problem method classification in optimization and control abstract there are many classifications in optimization and control continuous discrete integer problems convex or concave minimization steepest descent branch and bound path following space filling random search methods smooth or non smooth problems etc the aim of this short note is not to propose yet another classification but to clarify existing ambiguities and imprecisions in problem formulations and to emphasize certain fundamental distinctions in solution methods currently used

entropy like proximal methods in convex programming we study an extension of the proximal method for convex programming where the quadratic regularization kernel is substituted by a class of convex statistical distances called divergences which are typically entropy like in form after establishing several basic properties of these quasi distances we present a convergence analysis of the resulting entropy like proximal algorithm applying this algorithm to the dual of a convex program we recover a wide class of nonquadratic multiplier methods and prove their convergence

diffraction of elastic waves by a penny shaped crack consider an infinite elastic solid containing a penny shaped crack we suppose that time harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i in this paper we describe a new method for solving the corresponding linear boundary value problem for u i which we denote by s we begin by defining an elastic double layer we prove that any solution of s can be represented by an elastic double layer whose density satisfies certain conditions we then introduce various green functions and define a new crack green function g ij that is discontinuous across the crack next we use g ij to derive a fredholm integral equation of the second kind for the discontinuity in u i across the crack we prove that this equation always has a unique solution hence we are able to prove that the original boundary value problem s always possesses a unique solution and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind

embedding formulae in diffraction theory embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables angles of incidence and observation into those dependent upon fewer angular variables in terms of facilitating rapid computations across considerable parameter regimes this is a considerable advantage our aim is to derive embedding formulae for scattering and diffraction problems in acoustics electromagnetism and elasticity here we construct a general approach to formulating and using embedding formulae we do this using complementary approaches overly singular states and a physical interpretation in terms of sources the crucial point we identify is the form of the auxiliary state used in the reciprocal theorem this is unphysically singular at the edge and is reminiscent of weight function methods used in fracture mechanics illustrative implementations of our approach are given using wiener hopf techniques for semi infinite model problems in both elasticity and acoustics we also demonstrate our approach using a numerical example from acoustics and we make connections with high frequency asymptotic methods

discontinuity and measurability of robust functions in the integral global minimization abstract robustness and measurability of a set and of a function are the foundation of the integral approach to global minimization however a robust function may be discontinuous and nonmeasurable in this paper we show that the set of points of discontinuity of a robust function has empty interior and is of the first category a robust function on the interval 0 1 is constructed such that the lebesque measure of its set of points of discontinuity approaches 1 we also show that there is a robust set on 0 1 which is lebesque nonmeasurable and then a lebesque nonmeasurable robust function is constructed

on the solutions of a class of dual integral equations occurring in diffraction problems it is shown that there exists a category of two dimensional diffraction problems which can be put into a standard form of dual integral equations these diffraction problems include diffraction of electromagnetic waves by a finite strip a finite slit the diffraction of scalar or vector elastic waves by a rigid strip or crack etc a general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions the sufficient conditions for a given dual integral equations to be solvable in this manner are also determined hence the method forms a complement to the weiner hopf method to illustrate the method solutions are obtained for a bench mark problem the diffraction of light by a finite perfectly conducting strip or equivalently the diffraction of sh waves by a crack comparison with results obtained by others for low intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies both the near field and the far field are obtained the latter is shown to correspond to the fraunhoffer diffraction pattern for high frequency it is also shown that for the equivalent crack problem the stress intensity factor sif fluctuates rapidly with changes in the angle of incidence for high frequencies thus making the sif especially sensitive to angle of incidence at high frequencies

optimal coding for the binary deletion channel with small deletion probability the binary deletion channel is the simplest point to point communication channel that models lack of synchronization input bits are deleted independently with probability d and when they are not deleted they are not affected by the channel despite significant effort little is known about the capacity of this channel and even less about optimal coding schemes in this paper we develop a new systematic approach to this problem by demonstrating that capacity can be computed in a series expansion for small deletion probability we compute three leading terms of this expansion and find an input distribution that achieves capacity up to this order this constitutes the first optimal random coding result for the deletion channel the key idea employed is the following we understand perfectly the deletion channel with deletion probability d 0 it has capacity 1 and the optimal input distribution is iid bernoulli 1 2 it is natural to expect that the channel with small deletion probabilities has a capacity that varies smoothly with d and that the optimal input distribution is obtained by smoothly perturbing the iid bernoulli 1 2 process our results show that this is indeed the case

