system in two dimensions

exponential wealth distribution in a random market a rigorous explanation abstract in simulations of some economic gas like models the asymptotic regime shows an exponential wealth distribution independently of the initial wealth distribution given to the system the appearance of this statistical equilibrium for this type of gas like models is explained in a rigorous analytical way

illustrative example of the relationship between dissipation and relative entropy kawai parrondo and van den broeck phys rev lett 98 080602 2007 have recently established a quantitative relationship between dissipated work and a microscopic information theoretic measure of irreversibility we illustrate this result using the exactly solvable system of a brownian particle in a dragged harmonic trap

the cubic case of the main conjecture in vinogradov s mean value theorem abstract we apply a variant of the multigrade efficient congruencing method to estimate vinogradov s integral of degree 3 for moments of order 2 s establishing strongly diagonal behaviour for 1 s 6 consequently the main conjecture is now known to hold for the first time in a case of degree exceeding 2

a new approach to constrained total least squares image restoration abstract recently there has been a growing interest and progress in using total least squares tls methods for solving blind deconvolution problems arising in image restoration here the true image is to be estimated using only partial information about the blurring operator or point spread function psf which is subject to error and noise in this paper we present a new iterative regularized and constrained tls image restoration algorithm neumann boundary conditions are used to reduce the boundary artifacts that normally occur in restoration processes preliminary numerical tests are reported on some simulated optical imaging problems in order to illustrate the effectiveness of the approach as well as the fast convergence of our iterative scheme

decouplings for surfaces in r4 abstract we prove a sharp decoupling for nondegenerate surfaces in r 4 this puts the progress in 1 on the lindelof hypothesis into a more general perspective

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

generalization of thermodynamic potentials including information in this work we analyze the connection between information theory and thermodynamics three different potentials are presented all related to the kullback leibler divergence these potentials represent three different conditions of energy and information exchange between a reservoir and a subsystem the first potential describes a situation where the two systems are exchanging energy without any external control the second potential considers the situation where the reservoir can get information through the state of another secondary system about the internal state of the subsystem this information can then be used by a maxwell demon to increase the free energy of the subsystem in this second scenario it is assumed that the energy cost of the information probe and the maxwell demon are at the expense of the reservoir finally a third case is considered where the information obtained through the secondary system is used not only to manipulate an internal maxwell demon but to perform directly work over the subsystem this last case involves a direct modification of the hamiltonian of the subsystem

classical chaos and fluctuation dissipation relations for nonlinear response the classical nonlinear optical response is expressed in a form that closely resembles the fluctuation dissipation theorem the nth order response is shown to depend on interferences among n closely lying trajectories the relevant dynamical information on the vicinity of a given trajectory can be recast using the stability matrix related to the lyapunov exponents no such interference exists in the linear response and the nonlinear response is consequently a much more sensitive probe for classical chaos sequences of multiple femtosecond pulses can be designed to directly probe the stability matrix textcopyright 1996 the american physical society

lower bounds on dissipation upon coarse graining by different coarse graining procedures we derive lower bounds on the total mean work dissipated in brownian systems driven out of equilibrium with several analytically solvable examples we illustrate how when and where the information on the dissipation is captured

multidimensional measures of impulsively driven stochastic systems based on the kullback leibler distance by subjecting a dynamical system to a series of short pulses and varying several time delays we can obtain multidimensional characteristic measures of the system multidimensional kullback leibler response function klrf which are based on the kullback leibler distance between the initial and final states are defined we compare the klrf which are nonlinear in the probability density with ordinary response functions obtained from the expectation value of a dynamical variable which are linear we show that the klrf encode different level of information regarding the system s dynamics for overdamped stochastic dynamics two dimensional klrf shows a qualitatively different variation with the time delays between pulses depending on whether the system is initially in a steady state or in thermal equilibrium