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

discontinuous robust mappings are approximatable the concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization a set a of a topological space x is said to be robust iff cl a cl int a a mapping f x y is said tobe robust iffforeach open set uy of y f i uy is robust we prove that if x is a baire space and y satisfies the second axiom of countability then a mapping f x y is robust iff it is approximatable in the sense that the set of points of continuity of f is dense in x and that for any other point x e x x f x is the limit of x f x where for all a x is a continuous point of f this result justifies the notion of robustness

guess check codes for deletions insertions and synchronization we consider the problem of constructing codes that can correct delta deletions occurring in an arbitrary binary string of length n bits varshamov tenengolts vt codes dating back to 1965 are zero error single deletion delta 1 correcting codes and have an asymptotically optimal redundancy finding similar codes for delta geq 2 deletions remains an open problem in this paper we relax the standard zero error i e worst case decoding requirement by assuming that the positions of the delta deletions or insertions are independent of the code word our contribution is a new family of explicit codes that we call guess check gc codes that can correct with high probability up to a constant number of delta deletions or insertions gc codes are systematic and have deterministic polynomial time encoding and decoding algorithms we also describe the application of gc codes to file synchronization

shear and torsional impact of cracked viscoelastic bodies a numerical integral equation transform approach abstract the transient elastodynamic stress intensity factor was determined for a cracked linearly viscoelastic body under impact loading two separate geometries along with associated loading conditions were considered in the first case the body is in the form of an infinite strip containing a central finite length crack and is subjected to anti plane shear tractions various strip heights are considered including the possibility of a body of nearly infinite extent in the second case the body is of infinite extent containing a finite length penny shaped crack and is subjected to radial shear and torsional twisting tractions the analytical parts of the solutions are either given by a previous analysis of h g georgiadis or are obtained from results by g c sih through the use of the correspondence principle the numerical procedure consists of solving integral equations and then inverting the laplace transformed solution by the dubner abate crump technique numerical results were given for the standard linear solid by considering several combinations of material constants

bounds on the capacity of discrete memoryless channels corrupted by synchronization and substitution errors we study the capacity of discrete memoryless channels with synchronization errors and additive noise we first show that with very large alphabets their capacity can be achieved by independent and identically distributed input sources and establish proven tight lower and upper capacity bounds we also derive tight numerical capacity bounds for channels where the synchronization between the input and output is partly preserved for instance using incorruptible synchronization markers such channels include channels with duplication errors channels that only insert or delete zeros and channels with bitshift errors studied in magnetic recording channels with small alphabets and corrupted by synchronization errors have an infinite memory revisiting the theoretical work of dobrushin and adapting techniques used to compute capacity bounds for finite state source channel models we compute improved numerical capacity lower bounds for discrete memoryless channels with small alphabets synchronization errors and memoryless noise an interesting and some what surprising result is that as long as the input sequences are not completely deleted the capacity of channels corrupted by discrete timing errors is always nonzero even if all the symbols are corrupted

directly lower bounding the information capacity for channels with i i d deletions and duplications in this paper we directly lower bound the information capacity for channels with independent identically distributed i i d deletions and duplications our approach differs from previous work in that we focus on the information capacity using ideas from renewal theory rather than focusing on the transmission capacity by analyzing the error probability of some randomly generated code using a combinatorial argument of course the transmission and information capacities are equal but our change of perspective allows for a much simpler analysis that gives more general theoretical results we then apply these results to the binary deletion channel to improve existing lower bounds on its capacity

pseudo differential operators for embedding formulae a new method is proposed for deriving embedding formulae in 2d diffraction problems in contrast to the approach developed in craster and shanin 2005 7 which is based on a differential operator here a pseudo differential i e a non local operator is applied to the wave field using this non local operator a new embedding formula is derived for scattering by a single wedge the formula has uniform structure for all opening angles including angles irrational with respect to p the earlier theory craster and shanin 2005 7 was valid only for rational angles

improved lower bounds for the capacity of i i d deletion and duplication channels this paper considers the capacity of binary deletion channels where bits are deleted independently with probability d it improves significantly upon the best previous framework used to obtain provable lower bounds on this capacity by utilizing a stronger definition of a typical output from the channel the new results give the best known provable bounds on the capacity for all values of d moreover the techniques presented here extend to yield lower bounds for channels with certain types of random insertions namely duplications or combinations of duplications and deletions to demonstrate these techniques in this context two binary channels are analyzed a channel where each transmitted bit is copied with probability nu and a channel where each transmitted bit is copied a geometrically distributed number of times

bounds on the capacity of random insertion and deletion additive noise channels we develop several analytical lower bounds on the capacity of binary insertion and deletion channels by considering independent uniformly distributed i u d inputs and computing lower bounds on the mutual information between the input and output sequences for the deletion channel we consider two different models i i d deletion substitution channel and i i d deletion channel with additive white gaussian noise awgn these two models are considered to incorporate effects of the channel noise along with the synchronization errors for the insertion channel case we consider gallager s model in which the transmitted bits are replaced with two random bits and uniform over the four possibilities independently of any other insertion events the general approach taken is similar in all cases however the specific computations differ furthermore the approach yields a useful lower bound on the capacity for a wide range of deletion probabilities of the deletion channels while it provides a beneficial bound only for small insertion probabilities less than 0 25 of the insertion model adopted we emphasize the importance of these results by noting that 1 our results are the first analytical bounds on the capacity of deletion awgn channels 2 the results developed are the best available analytical lower bounds on the deletion substitution case 3 for the gallager insertion channel model the new lower bound improves the existing results for small insertion probabilities

scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation in this paper the composite boundary integral equation bie formulation is applied to scattering of elastic waves from thin shapes with small but finite thickness open cracks or thin voids thin inclusions thin layer interfaces etc which are modeled with two surfaces this composite bie formulation which is an extension of the burton and miller s formulation for acoustic waves uses a linear combination of the conventional bie and the hypersingular bie for thin shapes the conventional bie as well as the hypersingular bie will degenerate or nearly degenerate if they are applied individually on the two surfaces the composite bie formulation however will not degenerate for such problems as demonstrated in this paper nearly singular and hypersingular integrals which arise in problems involving thin shapes modeled with two surfaces are transformed into sums of weakly singular integrals and nonsingular line integrals thus no finer mesh is needed to compute these nearly singular integrals

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

on the projected subgradient method for nonsmooth convex optimization in a hilbert space we consider the method for constrained convex optimization in a hilbert space consisting of a step in the direction opposite to anek subgradient of the objective at a current iterate followed by an orthogonal projection onto the feasible set the normalized stepsizesek are exogenously given satisfying k 0 k k 0 k2 0 we prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions and is unbounded otherwise among the features of our convergence analysis we mention that it covers the nonsmooth case in the sense that we make no assumption of differentiability off and much less of lipschitz continuity of its gradient also we prove weak convergence of the whole sequence rather than just boundedness of the sequence and optimality of its weak accumulation points thus improving over all previously known convergence results we present also convergence rate results 1998 the mathematical programming society inc published by elsevier science b v

scattering by two penny shaped cracks with spring boundary conditions the elastodynamic scattering by a penny shaped crack with spring boundary conditions is investigated the transition t matrix of the crack is determined and the usefulness of this is illustrated by considering also the scattering by two cracks the t matrix of a single crack is first determined by a direct integral equation method which gives the crack opening displacement and the integral representation which subsequently gives the scattered field expanded in spherical waves two cracks are considered by a multi centred t matrix approach where matrix inverses are expanded in neuman series rotation matrices are employed so that the cracks may have an arbitrary rotation the back scattered longitudinal far field amplitude is computed both in the frequency and time domain in a few cases and the effects due to multiple scattering are in particular explored

sharp analytical capacity upper bounds for sticky and related channels we study natural examples of binary channels with synchronization errors these include the duplication channel which independently outputs a given bit once or twice and geometric channels that repeat a given bit according to a geometric rule with or without the possibility of bit deletion we apply the general framework of cheraghchi journal of the acm 2019 to obtain sharp analytical upper bounds on the capacity of these channels previously upper bounds were known via numerical computations involving the computation of finite approximations of the channels by a computer and then using the obtained numerical results to upper bound the actual capacity while leading to sharp numerical results further progress on the full understanding of the channel capacity inherently remains elusive using such methods our results can be regarded as a major step toward a complete understanding of the capacity curves quantitatively our upper bounds sharply approach and in some cases surpass the bounds that were previously only known by purely numerical methods among our results we notably give a completely analytical proof that when the number of repetitions per bit is geometric supported on the non negative integers with mean growing to infinity the channel capacity remains substantially bounded away from 1

polynomial time decodable codes for the binary deletion channel in the random deletion channel each bit is deleted independently with probability p for the random deletion channel the existence of codes of rate 1 p 9 and thus bounded away from 0 for any p has been known we give an explicit construction with polynomial time encoding and deletion correction algorithms with rate c 0 1 p for an absolute constant c 0 0