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

stabilized hyperbolic householder transformations a modification of the hyperbolic householder scheme is introduced which is demonstrably stable theoretically according to an established stability criterion and which exhibits superior numerical behavior in simulations the modified transform scheme effects downdating by applying conventional orthonormal rather than hyperbolic householder transformations to the data the latter have preferable numerical properties however the construction of these orthonormal operators itself requires hyperbolic computations thus the proposed method is in some sense half hyperbolic and half orthonormal there is no computational penalty incurred with these stabilized hyperbolic householder transforms they enjoy an operation count identical to their conventional counterparts

vinogradov s mean value theorem via efficient congruencing ii we apply the efficient congruencing method to estimate vino gradov s integral for moments of order 2s with 1 6 s 6 k 2 1 thereby we show that quasi diagonal behaviour holds when s o k 2 we obtain near optimal estimates for 1 6 s 6 1 k 2 k and optimal estimates for s k 2 1 in this way we come half way to proving the main conjecture in two different directions there are consequences for estimates of weyl type and in several allied applications thus for example the anticipated asymptotic formula in waring s problem is established for sums of s kth powers of natural numbers whenever s 2k 2 2k 8 k 6 estimates stemming from vinogradov s mean value theorem deliver bounds for exponential sums of large degree both in mean and pointwise beyond the competence of alternate approaches the ubiquity of such exponential sums in analytic number theory in the analysis for example of the riemann zeta function in waring s problem and beyond accounts for the high pro file of vinogradov s methods in the associated literature in recent work we established a version of vinogradov s mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible see 19 for systems of degree k previous estimates missed such a bound by a factor of order logk our earlier approach provides no upper bounds when the number of variables is smaller precluding the possibility of applications involving the finer features of these mean values our goal in this paper is to remedy this deficiency at the same time strengthening our previous conclusions it transpires that we are able to come within a hair s breadth of proving the main conjecture concerning vino gradov s mean value theorem in half of the basic interval of relevant moments such developments illustrate the flexibility of the new efficient congruencing method introduced in 19 we now introduce some notation when k n and r k define

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

implementation of the regularized structured total least squares algorithms for blind image deblurring abstract the structured total least squares stls problem has been introduced to handle problems involving structured matrices corrupted by noise often the problem is ill posed recently regularization has been proposed in the stls framework to solve ill posed blind deconvolution problems encountered in image deblurring when both the image and the blurring function have uncertainty the kernel of the regularized stls rstls problem is a least squares problem involving block toeplitz toeplitz block matrices in this paper an algorithm is described to solve this problem based on a particular implementation of the generalized schur algorithm gsa it is shown that this new implementation improves the computational efficiency of the straightforward implementation of gsa from o n 2 5 to o n 2 where n is the number of pixels in the image

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

physics of the shannon limits we provide a simple physical interpretation in the context of the second law of thermodynamics to the information inequality a k a the gibbs inequality which is also equivalent to the log sum inequality asserting that the relative entropy between two probability distributions cannot be negative since this inequality stands at the basis of the data processing theorem dpt and the dpt in turn is at the heart of most if not all proofs of converse theorems in shannon theory it is observed that conceptually the roots of fundamental limits of information theory can actually be attributed to the laws of physics in particular the second law of thermodynamics and indirectly also the law of energy conservation by the same token in the other direction one can view the second law as stemming from information theoretic principles

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

equilibrium distributions and relaxation times in gaslike economic models an analytical derivation a step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions pdfs of well known kinetic wealth exchange economic models is shown this technique gives a dynamical insight into the evolution of the pdf for example allowing the calculation of its relaxation times their equilibrium pdfs can also be calculated by finding its stationary solutions this gives as a result an integro differential equation which can be solved analytically in some cases and numerically in others this should provide some guidance into the type of pdfs that can be derived from particular economic agent exchange rules or for that matter any other kinetic model of gases with particular collision physics

a generalization of vinogradov s mean value theorem we obtain new upper bounds for the number of integral solutions of a complete system of symmetric equations which may be viewed as a multi dimensional version of the system considered in vinogradov s mean value theorem we then use these bounds to obtain weyl type estimates for an associated exponential sum in several variables finally we apply the hardy littlewood method to obtain asymptotic formulas for the number of solutions of the vinogradov type system and also of a related system connected to the problem of finding linear spaces on hypersurfaces