novel bounds on the capacity of the binary deletion channel we present novel bounds on the capacity of the independent and identically distributed binary deletion channel four upper bounds are obtained by providing the transmitter and the receiver with genie aided information on suitably defined random processes since some of the proposed bounds involve infinite series we also introduce provable inequalities that lead to more manageable results for most values of the deletion probability these bounds improve the existing ones and significantly narrow the gap with the available lower bounds exploiting the same auxiliary processes we also derive as a by product two simple lower bounds on the channel capacity which for low values of the deletion probability are almost as good as the best existing lower bounds

upper bounds on the capacity of deletion channels using channel fragmentation we study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop outs or symbol insertions with particular emphasis on the binary and non binary deletion channels we offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels the resulting output symbols are combined appropriately to come up with an equivalent input output representation of the original channel which allows for derivation of new upper bounds on the channel capacity we consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels with two specific applications of this idea a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels the resulting inequalities prove useful in deriving tighter capacity upper bounds for 1 independent identically distributed i i d deletion channels when the deletion probability exceeds 0 65 and 2 nonbinary deletion channels some extensions of these results for instance to the case of deletion substitution channels are also explored

a simple lower bound for the capacity of the deletion channel we present a simple proof that the capacity of the binary independent and identically distributed i i d deletion channel where each bit is deleted independently with probability d is at least 1 d 9 by developing a correspondence between the deletion channel and an insertion deletion channel that we call a poisson repeat channel

numerical resolution of the boundary integral equations for elastic scattering by a plane crack the problem of wave scattering by a plane crack is solved either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method a collocation method is often used to solve that equation but here we will use a variational method first writing the problem of fourier variables and then writing the associated integrals in the sesquilinear form with weak singularity kernels this representation is used in the numerical approach made with a finite element method in the surface of the crack numerical tests were made with circular and elliptical cracks but this method can be extended to other shapes with the same convergence profiles extensive results are given concerning the crack opening displacement the scattering cross section the back scattered amplitude and far field patterns

integral global optimization method in statistical applications abstract many statistical computations need a minimization method which can find global minima of a nonconvex nonsmooth and even discontinuous objective function conventional optimization methods hardly serve that the integral optimization method has been developed to satisfy such requirements in this paper we apply this method to several problems in statistics as nonlinear regression maximum likelihood estimation for three parameter weibull distribution mixture densities and sampling and show that the results obtained by a global minimization method are better than those by a gradient based one

on lower bounds for the capacity of deletion channels this correspondence considers binary deletion channels where bits are deleted independently with probability d it improves upon the framework used to analyze the capacity of binary deletion channels established by diggavi and grossglauser improving on their lower bounds diggavi and grossglauser considered codebooks with codewords generated by a first order markov chain they only consider typical outputs where an output is typical if an n bit input gives an n 1 d 1 epsi bit output the improvements in this correspondence arise from two considerations first a stronger notion of a typical output from the channel is used which yields better bounds even for the codebooks studied by diggavi and grossglauser second codewords generated by more general processes than first order markov chains are considered

weinstein s diffraction problem embedding formula and spectral equation in parabolic approximation a short wave problem of reflection and radiation by an open end of a two dimensional planar waveguide is studied the incident mode is assumed to have frequency close to the cut off the problem is studied in the parabolic approximation a recently developed approach based on the embedding formula and the spectral equation for the directivity of an edge green s function is applied to the problem

numerical simulations on resonance poles the trapping case of two plane cracks abstract in this paper a proof for the existence of acoustic resonance poles with robin boundary conditions is presented the analysis of the elastic crack scattering problem allows us to distinguish two types of resonance poles normal or tangential and we present some numerical results based on a variational boundary element method considering two plane cracks we present some estimates to justify the difference between simulation results when the cracks lay on the same plane or on different ones including a trapping case

bounds on the capacity of channels with insertions deletions and substitutions we present novel bounds on the capacity of binary channels with independent and identically distributed insertions deletions and substitutions the proposed bounds are obtained by exploiting an auxiliary system where the channel is the same as the one in the system of interest but the receiver is provided with partial genie aided information on the insertion deletion process in particular we show that when this information is revealed we obtain a memoryless channel whose capacity evaluated by means of the blahut arimoto algorithm gives an upper bound on the capacity of interest we also show that capacity lower bounds can be derived as well by exploiting the same auxiliary system and resorting to suitable information theoretical inequalities in most scenarios the proposed bounds improve the existing ones and significantly narrow the region to which the actual capacity can belong