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

a comment on the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank abstract the purpose of this comment is to correct mistaken assumptions and claims made in the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank t d frank stochastic feedback non linear families of markov processes and nonlinear fokker planck equations physica a 331 2004 391 our comment centers on the claims of a non linear markov process and a non linear fokker planck equation first memory in transition densities is misidentified as a markov process second the paper assumes that one can derive a fokker planck equation from a chapman kolmogorov equation but no proof was offered that a chapman kolmogorov equation exists for the memory dependent processes considered a non linear markov process is claimed on the basis of a non linear diffusion pde for a 1 point probability density we show that regardless of which initial value problem one may solve for the 1 point density the resulting stochastic process defined necessarily by the conditional probabilities the transition probabilities is either an ordinary linearly generated markovian one or else is a linearly generated non markovian process with memory we provide explicit examples of diffusion coefficients that reflect both the markovian and the memory dependent cases so there is neither a non linear markov process nor a non linear fokker planck equation for a conditional probability density the confusion rampant in the literature arises in part from labeling a non linear diffusion equation for a 1 point probability density as non linear fokker planck whereas neither a 1 point density nor an equation of motion for a 1 point density can define a stochastic process in a closely related context we point out that borland misidentified a translation invariant 1 point probability density derived from a non linear diffusion equation as a conditional probability density finally in the appendix a we present the theory of fokker planck pdes and chapman kolmogorov equations for stochastic processes with finite memory

entropy production and fluctuation theorems for langevin processes under continuous non markovian feedback control continuous feedback control of langevin processes may be non markovian due to a time lag between the measurement and the control action we show that this requires one to modify the basic relation between dissipation and time reversal and to include a contribution arising from the noncausal character of the reverse process we then propose a new definition of the quantity measuring the irreversibility of a path in a nonequilibrium stationary state which can also be regarded as the trajectory dependent total entropy production this leads to an extension of the second law which takes a simple form in the long time limit as an illustration we apply the general approach to linear systems that are both analytically tractable and experimentally relevant

testing ground for fluctuation theorems the one dimensional ising model in this paper we determine the nonequilibrium magnetic work performed on a ising model and relate it to the fluctuation theorem derived some years ago by jarzynski the basic idea behind this theorem is the relationship connecting the free energy difference between two thermodynamic states of a system and the average work performed by an external agent in a finite time through nonequilibrium paths between the same thermodynamic states we test the validity of this theorem by considering the one dimensional ising model where the free energy is exactly determined as a function of temperature and magnetic field we have found that the jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field applied to the system we have also determined the probability distribution function for the work performed on the system for the forward and reverse processes and verified that predictions based on the crooks relation are equally correct we also propose a method to calculate the lag between the current state of the system and that of the equilibrium based on macroscopic variables we have shown that the lag increases with the sweeping rate of the field at its final value for the reverse process while it decreases in the case of the forward process the lag increases linearly with the size of the chain and with a slope decreasing with the inverse of the rate of variation of the field

on vinogradov s mean value theorem strongly diagonal behaviour via efficient congruencing we enhance the efficient congruencing method for estimating vinogradov s integral for moments of order 2s with 1 leqslant s leqslant k 2 1 in this way we prove the main conjecture for such even moments when 1 leqslant s leqslant tfrac 1 4 k 1 2 showing that the moments exhibit strongly diagonal behaviour in this range there are improvements also for larger values of s these finding application to the asymptotic formula in waring s problem

dynamical entropies applied to stochastic resonance we will show that sr is accompanied by an extremal behavior of these quantities when sweeping noise intensity over the resonance region we will prove that an increase of applied noise at first leads to an increase of temporal order in the output sequence simultaneously the predictability of the output sequence will improve in the resonance region from the kullback entropy analysis we will learn that for a certain value of noise intensity the input and output distributions of binary subsequences maximally match moreover input and output subsequences registered at equal times exhibit maximum correlation finally we relate these results to the syn

gibbs jaynes entropy versus relative entropy the maximum entropy formalism developed by jaynes determines the relevant ensemble in nonequilibrium statistical mechanics by maximising the entropy functional subject to the constraints imposed by the available information we present an alternative derivation of the relevant ensemble based on the kullback leibler divergence from equilibrium if the equilibrium ensemble is already known then calculation of the relevant ensemble is considerably simplified the constraints must be chosen with care in order to avoid contradictions between the two alternative derivations the relative entropy functional measures how much a distribution departs from equilibrium therefore it provides a distinct approach to the calculation of statistical ensembles that might be applicable to situations in which the formalism presented by jaynes performs poorly such as non ergodic dynamical systems

time reversal and entropy there is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time reversal symmetry and the entropy production in such processes we explain on an elementary mathematical level the relations between entropy production phase space contraction and time reversal starting from a deterministic dynamics both closed and open systems in the transient and in the steady regime are considered the main result identifies under general conditions the statistical mechanical entropy production as the source term of time reversal breaking in the path space measure for the evolution of reduced variables this provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful in equalities we also discuss the markov approximation important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior

monotonic entropy growth for a nonlinear model of random exchanges we present a proof of the monotonic entropy growth for a nonlinear discrete time model of a random market this model based on binary collisions also may be viewed as a particular case of ulam s redistribution of energy problem we represent each step of this dynamics as a combination of two processes the first one is a linear energy conserving evolution of the two particle distribution for which the entropy growth can be easily verified the original nonlinear process is actually a result of a specific coarse graining of this linear evolution when after the collision one variable is integrated away this coarse graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth the combination of these two factors produces the required result which is obtained only by means of information theory inequalities

nonequilibrium thermodynamics of feedback control we establish a general theory of feedback control on classical stochastic thermodynamic systems and generalize nonequilibrium equalities such as the fluctuation theorem and the jarzynski equality in the presence of feedback control with multiple measurements our results are generalizations of the previous relevant works to the situations with general measurements and multiple heat baths the obtained equalities involve additional terms that characterize the information obtained by measurements or the efficacy of feedback control a generalized szilard engine and a feedback controlled ratchet are shown to satisfy the derived equalities

stationary and transient work fluctuation theorems for a dragged brownian particle recently wang et al carried out a laboratory experiment where a brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center this experiment confirmed a theoretically predicted work related integrated transient fluctuation theorem itft which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state the corresponding integrated stationary state fluctuation theorem issft was not observed using an overdamped langevin equation and an arbitrary motion for the center of the harmonic force all quantities of interest for these theorems and the corresponding nonintegrated ones tft and ssft respectively are theoretically explicitly obtained in this paper while the tft and the itft are satisfied for all times the ssft and the issft only hold asymptotically in time suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the issft could be observed are given in addition a nontrivial long time relation between the itft and the issft was discovered which could be observed experimentally especially in the case of a resonant circular motion of the center of the harmonic force

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

physical origins of entropy production free energy dissipation and their mathematical representations a unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented as generalizations of isothermal entropy and free energy two functions of state play central roles the gibbs entropy s and the relative entropy f which are related via the stationary distribution of the stochastic dynamics s satisfies the fundamental entropy balance equation ds dt e p ensuremath h d t with entropy production rate e p ensuremath ge 0 and heat dissipation rate h d while df dt ensuremath f d ensuremath le 0 for closed systems that satisfy detailed balance t e p t f d t for open systems one has t e p t f d t q hk t where the housekeeping heat q hk ensuremath ge 0 was first introduced in the phenomenological nonequilibrium steady state thermodynamics put forward by oono and paniconi q hk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium hence entropy production e p consists of free energy dissipation associated with spontaneous relaxation i e self organization f d and active energy pumping that sustains the open system q hk the amount of excess heat involved in the relaxation q ex h d ensuremath q hk f d ensuremath t ds dt two kinds of irreversibility and the meaning of the arrow of time emerge quasistationary processes adiabaticity and maximum principle for entropy are also generalized to nonequilibrium settings