a penny shaped crack in a transversely isotropic material under non axisymmetric impact loads abstract a solution is given for problems involving non axisymmetric dynamic impact loading of a penny shaped crack in a transversely isotropic medium laplace and hankel transforms are used to reduce the equations of elasticity to integral equations and solutions are obtained for the three modes of fracture the stress intensity factors are determined for a penny shaped crack loaded by concentrated normal impact forces and concentrated radial shear impact forces the integral equations are solved by numerical methods and the results are plotted showing how the dynamic stress intensity factors are influenced by the asymmetric loading

optimality condition and algorithm with deviation integral for global optimization to study the integral global minimization a general form of deviation integral is introduced and its properties are examined in this work in terms of the deviation integral optimality condition and algorithms are given algorithms are implemented by a properly designed monte carlo simulation numerical tests are given to show the effectiveness of the method

diffraction of elastic waves by a penny shaped crack analytical and numerical results the diffraction of time harmonic stress waves by a penny shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials martin proc r soc lond a 378 263 1981 has proved that the corresponding linear boundary value problem has precisely one solution and that this solution can be constructed by solving a two dimensional fredholm integral equation of the second kind however this integral equation has a complicated matrix kernel and the components of its vector solution are coupled the main purpose of the present paper is to show how martin s integral equation can be explicitly solved in terms of a sequence of functions each of which satisfies a very simple scalar integral equation of the second kind this simplification may be made for any incident wave for an incident plane wave further simplifications are possible we show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence namely grazing incidence the resulting computational procedure is especially attractive if only the stressintensity factors or the far field displacements are required finally we present some numerical results for the scattering of a p wave at normal incidence and an sv wave at oblique incidence and compare these with those of other authors

capacity upper bounds for deletion type channels we develop a systematic approach based on convex programming and real analysis for obtaining upper bounds on the capacity of the binary deletion channel and more generally channels with i i d insertions and deletions other than the classical deletion channel we give special attention to the poisson repeat channel introduced by mitzenmacher and drinea ieee transactions on information theory 2006 our framework can be applied to obtain capacity upper bounds for any repetition distribution the deletion and poisson repeat channels corresponding to the special cases of bernoulli and poisson distributions our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate real valued and often concave function over a bounded interval the corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function e g being only efficiently computable versus having an explicit closed form expression in terms of elementary or common special functions among our results we show the following 1 the capacity of the binary deletion channel with deletion probability d is at most 1 d p for d 1 2 and assuming that the capacity function is convex is at most 1 d log 4 p for d 2 we derive the first set of capacity upper bounds for the poisson repeat channel our results uncover further striking connections between this channel and the deletion channel and suggest somewhat counter intuitively that the poisson repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel 3 we derive several novel upper bounds on the capacity of the deletion channel all upper bounds are maximums of efficiently computable and concave univariate real functions over a bounded domain in turn we upper bound these functions in terms of explicit elementary and standard special functions whose maximums can be found even more efficiently and sometimes analytically for example for d e 1 2

nonsmooth nonconvex global optimization in a banach space with a basis following paper 1 significant improvements are proposed as concerns the class of problems that can be solved the quality of the solution and the numerical implementability of the method optimization is considered in a banach space over a nonconvex inf compact finitely infrobust set for a multivalued nonconvex continuous functional the iterative method is monotonic and represents generalized variant o setwise descent onto the set of all global minimizers a real life example of the vertical soft landing of an aircraft is considered to illustrate the method

level continuity of functions and applications abstract in this paper we study the concepts of level continuity and proper local maximum points of functions defined on a topological space x and on one hand we establish that under adequate conditions f is level continuous if f is without proper local maximum points and on the other we prove that level convergence and variational convergence convergence of functions are equivalents when the limit function is level continuous

time harmonic body force loading of a mode i penny shaped crack abstract the present paper is concerned with the problem of determining dynamic sif of a penny shaped crack in an infinite elastic medium which is subjected to the action of time harmonic axial body forces placed symmetrically with respect to the crack plane the solution of the problem is obtained by superposition of the solutions of two simpler problems the first of these problems is related to the unperturbed crackless space under the prescribed axial body forces while the second problem consists in finding the dynamic sif of the penny shaped crack whose faces are directly acted upon by some axial stresses the form of these axial stresses is determined from the solution of the first problem fourier and hankel transforms have been used to solve the first problem next by means of the hankel transform the second problem has been reduced to a pair of dual integral equations which have been subsequently transformed into a fredholm integral equation of the second kind via an auxiliary function the integral equation has been solved numerically in order to determine the variations of the dynamic stress intensity factor at the rim of the penny shaped crack for some particular body force loading cases