decoupling for moment manifolds associated to arkhipov chubarikov karatsuba systems abstract we prove l p l p decoupling inequalities for a class of moment manifolds these inequalities imply optimal mean value estimates for multidimensional weyl sums of the kind considered by arkhipov chubarikov and karatsuba and by parsell in our proofs we take a new point of view on the bourgain demeter guth induction on scales argument this point of view substantially simplifies even the proof of l 2 l p decoupling for the moment curve

generalized jarzynski equality under nonequilibrium feedback control the jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control the new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control our results lead to a generalized fluctuation dissipation theorem that reflects the readout information and they can be experimentally tested using small thermodynamic systems we illustrate our general results by introducing an information ratchet which can transport a brownian particle in one direction and extract positive work from the particle

fast deconvolution with approximated psf by rstls with antireflective boundary conditions the problem of reconstructing signals and images from degraded ones is considered in this paper the latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image moreover the coefficient matrix is very ill conditioned requiring an additional regularization term different boundary conditions can be proposed in this paper antireflective boundary conditions are considered since both sides of the linear system have uncertainties and the coefficient matrix is highly structured the regularized structured total least squares approach seems to be the more appropriate one to compute an approximation of the true signal image with the latter approach the original problem is formulated as an highly nonconvex one and seldom can the global minimum be computed it is shown that regularized structured total least squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform such subproblems are then transformed into one dimensional unimodal real valued minimization problems which can be solved globally some numerical examples show the effectiveness of the proposed approach

rare events and the convergence of exponentially averaged work values equilibrium free energy differences are given by exponential averages of nonequilibrium work values such averages however often converge poorly as they are dominated by rare realizations i show that there is a simple and intuitively appealing description of these rare but dominant realizations this description is expressed as a duality between forward and reverse processes and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average analogous results apply to the equilibrium perturbation method of estimating free energy differences the pedagogical example of a piston and gas r c lua and a y grosberg j phys chem b 109 6805 2005 is used to illustrate the general discussion

a regularized structured total least squares algorithm for high resolution image reconstruction abstract high resolution image reconstruction is an important problem in image processing in general the blurring matrices are ill conditioned and it is necessary to compute a regularized solution moreover error exists not only in the blurred image but also the blurring matrix thus the total least squares method tends to give better results than the ordinary least squares method since the blurring matrices are also structured it is more appropriate to apply structured total least squares stls ng et al int j imaging systems technol 12 2002 35 recently proposed a regularized constrained total least squares rctls algorithm for this problem rctls is essentially a different name for regularized structured total least squares rstls however ng et al solved a problem that approximates the rctls problem the algorithm proposed in this paper solves the exact regularization of the stls problem and it is a faster algorithm also proposed is a preconditioner for the linear systems encountered in our rstls algorithm

vinogradov s integral and bounds for the riemann zeta function the main result is an upper bound for the riemann zeta function in the critical strip zeta sigma it le a t b 1 sigma 3 2 log 2 3 t with a 76 2 and b 4 45 valid for frac12 le sigma le 1 and t ge 3 the previous best constant b was 18 5 tools include a variant of the korobov vinogradov method of bounding exponential sums an explicit version of t d wooley s bounds for vinogradov s integral and explicit bounds for mean values of exponential sums over numbers without small prime factors also using methods of wooley an auxiliary result is the exponential sum bound s n t le 9 463 n 1 1 133 66 lambda 2 where n is a positive integer t is a real number lambda log t log n and s n t max 0 2000 mathematical subject classification primary 11m06 11n05 11l15 secondary 11d72 11m35